1,1,107,110,0.1434158,"\int x^6 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^6*(a + b*ArcCsch[c*x]),x]","\frac{a x^7}{7}-\frac{5 b \log \left(x \left(\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right)\right)}{112 c^7}+b \sqrt{\frac{c^2 x^2+1}{c^2 x^2}} \left(\frac{5 x^2}{112 c^5}-\frac{5 x^4}{168 c^3}+\frac{x^6}{42 c}\right)+\frac{1}{7} b x^7 \text{csch}^{-1}(c x)","\frac{1}{7} x^7 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b x^6 \sqrt{\frac{1}{c^2 x^2}+1}}{42 c}-\frac{5 b \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{112 c^7}+\frac{5 b x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{112 c^5}-\frac{5 b x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{168 c^3}",1,"(a*x^7)/7 + b*Sqrt[(1 + c^2*x^2)/(c^2*x^2)]*((5*x^2)/(112*c^5) - (5*x^4)/(168*c^3) + x^6/(42*c)) + (b*x^7*ArcCsch[c*x])/7 - (5*b*Log[x*(1 + Sqrt[(1 + c^2*x^2)/(c^2*x^2)])])/(112*c^7)","A",1
2,1,72,86,0.1353071,"\int x^5 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^5*(a + b*ArcCsch[c*x]),x]","\frac{a x^6}{6}+b \sqrt{\frac{c^2 x^2+1}{c^2 x^2}} \left(\frac{4 x}{45 c^5}-\frac{2 x^3}{45 c^3}+\frac{x^5}{30 c}\right)+\frac{1}{6} b x^6 \text{csch}^{-1}(c x)","\frac{1}{6} x^6 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b x^5 \sqrt{\frac{1}{c^2 x^2}+1}}{30 c}+\frac{4 b x \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^5}-\frac{2 b x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{45 c^3}",1,"(a*x^6)/6 + b*Sqrt[(1 + c^2*x^2)/(c^2*x^2)]*((4*x)/(45*c^5) - (2*x^3)/(45*c^3) + x^5/(30*c)) + (b*x^6*ArcCsch[c*x])/6","A",1
3,1,97,86,0.0475442,"\int x^4 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^4*(a + b*ArcCsch[c*x]),x]","\frac{a x^5}{5}+\frac{3 b \log \left(x \left(\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right)\right)}{40 c^5}+b \sqrt{\frac{c^2 x^2+1}{c^2 x^2}} \left(\frac{x^4}{20 c}-\frac{3 x^2}{40 c^3}\right)+\frac{1}{5} b x^5 \text{csch}^{-1}(c x)","\frac{1}{5} x^5 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b x^4 \sqrt{\frac{1}{c^2 x^2}+1}}{20 c}+\frac{3 b \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{40 c^5}-\frac{3 b x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{40 c^3}",1,"(a*x^5)/5 + b*Sqrt[(1 + c^2*x^2)/(c^2*x^2)]*((-3*x^2)/(40*c^3) + x^4/(20*c)) + (b*x^5*ArcCsch[c*x])/5 + (3*b*Log[x*(1 + Sqrt[(1 + c^2*x^2)/(c^2*x^2)])])/(40*c^5)","A",1
4,1,62,62,0.1023037,"\int x^3 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^3*(a + b*ArcCsch[c*x]),x]","\frac{a x^4}{4}+b \sqrt{\frac{c^2 x^2+1}{c^2 x^2}} \left(\frac{x^3}{12 c}-\frac{x}{6 c^3}\right)+\frac{1}{4} b x^4 \text{csch}^{-1}(c x)","\frac{1}{4} x^4 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{12 c}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1}}{6 c^3}",1,"(a*x^4)/4 + b*Sqrt[(1 + c^2*x^2)/(c^2*x^2)]*(-1/6*x/c^3 + x^3/(12*c)) + (b*x^4*ArcCsch[c*x])/4","A",1
5,1,85,62,0.051918,"\int x^2 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^2*(a + b*ArcCsch[c*x]),x]","\frac{a x^3}{3}+\frac{b x^2 \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{6 c}-\frac{b \log \left(x \left(\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right)\right)}{6 c^3}+\frac{1}{3} b x^3 \text{csch}^{-1}(c x)","\frac{1}{3} x^3 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{6 c}-\frac{b \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{6 c^3}",1,"(a*x^3)/3 + (b*x^2*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(6*c) + (b*x^3*ArcCsch[c*x])/3 - (b*Log[x*(1 + Sqrt[(1 + c^2*x^2)/(c^2*x^2)])])/(6*c^3)","A",1
6,1,50,38,0.0245818,"\int x \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x*(a + b*ArcCsch[c*x]),x]","\frac{a x^2}{2}+\frac{b x \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{2 c}+\frac{1}{2} b x^2 \text{csch}^{-1}(c x)","\frac{1}{2} x^2 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c}",1,"(a*x^2)/2 + (b*x*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(2*c) + (b*x^2*ArcCsch[c*x])/2","A",1
7,1,44,30,0.0604974,"\int \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[a + b*ArcCsch[c*x],x]","a x+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+b x \text{csch}^{-1}(c x)","a x+\frac{b \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{c}+b x \text{csch}^{-1}(c x)",1,"a*x + b*x*ArcCsch[c*x] + (b*Sqrt[1 + 1/(c^2*x^2)]*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2]","A",1
8,1,47,56,0.0447605,"\int \frac{a+b \text{csch}^{-1}(c x)}{x} \, dx","Integrate[(a + b*ArcCsch[c*x])/x,x]","a \log (x)+\frac{1}{2} b \left(\text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)-\text{csch}^{-1}(c x) \left(\text{csch}^{-1}(c x)+2 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)\right)\right)","-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{2 b}-\log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{2} b \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)",1,"a*Log[x] + (b*(-(ArcCsch[c*x]*(ArcCsch[c*x] + 2*Log[1 - E^(-2*ArcCsch[c*x])])) + PolyLog[2, E^(-2*ArcCsch[c*x])]))/2","A",0
9,1,40,30,0.0312435,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^2} \, dx","Integrate[(a + b*ArcCsch[c*x])/x^2,x]","-\frac{a}{x}+b c \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}-\frac{b \text{csch}^{-1}(c x)}{x}","b c \sqrt{\frac{1}{c^2 x^2}+1}-\frac{a+b \text{csch}^{-1}(c x)}{x}",1,"-(a/x) + b*c*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsch[c*x])/x","A",1
10,1,66,50,0.0355613,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^3} \, dx","Integrate[(a + b*ArcCsch[c*x])/x^3,x]","-\frac{a}{2 x^2}+\frac{b c \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{4 x}-\frac{1}{4} b c^2 \sinh ^{-1}\left(\frac{1}{c x}\right)-\frac{b \text{csch}^{-1}(c x)}{2 x^2}","-\frac{a+b \text{csch}^{-1}(c x)}{2 x^2}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{4 x}-\frac{1}{4} b c^2 \text{csch}^{-1}(c x)",1,"-1/2*a/x^2 + (b*c*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(4*x) - (b*ArcCsch[c*x])/(2*x^2) - (b*c^2*ArcSinh[1/(c*x)])/4","A",1
11,1,59,58,0.0471276,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^4} \, dx","Integrate[(a + b*ArcCsch[c*x])/x^4,x]","-\frac{a}{3 x^3}+b \left(\frac{c}{9 x^2}-\frac{2 c^3}{9}\right) \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}-\frac{b \text{csch}^{-1}(c x)}{3 x^3}","-\frac{a+b \text{csch}^{-1}(c x)}{3 x^3}+\frac{1}{9} b c^3 \left(\frac{1}{c^2 x^2}+1\right)^{3/2}-\frac{1}{3} b c^3 \sqrt{\frac{1}{c^2 x^2}+1}",1,"-1/3*a/x^3 + b*((-2*c^3)/9 + c/(9*x^2))*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsch[c*x])/(3*x^3)","A",1
12,1,78,74,0.0427138,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^5} \, dx","Integrate[(a + b*ArcCsch[c*x])/x^5,x]","-\frac{a}{4 x^4}+\frac{3}{32} b c^4 \sinh ^{-1}\left(\frac{1}{c x}\right)+b \left(\frac{c}{16 x^3}-\frac{3 c^3}{32 x}\right) \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}-\frac{b \text{csch}^{-1}(c x)}{4 x^4}","-\frac{a+b \text{csch}^{-1}(c x)}{4 x^4}+\frac{3}{32} b c^4 \text{csch}^{-1}(c x)+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{16 x^3}-\frac{3 b c^3 \sqrt{\frac{1}{c^2 x^2}+1}}{32 x}",1,"-1/4*a/x^4 + b*(c/(16*x^3) - (3*c^3)/(32*x))*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsch[c*x])/(4*x^4) + (3*b*c^4*ArcSinh[1/(c*x)])/32","A",1
13,1,69,79,0.0631021,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^6} \, dx","Integrate[(a + b*ArcCsch[c*x])/x^6,x]","-\frac{a}{5 x^5}+b \left(\frac{8 c^5}{75}-\frac{4 c^3}{75 x^2}+\frac{c}{25 x^4}\right) \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}-\frac{b \text{csch}^{-1}(c x)}{5 x^5}","-\frac{a+b \text{csch}^{-1}(c x)}{5 x^5}+\frac{1}{25} b c^5 \left(\frac{1}{c^2 x^2}+1\right)^{5/2}-\frac{2}{15} b c^5 \left(\frac{1}{c^2 x^2}+1\right)^{3/2}+\frac{1}{5} b c^5 \sqrt{\frac{1}{c^2 x^2}+1}",1,"-1/5*a/x^5 + b*((8*c^5)/75 + c/(25*x^4) - (4*c^3)/(75*x^2))*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsch[c*x])/(5*x^5)","A",1
14,1,88,98,0.0743799,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^7} \, dx","Integrate[(a + b*ArcCsch[c*x])/x^7,x]","-\frac{a}{6 x^6}-\frac{5}{96} b c^6 \sinh ^{-1}\left(\frac{1}{c x}\right)+b \left(\frac{5 c^5}{96 x}-\frac{5 c^3}{144 x^3}+\frac{c}{36 x^5}\right) \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}-\frac{b \text{csch}^{-1}(c x)}{6 x^6}","-\frac{a+b \text{csch}^{-1}(c x)}{6 x^6}-\frac{5}{96} b c^6 \text{csch}^{-1}(c x)+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{36 x^5}+\frac{5 b c^5 \sqrt{\frac{1}{c^2 x^2}+1}}{96 x}-\frac{5 b c^3 \sqrt{\frac{1}{c^2 x^2}+1}}{144 x^3}",1,"-1/6*a/x^6 + b*(c/(36*x^5) - (5*c^3)/(144*x^3) + (5*c^5)/(96*x))*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsch[c*x])/(6*x^6) - (5*b*c^6*ArcSinh[1/(c*x)])/96","A",1
15,1,122,105,0.2113858,"\int x^3 \left(a+b \text{csch}^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(a + b*ArcCsch[c*x])^2,x]","\frac{c x \left(3 a^2 c^3 x^3+2 a b \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 x^2-2\right)+b^2 c x\right)+2 b c x \text{csch}^{-1}(c x) \left(3 a c^3 x^3+b \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 x^2-2\right)\right)+3 b^2 c^4 x^4 \text{csch}^{-1}(c x)^2-4 b^2 \log (x)}{12 c^4}","\frac{b x^3 \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)}{6 c}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)}{3 c^3}+\frac{1}{4} x^4 \left(a+b \text{csch}^{-1}(c x)\right)^2-\frac{b^2 \log (x)}{3 c^4}+\frac{b^2 x^2}{12 c^2}",1,"(c*x*(b^2*c*x + 3*a^2*c^3*x^3 + 2*a*b*Sqrt[1 + 1/(c^2*x^2)]*(-2 + c^2*x^2)) + 2*b*c*x*(3*a*c^3*x^3 + b*Sqrt[1 + 1/(c^2*x^2)]*(-2 + c^2*x^2))*ArcCsch[c*x] + 3*b^2*c^4*x^4*ArcCsch[c*x]^2 - 4*b^2*Log[x])/(12*c^4)","A",1
16,1,211,122,1.4272935,"\int x^2 \left(a+b \text{csch}^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(a + b*ArcCsch[c*x])^2,x]","\frac{a^2 c^3 x^3+2 a b c^3 x^3 \text{csch}^{-1}(c x)+a b c^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1}-\frac{a b c x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+b^2 c^3 x^3 \text{csch}^{-1}(c x)^2+b^2 c^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1} \text{csch}^{-1}(c x)+b^2 \text{Li}_2\left(-e^{-\text{csch}^{-1}(c x)}\right)-b^2 \text{Li}_2\left(e^{-\text{csch}^{-1}(c x)}\right)+b^2 c x+b^2 \text{csch}^{-1}(c x) \log \left(1-e^{-\text{csch}^{-1}(c x)}\right)-b^2 \text{csch}^{-1}(c x) \log \left(e^{-\text{csch}^{-1}(c x)}+1\right)}{3 c^3}","-\frac{2 b \tanh ^{-1}\left(e^{\text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{3 c^3}+\frac{b x^2 \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)}{3 c}+\frac{1}{3} x^3 \left(a+b \text{csch}^{-1}(c x)\right)^2-\frac{b^2 \text{Li}_2\left(-e^{\text{csch}^{-1}(c x)}\right)}{3 c^3}+\frac{b^2 \text{Li}_2\left(e^{\text{csch}^{-1}(c x)}\right)}{3 c^3}+\frac{b^2 x}{3 c^2}",1,"(b^2*c*x + a*b*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2 + a^2*c^3*x^3 + b^2*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2*ArcCsch[c*x] + 2*a*b*c^3*x^3*ArcCsch[c*x] + b^2*c^3*x^3*ArcCsch[c*x]^2 - (a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + b^2*ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])] - b^2*ArcCsch[c*x]*Log[1 + E^(-ArcCsch[c*x])] + b^2*PolyLog[2, -E^(-ArcCsch[c*x])] - b^2*PolyLog[2, E^(-ArcCsch[c*x])])/(3*c^3)","A",0
17,1,87,54,0.1427065,"\int x \left(a+b \text{csch}^{-1}(c x)\right)^2 \, dx","Integrate[x*(a + b*ArcCsch[c*x])^2,x]","\frac{a c x \left(a c x+2 b \sqrt{\frac{1}{c^2 x^2}+1}\right)+2 b c x \text{csch}^{-1}(c x) \left(a c x+b \sqrt{\frac{1}{c^2 x^2}+1}\right)+b^2 c^2 x^2 \text{csch}^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2}","\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)}{c}+\frac{1}{2} x^2 \left(a+b \text{csch}^{-1}(c x)\right)^2+\frac{b^2 \log (x)}{c^2}",1,"(a*c*x*(2*b*Sqrt[1 + 1/(c^2*x^2)] + a*c*x) + 2*b*c*x*(b*Sqrt[1 + 1/(c^2*x^2)] + a*c*x)*ArcCsch[c*x] + b^2*c^2*x^2*ArcCsch[c*x]^2 + 2*b^2*Log[c*x])/(2*c^2)","A",1
18,1,121,68,0.2363452,"\int \left(a+b \text{csch}^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcCsch[c*x])^2,x]","\frac{a^2 c x+2 a b c x \text{csch}^{-1}(c x)-2 a b \log \left(\tanh \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)\right)-2 b^2 \text{Li}_2\left(-e^{-\text{csch}^{-1}(c x)}\right)+2 b^2 \text{Li}_2\left(e^{-\text{csch}^{-1}(c x)}\right)+b^2 c x \text{csch}^{-1}(c x)^2-2 b^2 \text{csch}^{-1}(c x) \log \left(1-e^{-\text{csch}^{-1}(c x)}\right)+2 b^2 \text{csch}^{-1}(c x) \log \left(e^{-\text{csch}^{-1}(c x)}+1\right)}{c}","x \left(a+b \text{csch}^{-1}(c x)\right)^2+\frac{4 b \tanh ^{-1}\left(e^{\text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{c}+\frac{2 b^2 \text{Li}_2\left(-e^{\text{csch}^{-1}(c x)}\right)}{c}-\frac{2 b^2 \text{Li}_2\left(e^{\text{csch}^{-1}(c x)}\right)}{c}",1,"(a^2*c*x + 2*a*b*c*x*ArcCsch[c*x] + b^2*c*x*ArcCsch[c*x]^2 - 2*b^2*ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])] + 2*b^2*ArcCsch[c*x]*Log[1 + E^(-ArcCsch[c*x])] - 2*a*b*Log[Tanh[ArcCsch[c*x]/2]] - 2*b^2*PolyLog[2, -E^(-ArcCsch[c*x])] + 2*b^2*PolyLog[2, E^(-ArcCsch[c*x])])/c","A",0
19,1,115,81,0.1334153,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{x} \, dx","Integrate[(a + b*ArcCsch[c*x])^2/x,x]","a^2 \log (c x)+a b \left(\text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)-\text{csch}^{-1}(c x) \left(\text{csch}^{-1}(c x)+2 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)\right)\right)+b^2 \left(-\text{csch}^{-1}(c x) \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(e^{2 \text{csch}^{-1}(c x)}\right)+\frac{1}{3} \text{csch}^{-1}(c x)^3-\text{csch}^{-1}(c x)^2 \log \left(1-e^{2 \text{csch}^{-1}(c x)}\right)\right)","-b \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)+\frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{3 b}-\log \left(1-e^{2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)^2+\frac{1}{2} b^2 \text{Li}_3\left(e^{2 \text{csch}^{-1}(c x)}\right)",1,"a^2*Log[c*x] + a*b*(-(ArcCsch[c*x]*(ArcCsch[c*x] + 2*Log[1 - E^(-2*ArcCsch[c*x])])) + PolyLog[2, E^(-2*ArcCsch[c*x])]) + b^2*(ArcCsch[c*x]^3/3 - ArcCsch[c*x]^2*Log[1 - E^(2*ArcCsch[c*x])] - ArcCsch[c*x]*PolyLog[2, E^(2*ArcCsch[c*x])] + PolyLog[3, E^(2*ArcCsch[c*x])]/2)","A",0
20,1,70,49,0.1682254,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcCsch[c*x])^2/x^2,x]","-\frac{a^2-2 a b c x \sqrt{\frac{1}{c^2 x^2}+1}+2 b \text{csch}^{-1}(c x) \left(a-b c x \sqrt{\frac{1}{c^2 x^2}+1}\right)+b^2 \text{csch}^{-1}(c x)^2+2 b^2}{x}","2 b c \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{x}-\frac{2 b^2}{x}",1,"-((a^2 + 2*b^2 - 2*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x + 2*b*(a - b*c*Sqrt[1 + 1/(c^2*x^2)]*x)*ArcCsch[c*x] + b^2*ArcCsch[c*x]^2)/x)","A",1
21,1,100,87,0.1434792,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcCsch[c*x])^2/x^3,x]","-\frac{2 a^2-2 a b c x \sqrt{\frac{1}{c^2 x^2}+1}+2 a b c^2 x^2 \sinh ^{-1}\left(\frac{1}{c x}\right)-2 b \text{csch}^{-1}(c x) \left(b c x \sqrt{\frac{1}{c^2 x^2}+1}-2 a\right)+b^2 \left(c^2 x^2+2\right) \text{csch}^{-1}(c x)^2+b^2}{4 x^2}","\frac{b c \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)}{2 x}-\frac{1}{2} a b c^2 \text{csch}^{-1}(c x)-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{2 x^2}-\frac{1}{4} b^2 c^2 \text{csch}^{-1}(c x)^2-\frac{b^2}{4 x^2}",1,"-1/4*(2*a^2 + b^2 - 2*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x - 2*b*(-2*a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x)*ArcCsch[c*x] + b^2*(2 + c^2*x^2)*ArcCsch[c*x]^2 + 2*a*b*c^2*x^2*ArcSinh[1/(c*x)])/x^2","A",1
22,1,106,100,0.1941421,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcCsch[c*x])^2/x^4,x]","\frac{-9 a^2+6 a b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(1-2 c^2 x^2\right)-6 b \text{csch}^{-1}(c x) \left(3 a+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(2 c^2 x^2-1\right)\right)+2 b^2 \left(6 c^2 x^2-1\right)-9 b^2 \text{csch}^{-1}(c x)^2}{27 x^3}","\frac{2 b c \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)}{9 x^2}-\frac{4}{9} b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{3 x^3}+\frac{4 b^2 c^2}{9 x}-\frac{2 b^2}{27 x^3}",1,"(-9*a^2 + 6*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1 - 2*c^2*x^2) + 2*b^2*(-1 + 6*c^2*x^2) - 6*b*(3*a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-1 + 2*c^2*x^2))*ArcCsch[c*x] - 9*b^2*ArcCsch[c*x]^2)/(27*x^3)","A",1
23,1,147,132,0.1812579,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{x^5} \, dx","Integrate[(a + b*ArcCsch[c*x])^2/x^5,x]","\frac{-8 a^2+6 a b c^4 x^4 \sinh ^{-1}\left(\frac{1}{c x}\right)+4 a b c x \sqrt{\frac{1}{c^2 x^2}+1}-2 b \text{csch}^{-1}(c x) \left(8 a+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(3 c^2 x^2-2\right)\right)-6 a b c^3 x^3 \sqrt{\frac{1}{c^2 x^2}+1}+b^2 \left(3 c^4 x^4-8\right) \text{csch}^{-1}(c x)^2+3 b^2 c^2 x^2-b^2}{32 x^4}","\frac{3}{16} a b c^4 \text{csch}^{-1}(c x)+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)}{8 x^3}-\frac{3 b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)}{16 x}-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{4 x^4}+\frac{3}{32} b^2 c^4 \text{csch}^{-1}(c x)^2+\frac{3 b^2 c^2}{32 x^2}-\frac{b^2}{32 x^4}",1,"(-8*a^2 - b^2 + 4*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x + 3*b^2*c^2*x^2 - 6*a*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*x^3 - 2*b*(8*a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-2 + 3*c^2*x^2))*ArcCsch[c*x] + b^2*(-8 + 3*c^4*x^4)*ArcCsch[c*x]^2 + 6*a*b*c^4*x^4*ArcSinh[1/(c*x)])/(32*x^4)","A",1
24,1,271,195,1.0189453,"\int x^3 \left(a+b \text{csch}^{-1}(c x)\right)^3 \, dx","Integrate[x^3*(a + b*ArcCsch[c*x])^3,x]","\frac{a^3 c^4 x^4+b \text{csch}^{-1}(c x) \left(c x \left(3 a^2 c^3 x^3+2 a b \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 x^2-2\right)+b^2 c x\right)+4 b^2 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)\right)-2 a^2 b c x \sqrt{\frac{1}{c^2 x^2}+1}+a^2 b c^3 x^3 \sqrt{\frac{1}{c^2 x^2}+1}+a b^2 c^2 x^2+b^2 \text{csch}^{-1}(c x)^2 \left(3 a c^4 x^4+b \left(-2 c x \sqrt{\frac{1}{c^2 x^2}+1}+c^3 x^3 \sqrt{\frac{1}{c^2 x^2}+1}+2\right)\right)+4 a b^2 \log \left(\frac{1}{c x}\right)+b^3 c^4 x^4 \text{csch}^{-1}(c x)^3+b^3 c x \sqrt{\frac{1}{c^2 x^2}+1}-2 b^3 \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)}{4 c^4}","\frac{b^2 \log \left(1-e^{2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{c^4}+\frac{b^2 x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{4 c^2}-\frac{b \left(a+b \text{csch}^{-1}(c x)\right)^2}{2 c^4}+\frac{b x^3 \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2}{4 c}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2}{2 c^3}+\frac{1}{4} x^4 \left(a+b \text{csch}^{-1}(c x)\right)^3+\frac{b^3 \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)}{2 c^4}+\frac{b^3 x \sqrt{\frac{1}{c^2 x^2}+1}}{4 c^3}",1,"(-2*a^2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x + b^3*c*Sqrt[1 + 1/(c^2*x^2)]*x + a*b^2*c^2*x^2 + a^2*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*x^3 + a^3*c^4*x^4 + b^2*(3*a*c^4*x^4 + b*(2 - 2*c*Sqrt[1 + 1/(c^2*x^2)]*x + c^3*Sqrt[1 + 1/(c^2*x^2)]*x^3))*ArcCsch[c*x]^2 + b^3*c^4*x^4*ArcCsch[c*x]^3 + b*ArcCsch[c*x]*(c*x*(b^2*c*x + 3*a^2*c^3*x^3 + 2*a*b*Sqrt[1 + 1/(c^2*x^2)]*(-2 + c^2*x^2)) + 4*b^2*Log[1 - E^(-2*ArcCsch[c*x])]) + 4*a*b^2*Log[1/(c*x)] - 2*b^3*PolyLog[2, E^(-2*ArcCsch[c*x])])/(4*c^4)","A",0
25,1,548,194,7.5199214,"\int x^2 \left(a+b \text{csch}^{-1}(c x)\right)^3 \, dx","Integrate[x^2*(a + b*ArcCsch[c*x])^3,x]","\frac{a^3 x^3}{3}+\frac{a^2 b x^2 \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{2 c}-\frac{a^2 b \log \left(x \left(\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right)\right)}{2 c^3}+a^2 b x^3 \text{csch}^{-1}(c x)+\frac{a b^2 \left(2 c^3 x^3 \left(-\frac{4 \text{Li}_2\left(e^{-\text{csch}^{-1}(c x)}\right)}{c^3 x^3}+4 \text{csch}^{-1}(c x)^2+2 \cosh \left(2 \text{csch}^{-1}(c x)\right)-\frac{3 \text{csch}^{-1}(c x) \log \left(1-e^{-\text{csch}^{-1}(c x)}\right)}{c x}+\frac{3 \text{csch}^{-1}(c x) \log \left(e^{-\text{csch}^{-1}(c x)}+1\right)}{c x}+2 \text{csch}^{-1}(c x) \sinh \left(2 \text{csch}^{-1}(c x)\right)+\text{csch}^{-1}(c x) \log \left(1-e^{-\text{csch}^{-1}(c x)}\right) \sinh \left(3 \text{csch}^{-1}(c x)\right)-\text{csch}^{-1}(c x) \log \left(e^{-\text{csch}^{-1}(c x)}+1\right) \sinh \left(3 \text{csch}^{-1}(c x)\right)-2\right)+8 \text{Li}_2\left(-e^{-\text{csch}^{-1}(c x)}\right)\right)}{8 c^3}+\frac{b^3 \left(16 c^3 x^3 \text{csch}^{-1}(c x)^3 \sinh ^4\left(\frac{1}{2} \text{csch}^{-1}(c x)\right)+48 \text{csch}^{-1}(c x) \text{Li}_2\left(-e^{-\text{csch}^{-1}(c x)}\right)-48 \text{csch}^{-1}(c x) \text{Li}_2\left(e^{-\text{csch}^{-1}(c x)}\right)+48 \text{Li}_3\left(-e^{-\text{csch}^{-1}(c x)}\right)-48 \text{Li}_3\left(e^{-\text{csch}^{-1}(c x)}\right)+\frac{\text{csch}^{-1}(c x)^3 \text{csch}^4\left(\frac{1}{2} \text{csch}^{-1}(c x)\right)}{c x}+6 \text{csch}^{-1}(c x)^2 \text{csch}^2\left(\frac{1}{2} \text{csch}^{-1}(c x)\right)-4 \text{csch}^{-1}(c x)^3 \coth \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)+24 \text{csch}^{-1}(c x) \coth \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)+24 \text{csch}^{-1}(c x)^2 \log \left(1-e^{-\text{csch}^{-1}(c x)}\right)-24 \text{csch}^{-1}(c x)^2 \log \left(e^{-\text{csch}^{-1}(c x)}+1\right)+4 \text{csch}^{-1}(c x)^3 \tanh \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)-24 \text{csch}^{-1}(c x) \tanh \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)+6 \text{csch}^{-1}(c x)^2 \text{sech}^2\left(\frac{1}{2} \text{csch}^{-1}(c x)\right)-48 \log \left(\tanh \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)\right)\right)}{48 c^3}","-\frac{b^2 \text{Li}_2\left(-e^{\text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{c^3}+\frac{b^2 \text{Li}_2\left(e^{\text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{c^3}+\frac{b^2 x \left(a+b \text{csch}^{-1}(c x)\right)}{c^2}-\frac{b \tanh ^{-1}\left(e^{\text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)^2}{c^3}+\frac{b x^2 \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2}{2 c}+\frac{1}{3} x^3 \left(a+b \text{csch}^{-1}(c x)\right)^3+\frac{b^3 \text{Li}_3\left(-e^{\text{csch}^{-1}(c x)}\right)}{c^3}-\frac{b^3 \text{Li}_3\left(e^{\text{csch}^{-1}(c x)}\right)}{c^3}+\frac{b^3 \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{c^3}",1,"(a^3*x^3)/3 + (a^2*b*x^2*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(2*c) + a^2*b*x^3*ArcCsch[c*x] - (a^2*b*Log[x*(1 + Sqrt[(1 + c^2*x^2)/(c^2*x^2)])])/(2*c^3) + (a*b^2*(8*PolyLog[2, -E^(-ArcCsch[c*x])] + 2*c^3*x^3*(-2 + 4*ArcCsch[c*x]^2 + 2*Cosh[2*ArcCsch[c*x]] - (3*ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])])/(c*x) + (3*ArcCsch[c*x]*Log[1 + E^(-ArcCsch[c*x])])/(c*x) - (4*PolyLog[2, E^(-ArcCsch[c*x])])/(c^3*x^3) + 2*ArcCsch[c*x]*Sinh[2*ArcCsch[c*x]] + ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])]*Sinh[3*ArcCsch[c*x]] - ArcCsch[c*x]*Log[1 + E^(-ArcCsch[c*x])]*Sinh[3*ArcCsch[c*x]])))/(8*c^3) + (b^3*(24*ArcCsch[c*x]*Coth[ArcCsch[c*x]/2] - 4*ArcCsch[c*x]^3*Coth[ArcCsch[c*x]/2] + 6*ArcCsch[c*x]^2*Csch[ArcCsch[c*x]/2]^2 + (ArcCsch[c*x]^3*Csch[ArcCsch[c*x]/2]^4)/(c*x) + 24*ArcCsch[c*x]^2*Log[1 - E^(-ArcCsch[c*x])] - 24*ArcCsch[c*x]^2*Log[1 + E^(-ArcCsch[c*x])] - 48*Log[Tanh[ArcCsch[c*x]/2]] + 48*ArcCsch[c*x]*PolyLog[2, -E^(-ArcCsch[c*x])] - 48*ArcCsch[c*x]*PolyLog[2, E^(-ArcCsch[c*x])] + 48*PolyLog[3, -E^(-ArcCsch[c*x])] - 48*PolyLog[3, E^(-ArcCsch[c*x])] + 6*ArcCsch[c*x]^2*Sech[ArcCsch[c*x]/2]^2 + 16*c^3*x^3*ArcCsch[c*x]^3*Sinh[ArcCsch[c*x]/2]^4 - 24*ArcCsch[c*x]*Tanh[ArcCsch[c*x]/2] + 4*ArcCsch[c*x]^3*Tanh[ArcCsch[c*x]/2]))/(48*c^3)","B",0
26,1,171,117,0.4711256,"\int x \left(a+b \text{csch}^{-1}(c x)\right)^3 \, dx","Integrate[x*(a + b*ArcCsch[c*x])^3,x]","\frac{a \left(a c x \left(a c x+3 b \sqrt{\frac{1}{c^2 x^2}+1}\right)-6 b^2 \log \left(\frac{1}{c x}\right)\right)+3 b^2 \text{csch}^{-1}(c x)^2 \left(a c^2 x^2+b \left(c x \sqrt{\frac{1}{c^2 x^2}+1}-1\right)\right)+3 b \text{csch}^{-1}(c x) \left(a c x \left(a c x+2 b \sqrt{\frac{1}{c^2 x^2}+1}\right)-2 b^2 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)\right)+b^3 c^2 x^2 \text{csch}^{-1}(c x)^3+3 b^3 \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)}{2 c^2}","-\frac{3 b^2 \log \left(1-e^{2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{c^2}+\frac{3 b x \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2}{2 c}+\frac{3 b \left(a+b \text{csch}^{-1}(c x)\right)^2}{2 c^2}+\frac{1}{2} x^2 \left(a+b \text{csch}^{-1}(c x)\right)^3-\frac{3 b^3 \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)}{2 c^2}",1,"(3*b^2*(a*c^2*x^2 + b*(-1 + c*Sqrt[1 + 1/(c^2*x^2)]*x))*ArcCsch[c*x]^2 + b^3*c^2*x^2*ArcCsch[c*x]^3 + 3*b*ArcCsch[c*x]*(a*c*x*(2*b*Sqrt[1 + 1/(c^2*x^2)] + a*c*x) - 2*b^2*Log[1 - E^(-2*ArcCsch[c*x])]) + a*(a*c*x*(3*b*Sqrt[1 + 1/(c^2*x^2)] + a*c*x) - 6*b^2*Log[1/(c*x)]) + 3*b^3*PolyLog[2, E^(-2*ArcCsch[c*x])])/(2*c^2)","A",0
27,1,246,120,0.3410578,"\int \left(a+b \text{csch}^{-1}(c x)\right)^3 \, dx","Integrate[(a + b*ArcCsch[c*x])^3,x]","a^3 x+\frac{3 a^2 b \log \left(c x \left(\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right)\right)}{c}+3 a^2 b x \text{csch}^{-1}(c x)+\frac{3 a b^2 \left(-2 \text{Li}_2\left(-e^{-\text{csch}^{-1}(c x)}\right)+2 \text{Li}_2\left(e^{-\text{csch}^{-1}(c x)}\right)+\text{csch}^{-1}(c x) \left(c x \text{csch}^{-1}(c x)-2 \log \left(1-e^{-\text{csch}^{-1}(c x)}\right)+2 \log \left(e^{-\text{csch}^{-1}(c x)}+1\right)\right)\right)}{c}+\frac{b^3 \left(-6 \text{csch}^{-1}(c x) \text{Li}_2\left(-e^{-\text{csch}^{-1}(c x)}\right)+6 \text{csch}^{-1}(c x) \text{Li}_2\left(e^{-\text{csch}^{-1}(c x)}\right)-6 \text{Li}_3\left(-e^{-\text{csch}^{-1}(c x)}\right)+6 \text{Li}_3\left(e^{-\text{csch}^{-1}(c x)}\right)+c x \text{csch}^{-1}(c x)^3-3 \text{csch}^{-1}(c x)^2 \log \left(1-e^{-\text{csch}^{-1}(c x)}\right)+3 \text{csch}^{-1}(c x)^2 \log \left(e^{-\text{csch}^{-1}(c x)}+1\right)\right)}{c}","\frac{6 b^2 \text{Li}_2\left(-e^{\text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{c}-\frac{6 b^2 \text{Li}_2\left(e^{\text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{c}+x \left(a+b \text{csch}^{-1}(c x)\right)^3+\frac{6 b \tanh ^{-1}\left(e^{\text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)^2}{c}-\frac{6 b^3 \text{Li}_3\left(-e^{\text{csch}^{-1}(c x)}\right)}{c}+\frac{6 b^3 \text{Li}_3\left(e^{\text{csch}^{-1}(c x)}\right)}{c}",1,"a^3*x + 3*a^2*b*x*ArcCsch[c*x] + (3*a^2*b*Log[c*x*(1 + Sqrt[(1 + c^2*x^2)/(c^2*x^2)])])/c + (3*a*b^2*(ArcCsch[c*x]*(c*x*ArcCsch[c*x] - 2*Log[1 - E^(-ArcCsch[c*x])] + 2*Log[1 + E^(-ArcCsch[c*x])]) - 2*PolyLog[2, -E^(-ArcCsch[c*x])] + 2*PolyLog[2, E^(-ArcCsch[c*x])]))/c + (b^3*(c*x*ArcCsch[c*x]^3 - 3*ArcCsch[c*x]^2*Log[1 - E^(-ArcCsch[c*x])] + 3*ArcCsch[c*x]^2*Log[1 + E^(-ArcCsch[c*x])] - 6*ArcCsch[c*x]*PolyLog[2, -E^(-ArcCsch[c*x])] + 6*ArcCsch[c*x]*PolyLog[2, E^(-ArcCsch[c*x])] - 6*PolyLog[3, -E^(-ArcCsch[c*x])] + 6*PolyLog[3, E^(-ArcCsch[c*x])]))/c","B",0
