1,1,91,151,0.1679002,"\int x^4 \left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]","\frac{d \left(-105 a x^5 \left(5 c^2 x^2-7\right)-105 b x^5 \left(5 c^2 x^2-7\right) \cosh ^{-1}(c x)+\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(75 c^6 x^6-57 c^4 x^4-76 c^2 x^2-152\right)}{c^5}\right)}{3675}","-\frac{1}{7} c^2 d x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{5} d x^5 \left(a+b \cosh ^{-1}(c x)\right)-\frac{152 b d \sqrt{c x-1} \sqrt{c x+1}}{3675 c^5}-\frac{76 b d x^2 \sqrt{c x-1} \sqrt{c x+1}}{3675 c^3}+\frac{1}{49} b c d x^6 \sqrt{c x-1} \sqrt{c x+1}-\frac{19 b d x^4 \sqrt{c x-1} \sqrt{c x+1}}{1225 c}",1,"(d*(-105*a*x^5*(-7 + 5*c^2*x^2) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-152 - 76*c^2*x^2 - 57*c^4*x^4 + 75*c^6*x^6))/c^5 - 105*b*x^5*(-7 + 5*c^2*x^2)*ArcCosh[c*x]))/3675","A",1
2,1,166,135,0.0997783,"\int x^3 \left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]","-\frac{1}{6} a c^2 d x^6+\frac{1}{4} a d x^4-\frac{b d \tanh ^{-1}\left(\frac{\sqrt{c x-1}}{\sqrt{c x+1}}\right)}{12 c^4}-\frac{b d x \sqrt{c x-1} \sqrt{c x+1}}{24 c^3}-\frac{1}{6} b c^2 d x^6 \cosh ^{-1}(c x)+\frac{1}{36} b c d x^5 \sqrt{c x-1} \sqrt{c x+1}+\frac{1}{4} b d x^4 \cosh ^{-1}(c x)-\frac{b d x^3 \sqrt{c x-1} \sqrt{c x+1}}{36 c}","-\frac{1}{6} c^2 d x^6 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{4} d x^4 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b d \cosh ^{-1}(c x)}{24 c^4}-\frac{b d x \sqrt{c x-1} \sqrt{c x+1}}{24 c^3}+\frac{1}{36} b c d x^5 \sqrt{c x-1} \sqrt{c x+1}-\frac{b d x^3 \sqrt{c x-1} \sqrt{c x+1}}{36 c}",1,"(a*d*x^4)/4 - (a*c^2*d*x^6)/6 - (b*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(24*c^3) - (b*d*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(36*c) + (b*c*d*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/36 + (b*d*x^4*ArcCosh[c*x])/4 - (b*c^2*d*x^6*ArcCosh[c*x])/6 - (b*d*ArcTanh[Sqrt[-1 + c*x]/Sqrt[1 + c*x]])/(12*c^4)","A",1
3,1,89,121,0.1107362,"\int x^2 \left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]","-\frac{d \left(15 a c^3 x^3 \left(3 c^2 x^2-5\right)+b \sqrt{c x-1} \sqrt{c x+1} \left(-9 c^4 x^4+13 c^2 x^2+26\right)+15 b c^3 x^3 \left(3 c^2 x^2-5\right) \cosh ^{-1}(c x)\right)}{225 c^3}","-\frac{1}{5} c^2 d x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} d x^3 \left(a+b \cosh ^{-1}(c x)\right)-\frac{26 b d \sqrt{c x-1} \sqrt{c x+1}}{225 c^3}+\frac{1}{25} b c d x^4 \sqrt{c x-1} \sqrt{c x+1}-\frac{13 b d x^2 \sqrt{c x-1} \sqrt{c x+1}}{225 c}",1,"-1/225*(d*(15*a*c^3*x^3*(-5 + 3*c^2*x^2) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(26 + 13*c^2*x^2 - 9*c^4*x^4) + 15*b*c^3*x^3*(-5 + 3*c^2*x^2)*ArcCosh[c*x]))/c^3","A",1
4,1,100,98,0.1578733,"\int x \left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]","-\frac{d \left(c x \left(8 a c x \left(c^2 x^2-2\right)+b \sqrt{c x-1} \sqrt{c x+1} \left(5-2 c^2 x^2\right)\right)+8 b c^2 x^2 \left(c^2 x^2-2\right) \cosh ^{-1}(c x)+10 b \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{32 c^2}","-\frac{d \left(1-c^2 x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{4 c^2}+\frac{3 b d \cosh ^{-1}(c x)}{32 c^2}+\frac{b d x (c x-1)^{3/2} (c x+1)^{3/2}}{16 c}-\frac{3 b d x \sqrt{c x-1} \sqrt{c x+1}}{32 c}",1,"-1/32*(d*(c*x*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(5 - 2*c^2*x^2) + 8*a*c*x*(-2 + c^2*x^2)) + 8*b*c^2*x^2*(-2 + c^2*x^2)*ArcCosh[c*x] + 10*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/c^2","A",0
5,1,71,86,0.0907276,"\int \left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]","\frac{d \left(a \left(9 c x-3 c^3 x^3\right)+b \sqrt{c x-1} \sqrt{c x+1} \left(c^2 x^2-7\right)-3 b c x \left(c^2 x^2-3\right) \cosh ^{-1}(c x)\right)}{9 c}","-\frac{1}{3} c^2 d x^3 \left(a+b \cosh ^{-1}(c x)\right)+d x \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{9} b c d x^2 \sqrt{c x-1} \sqrt{c x+1}-\frac{7 b d \sqrt{c x-1} \sqrt{c x+1}}{9 c}",1,"(d*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-7 + c^2*x^2) + a*(9*c*x - 3*c^3*x^3) - 3*b*c*x*(-3 + c^2*x^2)*ArcCosh[c*x]))/(9*c)","A",1
6,1,130,117,0.1039002,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x,x]","-\frac{1}{2} a c^2 d x^2+a d \log (x)-\frac{1}{2} b c^2 d x^2 \cosh ^{-1}(c x)+\frac{1}{2} b d \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)-\text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)\right)+\frac{1}{4} b c d x \sqrt{c x-1} \sqrt{c x+1}+\frac{1}{2} b d \tanh ^{-1}\left(\frac{\sqrt{c x-1}}{\sqrt{c x+1}}\right)","\frac{1}{2} d \left(1-c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)+\frac{d \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b}+d \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)-\frac{1}{2} b d \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+\frac{1}{4} b c d x \sqrt{c x-1} \sqrt{c x+1}-\frac{1}{4} b d \cosh ^{-1}(c x)",1,"-1/2*(a*c^2*d*x^2) + (b*c*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/4 - (b*c^2*d*x^2*ArcCosh[c*x])/2 + (b*d*ArcTanh[Sqrt[-1 + c*x]/Sqrt[1 + c*x]])/2 + a*d*Log[x] + (b*d*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]))/2","A",0
7,1,110,76,0.1919899,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x^2,x]","-a c^2 d x-\frac{a d}{x}+\frac{b c d \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-b c^2 d x \cosh ^{-1}(c x)+b c d \sqrt{c x-1} \sqrt{c x+1}-\frac{b d \cosh ^{-1}(c x)}{x}","c^2 (-d) x \left(a+b \cosh ^{-1}(c x)\right)-\frac{d \left(a+b \cosh ^{-1}(c x)\right)}{x}+b c d \sqrt{c x-1} \sqrt{c x+1}+b c d \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)",1,"-((a*d)/x) - a*c^2*d*x + b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (b*d*ArcCosh[c*x])/x - b*c^2*d*x*ArcCosh[c*x] + (b*c*d*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
8,1,107,135,0.0882993,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x^3,x]","-a c^2 d \log (x)-\frac{a d}{2 x^2}-\frac{1}{2} b c^2 d \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)-\text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)\right)-\frac{b d \cosh ^{-1}(c x)}{2 x^2}+\frac{b c d \sqrt{c x-1} \sqrt{c x+1}}{2 x}","-\frac{d \left(1-c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{2 x^2}-\frac{c^2 d \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b}-c^2 d \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{2} b c^2 d \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-\frac{1}{2} b c^2 d \cosh ^{-1}(c x)+\frac{b c d \sqrt{c x-1} \sqrt{c x+1}}{2 x}",1,"-1/2*(a*d)/x^2 + (b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*x) - (b*d*ArcCosh[c*x])/(2*x^2) - a*c^2*d*Log[x] - (b*c^2*d*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]))/2","A",0
9,1,127,90,0.259223,"\int \frac{\left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)*(a + b*ArcCosh[c*x]))/x^4,x]","\frac{a c^2 d}{x}-\frac{a d}{3 x^3}+\frac{b c^2 d \cosh ^{-1}(c x)}{x}-\frac{5 b c^3 d \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b d \cosh ^{-1}(c x)}{3 x^3}+\frac{b c d \sqrt{c x-1} \sqrt{c x+1}}{6 x^2}","\frac{c^2 d \left(a+b \cosh ^{-1}(c x)\right)}{x}-\frac{d \left(a+b \cosh ^{-1}(c x)\right)}{3 x^3}-\frac{5}{6} b c^3 d \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)+\frac{b c d \sqrt{c x-1} \sqrt{c x+1}}{6 x^2}",1,"-1/3*(a*d)/x^3 + (a*c^2*d)/x + (b*c*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(6*x^2) - (b*d*ArcCosh[c*x])/(3*x^3) + (b*c^2*d*ArcCosh[c*x])/x - (5*b*c^3*d*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