28,1,198,110,0.2270821,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{x} \, dx","Integrate[(a + b*ArcCsch[c*x])^3/x,x]","\frac{1}{4} \left(4 a^3 \log (c x)+6 a^2 b \left(\text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)-\text{csch}^{-1}(c x) \left(\text{csch}^{-1}(c x)+2 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)\right)\right)+2 a b^2 \left(-6 \text{csch}^{-1}(c x) \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)+3 \text{Li}_3\left(e^{2 \text{csch}^{-1}(c x)}\right)+2 \text{csch}^{-1}(c x)^2 \left(\text{csch}^{-1}(c x)-3 \log \left(1-e^{2 \text{csch}^{-1}(c x)}\right)\right)\right)+b^3 \left(-6 \text{csch}^{-1}(c x)^2 \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)+6 \text{csch}^{-1}(c x) \text{Li}_3\left(e^{2 \text{csch}^{-1}(c x)}\right)-3 \text{Li}_4\left(e^{2 \text{csch}^{-1}(c x)}\right)+\text{csch}^{-1}(c x)^4-4 \text{csch}^{-1}(c x)^3 \log \left(1-e^{2 \text{csch}^{-1}(c x)}\right)\right)\right)","\frac{3}{2} b^2 \text{Li}_3\left(e^{2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)-\frac{3}{2} b \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)^2+\frac{\left(a+b \text{csch}^{-1}(c x)\right)^4}{4 b}-\log \left(1-e^{2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)^3-\frac{3}{4} b^3 \text{Li}_4\left(e^{2 \text{csch}^{-1}(c x)}\right)",1,"(4*a^3*Log[c*x] + 6*a^2*b*(-(ArcCsch[c*x]*(ArcCsch[c*x] + 2*Log[1 - E^(-2*ArcCsch[c*x])])) + PolyLog[2, E^(-2*ArcCsch[c*x])]) + 2*a*b^2*(2*ArcCsch[c*x]^2*(ArcCsch[c*x] - 3*Log[1 - E^(2*ArcCsch[c*x])]) - 6*ArcCsch[c*x]*PolyLog[2, E^(2*ArcCsch[c*x])] + 3*PolyLog[3, E^(2*ArcCsch[c*x])]) + b^3*(ArcCsch[c*x]^4 - 4*ArcCsch[c*x]^3*Log[1 - E^(2*ArcCsch[c*x])] - 6*ArcCsch[c*x]^2*PolyLog[2, E^(2*ArcCsch[c*x])] + 6*ArcCsch[c*x]*PolyLog[3, E^(2*ArcCsch[c*x])] - 3*PolyLog[4, E^(2*ArcCsch[c*x])]))/4","A",0
29,1,132,78,0.2492631,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcCsch[c*x])^3/x^2,x]","-\frac{a^3+3 b \text{csch}^{-1}(c x) \left(a^2-2 a b c x \sqrt{\frac{1}{c^2 x^2}+1}+2 b^2\right)-3 a^2 b c x \sqrt{\frac{1}{c^2 x^2}+1}+3 b^2 \text{csch}^{-1}(c x)^2 \left(a-b c x \sqrt{\frac{1}{c^2 x^2}+1}\right)+6 a b^2-6 b^3 c x \sqrt{\frac{1}{c^2 x^2}+1}+b^3 \text{csch}^{-1}(c x)^3}{x}","-\frac{6 b^2 \left(a+b \text{csch}^{-1}(c x)\right)}{x}+3 b c \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{x}+6 b^3 c \sqrt{\frac{1}{c^2 x^2}+1}",1,"-((a^3 + 6*a*b^2 - 3*a^2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x - 6*b^3*c*Sqrt[1 + 1/(c^2*x^2)]*x + 3*b*(a^2 + 2*b^2 - 2*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x)*ArcCsch[c*x] + 3*b^2*(a - b*c*Sqrt[1 + 1/(c^2*x^2)]*x)*ArcCsch[c*x]^2 + b^3*ArcCsch[c*x]^3)/x)","A",1
30,1,182,123,0.3260896,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcCsch[c*x])^3/x^3,x]","-\frac{4 a^3+3 b c^2 x^2 \left(2 a^2+b^2\right) \sinh ^{-1}\left(\frac{1}{c x}\right)+6 b \text{csch}^{-1}(c x) \left(2 a^2-2 a b c x \sqrt{\frac{1}{c^2 x^2}+1}+b^2\right)-6 a^2 b c x \sqrt{\frac{1}{c^2 x^2}+1}+6 b^2 \text{csch}^{-1}(c x)^2 \left(a \left(c^2 x^2+2\right)-b c x \sqrt{\frac{1}{c^2 x^2}+1}\right)+6 a b^2-3 b^3 c x \sqrt{\frac{1}{c^2 x^2}+1}+2 b^3 \left(c^2 x^2+2\right) \text{csch}^{-1}(c x)^3}{8 x^2}","-\frac{3 b^2 \left(a+b \text{csch}^{-1}(c x)\right)}{4 x^2}+\frac{3 b c \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2}{4 x}-\frac{1}{4} c^2 \left(a+b \text{csch}^{-1}(c x)\right)^3-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{2 x^2}+\frac{3 b^3 c \sqrt{\frac{1}{c^2 x^2}+1}}{8 x}-\frac{3}{8} b^3 c^2 \text{csch}^{-1}(c x)",1,"-1/8*(4*a^3 + 6*a*b^2 - 6*a^2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x - 3*b^3*c*Sqrt[1 + 1/(c^2*x^2)]*x + 6*b*(2*a^2 + b^2 - 2*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x)*ArcCsch[c*x] + 6*b^2*(-(b*c*Sqrt[1 + 1/(c^2*x^2)]*x) + a*(2 + c^2*x^2))*ArcCsch[c*x]^2 + 2*b^3*(2 + c^2*x^2)*ArcCsch[c*x]^3 + 3*b*(2*a^2 + b^2)*c^2*x^2*ArcSinh[1/(c*x)])/x^2","A",1
31,1,200,166,0.3139231,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{x^4} \, dx","Integrate[(a + b*ArcCsch[c*x])^3/x^4,x]","\frac{-9 a^3+3 b \text{csch}^{-1}(c x) \left(-9 a^2+6 a b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(1-2 c^2 x^2\right)+2 b^2 \left(6 c^2 x^2-1\right)\right)+9 a^2 b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(1-2 c^2 x^2\right)+6 a b^2 \left(6 c^2 x^2-1\right)-9 b^2 \text{csch}^{-1}(c x)^2 \left(3 a+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(2 c^2 x^2-1\right)\right)+2 b^3 c x \sqrt{\frac{1}{c^2 x^2}+1} \left(1-20 c^2 x^2\right)-9 b^3 \text{csch}^{-1}(c x)^3}{27 x^3}","\frac{4 b^2 c^2 \left(a+b \text{csch}^{-1}(c x)\right)}{3 x}-\frac{2 b^2 \left(a+b \text{csch}^{-1}(c x)\right)}{9 x^3}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2}{3 x^2}-\frac{2}{3} b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{3 x^3}+\frac{2}{27} b^3 c^3 \left(\frac{1}{c^2 x^2}+1\right)^{3/2}-\frac{14}{9} b^3 c^3 \sqrt{\frac{1}{c^2 x^2}+1}",1,"(-9*a^3 + 2*b^3*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1 - 20*c^2*x^2) + 9*a^2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1 - 2*c^2*x^2) + 6*a*b^2*(-1 + 6*c^2*x^2) + 3*b*(-9*a^2 + 6*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1 - 2*c^2*x^2) + 2*b^2*(-1 + 6*c^2*x^2))*ArcCsch[c*x] - 9*b^2*(3*a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-1 + 2*c^2*x^2))*ArcCsch[c*x]^2 - 9*b^3*ArcCsch[c*x]^3)/(27*x^3)","A",1
32,1,277,204,0.3677171,"\int \frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{x^5} \, dx","Integrate[(a + b*ArcCsch[c*x])^3/x^5,x]","\frac{-64 a^3+9 b c^4 x^4 \left(8 a^2+5 b^2\right) \sinh ^{-1}\left(\frac{1}{c x}\right)-24 b \text{csch}^{-1}(c x) \left(8 a^2+2 a b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(3 c^2 x^2-2\right)+b^2 \left(1-3 c^2 x^2\right)\right)+48 a^2 b c x \sqrt{\frac{1}{c^2 x^2}+1}-72 a^2 b c^3 x^3 \sqrt{\frac{1}{c^2 x^2}+1}+72 a b^2 c^2 x^2+24 b^2 \text{csch}^{-1}(c x)^2 \left(a \left(3 c^4 x^4-8\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(2-3 c^2 x^2\right)\right)-24 a b^2+8 b^3 \left(3 c^4 x^4-8\right) \text{csch}^{-1}(c x)^3+6 b^3 c x \sqrt{\frac{1}{c^2 x^2}+1}-45 b^3 c^3 x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{256 x^4}","\frac{9 b^2 c^2 \left(a+b \text{csch}^{-1}(c x)\right)}{32 x^2}-\frac{3 b^2 \left(a+b \text{csch}^{-1}(c x)\right)}{32 x^4}+\frac{3}{32} c^4 \left(a+b \text{csch}^{-1}(c x)\right)^3+\frac{3 b c \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2}{16 x^3}-\frac{9 b c^3 \sqrt{\frac{1}{c^2 x^2}+1} \left(a+b \text{csch}^{-1}(c x)\right)^2}{32 x}-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^3}{4 x^4}+\frac{45}{256} b^3 c^4 \text{csch}^{-1}(c x)+\frac{3 b^3 c \sqrt{\frac{1}{c^2 x^2}+1}}{128 x^3}-\frac{45 b^3 c^3 \sqrt{\frac{1}{c^2 x^2}+1}}{256 x}",1,"(-64*a^3 - 24*a*b^2 + 48*a^2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x + 6*b^3*c*Sqrt[1 + 1/(c^2*x^2)]*x + 72*a*b^2*c^2*x^2 - 72*a^2*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*x^3 - 45*b^3*c^3*Sqrt[1 + 1/(c^2*x^2)]*x^3 - 24*b*(8*a^2 + b^2*(1 - 3*c^2*x^2) + 2*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-2 + 3*c^2*x^2))*ArcCsch[c*x] + 24*b^2*(b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(2 - 3*c^2*x^2) + a*(-8 + 3*c^4*x^4))*ArcCsch[c*x]^2 + 8*b^3*(-8 + 3*c^4*x^4)*ArcCsch[c*x]^3 + 9*b*(8*a^2 + 5*b^2)*c^4*x^4*ArcSinh[1/(c*x)])/(256*x^4)","A",1
33,0,0,15,2.9312594,"\int \frac{x}{a+b \text{csch}^{-1}(c x)} \, dx","Integrate[x/(a + b*ArcCsch[c*x]),x]","\int \frac{x}{a+b \text{csch}^{-1}(c x)} \, dx","\text{Int}\left(\frac{x}{a+b \text{csch}^{-1}(c x)},x\right)",0,"Integrate[x/(a + b*ArcCsch[c*x]), x]","A",-1
34,0,0,13,2.8210438,"\int \frac{1}{a+b \text{csch}^{-1}(c x)} \, dx","Integrate[(a + b*ArcCsch[c*x])^(-1),x]","\int \frac{1}{a+b \text{csch}^{-1}(c x)} \, dx","\text{Int}\left(\frac{1}{a+b \text{csch}^{-1}(c x)},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])^(-1), x]","A",-1
35,0,0,17,0.306605,"\int \frac{1}{x \left(a+b \text{csch}^{-1}(c x)\right)} \, dx","Integrate[1/(x*(a + b*ArcCsch[c*x])),x]","\int \frac{1}{x \left(a+b \text{csch}^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \text{csch}^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*(a + b*ArcCsch[c*x])), x]","A",-1
36,1,44,46,0.0755897,"\int \frac{1}{x^2 \left(a+b \text{csch}^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*(a + b*ArcCsch[c*x])),x]","-\frac{c \left(\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)-\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)\right)}{b}","\frac{c \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)}{b}-\frac{c \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)}{b}",1,"-((c*(Cosh[a/b]*CoshIntegral[a/b + ArcCsch[c*x]] - Sinh[a/b]*SinhIntegral[a/b + ArcCsch[c*x]]))/b)","A",1
37,1,56,63,0.0764978,"\int \frac{1}{x^3 \left(a+b \text{csch}^{-1}(c x)\right)} \, dx","Integrate[1/(x^3*(a + b*ArcCsch[c*x])),x]","\frac{c^2 \left(\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right)-\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right)\right)}{2 b}","\frac{c^2 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right)}{2 b}-\frac{c^2 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \text{csch}^{-1}(c x)\right)}{2 b}",1,"(c^2*(CoshIntegral[(2*a)/b + 2*ArcCsch[c*x]]*Sinh[(2*a)/b] - Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCsch[c*x]]))/(2*b)","A",1
38,1,91,117,0.1653884,"\int \frac{1}{x^4 \left(a+b \text{csch}^{-1}(c x)\right)} \, dx","Integrate[1/(x^4*(a + b*ArcCsch[c*x])),x]","-\frac{c^3 \left(-\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)\right)+\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)\right)\right)}{4 b}","\frac{c^3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)}{4 b}-\frac{c^3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 a}{b}+3 \text{csch}^{-1}(c x)\right)}{4 b}-\frac{c^3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\text{csch}^{-1}(c x)\right)}{4 b}+\frac{c^3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 a}{b}+3 \text{csch}^{-1}(c x)\right)}{4 b}",1,"-1/4*(c^3*(-(Cosh[a/b]*CoshIntegral[a/b + ArcCsch[c*x]]) + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCsch[c*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcCsch[c*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCsch[c*x])]))/b","A",1
39,0,0,19,6.8703045,"\int (d x)^m \left(a+b \text{csch}^{-1}(c x)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcCsch[c*x])^3,x]","\int (d x)^m \left(a+b \text{csch}^{-1}(c x)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \text{csch}^{-1}(c x)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcCsch[c*x])^3, x]","A",-1
40,0,0,19,4.4894785,"\int (d x)^m \left(a+b \text{csch}^{-1}(c x)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcCsch[c*x])^2,x]","\int (d x)^m \left(a+b \text{csch}^{-1}(c x)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \text{csch}^{-1}(c x)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcCsch[c*x])^2, x]","A",-1
41,1,81,67,0.2434332,"\int (d x)^m \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcCsch[c*x]),x]","\frac{x (d x)^m \left((m+1) \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right)}{\sqrt{c^2 x^2+1}}\right)}{(m+1)^2}","\frac{(d x)^{m+1} \left(a+b \text{csch}^{-1}(c x)\right)}{d (m+1)}+\frac{b (d x)^m \, _2F_1\left(\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};-\frac{1}{c^2 x^2}\right)}{c m (m+1)}",1,"(x*(d*x)^m*((1 + m)*(a + b*ArcCsch[c*x]) + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)])/Sqrt[1 + c^2*x^2]))/(1 + m)^2","A",1
42,0,0,19,0.9306666,"\int \frac{(d x)^m}{a+b \text{csch}^{-1}(c x)} \, dx","Integrate[(d*x)^m/(a + b*ArcCsch[c*x]),x]","\int \frac{(d x)^m}{a+b \text{csch}^{-1}(c x)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \text{csch}^{-1}(c x)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcCsch[c*x]), x]","A",-1
43,0,0,19,1.8991357,"\int \frac{(d x)^m}{\left(a+b \text{csch}^{-1}(c x)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcCsch[c*x])^2,x]","\int \frac{(d x)^m}{\left(a+b \text{csch}^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \text{csch}^{-1}(c x)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcCsch[c*x])^2, x]","A",-1
44,1,165,167,0.2944823,"\int (d+e x)^3 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(a + b*ArcCsch[c*x]),x]","\frac{3 a c^3 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+3 b c^3 x \text{csch}^{-1}(c x) \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 \left(18 d^2+6 d e x+e^2 x^2\right)-2 e^2\right)+6 b d \left(2 c^2 d^2-e^2\right) \log \left(x \left(\sqrt{\frac{1}{c^2 x^2}+1}+1\right)\right)}{12 c^3}","\frac{(d+e x)^4 \left(a+b \text{csch}^{-1}(c x)\right)}{4 e}+\frac{b d e^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{2 c}+\frac{b e^3 x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{12 c}+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(9 c^2 d^2-e^2\right)}{6 c^3}+\frac{b d \left(2 c^2 d^2-e^2\right) \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{2 c^3}-\frac{b d^4 \text{csch}^{-1}(c x)}{4 e}",1,"(3*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(-2*e^2 + c^2*(18*d^2 + 6*d*e*x + e^2*x^2)) + 3*b*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcCsch[c*x] + 6*b*d*(2*c^2*d^2 - e^2)*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(12*c^3)","A",1
45,1,122,122,0.191046,"\int (d+e x)^2 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcCsch[c*x]),x]","\frac{c^2 x \left(2 a c \left(3 d^2+3 d e x+e^2 x^2\right)+b e \sqrt{\frac{1}{c^2 x^2}+1} (6 d+e x)\right)+2 b c^3 x \text{csch}^{-1}(c x) \left(3 d^2+3 d e x+e^2 x^2\right)+b \left(6 c^2 d^2-e^2\right) \log \left(x \left(\sqrt{\frac{1}{c^2 x^2}+1}+1\right)\right)}{6 c^3}","\frac{(d+e x)^3 \left(a+b \text{csch}^{-1}(c x)\right)}{3 e}+\frac{b d e x \sqrt{\frac{1}{c^2 x^2}+1}}{c}+\frac{b e^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1}}{6 c}+\frac{b \left(6 c^2 d^2-e^2\right) \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{6 c^3}-\frac{b d^3 \text{csch}^{-1}(c x)}{3 e}",1,"(c^2*x*(b*e*Sqrt[1 + 1/(c^2*x^2)]*(6*d + e*x) + 2*a*c*(3*d^2 + 3*d*e*x + e^2*x^2)) + 2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcCsch[c*x] + b*(6*c^2*d^2 - e^2)*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(6*c^3)","A",1
46,1,99,81,0.2027759,"\int (d+e x) \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcCsch[c*x]),x]","a d x+\frac{1}{2} a e x^2+\frac{b d x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{b e x \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{2 c}+b d x \text{csch}^{-1}(c x)+\frac{1}{2} b e x^2 \text{csch}^{-1}(c x)","\frac{(d+e x)^2 \left(a+b \text{csch}^{-1}(c x)\right)}{2 e}+\frac{b d \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{c}+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c}-\frac{b d^2 \text{csch}^{-1}(c x)}{2 e}",1,"a*d*x + (a*e*x^2)/2 + (b*e*x*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(2*c) + b*d*x*ArcCsch[c*x] + (b*e*x^2*ArcCsch[c*x])/2 + (b*d*Sqrt[1 + 1/(c^2*x^2)]*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2]","A",1
47,1,44,30,0.0499815,"\int \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[a + b*ArcCsch[c*x],x]","a x+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+b x \text{csch}^{-1}(c x)","a x+\frac{b \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{c}+b x \text{csch}^{-1}(c x)",1,"a*x + b*x*ArcCsch[c*x] + (b*Sqrt[1 + 1/(c^2*x^2)]*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2]","A",1
48,1,506,215,0.6637642,"\int \frac{a+b \text{csch}^{-1}(c x)}{d+e x} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x),x]","\frac{a \log (d+e x)}{e}+\frac{b \left(8 \text{Li}_2\left(\frac{\left(e-\sqrt{c^2 d^2+e^2}\right) e^{\text{csch}^{-1}(c x)}}{c d}\right)+8 \text{Li}_2\left(\frac{\left(e+\sqrt{c^2 d^2+e^2}\right) e^{\text{csch}^{-1}(c x)}}{c d}\right)+8 \text{csch}^{-1}(c x) \log \left(\frac{\left(\sqrt{c^2 d^2+e^2}-e\right) e^{\text{csch}^{-1}(c x)}}{c d}+1\right)+8 \text{csch}^{-1}(c x) \log \left(1-\frac{\left(\sqrt{c^2 d^2+e^2}+e\right) e^{\text{csch}^{-1}(c x)}}{c d}\right)+4 i \pi  \log \left(\frac{\left(\sqrt{c^2 d^2+e^2}-e\right) e^{\text{csch}^{-1}(c x)}}{c d}+1\right)+4 i \pi  \log \left(1-\frac{\left(\sqrt{c^2 d^2+e^2}+e\right) e^{\text{csch}^{-1}(c x)}}{c d}\right)+16 i \sin ^{-1}\left(\frac{\sqrt{1+\frac{i e}{c d}}}{\sqrt{2}}\right) \log \left(\frac{\left(\sqrt{c^2 d^2+e^2}-e\right) e^{\text{csch}^{-1}(c x)}}{c d}+1\right)-16 i \sin ^{-1}\left(\frac{\sqrt{1+\frac{i e}{c d}}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{c^2 d^2+e^2}+e\right) e^{\text{csch}^{-1}(c x)}}{c d}\right)-32 \sin ^{-1}\left(\frac{\sqrt{1+\frac{i e}{c d}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{(e+i c d) \cot \left(\frac{1}{4} \left(\pi +2 i \text{csch}^{-1}(c x)\right)\right)}{\sqrt{c^2 d^2+e^2}}\right)+4 \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)-8 \text{csch}^{-1}(c x)^2-4 i \pi  \text{csch}^{-1}(c x)-8 \text{csch}^{-1}(c x) \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)-4 i \pi  \log \left(\frac{d}{x}+e\right)+\pi ^2\right)}{8 e}","\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{\left(e-\sqrt{c^2 d^2+e^2}\right) e^{\text{csch}^{-1}(c x)}}{c d}\right)}{e}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{\left(\sqrt{c^2 d^2+e^2}+e\right) e^{\text{csch}^{-1}(c x)}}{c d}\right)}{e}-\frac{\log \left(1-e^{2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{e}+\frac{b \text{Li}_2\left(\frac{\left(e-\sqrt{c^2 d^2+e^2}\right) e^{\text{csch}^{-1}(c x)}}{c d}\right)}{e}+\frac{b \text{Li}_2\left(\frac{\left(e+\sqrt{c^2 d^2+e^2}\right) e^{\text{csch}^{-1}(c x)}}{c d}\right)}{e}-\frac{b \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)}{2 e}",1,"(a*Log[d + e*x])/e + (b*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 - 32*ArcSin[Sqrt[1 + (I*e)/(c*d)]/Sqrt[2]]*ArcTan[((I*c*d + e)*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[c^2*d^2 + e^2]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + ((-e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] + 8*ArcCsch[c*x]*Log[1 + ((-e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] + (16*I)*ArcSin[Sqrt[1 + (I*e)/(c*d)]/Sqrt[2]]*Log[1 + ((-e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] + (4*I)*Pi*Log[1 - ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] + 8*ArcCsch[c*x]*Log[1 - ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] - (16*I)*ArcSin[Sqrt[1 + (I*e)/(c*d)]/Sqrt[2]]*Log[1 - ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] - (4*I)*Pi*Log[e + d/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, ((e - Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)] + 8*PolyLog[2, ((e + Sqrt[c^2*d^2 + e^2])*E^ArcCsch[c*x])/(c*d)]))/(8*e)","C",0
49,1,134,98,0.2186798,"\int \frac{a+b \text{csch}^{-1}(c x)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x)^2,x]","-\frac{a}{e (d+e x)}-\frac{b \log \left(c x \left(\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}-c d\right)+e\right)}{d \sqrt{c^2 d^2+e^2}}+\frac{b \log (d+e x)}{d \sqrt{c^2 d^2+e^2}}+\frac{b \sinh ^{-1}\left(\frac{1}{c x}\right)}{d e}-\frac{b \text{csch}^{-1}(c x)}{e (d+e x)}","-\frac{a+b \text{csch}^{-1}(c x)}{e (d+e x)}+\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{e}{x}}{c \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}}\right)}{d \sqrt{c^2 d^2+e^2}}+\frac{b \text{csch}^{-1}(c x)}{d e}",1,"-(a/(e*(d + e*x))) - (b*ArcCsch[c*x])/(e*(d + e*x)) + (b*ArcSinh[1/(c*x)])/(d*e) + (b*Log[d + e*x])/(d*Sqrt[c^2*d^2 + e^2]) - (b*Log[e + c*(-(c*d) + Sqrt[c^2*d^2 + e^2]*Sqrt[1 + 1/(c^2*x^2)])*x])/(d*Sqrt[c^2*d^2 + e^2])","A",1
50,1,204,163,0.4780326,"\int \frac{a+b \text{csch}^{-1}(c x)}{(d+e x)^3} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x)^3,x]","\frac{1}{2} \left(-\frac{a}{e (d+e x)^2}-\frac{b c e x \sqrt{\frac{1}{c^2 x^2}+1}}{d \left(c^2 d^2+e^2\right) (d+e x)}-\frac{b \left(2 c^2 d^2+e^2\right) \log \left(c x \left(\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}-c d\right)+e\right)}{d^2 \left(c^2 d^2+e^2\right)^{3/2}}+\frac{b \left(2 c^2 d^2+e^2\right) \log (d+e x)}{d^2 \left(c^2 d^2+e^2\right)^{3/2}}+\frac{b \sinh ^{-1}\left(\frac{1}{c x}\right)}{d^2 e}-\frac{b \text{csch}^{-1}(c x)}{e (d+e x)^2}\right)","-\frac{a+b \text{csch}^{-1}(c x)}{2 e (d+e x)^2}-\frac{b c e \sqrt{\frac{1}{c^2 x^2}+1}}{2 d \left(c^2 d^2+e^2\right) \left(\frac{d}{x}+e\right)}+\frac{b \left(2 c^2 d^2+e^2\right) \tanh ^{-1}\left(\frac{c^2 d-\frac{e}{x}}{c \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d^2+e^2}}\right)}{2 d^2 \left(c^2 d^2+e^2\right)^{3/2}}+\frac{b \text{csch}^{-1}(c x)}{2 d^2 e}",1,"(-(a/(e*(d + e*x)^2)) - (b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x)/(d*(c^2*d^2 + e^2)*(d + e*x)) - (b*ArcCsch[c*x])/(e*(d + e*x)^2) + (b*ArcSinh[1/(c*x)])/(d^2*e) + (b*(2*c^2*d^2 + e^2)*Log[d + e*x])/(d^2*(c^2*d^2 + e^2)^(3/2)) - (b*(2*c^2*d^2 + e^2)*Log[e + c*(-(c*d) + Sqrt[c^2*d^2 + e^2]*Sqrt[1 + 1/(c^2*x^2)])*x])/(d^2*(c^2*d^2 + e^2)^(3/2)))/2","A",1
51,1,1094,918,14.3951345,"\int x^2 \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]),x]","\frac{b \left(-\frac{c \left(\frac{d}{x}+e\right) x \left(-\frac{16 c^3 \text{csch}^{-1}(c x) d^3}{105 e^3}-\frac{2}{7} c^3 x^3 \text{csch}^{-1}(c x)-\frac{2 c^2 x^2 \left(2 \sqrt{1+\frac{1}{c^2 x^2}} e+c d \text{csch}^{-1}(c x)\right)}{35 e}-\frac{8 c x \left(c d e \sqrt{1+\frac{1}{c^2 x^2}}-c^2 d^2 \text{csch}^{-1}(c x)\right)}{105 e^2}+\frac{4 \left(5 c^2 d^2+9 e^2\right) \sqrt{1+\frac{1}{c^2 x^2}}}{105 e^2}\right)}{\sqrt{d+e x}}-\frac{2 \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(-\frac{\sqrt{2} \left(9 c^3 e d^3+c e^3 d\right) \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left(8 c^4 d^4-5 c^2 e^2 d^2-9 e^4\right) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{2 \left(-5 c^3 e d^3-9 c e^3 d\right) \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right)}{c d \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}\right)}{105 e^3 \sqrt{d+e x}}\right)}{c^4}-\frac{a d^3 \sqrt{d+e x} B_{-\frac{e x}{d}}\left(3,\frac{3}{2}\right)}{e^3 \sqrt{\frac{e x}{d}+1}}","-\frac{32 b \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \sqrt{c^2 x^2+1} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right) d^4}{105 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{32 b c \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right) d^3}{105 \left(-c^2\right)^{3/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) d^2}{3 e^3}-\frac{32 b c \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right) d^2}{105 \left(-c^2\right)^{3/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{4 b \sqrt{d+e x} \left(c^2 x^2+1\right) d}{105 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{4 (d+e x)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right) d}{5 e^3}-\frac{4 b c \left(c^2 d^2+e^2\right) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right) d}{105 \left(-c^2\right)^{5/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b (d+e x)^{3/2} \left(c^2 x^2+1\right)}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{2 (d+e x)^{7/2} \left(a+b \text{csch}^{-1}(c x)\right)}{7 e^3}-\frac{4 b c \left(c^2 d^2-3 e^2\right) \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{35 \left(-c^2\right)^{5/2} e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}",1,"-((a*d^3*Sqrt[d + e*x]*Beta[-((e*x)/d), 3, 3/2])/(e^3*Sqrt[1 + (e*x)/d])) + (b*(-((c*(e + d/x)*x*((4*(5*c^2*d^2 + 9*e^2)*Sqrt[1 + 1/(c^2*x^2)])/(105*e^2) - (16*c^3*d^3*ArcCsch[c*x])/(105*e^3) - (2*c^3*x^3*ArcCsch[c*x])/7 - (2*c^2*x^2*(2*e*Sqrt[1 + 1/(c^2*x^2)] + c*d*ArcCsch[c*x]))/(35*e) - (8*c*x*(c*d*e*Sqrt[1 + 1/(c^2*x^2)] - c^2*d^2*ArcCsch[c*x]))/(105*e^2)))/Sqrt[d + e*x]) - (2*Sqrt[e + d/x]*Sqrt[c*x]*(-((Sqrt[2]*(9*c^3*d^3*e + c*d*e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(8*c^4*d^4 - 5*c^2*d^2*e^2 - 9*e^4)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*(-5*c^3*d^3*e - 9*c*d*e^3)*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(c*d*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(105*e^3*Sqrt[d + e*x])))/c^4","C",0
52,1,418,679,1.678277,"\int x \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]),x]","\frac{1}{15} \left(\frac{2 a \sqrt{d+e x} \left(-2 d^2+d e x+3 e^2 x^2\right)}{e^2}+\frac{4 b x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}{c}+\frac{4 i b \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left(\left(c^2 d^2-2 i c d e+e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)-2 c^2 d^2 \Pi \left(1-\frac{i e}{c d};i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)+2 c d (c d+i e) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)\right)}{c^3 e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{c}{c d-i e}}}+\frac{2 b \text{csch}^{-1}(c x) \sqrt{d+e x} \left(-2 d^2+d e x+3 e^2 x^2\right)}{e^2}\right)","-\frac{2 d (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^2}+\frac{2 (d+e x)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 e^2}+\frac{8 b d^3 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{15 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \left(c^2 d^2+e^2\right) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 \left(-c^2\right)^{5/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{8 b c d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 \left(-c^2\right)^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{8 b c d \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 \left(-c^2\right)^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{4 b \left(c^2 x^2+1\right) \sqrt{d+e x}}{15 c^3 x \sqrt{\frac{1}{c^2 x^2}+1}}",1,"((4*b*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + (2*a*Sqrt[d + e*x]*(-2*d^2 + d*e*x + 3*e^2*x^2))/e^2 + (2*b*Sqrt[d + e*x]*(-2*d^2 + d*e*x + 3*e^2*x^2)*ArcCsch[c*x])/e^2 + ((4*I)*b*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(2*c*d*(c*d + I*e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + (c^2*d^2 - (2*I)*c*d*e + e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] - 2*c^2*d^2*EllipticPi[1 - (I*e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)]))/(c^3*Sqrt[-(c/(c*d - I*e))]*e^2*Sqrt[1 + 1/(c^2*x^2)]*x))/15","C",1
53,1,926,429,14.0770574,"\int \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x]*(a + b*ArcCsch[c*x]),x]","\frac{2 a (d+e x)^{3/2}}{3 e}+\frac{b \left(-\frac{(c d+c e x) \left(-\frac{2}{3} c x \text{csch}^{-1}(c x)-\frac{2 c d \text{csch}^{-1}(c x)}{3 e}-\frac{4}{3} \sqrt{1+\frac{1}{c^2 x^2}}\right)}{\sqrt{d+e x}}-\frac{2 (c d+c e x) \left(-\frac{\sqrt{2} c d e \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left(c^2 d^2+e^2\right) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{2 e \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}\right)}{3 e \sqrt{\frac{d}{x}+e} \sqrt{c x} \sqrt{d+e x}}\right)}{c^2}","\frac{2 (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e}-\frac{4 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{3 c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 \left(-c^2\right)^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 \left(-c^2\right)^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}}",1,"(2*a*(d + e*x)^(3/2))/(3*e) + (b*(-(((c*d + c*e*x)*((-4*Sqrt[1 + 1/(c^2*x^2)])/3 - (2*c*d*ArcCsch[c*x])/(3*e) - (2*c*x*ArcCsch[c*x])/3))/Sqrt[d + e*x]) - (2*(c*d + c*e*x)*(-((Sqrt[2]*c*d*e*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(c^2*d^2 + e^2)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*e*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(3*e*Sqrt[e + d/x]*Sqrt[c*x]*Sqrt[d + e*x])))/c^2","C",1