10,1,124,206,0.2139877,"\int x^4 \left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{d^2 \left(315 a c^5 x^5 \left(35 c^4 x^4-90 c^2 x^2+63\right)+315 b c^5 x^5 \left(35 c^4 x^4-90 c^2 x^2+63\right) \cosh ^{-1}(c x)-b \sqrt{c x-1} \sqrt{c x+1} \left(1225 c^8 x^8-2650 c^6 x^6+789 c^4 x^4+1052 c^2 x^2+2104\right)\right)}{99225 c^5}","\frac{1}{9} c^4 d^2 x^9 \left(a+b \cosh ^{-1}(c x)\right)-\frac{2}{7} c^2 d^2 x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{5} d^2 x^5 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b d^2 (c x-1)^{9/2} (c x+1)^{9/2}}{81 c^5}-\frac{10 b d^2 (c x-1)^{7/2} (c x+1)^{7/2}}{441 c^5}-\frac{b d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{525 c^5}+\frac{4 b d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{945 c^5}-\frac{8 b d^2 \sqrt{c x-1} \sqrt{c x+1}}{315 c^5}",1,"(d^2*(315*a*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2104 + 1052*c^2*x^2 + 789*c^4*x^4 - 2650*c^6*x^6 + 1225*c^8*x^8) + 315*b*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4)*ArcCosh[c*x]))/(99225*c^5)","A",1
11,1,194,200,0.2357336,"\int x^3 \left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{d^2 \left(1152 a c^8 x^8-3072 a c^6 x^6+2304 a c^4 x^4-144 b c^7 x^7 \sqrt{c x-1} \sqrt{c x+1}+344 b c^5 x^5 \sqrt{c x-1} \sqrt{c x+1}-146 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+384 b c^4 x^4 \left(3 c^4 x^4-8 c^2 x^2+6\right) \cosh ^{-1}(c x)-219 b c x \sqrt{c x-1} \sqrt{c x+1}-438 b \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{9216 c^4}","\frac{1}{8} c^4 d^2 x^8 \left(a+b \cosh ^{-1}(c x)\right)-\frac{1}{3} c^2 d^2 x^6 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{4} d^2 x^4 \left(a+b \cosh ^{-1}(c x)\right)-\frac{73 b d^2 \cosh ^{-1}(c x)}{3072 c^4}-\frac{1}{64} b c^3 d^2 x^7 \sqrt{c x-1} \sqrt{c x+1}-\frac{73 b d^2 x \sqrt{c x-1} \sqrt{c x+1}}{3072 c^3}+\frac{43 b c d^2 x^5 \sqrt{c x-1} \sqrt{c x+1}}{1152}-\frac{73 b d^2 x^3 \sqrt{c x-1} \sqrt{c x+1}}{4608 c}",1,"(d^2*(2304*a*c^4*x^4 - 3072*a*c^6*x^6 + 1152*a*c^8*x^8 - 219*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 146*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 344*b*c^5*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 144*b*c^7*x^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 384*b*c^4*x^4*(6 - 8*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x] - 438*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(9216*c^4)","A",0
12,1,116,177,0.1781078,"\int x^2 \left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{d^2 \left(105 a c^3 x^3 \left(15 c^4 x^4-42 c^2 x^2+35\right)-b \sqrt{c x-1} \sqrt{c x+1} \left(225 c^6 x^6-612 c^4 x^4+409 c^2 x^2+818\right)+105 b c^3 x^3 \left(15 c^4 x^4-42 c^2 x^2+35\right) \cosh ^{-1}(c x)\right)}{11025 c^3}","\frac{1}{7} c^4 d^2 x^7 \left(a+b \cosh ^{-1}(c x)\right)-\frac{2}{5} c^2 d^2 x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} d^2 x^3 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b d^2 (c x-1)^{7/2} (c x+1)^{7/2}}{49 c^3}-\frac{b d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{175 c^3}+\frac{4 b d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{315 c^3}-\frac{8 b d^2 \sqrt{c x-1} \sqrt{c x+1}}{105 c^3}",1,"(d^2*(105*a*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(818 + 409*c^2*x^2 - 612*c^4*x^4 + 225*c^6*x^6) + 105*b*c^3*x^3*(35 - 42*c^2*x^2 + 15*c^4*x^4)*ArcCosh[c*x]))/(11025*c^3)","A",1
13,1,126,136,0.2347487,"\int x \left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{d^2 \left(c x \left(48 a c x \left(c^4 x^4-3 c^2 x^2+3\right)+b \sqrt{c x-1} \sqrt{c x+1} \left(-8 c^4 x^4+26 c^2 x^2-33\right)\right)+48 b c^2 x^2 \left(c^4 x^4-3 c^2 x^2+3\right) \cosh ^{-1}(c x)-66 b \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{288 c^2}","-\frac{d^2 \left(1-c^2 x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{6 c^2}+\frac{5 b d^2 \cosh ^{-1}(c x)}{96 c^2}-\frac{b d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{36 c}+\frac{5 b d^2 x (c x-1)^{3/2} (c x+1)^{3/2}}{144 c}-\frac{5 b d^2 x \sqrt{c x-1} \sqrt{c x+1}}{96 c}",1,"(d^2*(c*x*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-33 + 26*c^2*x^2 - 8*c^4*x^4) + 48*a*c*x*(3 - 3*c^2*x^2 + c^4*x^4)) + 48*b*c^2*x^2*(3 - 3*c^2*x^2 + c^4*x^4)*ArcCosh[c*x] - 66*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(288*c^2)","A",0
14,1,99,143,0.1661327,"\int \left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{d^2 \left(15 a c x \left(3 c^4 x^4-10 c^2 x^2+15\right)+b \sqrt{c x-1} \sqrt{c x+1} \left(-9 c^4 x^4+38 c^2 x^2-149\right)+15 b c x \left(3 c^4 x^4-10 c^2 x^2+15\right) \cosh ^{-1}(c x)\right)}{225 c}","\frac{1}{5} c^4 d^2 x^5 \left(a+b \cosh ^{-1}(c x)\right)-\frac{2}{3} c^2 d^2 x^3 \left(a+b \cosh ^{-1}(c x)\right)+d^2 x \left(a+b \cosh ^{-1}(c x)\right)-\frac{b d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{25 c}+\frac{4 b d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{45 c}-\frac{8 b d^2 \sqrt{c x-1} \sqrt{c x+1}}{15 c}",1,"(d^2*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-149 + 38*c^2*x^2 - 9*c^4*x^4) + 15*a*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 15*b*c*x*(15 - 10*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x]))/(225*c)","A",1
15,1,162,184,0.2795137,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x,x]","\frac{1}{32} d^2 \left(8 a c^4 x^4-32 a c^2 x^2+32 a \log (x)-2 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+8 b \cosh ^{-1}(c x) \left(c^4 x^4-4 c^2 x^2+4 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)-16 b \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+13 b c x \sqrt{c x-1} \sqrt{c x+1}+16 b \cosh ^{-1}(c x)^2+26 b \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)","\frac{1}{4} d^2 \left(1-c^2 x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{2} d^2 \left(1-c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)+\frac{d^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b}+d^2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)-\frac{1}{2} b d^2 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-\frac{1}{16} b c d^2 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac{11}{32} b c d^2 x \sqrt{c x-1} \sqrt{c x+1}-\frac{11}{32} b d^2 \cosh ^{-1}(c x)",1,"(d^2*(-32*a*c^2*x^2 + 8*a*c^4*x^4 + 13*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 16*b*ArcCosh[c*x]^2 + 26*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]] + 8*b*ArcCosh[c*x]*(-4*c^2*x^2 + c^4*x^4 + 4*Log[1 + E^(-2*ArcCosh[c*x])]) + 32*a*Log[x] - 16*b*PolyLog[2, -E^(-2*ArcCosh[c*x])]))/32","A",0
16,1,131,135,0.1713315,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x^2,x]","\frac{d^2 \left(3 a c^4 x^4-18 a c^2 x^2-9 a-b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+3 b \left(c^4 x^4-6 c^2 x^2-3\right) \cosh ^{-1}(c x)+16 b c x \sqrt{c x-1} \sqrt{c x+1}-9 b c x \tan ^{-1}\left(\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right)\right)}{9 x}","\frac{1}{3} c^4 d^2 x^3 \left(a+b \cosh ^{-1}(c x)\right)-2 c^2 d^2 x \left(a+b \cosh ^{-1}(c x)\right)-\frac{d^2 \left(a+b \cosh ^{-1}(c x)\right)}{x}-\frac{1}{9} b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}+\frac{5}{3} b c d^2 \sqrt{c x-1} \sqrt{c x+1}+b c d^2 \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)",1,"(d^2*(-9*a - 18*a*c^2*x^2 + 3*a*c^4*x^4 + 16*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 3*b*(-3 - 6*c^2*x^2 + c^4*x^4)*ArcCosh[c*x] - 9*b*c*x*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]))/(9*x)","A",1