54,0,0,24,19.5114659,"\int \frac{\sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x,x]","\int \frac{\sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{x},x\right)",0,"Integrate[(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x, x]","A",-1
55,0,0,24,7.9670018,"\int \frac{\sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x^2,x]","\int \frac{\sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x^2, x]","A",-1
56,1,380,486,1.6073891,"\int (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]),x]","\frac{2 \left(3 a (d+e x)^{5/2}+\frac{2 b e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}{c}+\frac{2 i b \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left(\left(-9 c^2 d^2-7 i c d e+e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)+3 c^2 d^2 \Pi \left(1-\frac{i e}{c d};i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)+7 c d (c d+i e) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)\right)}{c^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{c}{c d-i e}}}+3 b \text{csch}^{-1}(c x) (d+e x)^{5/2}\right)}{15 e}","\frac{2 (d+e x)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 e}-\frac{4 b d^3 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{5 c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{4 b c \sqrt{c^2 x^2+1} \left(2 c^2 d^2-e^2\right) \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 \left(-c^2\right)^{5/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{28 b c d \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 \left(-c^2\right)^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}}+\frac{4 b e \left(c^2 x^2+1\right) \sqrt{d+e x}}{15 c^3 x \sqrt{\frac{1}{c^2 x^2}+1}}",1,"(2*((2*b*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + 3*a*(d + e*x)^(5/2) + 3*b*(d + e*x)^(5/2)*ArcCsch[c*x] + ((2*I)*b*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(7*c*d*(c*d + I*e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + (-9*c^2*d^2 - (7*I)*c*d*e + e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + 3*c^2*d^2*EllipticPi[1 - (I*e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)]))/(c^3*Sqrt[-(c/(c*d - I*e))]*Sqrt[1 + 1/(c^2*x^2)]*x)))/(15*e)","C",1
57,1,1098,939,14.3020204,"\int \frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x}} \, dx","Integrate[(x^3*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x],x]","\frac{a \sqrt{\frac{e x}{d}+1} B_{-\frac{e x}{d}}\left(4,\frac{1}{2}\right) d^4}{e^4 \sqrt{d+e x}}+\frac{b \left(\frac{2 \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(-\frac{\sqrt{2} \left(40 c^3 d^3 e-8 c d e^3\right) \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left(48 c^4 d^4-16 c^2 e^2 d^2+9 e^4\right) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{2 \left(9 c d e^3-16 c^3 d^3 e\right) \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right)}{c d \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}\right)}{105 e^4 \sqrt{d+e x}}-\frac{c \left(\frac{d}{x}+e\right) x \left(\frac{32 c^3 \text{csch}^{-1}(c x) d^3}{35 e^4}-\frac{2 c^3 x^3 \text{csch}^{-1}(c x)}{7 e}-\frac{4 c^2 x^2 \left(e \sqrt{1+\frac{1}{c^2 x^2}}-3 c d \text{csch}^{-1}(c x)\right)}{35 e^2}+\frac{4 c x \left(5 c d e \sqrt{1+\frac{1}{c^2 x^2}}-12 c^2 d^2 \text{csch}^{-1}(c x)\right)}{105 e^3}+\frac{4 \left(9 e^2-16 c^2 d^2\right) \sqrt{1+\frac{1}{c^2 x^2}}}{105 e^3}\right)}{\sqrt{d+e x}}\right)}{c^4}","\frac{64 b \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \sqrt{c^2 x^2+1} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right) d^4}{35 c e^4 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right) d^3}{e^4}-\frac{64 b c \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right) d^3}{35 \left(-c^2\right)^{3/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{2 (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) d^2}{e^4}+\frac{24 b c \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right) d^2}{35 \left(-c^2\right)^{3/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{4 b \sqrt{d+e x} \left(c^2 x^2+1\right) d}{21 c^3 e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}-\frac{6 (d+e x)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right) d}{5 e^4}-\frac{32 b c \left(c^2 d^2+e^2\right) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{c^2 x^2+1} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right) d}{105 \left(-c^2\right)^{5/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}+\frac{4 b \sqrt{d+e x} \left(c^2 x^2+1\right)}{35 c^3 e \sqrt{1+\frac{1}{c^2 x^2}}}+\frac{2 (d+e x)^{7/2} \left(a+b \text{csch}^{-1}(c x)\right)}{7 e^4}+\frac{4 b c \left(2 c^2 d^2+9 e^2\right) \sqrt{d+e x} \sqrt{c^2 x^2+1} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{105 \left(-c^2\right)^{5/2} e^3 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}",1,"(a*d^4*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 4, 1/2])/(e^4*Sqrt[d + e*x]) + (b*(-((c*(e + d/x)*x*((4*(-16*c^2*d^2 + 9*e^2)*Sqrt[1 + 1/(c^2*x^2)])/(105*e^3) + (32*c^3*d^3*ArcCsch[c*x])/(35*e^4) - (2*c^3*x^3*ArcCsch[c*x])/(7*e) - (4*c^2*x^2*(e*Sqrt[1 + 1/(c^2*x^2)] - 3*c*d*ArcCsch[c*x]))/(35*e^2) + (4*c*x*(5*c*d*e*Sqrt[1 + 1/(c^2*x^2)] - 12*c^2*d^2*ArcCsch[c*x]))/(105*e^3)))/Sqrt[d + e*x]) + (2*Sqrt[e + d/x]*Sqrt[c*x]*(-((Sqrt[2]*(40*c^3*d^3*e - 8*c*d*e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(48*c^4*d^4 - 16*c^2*d^2*e^2 + 9*e^4)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*(-16*c^3*d^3*e + 9*c*d*e^3)*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(c*d*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(105*e^4*Sqrt[d + e*x])))/c^4","C",0
58,1,1012,707,14.4256414,"\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x}} \, dx","Integrate[(x^2*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x],x]","\frac{b \left(-\frac{c \left(\frac{d}{x}+e\right) x \left(-\frac{16 c^2 \text{csch}^{-1}(c x) d^2}{15 e^3}+\frac{4 c \sqrt{1+\frac{1}{c^2 x^2}} d}{5 e^2}-\frac{2 c^2 x^2 \text{csch}^{-1}(c x)}{5 e}-\frac{4 c x \left(e \sqrt{1+\frac{1}{c^2 x^2}}-2 c d \text{csch}^{-1}(c x)\right)}{15 e^2}\right)}{\sqrt{d+e x}}-\frac{2 \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(-\frac{\sqrt{2} \left(7 c^2 d^2 e-e^3\right) \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left(8 c^3 d^3-3 c d e^2\right) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{6 c d e \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}\right)}{15 e^3 \sqrt{d+e x}}\right)}{c^3}-\frac{a d^3 \sqrt{\frac{e x}{d}+1} B_{-\frac{e x}{d}}\left(3,\frac{1}{2}\right)}{e^3 \sqrt{d+e x}}","\frac{2 d^2 \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{e^3}-\frac{4 d (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^3}+\frac{2 (d+e x)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 e^3}-\frac{32 b d^3 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{15 c e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{32 b c d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 \left(-c^2\right)^{3/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \left(c^2 d^2+e^2\right) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 \left(-c^2\right)^{5/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{4 b c d \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{5 \left(-c^2\right)^{3/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{4 b \left(c^2 x^2+1\right) \sqrt{d+e x}}{15 c^3 e x \sqrt{\frac{1}{c^2 x^2}+1}}",1,"-((a*d^3*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 3, 1/2])/(e^3*Sqrt[d + e*x])) + (b*(-((c*(e + d/x)*x*((4*c*d*Sqrt[1 + 1/(c^2*x^2)])/(5*e^2) - (16*c^2*d^2*ArcCsch[c*x])/(15*e^3) - (2*c^2*x^2*ArcCsch[c*x])/(5*e) - (4*c*x*(e*Sqrt[1 + 1/(c^2*x^2)] - 2*c*d*ArcCsch[c*x]))/(15*e^2)))/Sqrt[d + e*x]) - (2*Sqrt[e + d/x]*Sqrt[c*x]*(-((Sqrt[2]*(7*c^2*d^2*e - e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(8*c^3*d^3 - 3*c*d*e^2)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (6*c*d*e*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(15*e^3*Sqrt[d + e*x])))/c^3","C",1
59,1,343,474,1.3350392,"\int \frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x}} \, dx","Integrate[(x*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x],x]","\frac{2 \left(a \sqrt{d+e x} (e x-2 d)+\frac{2 b \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left((e+i c d) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)+(-e+i c d) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)-2 i c d \Pi \left(1-\frac{i e}{c d};i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)\right)}{c^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{c}{c d-i e}}}+b \text{csch}^{-1}(c x) \sqrt{d+e x} (e x-2 d)\right)}{3 e^2}","-\frac{2 d \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{e^2}+\frac{2 (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^2}+\frac{8 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{3 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{8 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 \left(-c^2\right)^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 \left(-c^2\right)^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}",1,"(2*(a*(-2*d + e*x)*Sqrt[d + e*x] + b*(-2*d + e*x)*Sqrt[d + e*x]*ArcCsch[c*x] + (2*b*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*((I*c*d - e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + (I*c*d + e)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] - (2*I)*c*d*EllipticPi[1 - (I*e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)]))/(c^2*Sqrt[-(c/(c*d - I*e))]*Sqrt[1 + 1/(c^2*x^2)]*x)))/(3*e^2)","C",1
60,1,307,284,5.2238777,"\int \frac{a+b \text{csch}^{-1}(c x)}{\sqrt{d+e x}} \, dx","Integrate[(a + b*ArcCsch[c*x])/Sqrt[d + e*x],x]","\frac{2 \left(a e (d+e x)-\frac{b \left(\frac{d}{x}+e\right) \left(-c e x \text{csch}^{-1}(c x)+\frac{\sqrt{2} \sqrt{1+i c x} \left(c d (e+i c d) \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{c e (c x+i) (d+e x)}{(e+i c d)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)-e^2 (c x+i) \sqrt{\frac{c (d+e x)}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{e (c x+i)}{c d-i e}} (c d+c e x)}\right)}{c}\right)}{e^2 \sqrt{d+e x}}","\frac{2 \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{e}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{\left(-c^2\right)^{3/2} x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{4 b d \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}",1,"(2*(a*e*(d + e*x) - (b*(e + d/x)*(-(c*e*x*ArcCsch[c*x]) + (Sqrt[2]*Sqrt[1 + I*c*x]*(-(e^2*(I + c*x)*Sqrt[(c*(d + e*x))/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]) + c*d*(I*c*d + e)*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(c*e*(I + c*x)*(d + e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(c*d + c*e*x))))/c))/(e^2*Sqrt[d + e*x])","C",1
61,0,0,24,6.5512163,"\int \frac{a+b \text{csch}^{-1}(c x)}{x \sqrt{d+e x}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*Sqrt[d + e*x]),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x \sqrt{d+e x}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x \sqrt{d+e x}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x*Sqrt[d + e*x]), x]","A",-1
62,0,0,24,9.2143335,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \sqrt{d+e x}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^2*Sqrt[d + e*x]),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \sqrt{d+e x}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x^2 \sqrt{d+e x}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x^2*Sqrt[d + e*x]), x]","A",-1
63,1,1042,731,14.6986164,"\int \frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{(d+e x)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x)^(3/2),x]","\frac{a \left(\frac{e x}{d}+1\right)^{3/2} B_{-\frac{e x}{d}}\left(4,-\frac{1}{2}\right) d^4}{e^4 (d+e x)^{3/2}}+\frac{b \left(-\frac{c^2 \left(\frac{d}{x}+e\right)^2 \left(\frac{2 c^2 \text{csch}^{-1}(c x) d^2}{e^3 \left(\frac{d}{x}+e\right)}-\frac{32 c^2 \text{csch}^{-1}(c x) d^2}{5 e^4}+\frac{32 c \sqrt{1+\frac{1}{c^2 x^2}} d}{15 e^3}-\frac{2 c^2 x^2 \text{csch}^{-1}(c x)}{5 e^2}-\frac{2 c x \left(2 e \sqrt{1+\frac{1}{c^2 x^2}}-9 c d \text{csch}^{-1}(c x)\right)}{15 e^3}\right) x^2}{(d+e x)^{3/2}}-\frac{2 \left(\frac{d}{x}+e\right)^{3/2} (c x)^{3/2} \left(-\frac{\sqrt{2} \left(32 c^2 d^2 e-e^3\right) \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left(48 c^3 d^3-8 c d e^2\right) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{16 c d e \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}\right)}{15 e^4 (d+e x)^{3/2}}\right)}{c^4}","\frac{2 d^3 \left(a+b \text{csch}^{-1}(c x)\right)}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{e^4}-\frac{2 d (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{e^4}+\frac{2 (d+e x)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 e^4}-\frac{64 b d^3 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{5 c e^4 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{8 b c d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{\left(-c^2\right)^{3/2} e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{4 b c \sqrt{c^2 x^2+1} \left(2 c^2 d^2-e^2\right) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 \left(-c^2\right)^{5/2} e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{32 b c d \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 \left(-c^2\right)^{3/2} e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{4 b \left(c^2 x^2+1\right) \sqrt{d+e x}}{15 c^3 e^2 x \sqrt{\frac{1}{c^2 x^2}+1}}",1,"(a*d^4*(1 + (e*x)/d)^(3/2)*Beta[-((e*x)/d), 4, -1/2])/(e^4*(d + e*x)^(3/2)) + (b*(-((c^2*(e + d/x)^2*x^2*((32*c*d*Sqrt[1 + 1/(c^2*x^2)])/(15*e^3) - (32*c^2*d^2*ArcCsch[c*x])/(5*e^4) + (2*c^2*d^2*ArcCsch[c*x])/(e^3*(e + d/x)) - (2*c^2*x^2*ArcCsch[c*x])/(5*e^2) - (2*c*x*(2*e*Sqrt[1 + 1/(c^2*x^2)] - 9*c*d*ArcCsch[c*x]))/(15*e^3)))/(d + e*x)^(3/2)) - (2*(e + d/x)^(3/2)*(c*x)^(3/2)*(-((Sqrt[2]*(32*c^2*d^2*e - e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(48*c^3*d^3 - 8*c*d*e^2)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (16*c*d*e*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(15*e^4*(d + e*x)^(3/2))))/c^4","C",1
64,1,979,499,14.4942841,"\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{(d+e x)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x)^(3/2),x]","\frac{b \left(\frac{2 \left(\frac{d}{x}+e\right)^{3/2} (c x)^{3/2} \left(-\frac{5 \sqrt{2} c d e \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left(8 c^2 d^2-e^2\right) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 e \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}\right)}{3 e^3 (d+e x)^{3/2}}-\frac{c^2 \left(\frac{d}{x}+e\right)^2 x^2 \left(-\frac{2 c x \text{csch}^{-1}(c x)}{3 e^2}-\frac{2 c d \text{csch}^{-1}(c x)}{e^2 \left(\frac{d}{x}+e\right)}+\frac{16 c d \text{csch}^{-1}(c x)}{3 e^3}-\frac{4 \sqrt{1+\frac{1}{c^2 x^2}}}{3 e^2}\right)}{(d+e x)^{3/2}}\right)}{c^3}-\frac{a d^3 \left(\frac{e x}{d}+1\right)^{3/2} B_{-\frac{e x}{d}}\left(3,-\frac{1}{2}\right)}{e^3 (d+e x)^{3/2}}","-\frac{2 d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{e^3}+\frac{2 (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^3}+\frac{32 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{3 c e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{20 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 \left(-c^2\right)^{3/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 \left(-c^2\right)^{3/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}",1,"-((a*d^3*(1 + (e*x)/d)^(3/2)*Beta[-((e*x)/d), 3, -1/2])/(e^3*(d + e*x)^(3/2))) + (b*(-((c^2*(e + d/x)^2*x^2*((-4*Sqrt[1 + 1/(c^2*x^2)])/(3*e^2) + (16*c*d*ArcCsch[c*x])/(3*e^3) - (2*c*d*ArcCsch[c*x])/(e^2*(e + d/x)) - (2*c*x*ArcCsch[c*x])/(3*e^2)))/(d + e*x)^(3/2)) + (2*(e + d/x)^(3/2)*(c*x)^(3/2)*((-5*Sqrt[2]*c*d*e*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)]) + (I*Sqrt[2]*(c*d - I*e)*(8*c^2*d^2 - e^2)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*e*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(3*e^3*(d + e*x)^(3/2))))/c^3","C",1
65,1,264,318,1.5849948,"\int \frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{(d+e x)^{3/2}} \, dx","Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x)^(3/2),x]","\frac{2 \left(\frac{a (2 d+e x)}{\sqrt{d+e x}}-\frac{2 i b \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left(F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)-2 \Pi \left(1-\frac{i e}{c d};i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)\right)}{c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{c}{c d-i e}}}+\frac{b \text{csch}^{-1}(c x) (2 d+e x)}{\sqrt{d+e x}}\right)}{e^2}","\frac{2 \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{e^2}+\frac{2 d \left(a+b \text{csch}^{-1}(c x)\right)}{e^2 \sqrt{d+e x}}-\frac{8 b d \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{\left(-c^2\right)^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}",1,"(2*((a*(2*d + e*x))/Sqrt[d + e*x] + (b*(2*d + e*x)*ArcCsch[c*x])/Sqrt[d + e*x] - ((2*I)*b*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] - 2*EllipticPi[1 - (I*e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)]))/(c*Sqrt[-(c/(c*d - I*e))]*Sqrt[1 + 1/(c^2*x^2)]*x)))/e^2","C",1
66,1,166,149,0.6816261,"\int \frac{a+b \text{csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x)^(3/2),x]","\frac{-2 e \left(c^2 x^2+1\right) \left(a+b \text{csch}^{-1}(c x)\right)+2 b c x \sqrt{\frac{2}{c^2 x^2}+2} \sqrt{1+i c x} (e+i c d) \sqrt{\frac{c e (c x+i) (d+e x)}{(e+i c d)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e^2 \left(c^2 x^2+1\right) \sqrt{d+e x}}","\frac{4 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{2 \left(a+b \text{csch}^{-1}(c x)\right)}{e \sqrt{d+e x}}",1,"(-2*e*(1 + c^2*x^2)*(a + b*ArcCsch[c*x]) + 2*b*c*(I*c*d + e)*Sqrt[2 + 2/(c^2*x^2)]*x*Sqrt[1 + I*c*x]*Sqrt[(c*e*(I + c*x)*(d + e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e^2*Sqrt[d + e*x]*(1 + c^2*x^2))","C",1
67,0,0,24,12.1687793,"\int \frac{a+b \text{csch}^{-1}(c x)}{x (d+e x)^{3/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x)^(3/2)),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x (d+e x)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x (d+e x)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x)^(3/2)), x]","A",-1
68,0,0,24,15.6780059,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(3/2)),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(3/2)), x]","A",-1
69,1,1108,777,14.4874314,"\int \frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{(d+e x)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x)^(5/2),x]","\frac{a \left(\frac{e x}{d}+1\right)^{5/2} B_{-\frac{e x}{d}}\left(4,-\frac{3}{2}\right) d^4}{e^4 (d+e x)^{5/2}}+\frac{b \left(\frac{2 \left(\frac{d}{x}+e\right)^{5/2} (c x)^{5/2} \left(\frac{2 \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right) e^3}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}-\frac{\sqrt{2} \left(8 c^3 e d^3+8 c e^3 d\right) \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left(16 c^4 d^4+16 c^2 e^2 d^2-e^4\right) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} e}\right)}{3 e^4 \left(c^2 d^2+e^2\right) (d+e x)^{5/2}}-\frac{c^3 \left(\frac{d}{x}+e\right)^3 x^3 \left(-\frac{2 c x \text{csch}^{-1}(c x)}{3 e^3}-\frac{2 c d \text{csch}^{-1}(c x)}{3 e^2 \left(\frac{d}{x}+e\right)^2}+\frac{32 c d \text{csch}^{-1}(c x)}{3 e^4}-\frac{2 \left(7 c^3 \text{csch}^{-1}(c x) d^3+2 c^2 e \sqrt{1+\frac{1}{c^2 x^2}} d^2+7 c e^2 \text{csch}^{-1}(c x) d\right)}{3 e^3 \left(c^2 d^2+e^2\right) \left(\frac{d}{x}+e\right)}-\frac{4 \sqrt{1+\frac{1}{c^2 x^2}}}{3 e \left(c^2 d^2+e^2\right)}\right)}{(d+e x)^{5/2}}\right)}{c^4}","\frac{2 d^3 \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^4 (d+e x)^{3/2}}-\frac{6 d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{e^4 \sqrt{d+e x}}-\frac{6 d \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{e^4}+\frac{2 (d+e x)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^4}+\frac{64 b d^2 \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{3 c e^4 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b d^2 \left(c^2 x^2+1\right)}{3 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{d+e x}}-\frac{8 b \sqrt{-c^2} d^2 \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 c e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{4 b c \sqrt{c^2 x^2+1} \left(2 c^2 d^2+e^2\right) \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 \left(-c^2\right)^{3/2} e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{32 b c d \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 \left(-c^2\right)^{3/2} e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}",1,"(a*d^4*(1 + (e*x)/d)^(5/2)*Beta[-((e*x)/d), 4, -3/2])/(e^4*(d + e*x)^(5/2)) + (b*(-((c^3*(e + d/x)^3*x^3*((-4*Sqrt[1 + 1/(c^2*x^2)])/(3*e*(c^2*d^2 + e^2)) + (32*c*d*ArcCsch[c*x])/(3*e^4) - (2*c*d*ArcCsch[c*x])/(3*e^2*(e + d/x)^2) - (2*c*x*ArcCsch[c*x])/(3*e^3) - (2*(2*c^2*d^2*e*Sqrt[1 + 1/(c^2*x^2)] + 7*c^3*d^3*ArcCsch[c*x] + 7*c*d*e^2*ArcCsch[c*x]))/(3*e^3*(c^2*d^2 + e^2)*(e + d/x))))/(d + e*x)^(5/2)) + (2*(e + d/x)^(5/2)*(c*x)^(5/2)*(-((Sqrt[2]*(8*c^3*d^3*e + 8*c*d*e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(16*c^4*d^4 + 16*c^2*d^2*e^2 - e^4)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*e^3*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(3*e^4*(c^2*d^2 + e^2)*(d + e*x)^(5/2))))/c^4","C",1
70,1,1076,569,14.3826591,"\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{(d+e x)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x)^(5/2),x]","\frac{b \left(-\frac{c^3 \left(\frac{d}{x}+e\right)^3 \left(-\frac{4 c \sqrt{1+\frac{1}{c^2 x^2}} d}{3 e^2 \left(c^2 d^2+e^2\right)}+\frac{2 \text{csch}^{-1}(c x)}{3 e \left(\frac{d}{x}+e\right)^2}-\frac{16 \text{csch}^{-1}(c x)}{3 e^3}+\frac{4 \left(2 c^2 \text{csch}^{-1}(c x) d^2+c e \sqrt{1+\frac{1}{c^2 x^2}} d+2 e^2 \text{csch}^{-1}(c x)\right)}{3 e^2 \left(c^2 d^2+e^2\right) \left(\frac{d}{x}+e\right)}\right) x^3}{(d+e x)^{5/2}}-\frac{2 \left(\frac{d}{x}+e\right)^{5/2} (c x)^{5/2} \left(-\frac{\sqrt{2} \left(3 e^3+3 c^2 d^2 e\right) \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left(8 c^3 d^3+9 c e^2 d\right) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{2 c d e \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}\right)}{3 e^3 \left(c^2 d^2+e^2\right) (d+e x)^{5/2}}\right)}{c^3}-\frac{a d^3 \left(\frac{e x}{d}+1\right)^{5/2} B_{-\frac{e x}{d}}\left(3,-\frac{3}{2}\right)}{e^3 (d+e x)^{5/2}}","-\frac{2 d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left(a+b \text{csch}^{-1}(c x)\right)}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left(a+b \text{csch}^{-1}(c x)\right)}{e^3}-\frac{4 b d \left(c^2 x^2+1\right)}{3 c e x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{d+e x}}+\frac{4 b \sqrt{-c^2} d \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}-\frac{32 b d \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{3 c e^3 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{\left(-c^2\right)^{3/2} e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}",1,"-((a*d^3*(1 + (e*x)/d)^(5/2)*Beta[-((e*x)/d), 3, -3/2])/(e^3*(d + e*x)^(5/2))) + (b*(-((c^3*(e + d/x)^3*x^3*((-4*c*d*Sqrt[1 + 1/(c^2*x^2)])/(3*e^2*(c^2*d^2 + e^2)) - (16*ArcCsch[c*x])/(3*e^3) + (2*ArcCsch[c*x])/(3*e*(e + d/x)^2) + (4*(c*d*e*Sqrt[1 + 1/(c^2*x^2)] + 2*c^2*d^2*ArcCsch[c*x] + 2*e^2*ArcCsch[c*x]))/(3*e^2*(c^2*d^2 + e^2)*(e + d/x))))/(d + e*x)^(5/2)) - (2*(e + d/x)^(5/2)*(c*x)^(5/2)*(-((Sqrt[2]*(3*c^2*d^2*e + 3*e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(8*c^3*d^3 + 9*c*d*e^2)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*c*d*e*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(3*e^3*(c^2*d^2 + e^2)*(d + e*x)^(5/2))))/c^3","C",1
71,1,390,393,2.5136563,"\int \frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{(d+e x)^{5/2}} \, dx","Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x)^(5/2),x]","\frac{2}{3} \left(-\frac{a (2 d+3 e x)}{e^2 (d+e x)^{3/2}}+\frac{2 b c x \sqrt{\frac{1}{c^2 x^2}+1}}{\left(c^2 d^2+e^2\right) \sqrt{d+e x}}+\frac{2 i b \sqrt{-\frac{c}{c d-i e}} \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left(-c d F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)+c d E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)+2 (c d-i e) \Pi \left(1-\frac{i e}{c d};i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right)|\frac{c d-i e}{c d+i e}\right)\right)}{c^2 d e^2 x \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{b \text{csch}^{-1}(c x) (2 d+3 e x)}{e^2 (d+e x)^{3/2}}\right)","-\frac{2 \left(a+b \text{csch}^{-1}(c x)\right)}{e^2 \sqrt{d+e x}}+\frac{2 d \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^2 (d+e x)^{3/2}}+\frac{4 b \left(c^2 x^2+1\right)}{3 c x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{d+e x}}-\frac{4 b \sqrt{-c^2} \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 c e x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}}}+\frac{8 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{3 c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}",1,"(2*((2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x)/((c^2*d^2 + e^2)*Sqrt[d + e*x]) - (a*(2*d + 3*e*x))/(e^2*(d + e*x)^(3/2)) - (b*(2*d + 3*e*x)*ArcCsch[c*x])/(e^2*(d + e*x)^(3/2)) + ((2*I)*b*Sqrt[-(c/(c*d - I*e))]*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(c*d*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] - c*d*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + 2*(c*d - I*e)*EllipticPi[1 - (I*e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)]))/(c^2*d*e^2*Sqrt[1 + 1/(c^2*x^2)]*x)))/3","C",1
72,1,892,369,14.1018392,"\int \frac{a+b \text{csch}^{-1}(c x)}{(d+e x)^{5/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x)^(5/2),x]","\frac{b \left(\frac{2 \left(\frac{d}{x}+e\right)^{5/2} (c x)^{5/2} \left(\frac{i \sqrt{2} c d (c d-i e) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 e \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right)}{c d \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}\right)}{3 e \left(c^2 d^2+e^2\right) (d+e x)^{5/2}}-\frac{c^3 \left(\frac{d}{x}+e\right)^3 x^3 \left(\frac{2 \text{csch}^{-1}(c x)}{3 c^2 d^2 e}+\frac{2 e \text{csch}^{-1}(c x)}{3 c^2 d^2 \left(\frac{d}{x}+e\right)^2}-\frac{4 \left(c^2 \text{csch}^{-1}(c x) d^2-c e \sqrt{1+\frac{1}{c^2 x^2}} d+e^2 \text{csch}^{-1}(c x)\right)}{3 c^2 d^2 \left(c^2 d^2+e^2\right) \left(\frac{d}{x}+e\right)}-\frac{4 \sqrt{1+\frac{1}{c^2 x^2}}}{3 c d \left(c^2 d^2+e^2\right)}\right)}{(d+e x)^{5/2}}\right)}{c}-\frac{2 a}{3 e (d+e x)^{3/2}}","-\frac{2 \left(a+b \text{csch}^{-1}(c x)\right)}{3 e (d+e x)^{3/2}}-\frac{4 b e \left(c^2 x^2+1\right)}{3 c d x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{d+e x}}+\frac{4 b \sqrt{-c^2} \sqrt{c^2 x^2+1} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{3 c d x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}}+\frac{4 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{3 c d e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}",1,"(-2*a)/(3*e*(d + e*x)^(3/2)) + (b*(-((c^3*(e + d/x)^3*x^3*((-4*Sqrt[1 + 1/(c^2*x^2)])/(3*c*d*(c^2*d^2 + e^2)) + (2*ArcCsch[c*x])/(3*c^2*d^2*e) + (2*e*ArcCsch[c*x])/(3*c^2*d^2*(e + d/x)^2) - (4*(-(c*d*e*Sqrt[1 + 1/(c^2*x^2)]) + c^2*d^2*ArcCsch[c*x] + e^2*ArcCsch[c*x]))/(3*c^2*d^2*(c^2*d^2 + e^2)*(e + d/x))))/(d + e*x)^(5/2)) + (2*(e + d/x)^(5/2)*(c*x)^(5/2)*((I*Sqrt[2]*c*d*(c*d - I*e)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*e*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(c*d*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(3*e*(c^2*d^2 + e^2)*(d + e*x)^(5/2))))/c","C",1