17,1,182,200,0.2656072,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x^3,x]","\frac{d^2 \left(2 a c^4 x^4-8 a c^2 x^2 \log (x)-2 a-b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+4 b c^2 x^2 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-4 b c^2 x^2 \cosh ^{-1}(c x)^2-2 b c^2 x^2 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)+2 b \cosh ^{-1}(c x) \left(c^4 x^4-4 c^2 x^2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)-1\right)+2 b c x \sqrt{c x-1} \sqrt{c x+1}\right)}{4 x^2}","-c^2 d^2 \left(1-c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)-\frac{d^2 \left(1-c^2 x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{2 x^2}-\frac{c^2 d^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{b}-2 c^2 d^2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{4} b c^3 d^2 x \sqrt{c x-1} \sqrt{c x+1}+b c^2 d^2 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-\frac{1}{4} b c^2 d^2 \cosh ^{-1}(c x)-\frac{b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{2 x}",1,"(d^2*(-2*a + 2*a*c^4*x^4 + 2*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 4*b*c^2*x^2*ArcCosh[c*x]^2 - 2*b*c^2*x^2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]] + 2*b*ArcCosh[c*x]*(-1 + c^4*x^4 - 4*c^2*x^2*Log[1 + E^(-2*ArcCosh[c*x])]) - 8*a*c^2*x^2*Log[x] + 4*b*c^2*x^2*PolyLog[2, -E^(-2*ArcCosh[c*x])]))/(4*x^2)","A",0
18,1,135,142,0.1735616,"\int \frac{\left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]))/x^4,x]","\frac{d^2 \left(6 a c^4 x^4+12 a c^2 x^2-2 a-6 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+11 b c^3 x^3 \tan ^{-1}\left(\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right)+2 b \left(3 c^4 x^4+6 c^2 x^2-1\right) \cosh ^{-1}(c x)+b c x \sqrt{c x-1} \sqrt{c x+1}\right)}{6 x^3}","c^4 d^2 x \left(a+b \cosh ^{-1}(c x)\right)+\frac{2 c^2 d^2 \left(a+b \cosh ^{-1}(c x)\right)}{x}-\frac{d^2 \left(a+b \cosh ^{-1}(c x)\right)}{3 x^3}-b c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}-\frac{11}{6} b c^3 d^2 \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)+\frac{b c d^2 \sqrt{c x-1} \sqrt{c x+1}}{6 x^2}",1,"(d^2*(-2*a + 12*a*c^2*x^2 + 6*a*c^4*x^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 6*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*(-1 + 6*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x] + 11*b*c^3*x^3*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]))/(6*x^3)","A",1
19,1,147,256,0.2796282,"\int x^4 \left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]","-\frac{d^3 \left(3465 a c^5 x^5 \left(105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right)+3465 b c^5 x^5 \left(105 c^6 x^6-385 c^4 x^4+495 c^2 x^2-231\right) \cosh ^{-1}(c x)+b \sqrt{c x-1} \sqrt{c x+1} \left(-33075 c^{10} x^{10}+111475 c^8 x^8-117625 c^6 x^6+18933 c^4 x^4+25244 c^2 x^2+50488\right)\right)}{4002075 c^5}","-\frac{1}{11} c^6 d^3 x^{11} \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} c^4 d^3 x^9 \left(a+b \cosh ^{-1}(c x)\right)-\frac{3}{7} c^2 d^3 x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{5} d^3 x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{b d^3 (c x-1)^{11/2} (c x+1)^{11/2}}{121 c^5}+\frac{4 b d^3 (c x-1)^{9/2} (c x+1)^{9/2}}{297 c^5}+\frac{b d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{1617 c^5}-\frac{2 b d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{1925 c^5}+\frac{8 b d^3 (c x-1)^{3/2} (c x+1)^{3/2}}{3465 c^5}-\frac{16 b d^3 \sqrt{c x-1} \sqrt{c x+1}}{1155 c^5}",1,"-1/4002075*(d^3*(3465*a*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(50488 + 25244*c^2*x^2 + 18933*c^4*x^4 - 117625*c^6*x^6 + 111475*c^8*x^8 - 33075*c^10*x^10) + 3465*b*c^5*x^5*(-231 + 495*c^2*x^2 - 385*c^4*x^4 + 105*c^6*x^6)*ArcCosh[c*x]))/c^5","A",1
20,1,162,230,0.3684699,"\int x^3 \left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]","-\frac{d^3 \left(1920 a c^4 x^4 \left(4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right)+1920 b c^4 x^4 \left(4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right) \cosh ^{-1}(c x)+b c x \sqrt{c x-1} \sqrt{c x+1} \left(-768 c^8 x^8+2736 c^6 x^6-3208 c^4 x^4+790 c^2 x^2+1185\right)+2370 b \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{76800 c^4}","-\frac{d^3 (c x-1)^5 (c x+1)^5 \left(a+b \cosh ^{-1}(c x)\right)}{10 c^4}-\frac{d^3 (c x-1)^4 (c x+1)^4 \left(a+b \cosh ^{-1}(c x)\right)}{8 c^4}+\frac{49 b d^3 \cosh ^{-1}(c x)}{5120 c^4}+\frac{b d^3 x (c x-1)^{9/2} (c x+1)^{9/2}}{100 c^3}+\frac{7 b d^3 x (c x-1)^{7/2} (c x+1)^{7/2}}{1600 c^3}-\frac{49 b d^3 x (c x-1)^{5/2} (c x+1)^{5/2}}{9600 c^3}+\frac{49 b d^3 x (c x-1)^{3/2} (c x+1)^{3/2}}{7680 c^3}-\frac{49 b d^3 x \sqrt{c x-1} \sqrt{c x+1}}{5120 c^3}",1,"-1/76800*(d^3*(1920*a*c^4*x^4*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*c^6*x^6) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1185 + 790*c^2*x^2 - 3208*c^4*x^4 + 2736*c^6*x^6 - 768*c^8*x^8) + 1920*b*c^4*x^4*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*c^6*x^6)*ArcCosh[c*x] + 2370*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/c^4","A",0
21,1,139,227,0.2904948,"\int x^2 \left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]","-\frac{d^3 \left(315 a c^3 x^3 \left(35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right)+b \sqrt{c x-1} \sqrt{c x+1} \left(-1225 c^8 x^8+4675 c^6 x^6-6297 c^4 x^4+2629 c^2 x^2+5258\right)+315 b c^3 x^3 \left(35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right) \cosh ^{-1}(c x)\right)}{99225 c^3}","-\frac{1}{9} c^6 d^3 x^9 \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{7} c^4 d^3 x^7 \left(a+b \cosh ^{-1}(c x)\right)-\frac{3}{5} c^2 d^3 x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{b d^3 (c x-1)^{9/2} (c x+1)^{9/2}}{81 c^3}+\frac{b d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{441 c^3}-\frac{2 b d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{525 c^3}+\frac{8 b d^3 (c x-1)^{3/2} (c x+1)^{3/2}}{945 c^3}-\frac{16 b d^3 \sqrt{c x-1} \sqrt{c x+1}}{315 c^3}",1,"-1/99225*(d^3*(315*a*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(5258 + 2629*c^2*x^2 - 6297*c^4*x^4 + 4675*c^6*x^6 - 1225*c^8*x^8) + 315*b*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6)*ArcCosh[c*x]))/c^3","A",1
22,1,150,166,0.3895377,"\int x \left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]","-\frac{d^3 \left(c x \left(384 a c x \left(c^6 x^6-4 c^4 x^4+6 c^2 x^2-4\right)+b \sqrt{c x-1} \sqrt{c x+1} \left(-48 c^6 x^6+200 c^4 x^4-326 c^2 x^2+279\right)\right)+384 b c^2 x^2 \left(c^6 x^6-4 c^4 x^4+6 c^2 x^2-4\right) \cosh ^{-1}(c x)+558 b \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{3072 c^2}","-\frac{d^3 \left(1-c^2 x^2\right)^4 \left(a+b \cosh ^{-1}(c x)\right)}{8 c^2}+\frac{35 b d^3 \cosh ^{-1}(c x)}{1024 c^2}+\frac{b d^3 x (c x-1)^{7/2} (c x+1)^{7/2}}{64 c}-\frac{7 b d^3 x (c x-1)^{5/2} (c x+1)^{5/2}}{384 c}+\frac{35 b d^3 x (c x-1)^{3/2} (c x+1)^{3/2}}{1536 c}-\frac{35 b d^3 x \sqrt{c x-1} \sqrt{c x+1}}{1024 c}",1,"-1/3072*(d^3*(c*x*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(279 - 326*c^2*x^2 + 200*c^4*x^4 - 48*c^6*x^6) + 384*a*c*x*(-4 + 6*c^2*x^2 - 4*c^4*x^4 + c^6*x^6)) + 384*b*c^2*x^2*(-4 + 6*c^2*x^2 - 4*c^4*x^4 + c^6*x^6)*ArcCosh[c*x] + 558*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/c^2","A",0