73,0,0,24,28.8448553,"\int \frac{a+b \text{csch}^{-1}(c x)}{x (d+e x)^{5/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x)^(5/2)),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x (d+e x)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x (d+e x)^{5/2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x)^(5/2)), x]","A",-1
74,0,0,24,28.471311,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(5/2)),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x^2 (d+e x)^{5/2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(5/2)), x]","A",-1
75,1,1217,648,14.4942877,"\int \frac{a+b \text{csch}^{-1}(c x)}{(d+e x)^{7/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x)^(7/2),x]","\frac{b \left(\frac{2 \left(\frac{d}{x}+e\right)^{7/2} (c x)^{7/2} \left(-\frac{\sqrt{2} \left(e^3+c^2 d^2 e\right) \sqrt{i c x+1} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{\sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2} \sqrt{\frac{e (1-i c x)}{i c d+e}}}+\frac{i \sqrt{2} (c d-i e) \left(3 c^3 d^3-c d e^2\right) \sqrt{i c x+1} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)}{e \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{2 \left(-3 e^3-7 c^2 d^2 e\right) \cosh \left(2 \text{csch}^{-1}(c x)\right) \left(\frac{c x \left(c d \sqrt{2 i c x+2} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} F\left(\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)+2 \sqrt{-\frac{e (c x-i)}{c d+i e}} (c x+i) \sqrt{\frac{c d+c e x}{c d-i e}} \left((c d+i e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)-i e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-i e}}\right)|\frac{c d-i e}{c d+i e}\right)\right)+(i c d+e) \sqrt{2 i c x+2} \sqrt{-\frac{e (c x+i)}{c d-i e}} \sqrt{\frac{e (c x+i) (c d+c e x)}{(i c d+e)^2}} \Pi \left(\frac{i c d}{e}+1;\sin ^{-1}\left(\sqrt{-\frac{e (c x+i)}{c d-i e}}\right)|\frac{i c d+e}{2 e}\right)\right)}{2 \sqrt{-\frac{e (c x+i)}{c d-i e}}}-(c d+c e x) \left(c^2 x^2+1\right)\right)}{c d \sqrt{1+\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2+2\right)}\right)}{15 c d e \left(c^2 d^2+e^2\right)^2 (d+e x)^{7/2}}-\frac{c^4 \left(\frac{d}{x}+e\right)^4 x^4 \left(-\frac{2 \text{csch}^{-1}(c x) e^2}{5 c^3 d^3 \left(\frac{d}{x}+e\right)^3}+\frac{2 \left(9 \text{csch}^{-1}(c x) e^3-2 c d \sqrt{1+\frac{1}{c^2 x^2}} e^2+9 c^2 d^2 \text{csch}^{-1}(c x) e\right)}{15 c^3 d^3 \left(c^2 d^2+e^2\right) \left(\frac{d}{x}+e\right)^2}-\frac{2 \left(9 c^4 \text{csch}^{-1}(c x) d^4-16 c^3 e \sqrt{1+\frac{1}{c^2 x^2}} d^3+18 c^2 e^2 \text{csch}^{-1}(c x) d^2-8 c e^3 \sqrt{1+\frac{1}{c^2 x^2}} d+9 e^4 \text{csch}^{-1}(c x)\right)}{15 c^3 d^3 \left(c^2 d^2+e^2\right)^2 \left(\frac{d}{x}+e\right)}-\frac{4 \left(7 c^2 d^2+3 e^2\right) \sqrt{1+\frac{1}{c^2 x^2}}}{15 c^2 d^2 \left(c^2 d^2+e^2\right)^2}+\frac{2 \text{csch}^{-1}(c x)}{5 c^3 d^3 e}\right)}{(d+e x)^{7/2}}\right)}{c}-\frac{2 a}{5 e (d+e x)^{5/2}}","-\frac{2 \left(a+b \text{csch}^{-1}(c x)\right)}{5 e (d+e x)^{5/2}}-\frac{4 b e \left(c^2 x^2+1\right)}{5 c d^2 x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{d+e x}}-\frac{16 b c e \left(c^2 x^2+1\right)}{15 x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right)^2 \sqrt{d+e x}}-\frac{4 b e \left(c^2 x^2+1\right)}{15 c d x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) (d+e x)^{3/2}}-\frac{4 b \sqrt{-c^2} \sqrt{c^2 x^2+1} \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right)}{15 c d x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right) \sqrt{d+e x}}-\frac{4 b c \sqrt{c^2 x^2+1} \left(7 c^2 d^2+3 e^2\right) \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 \sqrt{-c^2} e}{\sqrt{-c^2} e-c^2 d}\right)}{15 \sqrt{-c^2} d^2 x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 d^2+e^2\right)^2 \sqrt{\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}}}+\frac{4 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right)|\frac{2 e}{\sqrt{-c^2} d+e}\right)}{5 c d^2 e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}",1,"(-2*a)/(5*e*(d + e*x)^(5/2)) + (b*(-((c^4*(e + d/x)^4*x^4*((-4*(7*c^2*d^2 + 3*e^2)*Sqrt[1 + 1/(c^2*x^2)])/(15*c^2*d^2*(c^2*d^2 + e^2)^2) + (2*ArcCsch[c*x])/(5*c^3*d^3*e) - (2*e^2*ArcCsch[c*x])/(5*c^3*d^3*(e + d/x)^3) + (2*(-2*c*d*e^2*Sqrt[1 + 1/(c^2*x^2)] + 9*c^2*d^2*e*ArcCsch[c*x] + 9*e^3*ArcCsch[c*x]))/(15*c^3*d^3*(c^2*d^2 + e^2)*(e + d/x)^2) - (2*(-16*c^3*d^3*e*Sqrt[1 + 1/(c^2*x^2)] - 8*c*d*e^3*Sqrt[1 + 1/(c^2*x^2)] + 9*c^4*d^4*ArcCsch[c*x] + 18*c^2*d^2*e^2*ArcCsch[c*x] + 9*e^4*ArcCsch[c*x]))/(15*c^3*d^3*(c^2*d^2 + e^2)^2*(e + d/x))))/(d + e*x)^(7/2)) + (2*(e + d/x)^(7/2)*(c*x)^(7/2)*(-((Sqrt[2]*(c^2*d^2*e + e^3)*Sqrt[1 + I*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)*Sqrt[(e*(1 - I*c*x))/(I*c*d + e)])) + (I*Sqrt[2]*(c*d - I*e)*(3*c^3*d^3 - c*d*e^2)*Sqrt[1 + I*c*x]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)])/(e*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*(-7*c^2*d^2*e - 3*e^3)*Cosh[2*ArcCsch[c*x]]*(-((c*d + c*e*x)*(1 + c^2*x^2)) + (c*x*(c*d*Sqrt[2 + (2*I)*c*x]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*(I + c*x)*Sqrt[(c*d + c*e*x)/(c*d - I*e)]*((c*d + I*e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)]) + (I*c*d + e)*Sqrt[2 + (2*I)*c*x]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*Sqrt[(e*(I + c*x)*(c*d + c*e*x))/(I*c*d + e)^2]*EllipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(2*Sqrt[-((e*(I + c*x))/(c*d - I*e))])))/(c*d*Sqrt[1 + 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(2 + c^2*x^2))))/(15*c*d*e*(c^2*d^2 + e^2)^2*(d + e*x)^(7/2))))/c","C",0
76,1,138,214,0.2348739,"\int x^4 \left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)*(a + b*ArcCsch[c*x]),x]","\frac{48 a c^7 x^5 \left(7 d+5 e x^2\right)+48 b c^7 x^5 \text{csch}^{-1}(c x) \left(7 d+5 e x^2\right)+3 b \left(42 c^2 d-25 e\right) \log \left(x \left(\sqrt{\frac{1}{c^2 x^2}+1}+1\right)\right)+b c^2 x^2 \sqrt{\frac{1}{c^2 x^2}+1} \left(c^4 \left(84 d x^2+40 e x^4\right)-2 c^2 \left(63 d+25 e x^2\right)+75 e\right)}{1680 c^7}","\frac{1}{5} d x^5 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{7} e x^7 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b e x^6 \sqrt{-c^2 x^2-1}}{42 c \sqrt{-c^2 x^2}}-\frac{b x \left(42 c^2 d-25 e\right) \tan ^{-1}\left(\frac{c x}{\sqrt{-c^2 x^2-1}}\right)}{560 c^6 \sqrt{-c^2 x^2}}-\frac{b x^2 \sqrt{-c^2 x^2-1} \left(42 c^2 d-25 e\right)}{560 c^5 \sqrt{-c^2 x^2}}+\frac{b x^4 \sqrt{-c^2 x^2-1} \left(42 c^2 d-25 e\right)}{840 c^3 \sqrt{-c^2 x^2}}",1,"(48*a*c^7*x^5*(7*d + 5*e*x^2) + b*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2*(75*e - 2*c^2*(63*d + 25*e*x^2) + c^4*(84*d*x^2 + 40*e*x^4)) + 48*b*c^7*x^5*(7*d + 5*e*x^2)*ArcCsch[c*x] + 3*b*(42*c^2*d - 25*e)*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(1680*c^7)","A",1
77,1,119,167,0.1699564,"\int x^2 \left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)*(a + b*ArcCsch[c*x]),x]","\frac{c^2 x^2 \left(8 a c^3 x \left(5 d+3 e x^2\right)+b \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 \left(20 d+6 e x^2\right)-9 e\right)\right)+8 b c^5 x^3 \text{csch}^{-1}(c x) \left(5 d+3 e x^2\right)+b \left(9 e-20 c^2 d\right) \log \left(x \left(\sqrt{\frac{1}{c^2 x^2}+1}+1\right)\right)}{120 c^5}","\frac{1}{3} d x^3 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b e x^4 \sqrt{-c^2 x^2-1}}{20 c \sqrt{-c^2 x^2}}+\frac{b x \left(20 c^2 d-9 e\right) \tan ^{-1}\left(\frac{c x}{\sqrt{-c^2 x^2-1}}\right)}{120 c^4 \sqrt{-c^2 x^2}}+\frac{b x^2 \sqrt{-c^2 x^2-1} \left(20 c^2 d-9 e\right)}{120 c^3 \sqrt{-c^2 x^2}}",1,"(c^2*x^2*(8*a*c^3*x*(5*d + 3*e*x^2) + b*Sqrt[1 + 1/(c^2*x^2)]*(-9*e + c^2*(20*d + 6*e*x^2))) + 8*b*c^5*x^3*(5*d + 3*e*x^2)*ArcCsch[c*x] + b*(-20*c^2*d + 9*e)*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(120*c^5)","A",1
78,1,135,115,0.2606262,"\int \left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)*(a + b*ArcCsch[c*x]),x]","a d x+\frac{1}{3} a e x^3+\frac{b d x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{b e x^2 \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{6 c}-\frac{b e \log \left(x \left(\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right)\right)}{6 c^3}+b d x \text{csch}^{-1}(c x)+\frac{1}{3} b e x^3 \text{csch}^{-1}(c x)","d x \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \text{csch}^{-1}(c x)\right)-\frac{b x \left(6 c^2 d-e\right) \tan ^{-1}\left(\frac{c x}{\sqrt{-c^2 x^2-1}}\right)}{6 c^2 \sqrt{-c^2 x^2}}+\frac{b e x^2 \sqrt{-c^2 x^2-1}}{6 c \sqrt{-c^2 x^2}}",1,"a*d*x + (a*e*x^3)/3 + (b*e*x^2*Sqrt[(1 + c^2*x^2)/(c^2*x^2)])/(6*c) + b*d*x*ArcCsch[c*x] + (b*e*x^3*ArcCsch[c*x])/3 + (b*d*Sqrt[1 + 1/(c^2*x^2)]*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (b*e*Log[x*(1 + Sqrt[(1 + c^2*x^2)/(c^2*x^2)])])/(6*c^3)","A",1
79,1,89,91,0.1323327,"\int \frac{\left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^2,x]","-\frac{a d}{x}+a e x+b c d \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{b d \text{csch}^{-1}(c x)}{x}+b e x \text{csch}^{-1}(c x)","-\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{x}+e x \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b c d \sqrt{-c^2 x^2-1}}{\sqrt{-c^2 x^2}}-\frac{b e x \tan ^{-1}\left(\frac{c x}{\sqrt{-c^2 x^2-1}}\right)}{\sqrt{-c^2 x^2}}",1,"-((a*d)/x) + a*e*x + b*c*d*Sqrt[(1 + c^2*x^2)/(c^2*x^2)] - (b*d*ArcCsch[c*x])/x + b*e*x*ArcCsch[c*x] + (b*e*Sqrt[1 + 1/(c^2*x^2)]*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2]","A",1
80,1,68,109,0.083542,"\int \frac{\left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^4,x]","\frac{-3 a \left(d+3 e x^2\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(-2 c^2 d x^2+d+9 e x^2\right)-3 b \text{csch}^{-1}(c x) \left(d+3 e x^2\right)}{9 x^3}","-\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \text{csch}^{-1}(c x)\right)}{x}-\frac{b c \sqrt{-c^2 x^2-1} \left(2 c^2 d-9 e\right)}{9 \sqrt{-c^2 x^2}}+\frac{b c d \sqrt{-c^2 x^2-1}}{9 x^2 \sqrt{-c^2 x^2}}",1,"(-3*a*(d + 3*e*x^2) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(d - 2*c^2*d*x^2 + 9*e*x^2) - 3*b*(d + 3*e*x^2)*ArcCsch[c*x])/(9*x^3)","A",1
81,1,93,158,0.1243613,"\int \frac{\left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^6,x]","\frac{-15 a \left(3 d+5 e x^2\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(25 e x^2 \left(1-2 c^2 x^2\right)+3 d \left(8 c^4 x^4-4 c^2 x^2+3\right)\right)-15 b \text{csch}^{-1}(c x) \left(3 d+5 e x^2\right)}{225 x^5}","-\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{5 x^5}-\frac{e \left(a+b \text{csch}^{-1}(c x)\right)}{3 x^3}-\frac{b c \sqrt{-c^2 x^2-1} \left(12 c^2 d-25 e\right)}{225 x^2 \sqrt{-c^2 x^2}}+\frac{b c d \sqrt{-c^2 x^2-1}}{25 x^4 \sqrt{-c^2 x^2}}+\frac{2 b c^3 \sqrt{-c^2 x^2-1} \left(12 c^2 d-25 e\right)}{225 \sqrt{-c^2 x^2}}",1,"(-15*a*(3*d + 5*e*x^2) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(25*e*x^2*(1 - 2*c^2*x^2) + 3*d*(3 - 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d + 5*e*x^2)*ArcCsch[c*x])/(225*x^5)","A",1
82,1,109,205,0.1455957,"\int \frac{\left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^8,x]","\frac{-105 a \left(5 d+7 e x^2\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(49 e x^2 \left(8 c^4 x^4-4 c^2 x^2+3\right)-15 d \left(16 c^6 x^6-8 c^4 x^4+6 c^2 x^2-5\right)\right)-105 b \text{csch}^{-1}(c x) \left(5 d+7 e x^2\right)}{3675 x^7}","-\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{7 x^7}-\frac{e \left(a+b \text{csch}^{-1}(c x)\right)}{5 x^5}-\frac{b c \sqrt{-c^2 x^2-1} \left(30 c^2 d-49 e\right)}{1225 x^4 \sqrt{-c^2 x^2}}+\frac{b c d \sqrt{-c^2 x^2-1}}{49 x^6 \sqrt{-c^2 x^2}}-\frac{8 b c^5 \sqrt{-c^2 x^2-1} \left(30 c^2 d-49 e\right)}{3675 \sqrt{-c^2 x^2}}+\frac{4 b c^3 \sqrt{-c^2 x^2-1} \left(30 c^2 d-49 e\right)}{3675 x^2 \sqrt{-c^2 x^2}}",1,"(-105*a*(5*d + 7*e*x^2) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(49*e*x^2*(3 - 4*c^2*x^2 + 8*c^4*x^4) - 15*d*(-5 + 6*c^2*x^2 - 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(5*d + 7*e*x^2)*ArcCsch[c*x])/(3675*x^7)","A",1
83,1,114,204,0.2489399,"\int x^5 \left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^5*(d + e*x^2)*(a + b*ArcCsch[c*x]),x]","\frac{x \left(105 a x^5 \left(4 d+3 e x^2\right)+\frac{b \sqrt{\frac{1}{c^2 x^2}+1} \left(c^6 \left(84 d x^4+45 e x^6\right)-2 c^4 \left(56 d x^2+27 e x^4\right)+8 c^2 \left(28 d+9 e x^2\right)-144 e\right)}{c^7}+105 b x^5 \text{csch}^{-1}(c x) \left(4 d+3 e x^2\right)\right)}{2520}","\frac{1}{6} d x^6 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{8} e x^8 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b x \left(-c^2 x^2-1\right)^{5/2} \left(4 c^2 d-9 e\right)}{120 c^7 \sqrt{-c^2 x^2}}+\frac{b x \left(-c^2 x^2-1\right)^{3/2} \left(8 c^2 d-9 e\right)}{72 c^7 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(4 c^2 d-3 e\right)}{24 c^7 \sqrt{-c^2 x^2}}-\frac{b e x \left(-c^2 x^2-1\right)^{7/2}}{56 c^7 \sqrt{-c^2 x^2}}",1,"(x*(105*a*x^5*(4*d + 3*e*x^2) + (b*Sqrt[1 + 1/(c^2*x^2)]*(-144*e + 8*c^2*(28*d + 9*e*x^2) - 2*c^4*(56*d*x^2 + 27*e*x^4) + c^6*(84*d*x^4 + 45*e*x^6)))/c^7 + 105*b*x^5*(4*d + 3*e*x^2)*ArcCsch[c*x]))/2520","A",1
84,1,97,159,0.2107826,"\int x^3 \left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)*(a + b*ArcCsch[c*x]),x]","\frac{1}{180} x \left(15 a x^3 \left(3 d+2 e x^2\right)+\frac{b \sqrt{\frac{1}{c^2 x^2}+1} \left(3 c^4 \left(5 d x^2+2 e x^4\right)-2 c^2 \left(15 d+4 e x^2\right)+16 e\right)}{c^5}+15 b x^3 \text{csch}^{-1}(c x) \left(3 d+2 e x^2\right)\right)","\frac{1}{4} d x^4 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \text{csch}^{-1}(c x)\right)-\frac{b x \left(-c^2 x^2-1\right)^{3/2} \left(3 c^2 d-4 e\right)}{36 c^5 \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left(3 c^2 d-2 e\right)}{12 c^5 \sqrt{-c^2 x^2}}+\frac{b e x \left(-c^2 x^2-1\right)^{5/2}}{30 c^5 \sqrt{-c^2 x^2}}",1,"(x*(15*a*x^3*(3*d + 2*e*x^2) + (b*Sqrt[1 + 1/(c^2*x^2)]*(16*e - 2*c^2*(15*d + 4*e*x^2) + 3*c^4*(5*d*x^2 + 2*e*x^4)))/c^5 + 15*b*x^3*(3*d + 2*e*x^2)*ArcCsch[c*x]))/180","A",1
85,1,77,146,0.100574,"\int x \left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)*(a + b*ArcCsch[c*x]),x]","\frac{x \left(3 a c^3 x \left(2 d+e x^2\right)+3 b c^3 x \text{csch}^{-1}(c x) \left(2 d+e x^2\right)+b \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 \left(6 d+e x^2\right)-2 e\right)\right)}{12 c^3}","\frac{\left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right)}{4 e}-\frac{b c d^2 x \tan ^{-1}\left(\sqrt{-c^2 x^2-1}\right)}{4 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(2 c^2 d-e\right)}{4 c^3 \sqrt{-c^2 x^2}}-\frac{b e x \left(-c^2 x^2-1\right)^{3/2}}{12 c^3 \sqrt{-c^2 x^2}}",1,"(x*(3*a*c^3*x*(2*d + e*x^2) + b*Sqrt[1 + 1/(c^2*x^2)]*(-2*e + c^2*(6*d + e*x^2)) + 3*b*c^3*x*(2*d + e*x^2)*ArcCsch[c*x]))/(12*c^3)","A",1
86,1,93,115,0.1411499,"\int \frac{\left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x,x]","\frac{2 a c d \log (x)+a c e x^2+b e x \sqrt{\frac{1}{c^2 x^2}+1}+b c \text{csch}^{-1}(c x) \left(e x^2-2 d \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)\right)+b c d \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)-b c d \text{csch}^{-1}(c x)^2}{2 c}","-d \log \left(\frac{1}{x}\right) \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{2} e x^2 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c}-\frac{1}{2} b d \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)+\frac{1}{2} b d \text{csch}^{-1}(c x)^2-b d \text{csch}^{-1}(c x) \log \left(1-e^{2 \text{csch}^{-1}(c x)}\right)+b d \log \left(\frac{1}{x}\right) \text{csch}^{-1}(c x)",1,"(b*e*Sqrt[1 + 1/(c^2*x^2)]*x + a*c*e*x^2 - b*c*d*ArcCsch[c*x]^2 + b*c*ArcCsch[c*x]*(e*x^2 - 2*d*Log[1 - E^(-2*ArcCsch[c*x])]) + 2*a*c*d*Log[x] + b*c*d*PolyLog[2, E^(-2*ArcCsch[c*x])])/(2*c)","A",0
87,1,138,128,0.5668559,"\int \frac{\left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^3,x]","\frac{1}{4} \left(-\frac{2 a d}{x^2}+4 a e \log (x)-\frac{b d \left(-c^2 x^2+c^2 x^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-1\right)}{c x^3 \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{2 b d \text{csch}^{-1}(c x)}{x^2}+2 b e \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)-2 b e \text{csch}^{-1}(c x) \left(\text{csch}^{-1}(c x)+2 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)\right)\right)","-\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{2 x^2}-e \log \left(\frac{1}{x}\right) \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b c d \sqrt{\frac{1}{c^2 x^2}+1}}{4 x}-\frac{1}{4} b c^2 d \text{csch}^{-1}(c x)-\frac{1}{2} b e \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)+\frac{1}{2} b e \text{csch}^{-1}(c x)^2-b e \text{csch}^{-1}(c x) \log \left(1-e^{2 \text{csch}^{-1}(c x)}\right)+b e \log \left(\frac{1}{x}\right) \text{csch}^{-1}(c x)",1,"((-2*a*d)/x^2 - (2*b*d*ArcCsch[c*x])/x^2 - (b*d*(-1 - c^2*x^2 + c^2*x^2*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]]))/(c*Sqrt[1 + 1/(c^2*x^2)]*x^3) - 2*b*e*ArcCsch[c*x]*(ArcCsch[c*x] + 2*Log[1 - E^(-2*ArcCsch[c*x])]) + 4*a*e*Log[x] + 2*b*e*PolyLog[2, E^(-2*ArcCsch[c*x])])/4","A",0
88,1,182,260,0.3574086,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^2*(a + b*ArcCsch[c*x]),x]","\frac{c^2 x^2 \left(16 a c^5 x \left(35 d^2+42 d e x^2+15 e^2 x^4\right)+b \sqrt{\frac{1}{c^2 x^2}+1} \left(8 c^4 \left(35 d^2+21 d e x^2+5 e^2 x^4\right)-2 c^2 e \left(126 d+25 e x^2\right)+75 e^2\right)\right)+16 b c^7 x^3 \text{csch}^{-1}(c x) \left(35 d^2+42 d e x^2+15 e^2 x^4\right)+b \left(-280 c^4 d^2+252 c^2 d e-75 e^2\right) \log \left(x \left(\sqrt{\frac{1}{c^2 x^2}+1}+1\right)\right)}{1680 c^7}","\frac{1}{3} d^2 x^3 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{2}{5} d e x^5 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{7} e^2 x^7 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b e^2 x^6 \sqrt{-c^2 x^2-1}}{42 c \sqrt{-c^2 x^2}}+\frac{b e x^4 \sqrt{-c^2 x^2-1} \left(84 c^2 d-25 e\right)}{840 c^3 \sqrt{-c^2 x^2}}+\frac{b x \left(280 c^4 d^2-252 c^2 d e+75 e^2\right) \tan ^{-1}\left(\frac{c x}{\sqrt{-c^2 x^2-1}}\right)}{1680 c^6 \sqrt{-c^2 x^2}}+\frac{b x^2 \sqrt{-c^2 x^2-1} \left(280 c^4 d^2-252 c^2 d e+75 e^2\right)}{1680 c^5 \sqrt{-c^2 x^2}}",1,"(c^2*x^2*(16*a*c^5*x*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + b*Sqrt[1 + 1/(c^2*x^2)]*(75*e^2 - 2*c^2*e*(126*d + 25*e*x^2) + 8*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4))) + 16*b*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcCsch[c*x] + b*(-280*c^4*d^2 + 252*c^2*d*e - 75*e^2)*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(1680*c^7)","A",1
89,1,149,197,0.2282173,"\int \left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcCsch[c*x]),x]","\frac{c^2 x \left(8 a c^3 \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 \left(40 d+6 e x^2\right)-9 e\right)\right)+8 b c^5 x \text{csch}^{-1}(c x) \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+b \left(120 c^4 d^2-40 c^2 d e+9 e^2\right) \log \left(x \left(\sqrt{\frac{1}{c^2 x^2}+1}+1\right)\right)}{120 c^5}","d^2 x \left(a+b \text{csch}^{-1}(c x)\right)+\frac{2}{3} d e x^3 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b e^2 x^4 \sqrt{-c^2 x^2-1}}{20 c \sqrt{-c^2 x^2}}-\frac{b x \left(120 c^4 d^2-40 c^2 d e+9 e^2\right) \tan ^{-1}\left(\frac{c x}{\sqrt{-c^2 x^2-1}}\right)}{120 c^4 \sqrt{-c^2 x^2}}+\frac{b e x^2 \sqrt{-c^2 x^2-1} \left(40 c^2 d-9 e\right)}{120 c^3 \sqrt{-c^2 x^2}}",1,"(c^2*x*(8*a*c^3*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(-9*e + c^2*(40*d + 6*e*x^2))) + 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x] + b*(120*c^4*d^2 - 40*c^2*d*e + 9*e^2)*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(120*c^5)","A",1
90,1,134,170,0.218188,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^2,x]","\frac{c^2 \left(2 a c \left(-3 d^2+6 d e x^2+e^2 x^4\right)+b x \sqrt{\frac{1}{c^2 x^2}+1} \left(6 c^2 d^2+e^2 x^2\right)\right)+2 b c^3 \text{csch}^{-1}(c x) \left(-3 d^2+6 d e x^2+e^2 x^4\right)+b e x \left(12 c^2 d-e\right) \log \left(x \left(\sqrt{\frac{1}{c^2 x^2}+1}+1\right)\right)}{6 c^3 x}","-\frac{d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{x}+2 d e x \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{3} e^2 x^3 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b c d^2 \sqrt{-c^2 x^2-1}}{\sqrt{-c^2 x^2}}-\frac{b e x \left(12 c^2 d-e\right) \tan ^{-1}\left(\frac{c x}{\sqrt{-c^2 x^2-1}}\right)}{6 c^2 \sqrt{-c^2 x^2}}+\frac{b e^2 x^2 \sqrt{-c^2 x^2-1}}{6 c \sqrt{-c^2 x^2}}",1,"(c^2*(b*Sqrt[1 + 1/(c^2*x^2)]*x*(6*c^2*d^2 + e^2*x^2) + 2*a*c*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)) + 2*b*c^3*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*ArcCsch[c*x] + b*(12*c^2*d - e)*e*x*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(6*c^3*x)","A",1
91,1,123,164,0.2697992,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^4,x]","\frac{b c d x \sqrt{\frac{1}{c^2 x^2}+1} \left(-2 c^2 d x^2+d+18 e x^2\right)-3 a \left(d^2+6 d e x^2-3 e^2 x^4\right)}{9 x^3}+\frac{b e^2 \log \left(x \left(\sqrt{\frac{1}{c^2 x^2}+1}+1\right)\right)}{c}-\frac{b \text{csch}^{-1}(c x) \left(d^2+6 d e x^2-3 e^2 x^4\right)}{3 x^3}","-\frac{d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{3 x^3}-\frac{2 d e \left(a+b \text{csch}^{-1}(c x)\right)}{x}+e^2 x \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b c d^2 \sqrt{-c^2 x^2-1}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{2 b c d \sqrt{-c^2 x^2-1} \left(c^2 d-9 e\right)}{9 \sqrt{-c^2 x^2}}-\frac{b e^2 x \tan ^{-1}\left(\frac{c x}{\sqrt{-c^2 x^2-1}}\right)}{\sqrt{-c^2 x^2}}",1,"(b*c*d*Sqrt[1 + 1/(c^2*x^2)]*x*(d - 2*c^2*d*x^2 + 18*e*x^2) - 3*a*(d^2 + 6*d*e*x^2 - 3*e^2*x^4))/(9*x^3) - (b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcCsch[c*x])/(3*x^3) + (b*e^2*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/c","A",1
92,1,126,189,0.2247516,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^6,x]","\frac{-15 a \left(3 d^2+10 d e x^2+15 e^2 x^4\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(-50 d e x^2 \left(2 c^2 x^2-1\right)+3 d^2 \left(8 c^4 x^4-4 c^2 x^2+3\right)+225 e^2 x^4\right)-15 b \text{csch}^{-1}(c x) \left(3 d^2+10 d e x^2+15 e^2 x^4\right)}{225 x^5}","-\frac{d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{5 x^5}-\frac{2 d e \left(a+b \text{csch}^{-1}(c x)\right)}{3 x^3}-\frac{e^2 \left(a+b \text{csch}^{-1}(c x)\right)}{x}+\frac{b c d^2 \sqrt{-c^2 x^2-1}}{25 x^4 \sqrt{-c^2 x^2}}-\frac{2 b c d \sqrt{-c^2 x^2-1} \left(6 c^2 d-25 e\right)}{225 x^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left(24 c^4 d^2-100 c^2 d e+225 e^2\right)}{225 \sqrt{-c^2 x^2}}",1,"(-15*a*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(225*e^2*x^4 - 50*d*e*x^2*(-1 + 2*c^2*x^2) + 3*d^2*(3 - 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4)*ArcCsch[c*x])/(225*x^5)","A",1
93,1,152,249,0.262209,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^8,x]","\frac{-105 a \left(15 d^2+42 d e x^2+35 e^2 x^4\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(1225 e^2 x^4 \left(1-2 c^2 x^2\right)+294 d e x^2 \left(8 c^4 x^4-4 c^2 x^2+3\right)-45 d^2 \left(16 c^6 x^6-8 c^4 x^4+6 c^2 x^2-5\right)\right)-105 b \text{csch}^{-1}(c x) \left(15 d^2+42 d e x^2+35 e^2 x^4\right)}{11025 x^7}","-\frac{d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{7 x^7}-\frac{2 d e \left(a+b \text{csch}^{-1}(c x)\right)}{5 x^5}-\frac{e^2 \left(a+b \text{csch}^{-1}(c x)\right)}{3 x^3}+\frac{b c d^2 \sqrt{-c^2 x^2-1}}{49 x^6 \sqrt{-c^2 x^2}}-\frac{2 b c d \sqrt{-c^2 x^2-1} \left(15 c^2 d-49 e\right)}{1225 x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left(360 c^4 d^2-1176 c^2 d e+1225 e^2\right)}{11025 x^2 \sqrt{-c^2 x^2}}-\frac{2 b c^3 \sqrt{-c^2 x^2-1} \left(360 c^4 d^2-1176 c^2 d e+1225 e^2\right)}{11025 \sqrt{-c^2 x^2}}",1,"(-105*a*(15*d^2 + 42*d*e*x^2 + 35*e^2*x^4) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1225*e^2*x^4*(1 - 2*c^2*x^2) + 294*d*e*x^2*(3 - 4*c^2*x^2 + 8*c^4*x^4) - 45*d^2*(-5 + 6*c^2*x^2 - 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(15*d^2 + 42*d*e*x^2 + 35*e^2*x^4)*ArcCsch[c*x])/(11025*x^7)","A",1
94,1,159,250,0.3880811,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^2*(a + b*ArcCsch[c*x]),x]","\frac{x \left(105 a x^3 \left(6 d^2+8 d e x^2+3 e^2 x^4\right)+\frac{b \sqrt{\frac{1}{c^2 x^2}+1} \left(3 c^6 \left(70 d^2 x^2+56 d e x^4+15 e^2 x^6\right)-2 c^4 \left(210 d^2+112 d e x^2+27 e^2 x^4\right)+8 c^2 e \left(56 d+9 e x^2\right)-144 e^2\right)}{c^7}+105 b x^3 \text{csch}^{-1}(c x) \left(6 d^2+8 d e x^2+3 e^2 x^4\right)\right)}{2520}","\frac{1}{4} d^2 x^4 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{3} d e x^6 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{8} e^2 x^8 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b e x \left(-c^2 x^2-1\right)^{5/2} \left(8 c^2 d-9 e\right)}{120 c^7 \sqrt{-c^2 x^2}}-\frac{b e^2 x \left(-c^2 x^2-1\right)^{7/2}}{56 c^7 \sqrt{-c^2 x^2}}-\frac{b x \left(-c^2 x^2-1\right)^{3/2} \left(6 c^4 d^2-16 c^2 d e+9 e^2\right)}{72 c^7 \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left(6 c^4 d^2-8 c^2 d e+3 e^2\right)}{24 c^7 \sqrt{-c^2 x^2}}",1,"(x*(105*a*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + (b*Sqrt[1 + 1/(c^2*x^2)]*(-144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) - 2*c^4*(210*d^2 + 112*d*e*x^2 + 27*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/c^7 + 105*b*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x]))/2520","A",1
95,1,123,203,0.3114245,"\int x \left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^2*(a + b*ArcCsch[c*x]),x]","\frac{1}{90} x \left(15 a x \left(3 d^2+3 d e x^2+e^2 x^4\right)+\frac{b \sqrt{\frac{1}{c^2 x^2}+1} \left(3 c^4 \left(15 d^2+5 d e x^2+e^2 x^4\right)-2 c^2 e \left(15 d+2 e x^2\right)+8 e^2\right)}{c^5}+15 b x \text{csch}^{-1}(c x) \left(3 d^2+3 d e x^2+e^2 x^4\right)\right)","\frac{\left(d+e x^2\right)^3 \left(a+b \text{csch}^{-1}(c x)\right)}{6 e}-\frac{b c d^3 x \tan ^{-1}\left(\sqrt{-c^2 x^2-1}\right)}{6 e \sqrt{-c^2 x^2}}-\frac{b e x \left(-c^2 x^2-1\right)^{3/2} \left(3 c^2 d-2 e\right)}{18 c^5 \sqrt{-c^2 x^2}}+\frac{b e^2 x \left(-c^2 x^2-1\right)^{5/2}}{30 c^5 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(3 c^4 d^2-3 c^2 d e+e^2\right)}{6 c^5 \sqrt{-c^2 x^2}}",1,"(x*(15*a*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + (b*Sqrt[1 + 1/(c^2*x^2)]*(8*e^2 - 2*c^2*e*(15*d + 2*e*x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4)))/c^5 + 15*b*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcCsch[c*x]))/90","A",1
96,1,148,178,0.4232791,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x,x]","a d^2 \log (x)+a d e x^2+\frac{1}{4} a e^2 x^4+\frac{b d e x \left(\sqrt{\frac{1}{c^2 x^2}+1}+c x \text{csch}^{-1}(c x)\right)}{c}+\frac{b e^2 x \left(3 c^3 x^3 \text{csch}^{-1}(c x)+\sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 x^2-2\right)\right)}{12 c^3}+\frac{1}{2} b d^2 \left(\text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)-\text{csch}^{-1}(c x) \left(\text{csch}^{-1}(c x)+2 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)\right)\right)","-d^2 \log \left(\frac{1}{x}\right) \left(a+b \text{csch}^{-1}(c x)\right)+d e x^2 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{4} e^2 x^4 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b e^2 x^3 \sqrt{\frac{1}{c^2 x^2}+1}}{12 c}+\frac{b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(6 c^2 d-e\right)}{6 c^3}-\frac{1}{2} b d^2 \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)+\frac{1}{2} b d^2 \text{csch}^{-1}(c x)^2-b d^2 \text{csch}^{-1}(c x) \log \left(1-e^{2 \text{csch}^{-1}(c x)}\right)+b d^2 \log \left(\frac{1}{x}\right) \text{csch}^{-1}(c x)",1,"a*d*e*x^2 + (a*e^2*x^4)/4 + (b*d*e*x*(Sqrt[1 + 1/(c^2*x^2)] + c*x*ArcCsch[c*x]))/c + (b*e^2*x*(Sqrt[1 + 1/(c^2*x^2)]*(-2 + c^2*x^2) + 3*c^3*x^3*ArcCsch[c*x]))/(12*c^3) + a*d^2*Log[x] + (b*d^2*(-(ArcCsch[c*x]*(ArcCsch[c*x] + 2*Log[1 - E^(-2*ArcCsch[c*x])])) + PolyLog[2, E^(-2*ArcCsch[c*x])]))/2","A",0
97,1,187,178,0.9066719,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^3,x]","\frac{1}{4} \left(-\frac{2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2-\frac{b d^2 \left(-c^2 x^2+c^2 x^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-1\right)}{c x^3 \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{2 b e^2 x \left(\sqrt{\frac{1}{c^2 x^2}+1}+c x \text{csch}^{-1}(c x)\right)}{c}-\frac{2 b d^2 \text{csch}^{-1}(c x)}{x^2}+4 b d e \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)-4 b d e \text{csch}^{-1}(c x) \left(\text{csch}^{-1}(c x)+2 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)\right)\right)","-\frac{d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{2 x^2}-2 d e \log \left(\frac{1}{x}\right) \left(a+b \text{csch}^{-1}(c x)\right)+\frac{1}{2} e^2 x^2 \left(a+b \text{csch}^{-1}(c x)\right)+\frac{b c d^2 \sqrt{\frac{1}{c^2 x^2}+1}}{4 x}-\frac{1}{4} b c^2 d^2 \text{csch}^{-1}(c x)+\frac{b e^2 x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c}-b d e \text{Li}_2\left(e^{2 \text{csch}^{-1}(c x)}\right)+b d e \text{csch}^{-1}(c x)^2-2 b d e \text{csch}^{-1}(c x) \log \left(1-e^{2 \text{csch}^{-1}(c x)}\right)+2 b d e \log \left(\frac{1}{x}\right) \text{csch}^{-1}(c x)",1,"((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*d^2*ArcCsch[c*x])/x^2 + (2*b*e^2*x*(Sqrt[1 + 1/(c^2*x^2)] + c*x*ArcCsch[c*x]))/c - (b*d^2*(-1 - c^2*x^2 + c^2*x^2*Sqrt[1 + c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]]))/(c*Sqrt[1 + 1/(c^2*x^2)]*x^3) - 4*b*d*e*ArcCsch[c*x]*(ArcCsch[c*x] + 2*Log[1 - E^(-2*ArcCsch[c*x])]) + 8*a*d*e*Log[x] + 4*b*d*e*PolyLog[2, E^(-2*ArcCsch[c*x])])/4","A",0
98,1,1221,512,1.6660001,"\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2),x]","\frac{4 b c \sqrt{e} \text{csch}^{-1}(c x) x+4 a c \sqrt{e} x-4 a c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)-8 i b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)-8 i b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)-2 i b c \sqrt{d} \text{csch}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+b c \sqrt{d} \pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b c \sqrt{d} \text{csch}^{-1}(c x) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-b c \sqrt{d} \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 i b c \sqrt{d} \text{csch}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-b c \sqrt{d} \pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b c \sqrt{d} \text{csch}^{-1}(c x) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-4 b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+b c \sqrt{d} \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+b c \sqrt{d} \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)-b c \sqrt{d} \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)-4 b \sqrt{e} \log \left(\tanh \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)\right)+2 i b c \sqrt{d} \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b c \sqrt{d} \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b c \sqrt{d} \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b c \sqrt{d} \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{4 c e^{3/2}}","\frac{\sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{2 e^{3/2}}+\frac{\sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{2 e^{3/2}}+\frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{e}-\frac{b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 e^{3/2}}+\frac{b \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{c e}",1,"(4*a*c*Sqrt[e]*x + 4*b*c*Sqrt[e]*x*ArcCsch[c*x] - 4*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - (8*I)*b*c*Sqrt[d]*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - (8*I)*b*c*Sqrt[d]*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] + b*c*Sqrt[d]*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - b*c*Sqrt[d]*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - b*c*Sqrt[d]*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + b*c*Sqrt[d]*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + b*c*Sqrt[d]*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - b*c*Sqrt[d]*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] - 4*b*Sqrt[e]*Log[Tanh[ArcCsch[c*x]/2]] + (2*I)*b*c*Sqrt[d]*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*c*Sqrt[d]*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*c*Sqrt[d]*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/(4*c*e^(3/2))","C",0
99,1,1103,467,0.3510109,"\int \frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x^2),x]","\frac{-8 b \text{csch}^{-1}(c x)^2-8 b \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \text{csch}^{-1}(c x)+4 b \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)+4 b \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)+4 b \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)+4 b \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)-4 i b \pi  \text{csch}^{-1}(c x)+16 b \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)-16 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 i b \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-8 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-8 i b \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+2 i b \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-2 i b \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)-2 i b \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+4 a \log \left(e x^2+d\right)+4 b \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)+4 b \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 b \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+b \pi ^2}{8 e}","\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{2 e}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{2 e}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{2 e}-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{b e}-\frac{\log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{e}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 e}+\frac{b \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)}{2 e}",1,"(b*Pi^2 - (4*I)*b*Pi*ArcCsch[c*x] - 8*b*ArcCsch[c*x]^2 + 16*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 16*b*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*b*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (2*I)*b*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*b*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*b*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - (2*I)*b*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*a*Log[d + e*x^2] + 4*b*PolyLog[2, E^(-2*ArcCsch[c*x])] + 4*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/(8*e)","C",0
100,1,1055,477,0.5080198,"\int \frac{a+b \text{csch}^{-1}(c x)}{d+e x^2} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x^2),x]","\frac{4 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)+2 i b \text{csch}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-4 b \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-b \pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b \text{csch}^{-1}(c x) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+b \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 i b \text{csch}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+b \pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \text{csch}^{-1}(c x) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+4 b \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-b \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-b \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)+b \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)-2 i b \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{4 \sqrt{d} \sqrt{e}}","\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}",1,"(4*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + (8*I)*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] + (8*I)*b*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - b*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 4*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + b*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + b*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 4*b*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - b*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - b*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + b*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] - (2*I)*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/(4*Sqrt[d]*Sqrt[e])","C",0
101,1,387,425,0.978135,"\int \frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)),x]","\frac{-2 a \log \left(d+e x^2\right)+4 a \log (x)+b \left(\text{Li}_2\left(\frac{\left(d c^2-2 e+2 \sqrt{e \left(e-c^2 d\right)}\right) e^{-2 \text{csch}^{-1}(c x)}}{c^2 d}\right)+\text{Li}_2\left(\frac{\left(c^2 d-2 \left(e+\sqrt{e \left(e-c^2 d\right)}\right)\right) e^{-2 \text{csch}^{-1}(c x)}}{c^2 d}\right)-2 \left(i \sin ^{-1}\left(\sqrt{\frac{e}{c^2 d}}\right) \left(2 \tanh ^{-1}\left(\frac{\sqrt{e \left(e-c^2 d\right)}}{c e x \sqrt{\frac{1}{c^2 x^2}+1}}\right)-\log \left(\frac{e^{-2 \text{csch}^{-1}(c x)} \left(-2 \sqrt{e \left(e-c^2 d\right)}+c^2 d \left(e^{2 \text{csch}^{-1}(c x)}-1\right)+2 e\right)}{c^2 d}\right)+\log \left(\frac{e^{-2 \text{csch}^{-1}(c x)} \left(2 \left(\sqrt{e \left(e-c^2 d\right)}+e\right)+c^2 d \left(e^{2 \text{csch}^{-1}(c x)}-1\right)\right)}{c^2 d}\right)\right)+\text{csch}^{-1}(c x) \left(\log \left(\frac{e^{-2 \text{csch}^{-1}(c x)} \left(-2 \sqrt{e \left(e-c^2 d\right)}+c^2 d \left(e^{2 \text{csch}^{-1}(c x)}-1\right)+2 e\right)}{c^2 d}\right)+\log \left(\frac{e^{-2 \text{csch}^{-1}(c x)} \left(2 \left(\sqrt{e \left(e-c^2 d\right)}+e\right)+c^2 d \left(e^{2 \text{csch}^{-1}(c x)}-1\right)\right)}{c^2 d}\right)\right)+\text{csch}^{-1}(c x)^2\right)\right)}{4 d}","-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 d}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{2 d}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{2 d}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{2 d}+\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{2 b d}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 d}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 d}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 d}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 d}",1,"(4*a*Log[x] - 2*a*Log[d + e*x^2] + b*(-2*(ArcCsch[c*x]^2 + I*ArcSin[Sqrt[e/(c^2*d)]]*(2*ArcTanh[Sqrt[e*(-(c^2*d) + e)]/(c*e*Sqrt[1 + 1/(c^2*x^2)]*x)] - Log[(2*e - 2*Sqrt[e*(-(c^2*d) + e)] + c^2*d*(-1 + E^(2*ArcCsch[c*x])))/(c^2*d*E^(2*ArcCsch[c*x]))] + Log[(2*(e + Sqrt[e*(-(c^2*d) + e)]) + c^2*d*(-1 + E^(2*ArcCsch[c*x])))/(c^2*d*E^(2*ArcCsch[c*x]))]) + ArcCsch[c*x]*(Log[(2*e - 2*Sqrt[e*(-(c^2*d) + e)] + c^2*d*(-1 + E^(2*ArcCsch[c*x])))/(c^2*d*E^(2*ArcCsch[c*x]))] + Log[(2*(e + Sqrt[e*(-(c^2*d) + e)]) + c^2*d*(-1 + E^(2*ArcCsch[c*x])))/(c^2*d*E^(2*ArcCsch[c*x]))])) + PolyLog[2, (c^2*d - 2*e + 2*Sqrt[e*(-(c^2*d) + e)])/(c^2*d*E^(2*ArcCsch[c*x]))] + PolyLog[2, (c^2*d - 2*(e + Sqrt[e*(-(c^2*d) + e)]))/(c^2*d*E^(2*ArcCsch[c*x]))]))/(4*d)","C",0
102,1,1211,518,1.6869453,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^2*(d + e*x^2)),x]","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a}{d^{3/2}}-\frac{a}{d x}+b \left(\frac{c \sqrt{1+\frac{1}{c^2 x^2}}-\frac{\text{csch}^{-1}(c x)}{x}}{d}-\frac{i \sqrt{e} \left(-8 \text{csch}^{-1}(c x)^2-8 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \text{csch}^{-1}(c x)+8 \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)+8 \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)-4 i \pi  \text{csch}^{-1}(c x)+32 \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)+16 i \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-16 i \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-4 i \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+4 \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)+8 \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+8 \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+\pi ^2\right)}{16 d^{3/2}}+\frac{i \sqrt{e} \left(-8 \text{csch}^{-1}(c x)^2-8 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \text{csch}^{-1}(c x)+8 \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)+8 \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)-4 i \pi  \text{csch}^{-1}(c x)-32 \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)+16 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-16 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-4 i \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)+4 \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)+8 \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+8 \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+\pi ^2\right)}{16 d^{3/2}}\right)","\frac{\sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{2 (-d)^{3/2}}-\frac{a}{d x}-\frac{b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 (-d)^{3/2}}-\frac{b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 (-d)^{3/2}}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{d}-\frac{b \text{csch}^{-1}(c x)}{d x}",1,"-(a/(d*x)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + b*((c*Sqrt[1 + 1/(c^2*x^2)] - ArcCsch[c*x]/x)/d - ((I/16)*Sqrt[e]*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/d^(3/2) + ((I/16)*Sqrt[e]*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 - 32*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/d^(3/2))","C",0
103,1,1447,591,6.3649289,"\int \frac{x^5 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^5*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2,x]","-\frac{\frac{2 a d^2}{e x^2+d}+4 a \log \left(e x^2+d\right) d-2 a e x^2+b \left(-2 e \text{csch}^{-1}(c x) x^2-\frac{2 e \sqrt{1+\frac{1}{c^2 x^2}} x}{c}-8 d \text{csch}^{-1}(c x)^2+\frac{d^{3/2} \text{csch}^{-1}(c x)}{\sqrt{d}-i \sqrt{e} x}+\frac{d^{3/2} \text{csch}^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-4 i d \pi  \text{csch}^{-1}(c x)-2 d \sinh ^{-1}\left(\frac{1}{c x}\right)+16 d \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)-16 d \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)-8 d \text{csch}^{-1}(c x) \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right)+4 d \text{csch}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+8 i d \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i d \pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 d \text{csch}^{-1}(c x) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+8 i d \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 i d \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 d \text{csch}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-8 i d \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i d \pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 d \text{csch}^{-1}(c x) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-8 i d \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+2 i d \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-2 i d \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)-2 i d \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+\frac{d \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{d} c+i \sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\sqrt{e-c^2 d} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}+\frac{d \sqrt{e} \log \left(-\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{e-c^2 d} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}+4 d \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)+4 d \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 d \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 d \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 d \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+d \pi ^2\right)}{4 e^3}","-\frac{d \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{e^3}-\frac{d \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{e^3}-\frac{d \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{e^3}-\frac{d \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{e^3}+\frac{2 d \left(a+b \text{csch}^{-1}(c x)\right)^2}{b e^3}+\frac{2 d \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{e^3}+\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{2 e^2 \left(\frac{d}{x^2}+e\right)}+\frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{2 e^2}-\frac{b d \tan ^{-1}\left(\frac{\sqrt{c^2 d-e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}+1}}\right)}{2 e^{5/2} \sqrt{c^2 d-e}}-\frac{b d \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{e^3}-\frac{b d \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{e^3}-\frac{b d \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{e^3}-\frac{b d \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{e^3}+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1}}{2 c e^2}-\frac{b d \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)}{e^3}",1,"-1/4*(-2*a*e*x^2 + (2*a*d^2)/(d + e*x^2) + 4*a*d*Log[d + e*x^2] + b*(d*Pi^2 - (2*e*Sqrt[1 + 1/(c^2*x^2)]*x)/c - (4*I)*d*Pi*ArcCsch[c*x] - 2*e*x^2*ArcCsch[c*x] + (d^(3/2)*ArcCsch[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (d^(3/2)*ArcCsch[c*x])/(Sqrt[d] + I*Sqrt[e]*x) - 8*d*ArcCsch[c*x]^2 - 2*d*ArcSinh[1/(c*x)] + 16*d*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 16*d*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*d*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (2*I)*d*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*d*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*d*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*d*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*d*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*d*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*d*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*d*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*d*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - (2*I)*d*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + (d*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) + e] + (d*Sqrt[e]*Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) + e] + 4*d*PolyLog[2, E^(-2*ArcCsch[c*x])] + 4*d*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*d*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/e^3","C",0
104,1,1410,553,2.4138691,"\int \frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2,x]","\frac{-8 b \text{csch}^{-1}(c x)^2-8 b \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \text{csch}^{-1}(c x)+4 b \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)+4 b \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)+4 b \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)+4 b \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)+\frac{2 b \sqrt{d} \text{csch}^{-1}(c x)}{\sqrt{d}-i \sqrt{e} x}+\frac{2 b \sqrt{d} \text{csch}^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-4 i b \pi  \text{csch}^{-1}(c x)-4 b \sinh ^{-1}\left(\frac{1}{c x}\right)+16 b \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)-16 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 i b \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-8 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-8 i b \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+2 i b \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-2 i b \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)-2 i b \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+\frac{2 b \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{d} c+i \sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\sqrt{e-c^2 d} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}+\frac{2 b \sqrt{e} \log \left(-\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{e-c^2 d} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}+4 a \log \left(e x^2+d\right)+4 b \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)+4 b \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 b \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 b \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 b \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+\frac{4 a d}{e x^2+d}+b \pi ^2}{8 e^2}","\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e^2}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{2 e^2}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{2 e^2}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{2 e^2}-\frac{a+b \text{csch}^{-1}(c x)}{2 e \left(\frac{d}{x^2}+e\right)}-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{b e^2}-\frac{\log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{e^2}+\frac{b \tan ^{-1}\left(\frac{\sqrt{c^2 d-e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}+1}}\right)}{2 e^{3/2} \sqrt{c^2 d-e}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)}{2 e^2}",1,"(b*Pi^2 + (4*a*d)/(d + e*x^2) - (4*I)*b*Pi*ArcCsch[c*x] + (2*b*Sqrt[d]*ArcCsch[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (2*b*Sqrt[d]*ArcCsch[c*x])/(Sqrt[d] + I*Sqrt[e]*x) - 8*b*ArcCsch[c*x]^2 - 4*b*ArcSinh[1/(c*x)] + 16*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 16*b*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*b*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (2*I)*b*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (8*I)*b*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*b*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (2*I)*b*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (8*I)*b*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (2*I)*b*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - (2*I)*b*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + (2*b*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) + e] + (2*b*Sqrt[e]*Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) + e] + 4*a*Log[d + e*x^2] + 4*b*PolyLog[2, E^(-2*ArcCsch[c*x])] + 4*b*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 4*b*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/(8*e^2)","C",0
105,1,271,139,0.7917073,"\int \frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2,x]","-\frac{\frac{2 a}{d+e x^2}+\frac{b \sqrt{e} \log \left(-\frac{4 \left(c d \sqrt{e} x \left(c \sqrt{d}+i \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}\right)+i d e\right)}{b \sqrt{e-c^2 d} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{d \sqrt{e-c^2 d}}+\frac{b \sqrt{e} \log \left(\frac{4 i \left(d e+c d \sqrt{e} x \left(\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}+i c \sqrt{d}\right)\right)}{b \sqrt{e-c^2 d} \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{d \sqrt{e-c^2 d}}+\frac{2 b \text{csch}^{-1}(c x)}{d+e x^2}-\frac{2 b \sinh ^{-1}\left(\frac{1}{c x}\right)}{d}}{4 e}","-\frac{a+b \text{csch}^{-1}(c x)}{2 e \left(d+e x^2\right)}+\frac{b c x \tan ^{-1}\left(\sqrt{-c^2 x^2-1}\right)}{2 d e \sqrt{-c^2 x^2}}+\frac{b c x \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{\sqrt{c^2 d-e}}\right)}{2 d \sqrt{e} \sqrt{-c^2 x^2} \sqrt{c^2 d-e}}",1,"-1/4*((2*a)/(d + e*x^2) + (2*b*ArcCsch[c*x])/(d + e*x^2) - (2*b*ArcSinh[1/(c*x)])/d + (b*Sqrt[e]*Log[(-4*(I*d*e + c*d*Sqrt[e]*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*Sqrt[-(c^2*d) + e]*(Sqrt[d] - I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) + e]) + (b*Sqrt[e]*Log[((4*I)*(d*e + c*d*Sqrt[e]*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) + e]))/e","C",1
106,0,0,515,44.15478,"\int \frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^2),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 d^2}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{2 d^2}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{2 d^2}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{2 d^2}-\frac{e \left(a+b \text{csch}^{-1}(c x)\right)}{2 d^2 \left(\frac{d}{x^2}+e\right)}+\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{2 b d^2}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 d^2}+\frac{b \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{c^2 d-e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}+1}}\right)}{2 d^2 \sqrt{c^2 d-e}}",1,"Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^2), x]","F",-1