23,1,123,191,0.2710249,"\int \left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]","-\frac{d^3 \left(105 a c x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)+b \sqrt{c x-1} \sqrt{c x+1} \left(-75 c^6 x^6+351 c^4 x^4-757 c^2 x^2+2161\right)+105 b c x \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right) \cosh ^{-1}(c x)\right)}{3675 c}","-\frac{1}{7} c^6 d^3 x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{5} c^4 d^3 x^5 \left(a+b \cosh ^{-1}(c x)\right)-c^2 d^3 x^3 \left(a+b \cosh ^{-1}(c x)\right)+d^3 x \left(a+b \cosh ^{-1}(c x)\right)+\frac{b d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{49 c}-\frac{6 b d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{175 c}+\frac{8 b d^3 (c x-1)^{3/2} (c x+1)^{3/2}}{105 c}-\frac{16 b d^3 \sqrt{c x-1} \sqrt{c x+1}}{35 c}",1,"-1/3675*(d^3*(b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2161 - 757*c^2*x^2 + 351*c^4*x^4 - 75*c^6*x^6) + 105*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 105*b*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcCosh[c*x]))/c","A",1
24,1,207,239,0.3927229,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x,x]","-\frac{1}{144} d^3 \left(24 a c^6 x^6-108 a c^4 x^4+216 a c^2 x^2-144 a \log (x)-4 b c^5 x^5 \sqrt{c x-1} \sqrt{c x+1}+22 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}+12 b \cosh ^{-1}(c x) \left(2 c^6 x^6-9 c^4 x^4+18 c^2 x^2-12 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)+72 b \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-75 b c x \sqrt{c x-1} \sqrt{c x+1}-72 b \cosh ^{-1}(c x)^2-150 b \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)","\frac{1}{6} d^3 \left(1-c^2 x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{4} d^3 \left(1-c^2 x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{2} d^3 \left(1-c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)+\frac{d^3 \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b}+d^3 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)-\frac{1}{2} b d^3 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+\frac{1}{36} b c d^3 x (c x-1)^{5/2} (c x+1)^{5/2}-\frac{7}{72} b c d^3 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac{19}{48} b c d^3 x \sqrt{c x-1} \sqrt{c x+1}-\frac{19}{48} b d^3 \cosh ^{-1}(c x)",1,"-1/144*(d^3*(216*a*c^2*x^2 - 108*a*c^4*x^4 + 24*a*c^6*x^6 - 75*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 22*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 4*b*c^5*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 72*b*ArcCosh[c*x]^2 - 150*b*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]] + 12*b*ArcCosh[c*x]*(18*c^2*x^2 - 9*c^4*x^4 + 2*c^6*x^6 - 12*Log[1 + E^(-2*ArcCosh[c*x])]) - 144*a*Log[x] + 72*b*PolyLog[2, -E^(-2*ArcCosh[c*x])]))","A",0
25,1,136,180,0.3060825,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^2,x]","\frac{1}{25} d^3 \left(-5 a c^6 x^5+25 a c^4 x^3-75 a c^2 x-\frac{25 a}{x}+b c \sqrt{c x-1} \sqrt{c x+1} \left(c^4 x^4-7 c^2 x^2+61\right)-\frac{5 b \left(c^6 x^6-5 c^4 x^4+15 c^2 x^2+5\right) \cosh ^{-1}(c x)}{x}-25 b c \tan ^{-1}\left(\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right)\right)","-\frac{1}{5} c^6 d^3 x^5 \left(a+b \cosh ^{-1}(c x)\right)+c^4 d^3 x^3 \left(a+b \cosh ^{-1}(c x)\right)-3 c^2 d^3 x \left(a+b \cosh ^{-1}(c x)\right)-\frac{d^3 \left(a+b \cosh ^{-1}(c x)\right)}{x}+\frac{1}{25} b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}-\frac{1}{5} b c d^3 (c x-1)^{3/2} (c x+1)^{3/2}+\frac{11}{5} b c d^3 \sqrt{c x-1} \sqrt{c x+1}+b c d^3 \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)",1,"(d^3*((-25*a)/x - 75*a*c^2*x + 25*a*c^4*x^3 - 5*a*c^6*x^5 + b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(61 - 7*c^2*x^2 + c^4*x^4) - (5*b*(5 + 15*c^2*x^2 - 5*c^4*x^4 + c^6*x^6)*ArcCosh[c*x])/x - 25*b*c*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]))/25","A",1
26,1,226,267,0.3723994,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^3,x]","-\frac{d^3 \left(8 a c^6 x^6-48 a c^4 x^4+96 a c^2 x^2 \log (x)+16 a-2 b c^5 x^5 \sqrt{c x-1} \sqrt{c x+1}+21 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}-48 b c^2 x^2 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+48 b c^2 x^2 \cosh ^{-1}(c x)^2+42 b c^2 x^2 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)+8 b \cosh ^{-1}(c x) \left(c^6 x^6-6 c^4 x^4+12 c^2 x^2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)+2\right)-16 b c x \sqrt{c x-1} \sqrt{c x+1}\right)}{32 x^2}","-\frac{d^3 \left(1-c^2 x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{2 x^2}-\frac{3}{4} c^2 d^3 \left(1-c^2 x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)-\frac{3}{2} c^2 d^3 \left(1-c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)-\frac{3 c^2 d^3 \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b}-3 c^2 d^3 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)-\frac{7}{16} b c^3 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}-\frac{3}{32} b c^3 d^3 x \sqrt{c x-1} \sqrt{c x+1}+\frac{3}{2} b c^2 d^3 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+\frac{3}{32} b c^2 d^3 \cosh ^{-1}(c x)+\frac{b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{2 x}",1,"-1/32*(d^3*(16*a - 48*a*c^4*x^4 + 8*a*c^6*x^6 - 16*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 21*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*b*c^5*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 48*b*c^2*x^2*ArcCosh[c*x]^2 + 42*b*c^2*x^2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]] + 8*b*ArcCosh[c*x]*(2 - 6*c^4*x^4 + c^6*x^6 + 12*c^2*x^2*Log[1 + E^(-2*ArcCosh[c*x])]) + 96*a*c^2*x^2*Log[x] - 48*b*c^2*x^2*PolyLog[2, -E^(-2*ArcCosh[c*x])]))/x^2","A",0
27,1,142,195,0.3094213,"\int \frac{\left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]))/x^4,x]","\frac{d^3 \left(-6 a c^6 x^6+54 a c^4 x^4+54 a c^2 x^2-6 a+51 b c^3 x^3 \tan ^{-1}\left(\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right)+b c x \sqrt{c x-1} \sqrt{c x+1} \left(2 c^4 x^4-50 c^2 x^2+3\right)-6 b \left(c^6 x^6-9 c^4 x^4-9 c^2 x^2+1\right) \cosh ^{-1}(c x)\right)}{18 x^3}","-\frac{1}{3} c^6 d^3 x^3 \left(a+b \cosh ^{-1}(c x)\right)+3 c^4 d^3 x \left(a+b \cosh ^{-1}(c x)\right)+\frac{3 c^2 d^3 \left(a+b \cosh ^{-1}(c x)\right)}{x}-\frac{d^3 \left(a+b \cosh ^{-1}(c x)\right)}{3 x^3}+\frac{1}{9} b c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}-\frac{8}{3} b c^3 d^3 \sqrt{c x-1} \sqrt{c x+1}-\frac{17}{6} b c^3 d^3 \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)+\frac{b c d^3 \sqrt{c x-1} \sqrt{c x+1}}{6 x^2}",1,"(d^3*(-6*a + 54*a*c^2*x^2 + 54*a*c^4*x^4 - 6*a*c^6*x^6 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - 50*c^2*x^2 + 2*c^4*x^4) - 6*b*(1 - 9*c^2*x^2 - 9*c^4*x^4 + c^6*x^6)*ArcCosh[c*x] + 51*b*c^3*x^3*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]))/(18*x^3)","A",1