107,1,1583,756,6.1722596,"\int \frac{x^4 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2,x]","\frac{a d x}{2 e^2 \left(e x^2+d\right)}+\frac{a x}{e^2}-\frac{3 a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 e^{5/2}}+b \left(-\frac{d \left(-\frac{\text{csch}^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}-\frac{i \left(\frac{\sinh ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{d} c+i \sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\sqrt{e-c^2 d} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}\right)}{\sqrt{d}}\right)}{4 e^2}-\frac{d \left(\frac{i \left(\frac{\sinh ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(-\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{e-c^2 d} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}\right)}{\sqrt{d}}-\frac{\text{csch}^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}\right)}{4 e^2}-\frac{3 i \sqrt{d} \left(-8 \text{csch}^{-1}(c x)^2-8 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \text{csch}^{-1}(c x)+8 \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)+8 \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)-4 i \pi  \text{csch}^{-1}(c x)+32 \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)+16 i \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-16 i \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-4 i \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+4 \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)+8 \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+8 \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+\pi ^2\right)}{32 e^{5/2}}+\frac{3 i \sqrt{d} \left(-8 \text{csch}^{-1}(c x)^2-8 \log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \text{csch}^{-1}(c x)+8 \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)+8 \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)-4 i \pi  \text{csch}^{-1}(c x)-32 \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)+16 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-16 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-4 i \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)+4 \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)+8 \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+8 \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+\pi ^2\right)}{32 e^{5/2}}+\frac{\frac{1}{2} \text{csch}^{-1}(c x) \coth \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)-\log \left(\tanh \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)\right)-\frac{1}{2} \text{csch}^{-1}(c x) \tanh \left(\frac{1}{2} \text{csch}^{-1}(c x)\right)}{c e^2}\right)","\frac{3 \sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{4 e^{5/2}}-\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{4 e^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{4 e^2 \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}+\frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{e^2}-\frac{3 b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 e^{5/2}}+\frac{3 b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 e^{5/2}}-\frac{3 b \sqrt{-d} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{4 e^{5/2}}+\frac{3 b \sqrt{-d} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{4 e^{5/2}}+\frac{b \sqrt{d} \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right)}{4 e^2 \sqrt{c^2 d-e}}+\frac{b \sqrt{d} \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right)}{4 e^2 \sqrt{c^2 d-e}}+\frac{b \tanh ^{-1}\left(\sqrt{\frac{1}{c^2 x^2}+1}\right)}{c e^2}",1,"(a*x)/e^2 + (a*d*x)/(2*e^2*(d + e*x^2)) - (3*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*e^(5/2)) + b*(-1/4*(d*(-(ArcCsch[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d]))/e^2 - (d*(-(ArcCsch[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d]))/(4*e^2) - (((3*I)/32)*Sqrt[d]*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/e^(5/2) + (((3*I)/32)*Sqrt[d]*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 - 32*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/e^(5/2) + ((ArcCsch[c*x]*Coth[ArcCsch[c*x]/2])/2 - Log[Tanh[ArcCsch[c*x]/2]] - (ArcCsch[c*x]*Tanh[ArcCsch[c*x]/2])/2)/(c*e^2))","C",0
108,1,1442,719,2.7360447,"\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2,x]","\frac{-\frac{4 a \sqrt{e} x}{e x^2+d}+\frac{4 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}+b \left(\frac{2 i \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)}{\sqrt{d}}-\frac{2 i \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)}{\sqrt{d}}-\frac{2 i \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)}{\sqrt{d}}+\frac{2 i \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)}{\sqrt{d}}+\frac{2 \text{csch}^{-1}(c x)}{i \sqrt{d}-\sqrt{e} x}-\frac{2 \text{csch}^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}+\frac{8 i \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)}{\sqrt{d}}+\frac{8 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)}{\sqrt{d}}-\frac{4 \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}-\frac{\pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{4 \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}+\frac{\pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{4 \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{\pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{4 \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{\pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{\pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)}{\sqrt{d}}+\frac{\pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)}{\sqrt{d}}+\frac{2 i \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{d} c+i \sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\sqrt{e-c^2 d} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{d} \sqrt{e-c^2 d}}-\frac{2 i \sqrt{e} \log \left(-\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{e-c^2 d} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{d} \sqrt{e-c^2 d}}-\frac{2 i \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{2 i \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{2 i \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}-\frac{2 i \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}\right)}{8 e^{3/2}}","\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{4 \sqrt{-d} e^{3/2}}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{4 \sqrt{-d} e^{3/2}}+\frac{a+b \text{csch}^{-1}(c x)}{4 e \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{a+b \text{csch}^{-1}(c x)}{4 e \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right)}{4 \sqrt{d} e \sqrt{c^2 d-e}}-\frac{b \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right)}{4 \sqrt{d} e \sqrt{c^2 d-e}}",1,"((-4*a*Sqrt[e]*x)/(d + e*x^2) + (4*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + b*((2*ArcCsch[c*x])/(I*Sqrt[d] - Sqrt[e]*x) - (2*ArcCsch[c*x])/(I*Sqrt[d] + Sqrt[e]*x) + ((8*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]])/Sqrt[d] + ((8*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]])/Sqrt[d] - (Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + ((2*I)*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - (4*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + (Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - ((2*I)*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + (4*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + (Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - ((2*I)*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - (4*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - (Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + ((2*I)*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + (4*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - (Pi*Log[Sqrt[e] - (I*Sqrt[d])/x])/Sqrt[d] + (Pi*Log[Sqrt[e] + (I*Sqrt[d])/x])/Sqrt[d] + ((2*I)*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))])/(Sqrt[d]*Sqrt[-(c^2*d) + e]) - ((2*I)*Sqrt[e]*Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))])/(Sqrt[d]*Sqrt[-(c^2*d) + e]) - ((2*I)*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + ((2*I)*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + ((2*I)*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - ((2*I)*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d]))/(8*e^(3/2))","C",0
109,1,1437,713,3.6477195,"\int \frac{a+b \text{csch}^{-1}(c x)}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x^2)^2,x]","\frac{1}{2} \left(\frac{a x}{d^2+e x^2 d}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \sqrt{e}}+\frac{b \left(\frac{2 i \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)}{\sqrt{e}}-\frac{2 i \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)}{\sqrt{e}}-\frac{2 i \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right) \text{csch}^{-1}(c x)}{\sqrt{e}}+\frac{2 i \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right) \text{csch}^{-1}(c x)}{\sqrt{e}}+\frac{2 \sqrt{d} \text{csch}^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}+\frac{2 \sqrt{d} \text{csch}^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}+\frac{8 i \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)}{\sqrt{e}}+\frac{8 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)}{\sqrt{e}}-\frac{4 \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}-\frac{\pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{4 \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}+\frac{\pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{4 \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{\pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{4 \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{\pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{\pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)}{\sqrt{e}}+\frac{\pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)}{\sqrt{e}}-\frac{2 i \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{d} c+i \sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\sqrt{e-c^2 d} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}+\frac{2 i \log \left(-\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{e-c^2 d} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}-\frac{2 i \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{2 i \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{2 i \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}-\frac{2 i \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}\right)}{4 d^{3/2}}\right)","-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \text{csch}^{-1}(c x)}{4 d \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{a+b \text{csch}^{-1}(c x)}{4 d \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}+\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right)}{4 d^{3/2} \sqrt{c^2 d-e}}+\frac{b \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right)}{4 d^{3/2} \sqrt{c^2 d-e}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}",1,"((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (b*((2*Sqrt[d]*ArcCsch[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) + (2*Sqrt[d]*ArcCsch[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) + ((8*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]])/Sqrt[e] + ((8*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]])/Sqrt[e] - (Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + ((2*I)*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - (4*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + (Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - ((2*I)*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + (4*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + (Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - ((2*I)*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - (4*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - (Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + ((2*I)*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + (4*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - (Pi*Log[Sqrt[e] - (I*Sqrt[d])/x])/Sqrt[e] + (Pi*Log[Sqrt[e] + (I*Sqrt[d])/x])/Sqrt[e] - ((2*I)*Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) + e] + ((2*I)*Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) + e] - ((2*I)*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + ((2*I)*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + ((2*I)*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - ((2*I)*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e]))/(4*d^(3/2)))/2","C",0
110,1,1487,758,3.1188828,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^2*(d + e*x^2)^2),x]","\frac{-12 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a-\frac{8 \sqrt{d} a}{x}-\frac{4 \sqrt{d} e x a}{e x^2+d}+b \left(8 \sqrt{d} \sqrt{1+\frac{1}{c^2 x^2}} c-\frac{8 \sqrt{d} \text{csch}^{-1}(c x)}{x}-\frac{2 \sqrt{d} e \text{csch}^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}-\frac{2 \sqrt{d} e \text{csch}^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}-24 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(c \sqrt{d}-\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)-24 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 i \text{csch}^{-1}(c x)+\pi \right)\right)}{\sqrt{e-c^2 d}}\right)-6 i \sqrt{e} \text{csch}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+12 \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+3 \sqrt{e} \pi  \log \left(1-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+6 i \sqrt{e} \text{csch}^{-1}(c x) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-12 \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-3 \sqrt{e} \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e-c^2 d}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+6 i \sqrt{e} \text{csch}^{-1}(c x) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+12 \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{\sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-3 \sqrt{e} \pi  \log \left(1-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-6 i \sqrt{e} \text{csch}^{-1}(c x) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)-12 \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{\sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+3 \sqrt{e} \pi  \log \left(\frac{i e^{\text{csch}^{-1}(c x)} \left(\sqrt{e}+\sqrt{e-c^2 d}\right)}{c \sqrt{d}}+1\right)+3 \sqrt{e} \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)-3 \sqrt{e} \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+\frac{2 i e \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{d} c+i \sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+i \sqrt{e}\right)}{\sqrt{e-c^2 d} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}-\frac{2 i e \log \left(-\frac{2 \sqrt{d} \sqrt{e} \left(c \left(i \sqrt{d} c+\sqrt{e-c^2 d} \sqrt{1+\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{e-c^2 d} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{e-c^2 d}}+6 i \sqrt{e} \text{Li}_2\left(-\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-6 i \sqrt{e} \text{Li}_2\left(\frac{i \left(\sqrt{e-c^2 d}-\sqrt{e}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)-6 i \sqrt{e} \text{Li}_2\left(-\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)+6 i \sqrt{e} \text{Li}_2\left(\frac{i \left(\sqrt{e}+\sqrt{e-c^2 d}\right) e^{\text{csch}^{-1}(c x)}}{c \sqrt{d}}\right)\right)}{8 d^{5/2}}","-\frac{3 \sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{4 (-d)^{5/2}}+\frac{e \left(a+b \text{csch}^{-1}(c x)\right)}{4 d^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{e \left(a+b \text{csch}^{-1}(c x)\right)}{4 d^2 \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}-\frac{a}{d^2 x}-\frac{b e \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right)}{4 d^{5/2} \sqrt{c^2 d-e}}-\frac{b e \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{c^2 d-e}}\right)}{4 d^{5/2} \sqrt{c^2 d-e}}+\frac{b c \sqrt{\frac{1}{c^2 x^2}+1}}{d^2}+\frac{3 b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{4 (-d)^{5/2}}-\frac{b \text{csch}^{-1}(c x)}{d^2 x}",1,"((-8*a*Sqrt[d])/x - (4*a*Sqrt[d]*e*x)/(d + e*x^2) - 12*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(8*c*Sqrt[d]*Sqrt[1 + 1/(c^2*x^2)] - (8*Sqrt[d]*ArcCsch[c*x])/x - (2*Sqrt[d]*e*ArcCsch[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) - (2*Sqrt[d]*e*ArcCsch[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) - (24*I)*Sqrt[e]*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - (24*I)*Sqrt[e]*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] + 3*Sqrt[e]*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (6*I)*Sqrt[e]*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 12*Sqrt[e]*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 3*Sqrt[e]*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (6*I)*Sqrt[e]*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 12*Sqrt[e]*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 3*Sqrt[e]*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (6*I)*Sqrt[e]*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 12*Sqrt[e]*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 3*Sqrt[e]*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (6*I)*Sqrt[e]*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - 12*Sqrt[e]*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 3*Sqrt[e]*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - 3*Sqrt[e]*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + ((2*I)*e*Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) + e] - ((2*I)*e*Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) + e] + (6*I)*Sqrt[e]*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (6*I)*Sqrt[e]*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (6*I)*Sqrt[e]*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (6*I)*Sqrt[e]*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/(8*d^(5/2))","C",0
111,1,2023,694,7.8542336,"\int \frac{x^5 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^5*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]","\text{Result too large to show}","\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e^3}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{2 e^3}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{2 e^3}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{2 e^3}-\frac{a+b \text{csch}^{-1}(c x)}{2 e^2 \left(\frac{d}{x^2}+e\right)}-\frac{a+b \text{csch}^{-1}(c x)}{4 e \left(\frac{d}{x^2}+e\right)^2}-\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{b e^3}-\frac{\log \left(1-e^{-2 \text{csch}^{-1}(c x)}\right) \left(a+b \text{csch}^{-1}(c x)\right)}{e^3}+\frac{b \tan ^{-1}\left(\frac{\sqrt{c^2 d-e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}+1}}\right)}{2 e^{5/2} \sqrt{c^2 d-e}}+\frac{b \left(c^2 d-2 e\right) \tan ^{-1}\left(\frac{\sqrt{c^2 d-e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}+1}}\right)}{8 e^{5/2} \left(c^2 d-e\right)^{3/2}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e^3}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 e^3}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 e^3}+\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 e^3}+\frac{b c d \sqrt{\frac{1}{c^2 x^2}+1}}{8 e^2 x \left(c^2 d-e\right) \left(\frac{d}{x^2}+e\right)}+\frac{b \text{Li}_2\left(e^{-2 \text{csch}^{-1}(c x)}\right)}{2 e^3}",1,"-1/4*(a*d^2)/(e^3*(d + e*x^2)^2) + (a*d)/(e^3*(d + e*x^2)) + (a*Log[d + e*x^2])/(2*e^3) + b*(-1/16*(d*((I*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d - e)*Log[(4*d*Sqrt[c^2*d - e]*Sqrt[e]*(Sqrt[e] + I*c*(c*Sqrt[d] - Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/e^(5/2) - (d*(((-I)*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d - e)*Log[((4*I)*d*Sqrt[c^2*d - e]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] - I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/(16*e^(5/2)) - (((7*I)/16)*Sqrt[d]*(-(ArcCsch[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d]))/e^(5/2) + (((7*I)/16)*Sqrt[d]*(-(ArcCsch[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d]))/e^(5/2) + (Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/(16*e^3) + (Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 - 32*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/(16*e^3))","C",0
112,1,375,167,1.4759647,"\int \frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]","-\frac{\frac{8 a}{d+e x^2}-\frac{4 a d}{\left(d+e x^2\right)^2}+\frac{b \sqrt{e} \left(2 e-c^2 d\right) \log \left(\frac{16 d e^{3/2} \sqrt{e-c^2 d} \left(\sqrt{e}+c x \left(\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}-i c \sqrt{d}\right)\right)}{b \left(2 e-c^2 d\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d \left(e-c^2 d\right)^{3/2}}+\frac{b \sqrt{e} \left(2 e-c^2 d\right) \log \left(-\frac{16 i d e^{3/2} \sqrt{e-c^2 d} \left(\sqrt{e}+c x \left(\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}+i c \sqrt{d}\right)\right)}{b \left(c^2 d-2 e\right) \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{d \left(e-c^2 d\right)^{3/2}}-\frac{2 b c e x \sqrt{\frac{1}{c^2 x^2}+1}}{\left(e-c^2 d\right) \left(d+e x^2\right)}+\frac{4 b \text{csch}^{-1}(c x) \left(d+2 e x^2\right)}{\left(d+e x^2\right)^2}-\frac{4 b \sinh ^{-1}\left(\frac{1}{c x}\right)}{d}}{16 e^2}","\frac{x^4 \left(a+b \text{csch}^{-1}(c x)\right)}{4 d \left(d+e x^2\right)^2}+\frac{b c x \left(c^2 d-2 e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{\sqrt{c^2 d-e}}\right)}{8 d e^{3/2} \sqrt{-c^2 x^2} \left(c^2 d-e\right)^{3/2}}-\frac{b c x \sqrt{-c^2 x^2-1}}{8 e \sqrt{-c^2 x^2} \left(c^2 d-e\right) \left(d+e x^2\right)}",1,"-1/16*((-4*a*d)/(d + e*x^2)^2 + (8*a)/(d + e*x^2) - (2*b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x)/((-(c^2*d) + e)*(d + e*x^2)) + (4*b*(d + 2*e*x^2)*ArcCsch[c*x])/(d + e*x^2)^2 - (4*b*ArcSinh[1/(c*x)])/d + (b*Sqrt[e]*(-(c^2*d) + 2*e)*Log[(16*d*e^(3/2)*Sqrt[-(c^2*d) + e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*(-(c^2*d) + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(-(c^2*d) + e)^(3/2)) + (b*Sqrt[e]*(-(c^2*d) + 2*e)*Log[((-16*I)*d*e^(3/2)*Sqrt[-(c^2*d) + e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*(c^2*d - 2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(-(c^2*d) + e)^(3/2)))/e^2","C",1
113,1,368,205,0.9566489,"\int \frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]","\frac{1}{16} \left(-\frac{4 a}{e \left(d+e x^2\right)^2}+\frac{b \left(3 c^2 d-2 e\right) \log \left(\frac{16 d^2 \sqrt{e} \sqrt{e-c^2 d} \left(\sqrt{e}+c x \left(\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}-i c \sqrt{d}\right)\right)}{b \left(2 e-3 c^2 d\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d^2 \sqrt{e} \left(e-c^2 d\right)^{3/2}}+\frac{b \left(3 c^2 d-2 e\right) \log \left(-\frac{16 i d^2 \sqrt{e} \sqrt{e-c^2 d} \left(\sqrt{e}+c x \left(\sqrt{\frac{1}{c^2 x^2}+1} \sqrt{e-c^2 d}+i c \sqrt{d}\right)\right)}{b \left(3 c^2 d-2 e\right) \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{d^2 \sqrt{e} \left(e-c^2 d\right)^{3/2}}+\frac{2 b c x \sqrt{\frac{1}{c^2 x^2}+1}}{d \left(c^2 d-e\right) \left(d+e x^2\right)}+\frac{4 b \sinh ^{-1}\left(\frac{1}{c x}\right)}{d^2 e}-\frac{4 b \text{csch}^{-1}(c x)}{e \left(d+e x^2\right)^2}\right)","-\frac{a+b \text{csch}^{-1}(c x)}{4 e \left(d+e x^2\right)^2}+\frac{b c x \tan ^{-1}\left(\sqrt{-c^2 x^2-1}\right)}{4 d^2 e \sqrt{-c^2 x^2}}+\frac{b c x \left(3 c^2 d-2 e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{\sqrt{c^2 d-e}}\right)}{8 d^2 \sqrt{e} \sqrt{-c^2 x^2} \left(c^2 d-e\right)^{3/2}}+\frac{b c x \sqrt{-c^2 x^2-1}}{8 d \sqrt{-c^2 x^2} \left(c^2 d-e\right) \left(d+e x^2\right)}",1,"((-4*a)/(e*(d + e*x^2)^2) + (2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x)/(d*(c^2*d - e)*(d + e*x^2)) - (4*b*ArcCsch[c*x])/(e*(d + e*x^2)^2) + (4*b*ArcSinh[1/(c*x)])/(d^2*e) + (b*(3*c^2*d - 2*e)*Log[(16*d^2*Sqrt[e]*Sqrt[-(c^2*d) + e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*(-3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d^2*Sqrt[e]*(-(c^2*d) + e)^(3/2)) + (b*(3*c^2*d - 2*e)*Log[((-16*I)*d^2*Sqrt[e]*Sqrt[-(c^2*d) + e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(b*(3*c^2*d - 2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d^2*Sqrt[e]*(-(c^2*d) + e)^(3/2)))/16","C",1
114,0,0,657,70.3454672,"\int \frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^3),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 d^3}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{2 d^3}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}\right)}{2 d^3}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e-c^2 d}+\sqrt{e}}+1\right)}{2 d^3}+\frac{e^2 \left(a+b \text{csch}^{-1}(c x)\right)}{4 d^3 \left(\frac{d}{x^2}+e\right)^2}-\frac{e \left(a+b \text{csch}^{-1}(c x)\right)}{d^3 \left(\frac{d}{x^2}+e\right)}+\frac{\left(a+b \text{csch}^{-1}(c x)\right)^2}{2 b d^3}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 d^3}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{2 d^3}-\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 d^3}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{2 d^3}+\frac{b \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{c^2 d-e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}+1}}\right)}{d^3 \sqrt{c^2 d-e}}-\frac{b \sqrt{e} \left(c^2 d-2 e\right) \tan ^{-1}\left(\frac{\sqrt{c^2 d-e}}{c \sqrt{e} x \sqrt{\frac{1}{c^2 x^2}+1}}\right)}{8 d^3 \left(c^2 d-e\right)^{3/2}}-\frac{b c e \sqrt{\frac{1}{c^2 x^2}+1}}{8 d^2 x \left(c^2 d-e\right) \left(\frac{d}{x^2}+e\right)}",1,"Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^3), x]","F",-1
115,1,2045,1106,6.2144444,"\int \frac{x^4 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]","\text{Result too large to show}","-\frac{b \sqrt{-d} \sqrt{1+\frac{1}{c^2 x^2}} c}{16 \left(c^2 d-e\right) e^{3/2} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{b \sqrt{-d} \sqrt{1+\frac{1}{c^2 x^2}} c}{16 \left(c^2 d-e\right) e^{3/2} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{3 \left(a+b \text{csch}^{-1}(c x)\right)}{16 e^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{3 \left(a+b \text{csch}^{-1}(c x)\right)}{16 e^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{\sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right)}{16 e^{3/2} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{\sqrt{-d} \left(a+b \text{csch}^{-1}(c x)\right)}{16 e^{3/2} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}+\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} \left(c^2 d-e\right)^{3/2} e}-\frac{3 b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} \sqrt{c^2 d-e} e^2}+\frac{b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} \left(c^2 d-e\right)^{3/2} e}-\frac{3 b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} \sqrt{c^2 d-e} e^2}+\frac{3 \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{csch}^{-1}(c x)} c}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{csch}^{-1}(c x)} c}{\sqrt{e}+\sqrt{e-c^2 d}}+1\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{16 \sqrt{-d} e^{5/2}}",1,"(a*d*x)/(4*e^2*(d + e*x^2)^2) - (5*a*x)/(8*e^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(5/2)) + b*(((I/16)*Sqrt[d]*((I*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d - e)*Log[(4*d*Sqrt[c^2*d - e]*Sqrt[e]*(Sqrt[e] + I*c*(c*Sqrt[d] - Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/e^2 - ((I/16)*Sqrt[d]*(((-I)*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d - e)*Log[((4*I)*d*Sqrt[c^2*d - e]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] - I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/e^2 + (5*(-(ArcCsch[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d]))/(16*e^2) + (5*(-(ArcCsch[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d]))/(16*e^2) + (((3*I)/128)*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/(Sqrt[d]*e^(5/2)) - (((3*I)/128)*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 - 32*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/(Sqrt[d]*e^(5/2)))","C",0
116,1,2053,1106,6.148265,"\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^3,x]","\text{Result too large to show}","-\frac{b \sqrt{1+\frac{1}{c^2 x^2}} c}{16 \sqrt{-d} \left(c^2 d-e\right) \sqrt{e} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{b \sqrt{1+\frac{1}{c^2 x^2}} c}{16 \sqrt{-d} \left(c^2 d-e\right) \sqrt{e} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{a+b \text{csch}^{-1}(c x)}{16 d e \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{a+b \text{csch}^{-1}(c x)}{16 d e \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{a+b \text{csch}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{a+b \text{csch}^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}-\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} \sqrt{c^2 d-e} e}-\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} \left(c^2 d-e\right)^{3/2}}-\frac{b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} \sqrt{c^2 d-e} e}-\frac{b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} \left(c^2 d-e\right)^{3/2}}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{csch}^{-1}(c x)} c}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{csch}^{-1}(c x)} c}{\sqrt{e}+\sqrt{e-c^2 d}}+1\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{16 (-d)^{3/2} e^{3/2}}",1,"-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + b*(((-1/16*I)*((I*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d - e)*Log[(4*d*Sqrt[c^2*d - e]*Sqrt[e]*(Sqrt[e] + I*c*(c*Sqrt[d] - Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/(Sqrt[d]*e) + ((I/16)*(((-I)*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d - e)*Log[((4*I)*d*Sqrt[c^2*d - e]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] - I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/(Sqrt[d]*e) - (-(ArcCsch[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d])/(16*d*e) - (-(ArcCsch[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d])/(16*d*e) + ((I/128)*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/(d^(3/2)*e^(3/2)) - ((I/128)*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 - 32*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/(d^(3/2)*e^(3/2)))","C",0
117,1,2038,1096,6.0658567,"\int \frac{a+b \text{csch}^{-1}(c x)}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x^2)^3,x]","\text{Result too large to show}","-\frac{b \sqrt{e} \sqrt{1+\frac{1}{c^2 x^2}} c}{16 (-d)^{3/2} \left(c^2 d-e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{b \sqrt{e} \sqrt{1+\frac{1}{c^2 x^2}} c}{16 (-d)^{3/2} \left(c^2 d-e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}-\frac{5 \left(a+b \text{csch}^{-1}(c x)\right)}{16 d^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{5 \left(a+b \text{csch}^{-1}(c x)\right)}{16 d^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{\sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right)}{16 (-d)^{3/2} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{\sqrt{e} \left(a+b \text{csch}^{-1}(c x)\right)}{16 (-d)^{3/2} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}+\frac{b e \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \left(c^2 d-e\right)^{3/2}}+\frac{5 b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \sqrt{c^2 d-e}}+\frac{b e \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \left(c^2 d-e\right)^{3/2}}+\frac{5 b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{c^2 d-e} \sqrt{1+\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \sqrt{c^2 d-e}}+\frac{3 \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{csch}^{-1}(c x)} c}{\sqrt{e}-\sqrt{e-c^2 d}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left(a+b \text{csch}^{-1}(c x)\right) \log \left(1-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \text{csch}^{-1}(c x)\right) \log \left(\frac{\sqrt{-d} e^{\text{csch}^{-1}(c x)} c}{\sqrt{e}+\sqrt{e-c^2 d}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}-\sqrt{e-c^2 d}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{Li}_2\left(-\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{Li}_2\left(\frac{c \sqrt{-d} e^{\text{csch}^{-1}(c x)}}{\sqrt{e}+\sqrt{e-c^2 d}}\right)}{16 (-d)^{5/2} \sqrt{e}}",1,"(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + b*(((I/16)*((I*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d - e)*Log[(4*d*Sqrt[c^2*d - e]*Sqrt[e]*(Sqrt[e] + I*c*(c*Sqrt[d] - Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/d^(3/2) - ((I/16)*(((-I)*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d - e)*Log[((4*I)*d*Sqrt[c^2*d - e]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] - I*Sqrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/d^(3/2) - (3*(-(ArcCsch[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d]))/(16*d^2) - (3*(-(ArcCsch[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(-2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]))/Sqrt[d]))/(16*d^2) + (((3*I)/128)*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/(d^(5/2)*Sqrt[e]) - (((3*I)/128)*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 - 32*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + (4*I)*Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (16*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*PolyLog[2, ((-I)*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 8*PolyLog[2, (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]))/(d^(5/2)*Sqrt[e]))","C",0
118,1,345,413,0.7045614,"\int x^5 \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^5*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]),x]","\frac{\sqrt{d+e x^2} \left(16 a c^5 \left(8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right)+16 b c^5 \text{csch}^{-1}(c x) \left(8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right)+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^4 \left(-41 d^2+22 d e x^2+40 e^2 x^4\right)-2 c^2 e \left(19 d+25 e x^2\right)+75 e^2\right)\right)}{1680 c^5 e^3}-\frac{b \left(128 c^4 d^4 \sqrt{\frac{d}{e x^2}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;-\frac{1}{c^2 x^2},-\frac{d}{e x^2}\right)-\frac{e x^4 \sqrt{\frac{1}{c^2 x^2}+1} \left(105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right) \sqrt{\frac{e x^2}{d}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right)}{\sqrt{c^2 x^2+1}}\right)}{3360 c^5 e^3 x \sqrt{d+e x^2}}","\frac{d^2 \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^3}-\frac{2 d \left(d+e x^2\right)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 e^3}+\frac{\left(d+e x^2\right)^{7/2} \left(a+b \text{csch}^{-1}(c x)\right)}{7 e^3}+\frac{8 b c d^{7/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{105 e^3 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(d+e x^2\right)^{5/2}}{42 c e^2 \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left(29 c^2 d+25 e\right) \left(d+e x^2\right)^{3/2}}{840 c^3 e^2 \sqrt{-c^2 x^2}}+\frac{b x \left(105 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2-75 e^3\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{1680 c^6 e^{5/2} \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left(23 c^4 d^2-12 c^2 d e-75 e^2\right) \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{-c^2 x^2}}",1,"-1/3360*(b*(128*c^4*d^4*Sqrt[1 + d/(e*x^2)]*AppellF1[1, 1/2, 1/2, 2, -(1/(c^2*x^2)), -(d/(e*x^2))] - (e*(105*c^6*d^3 + 35*c^4*d^2*e + 63*c^2*d*e^2 - 75*e^3)*Sqrt[1 + 1/(c^2*x^2)]*x^4*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^2), -((e*x^2)/d)])/Sqrt[1 + c^2*x^2]))/(c^5*e^3*x*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(16*a*c^5*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6) + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(75*e^2 - 2*c^2*e*(19*d + 25*e*x^2) + c^4*(-41*d^2 + 22*d*e*x^2 + 40*e^2*x^4)) + 16*b*c^5*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6)*ArcCsch[c*x]))/(1680*c^5*e^3)","C",0
119,1,337,302,0.7521686,"\int x^3 \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^3*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]),x]","\frac{\sqrt{d+e x^2} \left(8 a c^3 \left(-2 d^2+d e x^2+3 e^2 x^4\right)+8 b c^3 \text{csch}^{-1}(c x) \left(-2 d^2+d e x^2+3 e^2 x^4\right)+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 \left(7 d+6 e x^2\right)-9 e\right)\right)}{120 c^3 e^2}+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(16 c^7 d^{5/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \left(15 c^4 d^2+10 c^2 d e-9 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d-e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right)\right)}{120 c^6 e^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","-\frac{d \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^2}+\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 e^2}-\frac{2 b c d^{5/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{15 e^2 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{20 c e \sqrt{-c^2 x^2}}-\frac{b x \left(15 c^4 d^2+10 c^2 d e-9 e^2\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{120 c^4 e^{3/2} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(c^2 d-9 e\right) \sqrt{d+e x^2}}{120 c^3 e \sqrt{-c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^3*(-2*d^2 + d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(-9*e + c^2*(7*d + 6*e*x^2)) + 8*b*c^3*(-2*d^2 + d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x]))/(120*c^3*e^2) + (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-(Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*(15*c^4*d^2 + 10*c^2*d*e - 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])]) + 16*c^7*d^(5/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(120*c^6*e^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
120,1,278,203,0.510634,"\int x \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]),x]","\frac{\sqrt{d+e x^2} \left(2 a c \left(d+e x^2\right)+b e x \sqrt{\frac{1}{c^2 x^2}+1}+2 b c \text{csch}^{-1}(c x) \left(d+e x^2\right)\right)}{6 c e}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(2 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \left(3 c^2 d-e\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d-e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right)\right)}{6 c^4 e \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e}+\frac{b c d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{3 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{6 c \sqrt{-c^2 x^2}}+\frac{b x \left(3 c^2 d-e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{6 c^2 \sqrt{e} \sqrt{-c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(b*e*Sqrt[1 + 1/(c^2*x^2)]*x + 2*a*c*(d + e*x^2) + 2*b*c*(d + e*x^2)*ArcCsch[c*x]))/(6*c*e) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-(Sqrt[c^2]*Sqrt[c^2*d - e]*(3*c^2*d - e)*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])]) + 2*c^5*d^(3/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(6*c^4*e*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
121,0,0,26,5.6245661,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x, x]","A",-1
122,0,0,26,5.3993031,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^3} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x^3,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^3} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^3},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x^3, x]","A",-1
123,0,0,26,9.3094611,"\int x^2 \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]),x]","\int x^2 \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","\text{Int}\left(x^2 \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right),x\right)",0,"Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x]","A",-1
124,0,0,23,2.6557611,"\int \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]),x]","\int \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right),x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x]","A",-1
125,0,0,26,1.7791486,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x^2,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x^2, x]","A",-1
126,1,237,389,0.6185536,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^4} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x^4,x]","-\frac{\sqrt{d+e x^2} \left(3 a \left(d+e x^2\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(2 c^2 d x^2-d-4 e x^2\right)+3 b \text{csch}^{-1}(c x) \left(d+e x^2\right)\right)}{9 d x^3}-\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left(2 c^2 d \left(c^2 d-2 e\right) E\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)+\left(-2 c^4 d^2+5 c^2 d e-3 e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)\right)}{9 \sqrt{c^2} d \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 d x^3}-\frac{b e x \left(c^2 d-3 e\right) \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{9 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}-\frac{2 b c \sqrt{-c^2 x^2-1} \left(c^2 d-2 e\right) \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{9 x^2 \sqrt{-c^2 x^2}}+\frac{2 b c^2 x \left(c^2 d-2 e\right) \sqrt{d+e x^2} E\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{9 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}-\frac{2 b c^3 x^2 \left(c^2 d-2 e\right) \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}",1,"-1/9*(Sqrt[d + e*x^2]*(b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-d + 2*c^2*d*x^2 - 4*e*x^2) + 3*a*(d + e*x^2) + 3*b*(d + e*x^2)*ArcCsch[c*x]))/(d*x^3) - ((I/9)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(2*c^2*d*(c^2*d - 2*e)*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] + (-2*c^4*d^2 + 5*c^2*d*e - 3*e^2)*EllipticF[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","C",1
127,1,314,527,0.6956984,"\int \frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^6} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/x^6,x]","\frac{\sqrt{d+e x^2} \left(-15 a \left(3 d^2+d e x^2-2 e^2 x^4\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(d e x^2 \left(8-19 c^2 x^2\right)+3 d^2 \left(8 c^4 x^4-4 c^2 x^2+3\right)-31 e^2 x^4\right)-15 b \text{csch}^{-1}(c x) \left(3 d^2+d e x^2-2 e^2 x^4\right)\right)}{225 d^2 x^5}+\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(24 c^4 d^2-19 c^2 d e-31 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)+\left(-24 c^6 d^3+31 c^4 d^2 e+23 c^2 d e^2-30 e^3\right) F\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)\right)}{225 \sqrt{c^2} d^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","\frac{2 e \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{15 d^2 x^3}-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 d x^5}-\frac{b c \sqrt{-c^2 x^2-1} \left(12 c^2 d+e\right) \sqrt{d+e x^2}}{225 d x^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{25 d x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left(24 c^4 d^2-19 c^2 d e-31 e^2\right) \sqrt{d+e x^2}}{225 d^2 \sqrt{-c^2 x^2}}-\frac{b c^2 x \left(24 c^4 d^2-19 c^2 d e-31 e^2\right) \sqrt{d+e x^2} E\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{225 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{2 b e x \left(6 c^4 d^2-4 c^2 d e-15 e^2\right) \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{225 d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c^3 x^2 \left(24 c^4 d^2-19 c^2 d e-31 e^2\right) \sqrt{d+e x^2}}{225 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}",1,"(Sqrt[d + e*x^2]*(-15*a*(3*d^2 + d*e*x^2 - 2*e^2*x^4) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-31*e^2*x^4 + d*e*x^2*(8 - 19*c^2*x^2) + 3*d^2*(3 - 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(3*d^2 + d*e*x^2 - 2*e^2*x^4)*ArcCsch[c*x]))/(225*d^2*x^5) + ((I/225)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(24*c^4*d^2 - 19*c^2*d*e - 31*e^2)*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] + (-24*c^6*d^3 + 31*c^4*d^2*e + 23*c^2*d*e^2 - 30*e^3)*EllipticF[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","C",1
128,1,318,384,0.7690122,"\int x^3 \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]),x]","\frac{\sqrt{d+e x^2} \left(-48 a c^5 \left(2 d-5 e x^2\right) \left(d+e x^2\right)^2-48 b c^5 \text{csch}^{-1}(c x) \left(2 d-5 e x^2\right) \left(d+e x^2\right)^2+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^4 \left(57 d^2+106 d e x^2+40 e^2 x^4\right)-2 c^2 e \left(82 d+25 e x^2\right)+75 e^2\right)\right)}{1680 c^5 e^2}-\frac{b \left(\frac{e x^4 \sqrt{\frac{1}{c^2 x^2}+1} \left(35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right) \sqrt{\frac{e x^2}{d}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right)}{\sqrt{c^2 x^2+1}}-32 c^4 d^4 \sqrt{\frac{d}{e x^2}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;-\frac{1}{c^2 x^2},-\frac{d}{e x^2}\right)\right)}{1120 c^5 e^2 x \sqrt{d+e x^2}}","-\frac{d \left(d+e x^2\right)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 e^2}+\frac{\left(d+e x^2\right)^{7/2} \left(a+b \text{csch}^{-1}(c x)\right)}{7 e^2}-\frac{2 b c d^{7/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{35 e^2 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(d+e x^2\right)^{5/2}}{42 c e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(13 c^2 d-25 e\right) \left(d+e x^2\right)^{3/2}}{840 c^3 e \sqrt{-c^2 x^2}}-\frac{b x \left(35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{560 c^6 e^{3/2} \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left(3 c^4 d^2+38 c^2 d e-25 e^2\right) \sqrt{d+e x^2}}{560 c^5 e \sqrt{-c^2 x^2}}",1,"-1/1120*(b*(-32*c^4*d^4*Sqrt[1 + d/(e*x^2)]*AppellF1[1, 1/2, 1/2, 2, -(1/(c^2*x^2)), -(d/(e*x^2))] + (e*(35*c^6*d^3 + 35*c^4*d^2*e - 63*c^2*d*e^2 + 25*e^3)*Sqrt[1 + 1/(c^2*x^2)]*x^4*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^2), -((e*x^2)/d)])/Sqrt[1 + c^2*x^2]))/(c^5*e^2*x*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(-48*a*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2 + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(75*e^2 - 2*c^2*e*(82*d + 25*e*x^2) + c^4*(57*d^2 + 106*d*e*x^2 + 40*e^2*x^4)) - 48*b*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2*ArcCsch[c*x]))/(1680*c^5*e^2)","C",0
129,1,314,270,0.7980088,"\int x \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]),x]","\frac{\sqrt{d+e x^2} \left(8 a c^3 \left(d+e x^2\right)^2+8 b c^3 \text{csch}^{-1}(c x) \left(d+e x^2\right)^2+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 \left(9 d+2 e x^2\right)-3 e\right)\right)}{40 c^3 e}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(8 c^7 d^{5/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \left(-15 c^4 d^2+10 c^2 d e-3 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d-e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right)\right)}{40 c^6 e \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 e}+\frac{b c d^{5/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{5 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{20 c \sqrt{-c^2 x^2}}+\frac{b x \left(15 c^4 d^2-10 c^2 d e+3 e^2\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{40 c^4 \sqrt{e} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(7 c^2 d-3 e\right) \sqrt{d+e x^2}}{40 c^3 \sqrt{-c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^3*(d + e*x^2)^2 + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(-3*e + c^2*(9*d + 2*e*x^2)) + 8*b*c^3*(d + e*x^2)^2*ArcCsch[c*x]))/(40*c^3*e) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*(-15*c^4*d^2 + 10*c^2*d*e - 3*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])] + 8*c^7*d^(5/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(40*c^6*e*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
130,0,0,26,6.5590365,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x, x]","A",-1
131,0,0,26,5.8604221,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^3,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^3} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^3},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^3, x]","A",-1
132,0,0,26,9.6686468,"\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]),x]","\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","\text{Int}\left(x^2 \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right),x\right)",0,"Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x]","A",-1
133,0,0,23,3.5232032,"\int \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]),x]","\int \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right),x\right)",0,"Integrate[(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x]","A",-1
134,0,0,26,5.793695,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^2,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^2, x]","A",-1
135,0,0,26,15.9688204,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^4,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^4} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^4},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^4, x]","A",-1
136,1,291,492,0.7025626,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^6,x]","\frac{\sqrt{d+e x^2} \left(-15 a \left(d+e x^2\right)^2+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(d e x^2 \left(11-23 c^2 x^2\right)+d^2 \left(8 c^4 x^4-4 c^2 x^2+3\right)+23 e^2 x^4\right)-15 b \text{csch}^{-1}(c x) \left(d+e x^2\right)^2\right)}{75 d x^5}+\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(8 c^4 d^2-23 c^2 d e+23 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)+\left(-8 c^6 d^3+27 c^4 d^2 e-34 c^2 d e^2+15 e^3\right) F\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)\right)}{75 \sqrt{c^2} d \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 d x^5}-\frac{4 b c \sqrt{-c^2 x^2-1} \left(c^2 d-2 e\right) \sqrt{d+e x^2}}{75 x^2 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{25 x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left(8 c^4 d^2-23 c^2 d e+23 e^2\right) \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2}}+\frac{b e x \left(4 c^4 d^2-11 c^2 d e+15 e^2\right) \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{75 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}-\frac{b c^2 x \left(8 c^4 d^2-23 c^2 d e+23 e^2\right) \sqrt{d+e x^2} E\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{75 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c^3 x^2 \left(8 c^4 d^2-23 c^2 d e+23 e^2\right) \sqrt{d+e x^2}}{75 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}",1,"(Sqrt[d + e*x^2]*(-15*a*(d + e*x^2)^2 + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(23*e^2*x^4 + d*e*x^2*(11 - 23*c^2*x^2) + d^2*(3 - 4*c^2*x^2 + 8*c^4*x^4)) - 15*b*(d + e*x^2)^2*ArcCsch[c*x]))/(75*d*x^5) + ((I/75)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(8*c^4*d^2 - 23*c^2*d*e + 23*e^2)*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] + (-8*c^6*d^3 + 27*c^4*d^2*e - 34*c^2*d*e^2 + 15*e^3)*EllipticF[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","C",1
137,1,372,643,0.8280754,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/x^8,x]","-\frac{\sqrt{d+e x^2} \left(105 a \left(5 d-2 e x^2\right) \left(d+e x^2\right)^2+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(d e^2 x^4 \left(193 c^2 x^2-71\right)-3 d^2 e x^2 \left(176 c^4 x^4-83 c^2 x^2+61\right)+15 d^3 \left(16 c^6 x^6-8 c^4 x^4+6 c^2 x^2-5\right)+247 e^3 x^6\right)+105 b \text{csch}^{-1}(c x) \left(5 d-2 e x^2\right) \left(d+e x^2\right)^2\right)}{3675 d^2 x^7}-\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right) E\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)-2 \left(120 c^8 d^4-324 c^6 d^3 e+221 c^4 d^2 e^2+88 c^2 d e^3-105 e^4\right) F\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)\right)}{3675 \sqrt{c^2} d^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","\frac{2 e \left(d+e x^2\right)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{35 d^2 x^5}-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{7 d x^7}+\frac{b c \sqrt{-c^2 x^2-1} \left(d+e x^2\right)^{5/2}}{49 d x^6 \sqrt{-c^2 x^2}}-\frac{b c \sqrt{-c^2 x^2-1} \left(30 c^2 d-11 e\right) \left(d+e x^2\right)^{3/2}}{1225 d x^4 \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \left(120 c^4 d^2-159 c^2 d e-37 e^2\right) \sqrt{d+e x^2}}{3675 d x^2 \sqrt{-c^2 x^2}}-\frac{b c \sqrt{-c^2 x^2-1} \left(240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right) \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2}}-\frac{b e x \left(120 c^6 d^3-249 c^4 d^2 e+71 c^2 d e^2+210 e^3\right) \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{3675 d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c^2 x \left(240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right) \sqrt{d+e x^2} E\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{3675 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}-\frac{b c^3 x^2 \left(240 c^6 d^3-528 c^4 d^2 e+193 c^2 d e^2+247 e^3\right) \sqrt{d+e x^2}}{3675 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}",1,"-1/3675*(Sqrt[d + e*x^2]*(105*a*(5*d - 2*e*x^2)*(d + e*x^2)^2 + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(247*e^3*x^6 + d*e^2*x^4*(-71 + 193*c^2*x^2) - 3*d^2*e*x^2*(61 - 83*c^2*x^2 + 176*c^4*x^4) + 15*d^3*(-5 + 6*c^2*x^2 - 8*c^4*x^4 + 16*c^6*x^6)) + 105*b*(5*d - 2*e*x^2)*(d + e*x^2)^2*ArcCsch[c*x]))/(d^2*x^7) - ((I/3675)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(240*c^6*d^3 - 528*c^4*d^2*e + 193*c^2*d*e^2 + 247*e^3)*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] - 2*(120*c^8*d^4 - 324*c^6*d^3*e + 221*c^4*d^2*e^2 + 88*c^2*d*e^3 - 105*e^4)*EllipticF[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","C",1
138,1,339,329,0.7890517,"\int \frac{x^5 \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^5*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(8 a c^3 \left(8 d^2-4 d e x^2+3 e^2 x^4\right)+8 b c^3 \text{csch}^{-1}(c x) \left(8 d^2-4 d e x^2+3 e^2 x^4\right)+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^2 \left(6 e x^2-13 d\right)-9 e\right)\right)}{120 c^3 e^3}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(64 c^7 d^{5/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \left(45 c^4 d^2+10 c^2 d e+9 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d-e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right)\right)}{120 c^6 e^3 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","\frac{d^2 \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{e^3}-\frac{2 d \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^3}+\frac{\left(d+e x^2\right)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{5 e^3}+\frac{8 b c d^{5/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{15 e^3 \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{20 c e^2 \sqrt{-c^2 x^2}}+\frac{b x \left(45 c^4 d^2+10 c^2 d e+9 e^2\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{120 c^4 e^{5/2} \sqrt{-c^2 x^2}}-\frac{b x \sqrt{-c^2 x^2-1} \left(19 c^2 d+9 e\right) \sqrt{d+e x^2}}{120 c^3 e^2 \sqrt{-c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^3*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(-9*e + c^2*(-13*d + 6*e*x^2)) + 8*b*c^3*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x]))/(120*c^3*e^3) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-(Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*(45*c^4*d^2 + 10*c^2*d*e + 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])]) + 64*c^7*d^(5/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(120*c^6*e^3*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
139,1,280,229,0.5649565,"\int \frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^3*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(-4 a c d+2 a c e x^2+b e x \sqrt{\frac{1}{c^2 x^2}+1}+2 b c \text{csch}^{-1}(c x) \left(e x^2-2 d\right)\right)}{6 c e^2}+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(4 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \left(3 c^2 d+e\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d-e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right)\right)}{6 c^4 e^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","-\frac{d \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{e^2}+\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^2}-\frac{2 b c d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{3 e^2 \sqrt{-c^2 x^2}}-\frac{b x \left(3 c^2 d+e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{6 c^2 e^{3/2} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{6 c e \sqrt{-c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(-4*a*c*d + b*e*Sqrt[1 + 1/(c^2*x^2)]*x + 2*a*c*e*x^2 + 2*b*c*(-2*d + e*x^2)*ArcCsch[c*x]))/(6*c*e^2) + (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-(Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*(3*c^2*d + e)*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])]) + 4*c^5*d^(3/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(6*c^4*e^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