28,1,227,158,0.3432688,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2),x]","-\frac{6 a c^3 x^3+18 a c x+9 a \log (1-c x)-9 a \log (c x+1)+6 b c^3 x^3 \cosh ^{-1}(c x)-2 b c^2 x^2 \sqrt{c x-1} \sqrt{c x+1}+18 b \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)+18 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)-18 b c x \sqrt{\frac{c x-1}{c x+1}}-18 b \sqrt{\frac{c x-1}{c x+1}}-4 b \sqrt{c x-1} \sqrt{c x+1}+18 b c x \cosh ^{-1}(c x)-9 b \cosh ^{-1}(c x)^2-18 b \cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right)+18 b \cosh ^{-1}(c x) \log \left(1-e^{\cosh ^{-1}(c x)}\right)}{18 c^5 d}","\frac{2 \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^5 d}-\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{c^4 d}-\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{3 c^2 d}+\frac{b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{c^5 d}-\frac{b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{c^5 d}+\frac{11 b \sqrt{c x-1} \sqrt{c x+1}}{9 c^5 d}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c^3 d}",1,"-1/18*(18*a*c*x + 6*a*c^3*x^3 - 18*b*Sqrt[(-1 + c*x)/(1 + c*x)] - 18*b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*b*c^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 18*b*c*x*ArcCosh[c*x] + 6*b*c^3*x^3*ArcCosh[c*x] - 9*b*ArcCosh[c*x]^2 - 18*b*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 18*b*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] + 9*a*Log[1 - c*x] - 9*a*Log[1 + c*x] + 18*b*PolyLog[2, -E^(-ArcCosh[c*x])] + 18*b*PolyLog[2, E^ArcCosh[c*x]])/(c^5*d)","A",0
29,1,151,140,0.3092171,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2),x]","-\frac{2 c^2 x^2 \left(a+b \cosh ^{-1}(c x)\right)-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)^2}{b}+4 \log \left(1-e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)+4 \log \left(e^{\cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)+4 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)+4 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)-b \left(c x \sqrt{c x-1} \sqrt{c x+1}+2 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{4 c^4 d}","\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{2 b c^4 d}-\frac{\log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^4 d}-\frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{2 c^2 d}-\frac{b \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{2 c^4 d}+\frac{b \cosh ^{-1}(c x)}{4 c^4 d}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{4 c^3 d}",1,"-1/4*(2*c^2*x^2*(a + b*ArcCosh[c*x]) - (2*(a + b*ArcCosh[c*x])^2)/b - b*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]) + 4*(a + b*ArcCosh[c*x])*Log[1 - E^ArcCosh[c*x]] + 4*(a + b*ArcCosh[c*x])*Log[1 + E^ArcCosh[c*x]] + 4*b*PolyLog[2, -E^ArcCosh[c*x]] + 4*b*PolyLog[2, E^ArcCosh[c*x]])/(c^4*d)","A",0
30,1,155,102,0.1610951,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2),x]","\frac{-2 a c x-a \log (1-c x)+a \log (c x+1)-2 b \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)-2 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)+2 b \sqrt{\frac{c x-1}{c x+1}}+2 b c x \sqrt{\frac{c x-1}{c x+1}}+b \cosh ^{-1}(c x)^2-2 b c x \cosh ^{-1}(c x)+2 b \cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right)-2 b \cosh ^{-1}(c x) \log \left(1-e^{\cosh ^{-1}(c x)}\right)}{2 c^3 d}","\frac{2 \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^3 d}-\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{c^2 d}+\frac{b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{c^3 d}-\frac{b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{c^3 d}+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c^3 d}",1,"(-2*a*c*x + 2*b*Sqrt[(-1 + c*x)/(1 + c*x)] + 2*b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*b*c*x*ArcCosh[c*x] + b*ArcCosh[c*x]^2 + 2*b*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] - 2*b*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] - a*Log[1 - c*x] + a*Log[1 + c*x] - 2*b*PolyLog[2, -E^(-ArcCosh[c*x])] - 2*b*PolyLog[2, E^ArcCosh[c*x]])/(2*c^3*d)","A",0
31,1,85,74,0.0885335,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[(x*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2),x]","\frac{\left(a+b \cosh ^{-1}(c x)\right) \left(a+b \cosh ^{-1}(c x)-2 b \log \left(1-e^{\cosh ^{-1}(c x)}\right)-2 b \log \left(e^{\cosh ^{-1}(c x)}+1\right)\right)-2 b^2 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)-2 b^2 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{2 b c^2 d}","\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{2 b c^2 d}-\frac{\log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^2 d}-\frac{b \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{2 c^2 d}",1,"((a + b*ArcCosh[c*x])*(a + b*ArcCosh[c*x] - 2*b*Log[1 - E^ArcCosh[c*x]] - 2*b*Log[1 + E^ArcCosh[c*x]]) - 2*b^2*PolyLog[2, -E^ArcCosh[c*x]] - 2*b^2*PolyLog[2, E^ArcCosh[c*x]])/(2*b*c^2*d)","A",0
32,1,64,59,0.0710922,"\int \frac{a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2),x]","\frac{-\left(\left(\log \left(1-e^{\cosh ^{-1}(c x)}\right)-\log \left(e^{\cosh ^{-1}(c x)}+1\right)\right) \left(a+b \cosh ^{-1}(c x)\right)\right)+b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)-b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{c d}","\frac{2 \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c d}+\frac{b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{c d}-\frac{b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{c d}",1,"(-((a + b*ArcCosh[c*x])*(Log[1 - E^ArcCosh[c*x]] - Log[1 + E^ArcCosh[c*x]])) + b*PolyLog[2, -E^ArcCosh[c*x]] - b*PolyLog[2, E^ArcCosh[c*x]])/(c*d)","A",0
33,1,93,61,0.1593001,"\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)),x]","-\frac{a \log \left(1-c^2 x^2\right)}{2 d}+\frac{a \log (x)}{d}-\frac{b \left(\text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)+2 \cosh ^{-1}(c x) \left(\log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)-\log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)\right)}{2 d}","\frac{2 \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d}+\frac{b \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{2 d}-\frac{b \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{2 d}",1,"(a*Log[x])/d - (a*Log[1 - c^2*x^2])/(2*d) - (b*(2*ArcCosh[c*x]*(Log[1 - E^(-2*ArcCosh[c*x])] - Log[1 + E^(-2*ArcCosh[c*x])]) + PolyLog[2, -E^(-2*ArcCosh[c*x])] - PolyLog[2, E^(-2*ArcCosh[c*x])]))/(2*d)","A",0
34,1,132,95,0.2983483,"\int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)),x]","\frac{-\frac{a+b \cosh ^{-1}(c x)}{x}-c \log \left(1-e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)+c \log \left(e^{\cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)+\frac{b c \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+b c \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)-b c \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{d}","-\frac{a+b \cosh ^{-1}(c x)}{d x}+\frac{2 c \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d}+\frac{b c \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{d}-\frac{b c \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{d}+\frac{b c \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{d}",1,"(-((a + b*ArcCosh[c*x])/x) + (b*c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - c*(a + b*ArcCosh[c*x])*Log[1 - E^ArcCosh[c*x]] + c*(a + b*ArcCosh[c*x])*Log[1 + E^ArcCosh[c*x]] + b*c*PolyLog[2, -E^ArcCosh[c*x]] - b*c*PolyLog[2, E^ArcCosh[c*x]])/d","A",0