140,1,223,135,0.2395353,"\int \frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{e}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(c^3 \sqrt{d} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d-e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right)\right)}{c^2 e \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","\frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{e}+\frac{b x \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{\sqrt{e} \sqrt{-c^2 x^2}}+\frac{b c \sqrt{d} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{e \sqrt{-c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-(Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])]) + c^3*Sqrt[d]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(c^2*e*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
141,0,0,26,1.7454291,"\int \frac{a+b \text{csch}^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*Sqrt[d + e*x^2]),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x \sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x*Sqrt[d + e*x^2]), x]","A",-1
142,0,0,26,23.2549024,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^3*Sqrt[d + e*x^2]),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x^3 \sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x^3*Sqrt[d + e*x^2]), x]","A",-1
143,0,0,26,6.3328301,"\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^2*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(x^2*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2], x]","A",-1
144,0,0,23,1.045851,"\int \frac{a+b \text{csch}^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/Sqrt[d + e*x^2],x]","\int \frac{a+b \text{csch}^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/Sqrt[d + e*x^2], x]","A",-1
145,1,139,294,0.2071793,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^2*Sqrt[d + e*x^2]),x]","\frac{\sqrt{d+e x^2} \left(-a+b c x \sqrt{\frac{1}{c^2 x^2}+1}-b \text{csch}^{-1}(c x)\right)}{d x}-\frac{b c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{e}{d}} x\right)|\frac{c^2 d}{e}\right)}{d \sqrt{c^2 x^2+1} \sqrt{-\frac{e}{d}} \sqrt{d+e x^2}}","-\frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{d x}+\frac{b e x \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{d \sqrt{-c^2 x^2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}",1,"(Sqrt[d + e*x^2]*(-a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x - b*ArcCsch[c*x]))/(d*x) - (b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], (c^2*d)/e])/(d*Sqrt[-(e/d)]*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
146,1,239,425,0.6360744,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^4 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^4*Sqrt[d + e*x^2]),x]","-\frac{\sqrt{d+e x^2} \left(3 a \left(d-2 e x^2\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(2 c^2 d x^2-d+5 e x^2\right)+3 b \text{csch}^{-1}(c x) \left(d-2 e x^2\right)\right)}{9 d^2 x^3}-\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(2 c^2 d+5 e\right) E\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)-2 \left(c^4 d^2+2 c^2 d e-3 e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)\right)}{9 \sqrt{c^2} d^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","\frac{2 e \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 d^2 x}-\frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 d x^3}-\frac{b e x \left(c^2 d+6 e\right) \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{9 d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}-\frac{b c \sqrt{-c^2 x^2-1} \left(2 c^2 d+5 e\right) \sqrt{d+e x^2}}{9 d^2 \sqrt{-c^2 x^2}}+\frac{b c^2 x \left(2 c^2 d+5 e\right) \sqrt{d+e x^2} E\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{9 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{9 d x^2 \sqrt{-c^2 x^2}}-\frac{b c^3 x^2 \left(2 c^2 d+5 e\right) \sqrt{d+e x^2}}{9 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}",1,"-1/9*(Sqrt[d + e*x^2]*(3*a*(d - 2*e*x^2) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-d + 2*c^2*d*x^2 + 5*e*x^2) + 3*b*(d - 2*e*x^2)*ArcCsch[c*x]))/(d^2*x^3) - ((I/9)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(2*c^2*d + 5*e)*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] - 2*(c^4*d^2 + 2*c^2*d*e - 3*e^2)*EllipticF[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","C",1
147,1,311,256,0.6444514,"\int \frac{x^5 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]","\frac{-2 a c \left(8 d^2+4 d e x^2-e^2 x^4\right)+b e x \sqrt{\frac{1}{c^2 x^2}+1} \left(d+e x^2\right)-2 b c \text{csch}^{-1}(c x) \left(8 d^2+4 d e x^2-e^2 x^4\right)}{6 c e^3 \sqrt{d+e x^2}}+\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(16 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \left(9 c^2 d+e\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d-e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right)\right)}{6 c^4 e^3 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","-\frac{d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{e^3}+\frac{\left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^3}-\frac{8 b c d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{3 e^3 \sqrt{-c^2 x^2}}-\frac{b x \left(9 c^2 d+e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{6 c^2 e^{5/2} \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{6 c e^2 \sqrt{-c^2 x^2}}",1,"(b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2) - 2*a*c*(8*d^2 + 4*d*e*x^2 - e^2*x^4) - 2*b*c*(8*d^2 + 4*d*e*x^2 - e^2*x^4)*ArcCsch[c*x])/(6*c*e^3*Sqrt[d + e*x^2]) + (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-(Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*(9*c^2*d + e)*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])]) + 16*c^5*d^(3/2)*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(6*c^4*e^3*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
148,1,233,160,0.3747236,"\int \frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]","\frac{\left(2 d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(2 c^3 \sqrt{d} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d-e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right)\right)}{c^2 e^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","\frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{e^2}+\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}+\frac{b x \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{e^{3/2} \sqrt{-c^2 x^2}}+\frac{2 b c \sqrt{d} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{e^2 \sqrt{-c^2 x^2}}",1,"((2*d + e*x^2)*(a + b*ArcCsch[c*x]))/(e^2*Sqrt[d + e*x^2]) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-(Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])]) + 2*c^3*Sqrt[d]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(c^2*e^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
149,1,122,82,0.1649192,"\int \frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]","\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)}{\sqrt{d} e \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}-\frac{a+b \text{csch}^{-1}(c x)}{e \sqrt{d+e x^2}}","-\frac{a+b \text{csch}^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{b c x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{\sqrt{d} e \sqrt{-c^2 x^2}}",1,"-((a + b*ArcCsch[c*x])/(e*Sqrt[d + e*x^2])) + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]])/(Sqrt[d]*e*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
150,0,0,26,32.9598155,"\int \frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^(3/2)), x]","A",-1
151,0,0,26,38.5937638,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(3/2)), x]","A",-1
152,0,0,26,8.804889,"\int \frac{x^4 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^4 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^4 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
153,0,0,26,5.019253,"\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
154,1,113,111,0.197239,"\int \frac{a+b \text{csch}^{-1}(c x)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x^2)^(3/2),x]","\frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{d \sqrt{d+e x^2}}+\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}\left(\sqrt{-c^2} x\right)|\frac{e}{c^2 d}\right)}{\sqrt{-c^2} d \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","\frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{d \sqrt{d+e x^2}}-\frac{b x \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}",1,"(x*(a + b*ArcCsch[c*x]))/(d*Sqrt[d + e*x^2]) + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[Sqrt[-c^2]*x], e/(c^2*d)])/(Sqrt[-c^2]*d*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
155,1,201,321,0.4570495,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^2*(d + e*x^2)^(3/2)),x]","\frac{-a \left(d+2 e x^2\right)+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left(d+e x^2\right)-b \text{csch}^{-1}(c x) \left(d+2 e x^2\right)}{d^2 x \sqrt{d+e x^2}}+\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left(\left(2 e-c^2 d\right) F\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)+c^2 d E\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)\right)}{\sqrt{c^2} d^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","-\frac{2 e x \left(a+b \text{csch}^{-1}(c x)\right)}{d^2 \sqrt{d+e x^2}}-\frac{a+b \text{csch}^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{2 b e x \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}",1,"(b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2) - a*(d + 2*e*x^2) - b*(d + 2*e*x^2)*ArcCsch[c*x])/(d^2*x*Sqrt[d + e*x^2]) + (I*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] + (-(c^2*d) + 2*e)*EllipticF[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","C",1
156,1,327,251,0.5622032,"\int \frac{x^5 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2),x]","\frac{a \left(c^2 d-e\right) \left(8 d^2+12 d e x^2+3 e^2 x^4\right)+b \left(c^2 d-e\right) \text{csch}^{-1}(c x) \left(8 d^2+12 d e x^2+3 e^2 x^4\right)+b c d e x \sqrt{\frac{1}{c^2 x^2}+1} \left(d+e x^2\right)}{3 e^3 \left(c^2 d-e\right) \left(d+e x^2\right)^{3/2}}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left(8 c^3 \sqrt{d} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)-3 \sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d-e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right)\right)}{3 c^2 e^3 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","-\frac{d^2 \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^3 \left(d+e x^2\right)^{3/2}}+\frac{2 d \left(a+b \text{csch}^{-1}(c x)\right)}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right)}{e^3}+\frac{b x \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{e^{5/2} \sqrt{-c^2 x^2}}+\frac{8 b c \sqrt{d} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{3 e^3 \sqrt{-c^2 x^2}}+\frac{b c d x \sqrt{-c^2 x^2-1}}{3 e^2 \sqrt{-c^2 x^2} \left(c^2 d-e\right) \sqrt{d+e x^2}}",1,"(b*c*d*e*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2) + a*(c^2*d - e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4) + b*(c^2*d - e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x])/(3*(c^2*d - e)*e^3*(d + e*x^2)^(3/2)) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-3*Sqrt[c^2]*Sqrt[c^2*d - e]*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d - e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])] + 8*c^3*Sqrt[d]*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(3*c^2*e^3*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
157,1,201,169,0.2978944,"\int \frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2),x]","\frac{a \left(c^2 d-e\right) \left(2 d+3 e x^2\right)+b c e x \sqrt{\frac{1}{c^2 x^2}+1} \left(d+e x^2\right)+b \left(c^2 d-e\right) \text{csch}^{-1}(c x) \left(2 d+3 e x^2\right)}{3 e^2 \left(e-c^2 d\right) \left(d+e x^2\right)^{3/2}}+\frac{2 b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)}{3 \sqrt{d} e^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","-\frac{a+b \text{csch}^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left(a+b \text{csch}^{-1}(c x)\right)}{3 e^2 \left(d+e x^2\right)^{3/2}}-\frac{2 b c x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{3 \sqrt{d} e^2 \sqrt{-c^2 x^2}}-\frac{b c x \sqrt{-c^2 x^2-1}}{3 e \sqrt{-c^2 x^2} \left(c^2 d-e\right) \sqrt{d+e x^2}}",1,"(b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2) + a*(c^2*d - e)*(2*d + 3*e*x^2) + b*(c^2*d - e)*(2*d + 3*e*x^2)*ArcCsch[c*x])/(3*e^2*(-(c^2*d) + e)*(d + e*x^2)^(3/2)) + (2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]])/(3*Sqrt[d]*e^2*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
158,1,185,144,0.2335164,"\int \frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2),x]","\frac{a d \left(e-c^2 d\right)+b c e x \sqrt{\frac{1}{c^2 x^2}+1} \left(d+e x^2\right)+b d \left(e-c^2 d\right) \text{csch}^{-1}(c x)}{3 d e \left(c^2 d-e\right) \left(d+e x^2\right)^{3/2}}+\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-d-e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right)}{3 d^{3/2} e \sqrt{c^2 x^2+1} \sqrt{d+e x^2}}","-\frac{a+b \text{csch}^{-1}(c x)}{3 e \left(d+e x^2\right)^{3/2}}-\frac{b c x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right)}{3 d^{3/2} e \sqrt{-c^2 x^2}}+\frac{b c x \sqrt{-c^2 x^2-1}}{3 d \sqrt{-c^2 x^2} \left(c^2 d-e\right) \sqrt{d+e x^2}}",1,"(a*d*(-(c^2*d) + e) + b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2) + b*d*(-(c^2*d) + e)*ArcCsch[c*x])/(3*d*(c^2*d - e)*e*(d + e*x^2)^(3/2)) + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[-d - e*x^2]*ArcTan[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]])/(3*d^(3/2)*e*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
159,0,0,26,45.8436203,"\int \frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^(5/2)), x]","A",-1
160,0,0,26,59.8356594,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x^3*(d + e*x^2)^(5/2)), x]","A",-1
161,0,0,26,13.402521,"\int \frac{x^6 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^6*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^6 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^6 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(x^6*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
162,0,0,26,12.6017742,"\int \frac{x^4 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^4 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^4 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
163,1,189,359,0.2964241,"\int \frac{x^2 \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(5/2),x]","\frac{x^2 \left(a x \left(c^2 d-e\right)+b c \sqrt{\frac{1}{c^2 x^2}+1} \left(d+e x^2\right)+b x \left(c^2 d-e\right) \text{csch}^{-1}(c x)\right)}{3 d \left(c^2 d-e\right) \left(d+e x^2\right)^{3/2}}-\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{e}{d}} x\right)|\frac{c^2 d}{e}\right)}{3 d \sqrt{c^2 x^2+1} \sqrt{-\frac{e}{d}} \left(c^2 d-e\right) \sqrt{d+e x^2}}","\frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{b x \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{3 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left(c^2 d-e\right) \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c x^2 \sqrt{-c^2 x^2-1}}{3 d \sqrt{-c^2 x^2} \left(c^2 d-e\right) \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{3 d e \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left(c^2 d-e\right) \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{3 d e \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left(c^2 d-e\right)}",1,"(x^2*(a*(c^2*d - e)*x + b*c*Sqrt[1 + 1/(c^2*x^2)]*(d + e*x^2) + b*(c^2*d - e)*x*ArcCsch[c*x]))/(3*d*(c^2*d - e)*(d + e*x^2)^(3/2)) - (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], (c^2*d)/e])/(3*d*(c^2*d - e)*Sqrt[-(e/d)]*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
164,1,248,278,0.5600128,"\int \frac{a+b \text{csch}^{-1}(c x)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(d + e*x^2)^(5/2),x]","\frac{x \left(a \left(c^2 d-e\right) \left(3 d+2 e x^2\right)-b c e x \sqrt{\frac{1}{c^2 x^2}+1} \left(d+e x^2\right)+b \left(c^2 d-e\right) \text{csch}^{-1}(c x) \left(3 d+2 e x^2\right)\right)}{3 d^2 \left(c^2 d-e\right) \left(d+e x^2\right)^{3/2}}-\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left(2 \left(c^2 d-e\right) F\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)+c^2 d E\left(i \sinh ^{-1}\left(\sqrt{c^2} x\right)|\frac{e}{c^2 d}\right)\right)}{3 \sqrt{c^2} d^2 \sqrt{c^2 x^2+1} \left(c^2 d-e\right) \sqrt{d+e x^2}}","\frac{2 x \left(a+b \text{csch}^{-1}(c x)\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left(a+b \text{csch}^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}-\frac{b c \sqrt{e} x \sqrt{-c^2 x^2-1} E\left(\tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)|1-\frac{c^2 d}{e}\right)}{3 d^{3/2} \sqrt{-c^2 x^2} \left(c^2 d-e\right) \sqrt{d+e x^2} \sqrt{\frac{d \left(c^2 x^2+1\right)}{d+e x^2}}}-\frac{b x \left(3 c^2 d-2 e\right) \sqrt{d+e x^2} F\left(\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right)}{3 d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \left(c^2 d-e\right) \sqrt{\frac{d+e x^2}{d \left(c^2 x^2+1\right)}}}",1,"(x*(-(b*c*e*Sqrt[1 + 1/(c^2*x^2)]*x*(d + e*x^2)) + a*(c^2*d - e)*(3*d + 2*e*x^2) + b*(c^2*d - e)*(3*d + 2*e*x^2)*ArcCsch[c*x]))/(3*d^2*(c^2*d - e)*(d + e*x^2)^(3/2)) - ((I/3)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*EllipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] + 2*(c^2*d - e)*EllipticF[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)]))/(Sqrt[c^2]*d^2*(c^2*d - e)*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])","C",1
165,0,0,596,0.2284952,"\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcCsch[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","\frac{d^3 (f x)^{m+1} \left(a+b \text{csch}^{-1}(c x)\right)}{f (m+1)}+\frac{3 d^2 e (f x)^{m+3} \left(a+b \text{csch}^{-1}(c x)\right)}{f^3 (m+3)}+\frac{3 d e^2 (f x)^{m+5} \left(a+b \text{csch}^{-1}(c x)\right)}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left(a+b \text{csch}^{-1}(c x)\right)}{f^7 (m+7)}+\frac{b e^3 x \sqrt{-c^2 x^2-1} (f x)^{m+5}}{c f^5 (m+6) (m+7) \sqrt{-c^2 x^2}}-\frac{b e^2 x \sqrt{-c^2 x^2-1} (f x)^{m+3} \left(e (m+5)^2-3 c^2 d \left(m^2+13 m+42\right)\right)}{c^3 f^3 (m+4) (m+5) (m+6) (m+7) \sqrt{-c^2 x^2}}+\frac{b e x \sqrt{-c^2 x^2-1} (f x)^{m+1} \left(3 c^4 d^2 \left(m^4+22 m^3+179 m^2+638 m+840\right)-3 c^2 d e (m+3)^2 \left(m^2+13 m+42\right)+e^2 \left(m^2+8 m+15\right)^2\right)}{c^5 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \sqrt{-c^2 x^2}}-\frac{b x \sqrt{c^2 x^2+1} (f x)^{m+1} \left(\frac{c^6 d^3 (m+2) (m+4) (m+6)}{m+1}-\frac{e (m+1) \left(3 c^4 d^2 \left(m^4+22 m^3+179 m^2+638 m+840\right)-3 c^2 d e (m+3)^2 \left(m^2+13 m+42\right)+e^2 \left(m^2+8 m+15\right)^2\right)}{(m+3) (m+5) (m+7)}\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right)}{c^5 f (m+1) (m+2) (m+4) (m+6) \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}",1,"Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcCsch[c*x]), x]","F",-1
166,0,0,379,0.1558457,"\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcCsch[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \text{csch}^{-1}(c x)\right) \, dx","\frac{d^2 (f x)^{m+1} \left(a+b \text{csch}^{-1}(c x)\right)}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left(a+b \text{csch}^{-1}(c x)\right)}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left(a+b \text{csch}^{-1}(c x)\right)}{f^5 (m+5)}+\frac{b e^2 x \sqrt{-c^2 x^2-1} (f x)^{m+3}}{c f^3 (m+4) (m+5) \sqrt{-c^2 x^2}}-\frac{b e x \sqrt{-c^2 x^2-1} (f x)^{m+1} \left(e (m+3)^2-2 c^2 d \left(m^2+9 m+20\right)\right)}{c^3 f (m+2) (m+3) (m+4) (m+5) \sqrt{-c^2 x^2}}-\frac{b x \sqrt{c^2 x^2+1} (f x)^{m+1} \left(c^4 d^2 (m+2) (m+3) (m+4) (m+5)+e (m+1)^2 \left(e (m+3)^2-2 c^2 d \left(m^2+9 m+20\right)\right)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right)}{c^3 f (m+1)^2 (m+2) (m+3) (m+4) (m+5) \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}",1,"Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcCsch[c*x]), x]","F",-1
167,0,0,220,0.1142319,"\int (f x)^m \left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcCsch[c*x]),x]","\int (f x)^m \left(d+e x^2\right) \left(a+b \text{csch}^{-1}(c x)\right) \, dx","\frac{d (f x)^{m+1} \left(a+b \text{csch}^{-1}(c x)\right)}{f (m+1)}+\frac{e (f x)^{m+3} \left(a+b \text{csch}^{-1}(c x)\right)}{f^3 (m+3)}+\frac{b x \sqrt{c^2 x^2+1} (f x)^{m+1} \left(e (m+1)^2-c^2 d (m+2) (m+3)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right)}{c f (m+1)^2 (m+2) (m+3) \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}+\frac{b e x \sqrt{-c^2 x^2-1} (f x)^{m+1}}{c f \left(m^2+5 m+6\right) \sqrt{-c^2 x^2}}",1,"Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcCsch[c*x]), x]","F",-1
168,0,0,26,1.8670929,"\int \frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2),x]","\int \frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{d+e x^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{d+e x^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2), x]","A",-1
169,0,0,26,6.9555734,"\int \frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2,x]","\int \frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2, x]","A",-1
170,0,0,28,1.1452878,"\int (f x)^m \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{3/2} (f x)^m \left(a+b \text{csch}^{-1}(c x)\right),x\right)",0,"Integrate[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]), x]","A",-1
171,0,0,28,0.1172623,"\int (f x)^m \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]),x]","\int (f x)^m \sqrt{d+e x^2} \left(a+b \text{csch}^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} (f x)^m \left(a+b \text{csch}^{-1}(c x)\right),x\right)",0,"Integrate[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]), x]","A",-1
172,0,0,28,1.4585765,"\int \frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2], x]","A",-1
173,0,0,28,1.8724544,"\int \frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \text{csch}^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCsch[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
174,1,214,395,0.3142898,"\int \frac{x^{11} \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^11*(a + b*ArcCsch[c*x]))/Sqrt[1 - c^4*x^4],x]","-\frac{105 a \sqrt{1-c^4 x^4} \left(3 c^8 x^8+4 c^4 x^4+8\right)+840 b \log \left(c^2 x^3+x\right)+105 b \sqrt{1-c^4 x^4} \left(3 c^8 x^8+4 c^4 x^4+8\right) \text{csch}^{-1}(c x)-840 b \log \left(c^2 x^2+c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{1-c^4 x^4}+1\right)+\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{1-c^4 x^4} \left(35 c^8 x^8-5 c^6 x^6+78 c^4 x^4-36 c^2 x^2+768\right)}{c^2 x^2+1}}{3150 c^{12}}","-\frac{\left(1-c^4 x^4\right)^{5/2} \left(a+b \text{csch}^{-1}(c x)\right)}{10 c^{12}}+\frac{\left(1-c^4 x^4\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left(a+b \text{csch}^{-1}(c x)\right)}{2 c^{12}}-\frac{b \sqrt{c^2 x^2+1} \left(1-c^2 x^2\right)^{9/2}}{90 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{3 b \sqrt{c^2 x^2+1} \left(1-c^2 x^2\right)^{7/2}}{70 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{13 b \sqrt{c^2 x^2+1} \left(1-c^2 x^2\right)^{5/2}}{150 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{7 b \sqrt{c^2 x^2+1} \left(1-c^2 x^2\right)^{3/2}}{90 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{4 b \sqrt{c^2 x^2+1} \sqrt{1-c^2 x^2}}{15 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{4 b \sqrt{c^2 x^2+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{15 c^{13} x \sqrt{\frac{1}{c^2 x^2}+1}}",1,"-1/3150*(105*a*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8) + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]*(768 - 36*c^2*x^2 + 78*c^4*x^4 - 5*c^6*x^6 + 35*c^8*x^8))/(1 + c^2*x^2) + 105*b*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8)*ArcCsch[c*x] + 840*b*Log[x + c^2*x^3] - 840*b*Log[1 + c^2*x^2 + c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]])/c^12","A",1
175,1,180,264,0.4175823,"\int \frac{x^7 \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^7*(a + b*ArcCsch[c*x]))/Sqrt[1 - c^4*x^4],x]","-\frac{15 a \sqrt{1-c^4 x^4} \left(c^4 x^4+2\right)+15 b \sqrt{1-c^4 x^4} \left(c^4 x^4+2\right) \text{csch}^{-1}(c x)+30 b \log \left(c^2 x^3+x\right)+\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{1-c^4 x^4} \left(3 c^4 x^4-c^2 x^2+28\right)}{c^2 x^2+1}-30 b \log \left(c^2 x^2+c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{1-c^4 x^4}+1\right)}{90 c^8}","\frac{\left(1-c^4 x^4\right)^{3/2} \left(a+b \text{csch}^{-1}(c x)\right)}{6 c^8}-\frac{\sqrt{1-c^4 x^4} \left(a+b \text{csch}^{-1}(c x)\right)}{2 c^8}-\frac{b \sqrt{c^2 x^2+1} \left(1-c^2 x^2\right)^{5/2}}{30 c^9 x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{b \sqrt{c^2 x^2+1} \left(1-c^2 x^2\right)^{3/2}}{18 c^9 x \sqrt{\frac{1}{c^2 x^2}+1}}-\frac{b \sqrt{c^2 x^2+1} \sqrt{1-c^2 x^2}}{3 c^9 x \sqrt{\frac{1}{c^2 x^2}+1}}+\frac{b \sqrt{c^2 x^2+1} \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{3 c^9 x \sqrt{\frac{1}{c^2 x^2}+1}}",1,"-1/90*(15*a*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4) + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]*(28 - c^2*x^2 + 3*c^4*x^4))/(1 + c^2*x^2) + 15*b*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4)*ArcCsch[c*x] + 30*b*Log[x + c^2*x^3] - 30*b*Log[1 + c^2*x^2 + c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]])/c^8","A",1
176,1,141,130,0.3369784,"\int \frac{x^3 \left(a+b \text{csch}^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^3*(a + b*ArcCsch[c*x]))/Sqrt[1 - c^4*x^4],x]","-\frac{a \sqrt{1-c^4 x^4}+b \sqrt{1-c^4 x^4} \text{csch}^{-1}(c x)+b \log \left(c^2 x^3+x\right)+\frac{b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{1-c^4 x^4}}{c^2 x^2+1}-b \log \left(c^2 x^2+c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{1-c^4 x^4}+1\right)}{2 c^4}","-\frac{\sqrt{1-c^4 x^4} \left(a+b \text{csch}^{-1}(c x)\right)}{2 c^4}+\frac{b x \sqrt{1-c^4 x^4}}{2 c^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}-\frac{b x \tan ^{-1}\left(\frac{\sqrt{1-c^4 x^4}}{\sqrt{-c^2 x^2-1}}\right)}{2 c^3 \sqrt{-c^2 x^2}}",1,"-1/2*(a*Sqrt[1 - c^4*x^4] + (b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4])/(1 + c^2*x^2) + b*Sqrt[1 - c^4*x^4]*ArcCsch[c*x] + b*Log[x + c^2*x^3] - b*Log[1 + c^2*x^2 + c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]])/c^4","A",1
177,0,0,29,0.4106941,"\int \frac{a+b \text{csch}^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x*Sqrt[1 - c^4*x^4]),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x \sqrt{1-c^4 x^4}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x*Sqrt[1 - c^4*x^4]), x]","A",-1
178,0,0,29,3.4867658,"\int \frac{a+b \text{csch}^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}} \, dx","Integrate[(a + b*ArcCsch[c*x])/(x^5*Sqrt[1 - c^4*x^4]),x]","\int \frac{a+b \text{csch}^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}} \, dx","\text{Int}\left(\frac{a+b \text{csch}^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}},x\right)",0,"Integrate[(a + b*ArcCsch[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]","A",-1