35,1,144,118,0.5669219,"\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)),x]","-\frac{a c^2 \log \left(1-c^2 x^2\right)-2 a c^2 \log (x)+\frac{a}{x^2}+b c^2 \left(\frac{\cosh ^{-1}(c x)}{c^2 x^2}+\text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+2 \cosh ^{-1}(c x) \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)-2 \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)}{2 d}","\frac{2 c^2 \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac{b c^2 \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{2 d}-\frac{b c^2 \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{2 d}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d x}",1,"-1/2*(a/x^2 - 2*a*c^2*Log[x] + a*c^2*Log[1 - c^2*x^2] + b*c^2*(-((Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x)) + ArcCosh[c*x]/(c^2*x^2) + 2*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 2*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] + PolyLog[2, -E^(-2*ArcCosh[c*x])] - PolyLog[2, E^(-2*ArcCosh[c*x])]))/d","A",0
36,1,223,157,0.3635227,"\int \frac{a+b \cosh ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)),x]","\frac{-6 a c^3 \log \left(1-e^{\cosh ^{-1}(c x)}\right)+6 a c^3 \log \left(e^{\cosh ^{-1}(c x)}+1\right)-\frac{6 a c^2}{x}-\frac{2 a}{x^3}+6 b c^3 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)-6 b c^3 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)-6 b c^3 \cosh ^{-1}(c x) \log \left(1-e^{\cosh ^{-1}(c x)}\right)+6 b c^3 \cosh ^{-1}(c x) \log \left(e^{\cosh ^{-1}(c x)}+1\right)-\frac{6 b c^2 \cosh ^{-1}(c x)}{x}+\frac{7 b c^3 \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b \cosh ^{-1}(c x)}{x^3}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{x^2}}{6 d}","\frac{2 c^3 \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d}-\frac{c^2 \left(a+b \cosh ^{-1}(c x)\right)}{d x}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{b c^3 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{d}-\frac{b c^3 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{d}+\frac{7 b c^3 \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{6 d}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 d x^2}",1,"((-2*a)/x^3 - (6*a*c^2)/x + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/x^2 - (2*b*ArcCosh[c*x])/x^3 - (6*b*c^2*ArcCosh[c*x])/x + (7*b*c^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - 6*a*c^3*Log[1 - E^ArcCosh[c*x]] - 6*b*c^3*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] + 6*a*c^3*Log[1 + E^ArcCosh[c*x]] + 6*b*c^3*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 6*b*c^3*PolyLog[2, -E^ArcCosh[c*x]] - 6*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/(6*d)","A",0
37,1,244,177,1.093474,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{-\frac{2 a c x}{c^2 x^2-1}+4 a c x+3 a \log (1-c x)-3 a \log (c x+1)-6 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)+6 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)-4 b c x \sqrt{\frac{c x-1}{c x+1}}+\frac{b c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{b \sqrt{\frac{c x-1}{c x+1}}}{1-c x}-3 b \sqrt{\frac{c x-1}{c x+1}}+4 b c x \cosh ^{-1}(c x)+\frac{b \cosh ^{-1}(c x)}{1-c x}-\frac{b \cosh ^{-1}(c x)}{c x+1}+6 b \cosh ^{-1}(c x) \log \left(1-e^{\cosh ^{-1}(c x)}\right)-6 b \cosh ^{-1}(c x) \log \left(e^{\cosh ^{-1}(c x)}+1\right)}{4 c^5 d^2}","-\frac{3 \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^5 d^2}+\frac{3 x \left(a+b \cosh ^{-1}(c x)\right)}{2 c^4 d^2}+\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{3 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{2 c^5 d^2}+\frac{3 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{2 c^5 d^2}-\frac{b \sqrt{c x-1} \sqrt{c x+1}}{2 c^5 d^2}-\frac{b x^2}{2 c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(4*a*c*x - 3*b*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] + (b*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + (b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (2*a*c*x)/(-1 + c^2*x^2) + 4*b*c*x*ArcCosh[c*x] + (b*ArcCosh[c*x])/(1 - c*x) - (b*ArcCosh[c*x])/(1 + c*x) + 6*b*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] - 6*b*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 3*a*Log[1 - c*x] - 3*a*Log[1 + c*x] - 6*b*PolyLog[2, -E^ArcCosh[c*x]] + 6*b*PolyLog[2, E^ArcCosh[c*x]])/(4*c^5*d^2)","A",0
38,1,209,179,0.680124,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{-\frac{2 a}{c^2 x^2-1}+2 a \log \left(1-c^2 x^2\right)+4 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)+4 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)-b \sqrt{\frac{c x-1}{c x+1}}+\frac{b \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{b c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}-2 b \cosh ^{-1}(c x)^2+\frac{b \cosh ^{-1}(c x)}{1-c x}+\frac{b \cosh ^{-1}(c x)}{c x+1}+4 b \cosh ^{-1}(c x) \log \left(1-e^{\cosh ^{-1}(c x)}\right)+4 b \cosh ^{-1}(c x) \log \left(e^{\cosh ^{-1}(c x)}+1\right)}{4 c^4 d^2}","-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{2 b c^4 d^2}+\frac{\log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^4 d^2}+\frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}+\frac{b \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{2 c^4 d^2}-\frac{b \sqrt{c x-1}}{2 c^4 d^2 \sqrt{c x+1}}-\frac{b}{2 c^4 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \cosh ^{-1}(c x)}{2 c^4 d^2}",1,"(-(b*Sqrt[(-1 + c*x)/(1 + c*x)]) + (b*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + (b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (2*a)/(-1 + c^2*x^2) + (b*ArcCosh[c*x])/(1 - c*x) + (b*ArcCosh[c*x])/(1 + c*x) - 2*b*ArcCosh[c*x]^2 + 4*b*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] + 4*b*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 2*a*Log[1 - c^2*x^2] + 4*b*PolyLog[2, -E^ArcCosh[c*x]] + 4*b*PolyLog[2, E^ArcCosh[c*x]])/(4*c^4*d^2)","A",0
39,1,206,124,0.7588186,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{-\frac{2 a c x}{c^2 x^2-1}+a \log (1-c x)-a \log (c x+1)-2 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)+2 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)+\frac{b c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{b \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+b \sqrt{\frac{c x-1}{c x+1}}+\frac{b \cosh ^{-1}(c x)}{1-c x}-\frac{b \cosh ^{-1}(c x)}{c x+1}+2 b \cosh ^{-1}(c x) \log \left(1-e^{\cosh ^{-1}(c x)}\right)-2 b \cosh ^{-1}(c x) \log \left(e^{\cosh ^{-1}(c x)}+1\right)}{4 c^3 d^2}","-\frac{\tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^3 d^2}+\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{2 c^3 d^2}+\frac{b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{2 c^3 d^2}-\frac{b}{2 c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(b*Sqrt[(-1 + c*x)/(1 + c*x)] + (b*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + (b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (2*a*c*x)/(-1 + c^2*x^2) + (b*ArcCosh[c*x])/(1 - c*x) - (b*ArcCosh[c*x])/(1 + c*x) + 2*b*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] - 2*b*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + a*Log[1 - c*x] - a*Log[1 + c*x] - 2*b*PolyLog[2, -E^ArcCosh[c*x]] + 2*b*PolyLog[2, E^ArcCosh[c*x]])/(4*c^3*d^2)","A",0
40,1,53,61,0.1425776,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^2,x]","\frac{a+b c x \sqrt{c x-1} \sqrt{c x+1}+b \cosh ^{-1}(c x)}{2 c^2 d^2-2 c^4 d^2 x^2}","\frac{a+b \cosh ^{-1}(c x)}{2 c^2 d^2 \left(1-c^2 x^2\right)}-\frac{b x}{2 c d^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(a + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*ArcCosh[c*x])/(2*c^2*d^2 - 2*c^4*d^2*x^2)","A",1
41,1,189,120,1.4078147,"\int \frac{a+b \cosh ^{-1}(c x)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^2,x]","\frac{\frac{a c^2 x^2 \log (c x+1)+\left(a-a c^2 x^2\right) \log (1-c x)-2 a c x-a \log (c x+1)-2 b \cosh ^{-1}(c x) \left(\left(c^2 x^2-1\right) \log \left(1-e^{\cosh ^{-1}(c x)}\right)+\left(1-c^2 x^2\right) \log \left(e^{\cosh ^{-1}(c x)}+1\right)+c x\right)-2 b c x \sqrt{\frac{c x-1}{c x+1}}-2 b \sqrt{\frac{c x-1}{c x+1}}}{c^2 x^2-1}+2 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)-2 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{4 c d^2}","\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}+\frac{\tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c d^2}+\frac{b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{2 c d^2}-\frac{b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{2 c d^2}-\frac{b}{2 c d^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"((-2*a*c*x - 2*b*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*b*ArcCosh[c*x]*(c*x + (-1 + c^2*x^2)*Log[1 - E^ArcCosh[c*x]] + (1 - c^2*x^2)*Log[1 + E^ArcCosh[c*x]]) + (a - a*c^2*x^2)*Log[1 - c*x] - a*Log[1 + c*x] + a*c^2*x^2*Log[1 + c*x])/(-1 + c^2*x^2) + 2*b*PolyLog[2, -E^ArcCosh[c*x]] - 2*b*PolyLog[2, E^ArcCosh[c*x]])/(4*c*d^2)","A",0
42,1,149,116,0.8076853,"\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^2),x]","\frac{\frac{a}{1-c^2 x^2}-a \log \left(1-c^2 x^2\right)+2 a \log (x)+b \left(\frac{\cosh ^{-1}(c x)}{1-c^2 x^2}-\text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+\text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)+\frac{c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}-2 \cosh ^{-1}(c x) \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)+2 \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)}{2 d^2}","\frac{a+b \cosh ^{-1}(c x)}{2 d^2 \left(1-c^2 x^2\right)}+\frac{2 \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2}+\frac{b \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{2 d^2}-\frac{b \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{2 d^2}-\frac{b c x}{2 d^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(a/(1 - c^2*x^2) + 2*a*Log[x] - a*Log[1 - c^2*x^2] + b*((c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + ArcCosh[c*x]/(1 - c^2*x^2) - 2*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] + 2*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] - PolyLog[2, -E^(-2*ArcCosh[c*x])] + PolyLog[2, E^(-2*ArcCosh[c*x])]))/(2*d^2)","A",0
43,1,283,170,0.7167788,"\int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^2),x]","\frac{-\frac{2 a c^2 x}{c^2 x^2-1}-3 a c \log (1-c x)+3 a c \log (c x+1)-\frac{4 a}{x}+\frac{4 b c \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^2 x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+6 b c \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)-6 b c \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)+b c \sqrt{\frac{c x-1}{c x+1}}+\frac{b c \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{b c \cosh ^{-1}(c x)}{1-c x}-\frac{b c \cosh ^{-1}(c x)}{c x+1}-\frac{4 b \cosh ^{-1}(c x)}{x}-6 b c \cosh ^{-1}(c x) \log \left(1-e^{\cosh ^{-1}(c x)}\right)+6 b c \cosh ^{-1}(c x) \log \left(e^{\cosh ^{-1}(c x)}+1\right)}{4 d^2}","\frac{3 c^2 x \left(a+b \cosh ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x \left(1-c^2 x^2\right)}+\frac{3 c \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2}+\frac{3 b c \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{2 d^2}-\frac{3 b c \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{2 d^2}-\frac{b c}{2 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{d^2}",1,"((-4*a)/x + b*c*Sqrt[(-1 + c*x)/(1 + c*x)] + (b*c*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + (b*c^2*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (2*a*c^2*x)/(-1 + c^2*x^2) - (4*b*ArcCosh[c*x])/x + (b*c*ArcCosh[c*x])/(1 - c*x) - (b*c*ArcCosh[c*x])/(1 + c*x) + (4*b*c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - 6*b*c*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] + 6*b*c*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] - 3*a*c*Log[1 - c*x] + 3*a*c*Log[1 + c*x] + 6*b*c*PolyLog[2, -E^ArcCosh[c*x]] - 6*b*c*PolyLog[2, E^ArcCosh[c*x]])/(4*d^2)","A",0
44,1,319,152,0.5977757,"\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^2),x]","\frac{4 a c^4 x^4 \log (x)-2 a c^2 x^2-4 a c^2 x^2 \log (x)+2 a c^2 x^2 \log \left(1-c^2 x^2\right)-2 a c^4 x^4 \log \left(1-c^2 x^2\right)+a-4 b c^4 x^4 \cosh ^{-1}(c x) \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)+4 b c^4 x^4 \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)-2 b c^2 x^2 \left(c^2 x^2-1\right) \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+2 b c^2 x^2 \left(c^2 x^2-1\right) \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)-b c^2 x^2 \sqrt{\frac{c x-1}{c x+1}}-2 b c^2 x^2 \cosh ^{-1}(c x)+4 b c^2 x^2 \cosh ^{-1}(c x) \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)-4 b c^2 x^2 \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)-b c x \sqrt{\frac{c x-1}{c x+1}}+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left(c^2 x^2-1\right)}","\frac{c^2 \left(a+b \cosh ^{-1}(c x)\right)}{d^2 \left(1-c^2 x^2\right)}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left(1-c^2 x^2\right)}+\frac{4 c^2 \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2}+\frac{b c^2 \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{d^2}-\frac{b c^2 \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{d^2}-\frac{b c}{2 d^2 x \sqrt{c x-1} \sqrt{c x+1}}",1,"(a - 2*a*c^2*x^2 - b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - b*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + b*ArcCosh[c*x] - 2*b*c^2*x^2*ArcCosh[c*x] + 4*b*c^2*x^2*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 4*b*c^4*x^4*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 4*b*c^2*x^2*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] + 4*b*c^4*x^4*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] - 4*a*c^2*x^2*Log[x] + 4*a*c^4*x^4*Log[x] + 2*a*c^2*x^2*Log[1 - c^2*x^2] - 2*a*c^4*x^4*Log[1 - c^2*x^2] - 2*b*c^2*x^2*(-1 + c^2*x^2)*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 2*b*c^2*x^2*(-1 + c^2*x^2)*PolyLog[2, E^(-2*ArcCosh[c*x])])/(2*d^2*x^2*(-1 + c^2*x^2))","B",0
45,1,377,248,1.7032393,"\int \frac{a+b \cosh ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^2),x]","-\frac{15 a c^3 \log (1-c x)-15 a c^3 \log (c x+1)+\frac{24 a c^2}{x}+\frac{6 a c^4 x}{c^2 x^2-1}+\frac{4 a}{x^3}+\frac{3 b c^4 x \sqrt{\frac{c x-1}{c x+1}}}{c x-1}-30 b c^3 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)+30 b c^3 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)-3 b c^3 \sqrt{\frac{c x-1}{c x+1}}+\frac{3 b c^3 \sqrt{\frac{c x-1}{c x+1}}}{c x-1}-\frac{2 b c^3}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 \cosh ^{-1}(c x)}{c x-1}+\frac{3 b c^3 \cosh ^{-1}(c x)}{c x+1}+30 b c^3 \cosh ^{-1}(c x) \log \left(1-e^{\cosh ^{-1}(c x)}\right)-30 b c^3 \cosh ^{-1}(c x) \log \left(e^{\cosh ^{-1}(c x)}+1\right)+\frac{24 b c^2 \cosh ^{-1}(c x)}{x}-\frac{26 b c^3 \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b \cosh ^{-1}(c x)}{x^3}+\frac{2 b c}{x^2 \sqrt{c x-1} \sqrt{c x+1}}}{12 d^2}","\frac{5 c^3 \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2}-\frac{5 c^2 \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 x \left(1-c^2 x^2\right)}-\frac{a+b \cosh ^{-1}(c x)}{3 d^2 x^3 \left(1-c^2 x^2\right)}+\frac{5 c^4 x \left(a+b \cosh ^{-1}(c x)\right)}{2 d^2 \left(1-c^2 x^2\right)}+\frac{5 b c^3 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{2 d^2}-\frac{5 b c^3 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{2 d^2}-\frac{b c^3}{3 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{13 b c^3 \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{6 d^2}-\frac{b c}{6 d^2 x^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/12*((4*a)/x^3 + (24*a*c^2)/x - 3*b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)] + (3*b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)])/(-1 + c*x) + (3*b*c^4*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(-1 + c*x) - (2*b*c^3)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c)/(x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (6*a*c^4*x)/(-1 + c^2*x^2) + (4*b*ArcCosh[c*x])/x^3 + (24*b*c^2*ArcCosh[c*x])/x + (3*b*c^3*ArcCosh[c*x])/(-1 + c*x) + (3*b*c^3*ArcCosh[c*x])/(1 + c*x) - (26*b*c^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 30*b*c^3*ArcCosh[c*x]*Log[1 - E^ArcCosh[c*x]] - 30*b*c^3*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 15*a*c^3*Log[1 - c*x] - 15*a*c^3*Log[1 + c*x] - 30*b*c^3*PolyLog[2, -E^ArcCosh[c*x]] + 30*b*c^3*PolyLog[2, E^ArcCosh[c*x]])/d^2","A",0
46,1,287,249,1.7879063,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3,x]","\frac{\frac{30 a c x}{c^2 x^2-1}+\frac{12 a c x}{\left(c^2 x^2-1\right)^2}-9 a \log (1-c x)+9 a \log (c x+1)+18 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)-18 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)+\frac{b \sqrt{c x-1} (c x+2)}{(c x+1)^{3/2}}-\frac{b (c x-2) \sqrt{c x+1}}{(c x-1)^{3/2}}+\frac{3 b \cosh ^{-1}(c x)}{(c x-1)^2}-\frac{3 b \cosh ^{-1}(c x)}{(c x+1)^2}-15 b \left(\frac{\cosh ^{-1}(c x)}{1-c x}-\frac{1}{\sqrt{\frac{c x-1}{c x+1}}}\right)-15 b \left(\sqrt{\frac{c x-1}{c x+1}}-\frac{\cosh ^{-1}(c x)}{c x+1}\right)+\frac{9}{2} b \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(1-e^{\cosh ^{-1}(c x)}\right)\right)-\frac{9}{2} b \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(e^{\cosh ^{-1}(c x)}+1\right)\right)}{48 c^5 d^3}","\frac{3 \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 c^5 d^3}+\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{4 c^2 d^3 \left(1-c^2 x^2\right)^2}-\frac{3 x \left(a+b \cosh ^{-1}(c x)\right)}{8 c^4 d^3 \left(1-c^2 x^2\right)}+\frac{3 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{8 c^5 d^3}-\frac{3 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{8 c^5 d^3}+\frac{3 b}{8 c^5 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b (c x-1)^{3/2}}{12 c^5 d^3 (c x+1)^{3/2}}+\frac{b}{4 c^5 d^3 \sqrt{c x-1} (c x+1)^{3/2}}+\frac{b x^3}{12 c^2 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}",1,"(-((b*(-2 + c*x)*Sqrt[1 + c*x])/(-1 + c*x)^(3/2)) + (b*Sqrt[-1 + c*x]*(2 + c*x))/(1 + c*x)^(3/2) + (12*a*c*x)/(-1 + c^2*x^2)^2 + (30*a*c*x)/(-1 + c^2*x^2) + (3*b*ArcCosh[c*x])/(-1 + c*x)^2 - (3*b*ArcCosh[c*x])/(1 + c*x)^2 - 15*b*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) - 15*b*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)) + (9*b*ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]))/2 - (9*b*ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]]))/2 - 9*a*Log[1 - c*x] + 9*a*Log[1 + c*x] + 18*b*PolyLog[2, -E^ArcCosh[c*x]] - 18*b*PolyLog[2, E^ArcCosh[c*x]])/(48*c^5*d^3)","A",0
47,1,83,136,0.2517947,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3,x]","\frac{a \left(6 c^2 x^2-3\right)+b c x \sqrt{c x-1} \sqrt{c x+1} \left(4 c^2 x^2-3\right)+3 b \left(2 c^2 x^2-1\right) \cosh ^{-1}(c x)}{12 c^4 d^3 \left(c^2 x^2-1\right)^2}","\frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{4 d^3 \left(1-c^2 x^2\right)^2}+\frac{b \sqrt{c x-1}}{4 c^4 d^3 \sqrt{c x+1}}+\frac{b}{4 c^4 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \cosh ^{-1}(c x)}{4 c^4 d^3}+\frac{b x^3}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}",1,"(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-3 + 4*c^2*x^2) + a*(-3 + 6*c^2*x^2) + 3*b*(-1 + 2*c^2*x^2)*ArcCosh[c*x])/(12*c^4*d^3*(-1 + c^2*x^2)^2)","A",1
48,1,287,186,1.6675187,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3,x]","\frac{\frac{6 a c x}{c^2 x^2-1}+\frac{12 a c x}{\left(c^2 x^2-1\right)^2}+3 a \log (1-c x)-3 a \log (c x+1)-6 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)+6 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)+\frac{b \sqrt{c x-1} (c x+2)}{(c x+1)^{3/2}}-\frac{b (c x-2) \sqrt{c x+1}}{(c x-1)^{3/2}}+\frac{3 b \cosh ^{-1}(c x)}{(c x-1)^2}-\frac{3 b \cosh ^{-1}(c x)}{(c x+1)^2}-3 b \left(\frac{\cosh ^{-1}(c x)}{1-c x}-\frac{1}{\sqrt{\frac{c x-1}{c x+1}}}\right)-3 b \left(\sqrt{\frac{c x-1}{c x+1}}-\frac{\cosh ^{-1}(c x)}{c x+1}\right)-\frac{3}{2} b \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(1-e^{\cosh ^{-1}(c x)}\right)\right)+\frac{3}{2} b \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(e^{\cosh ^{-1}(c x)}+1\right)\right)}{48 c^3 d^3}","-\frac{\tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 c^3 d^3}-\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{8 c^2 d^3 \left(1-c^2 x^2\right)}+\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{4 c^2 d^3 \left(1-c^2 x^2\right)^2}-\frac{b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{8 c^3 d^3}+\frac{b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{8 c^3 d^3}+\frac{b}{8 c^3 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b}{12 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}",1,"(-((b*(-2 + c*x)*Sqrt[1 + c*x])/(-1 + c*x)^(3/2)) + (b*Sqrt[-1 + c*x]*(2 + c*x))/(1 + c*x)^(3/2) + (12*a*c*x)/(-1 + c^2*x^2)^2 + (6*a*c*x)/(-1 + c^2*x^2) + (3*b*ArcCosh[c*x])/(-1 + c*x)^2 - (3*b*ArcCosh[c*x])/(1 + c*x)^2 - 3*b*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) - 3*b*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)) - (3*b*ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]))/2 + (3*b*ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]]))/2 + 3*a*Log[1 - c*x] - 3*a*Log[1 + c*x] - 6*b*PolyLog[2, -E^ArcCosh[c*x]] + 6*b*PolyLog[2, E^ArcCosh[c*x]])/(48*c^3*d^3)","A",0
49,1,64,91,0.2189977,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3,x]","\frac{3 a+b c x \sqrt{c x-1} \sqrt{c x+1} \left(3-2 c^2 x^2\right)+3 b \cosh ^{-1}(c x)}{12 c^2 d^3 \left(c^2 x^2-1\right)^2}","\frac{a+b \cosh ^{-1}(c x)}{4 c^2 d^3 \left(1-c^2 x^2\right)^2}-\frac{b x}{6 c d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}",1,"(3*a + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - 2*c^2*x^2) + 3*b*ArcCosh[c*x])/(12*c^2*d^3*(-1 + c^2*x^2)^2)","A",1
50,1,316,180,1.1748526,"\int \frac{a+b \cosh ^{-1}(c x)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^3,x]","\frac{-\frac{6 a x}{c^2 x^2-1}+\frac{4 a x}{\left(c^2 x^2-1\right)^2}-\frac{3 a \log (1-c x)}{c}+\frac{3 a \log (c x+1)}{c}-\frac{3 b \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(e^{\cosh ^{-1}(c x)}+1\right)\right)-4 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)\right)}{2 c}+\frac{3 b \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(1-e^{\cosh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)\right)}{2 c}+\frac{b \left(\sqrt{c x-1} \sqrt{c x+1} (c x+2)-3 \cosh ^{-1}(c x)\right)}{3 c (c x+1)^2}+\frac{b \left(\sqrt{c x-1} \sqrt{c x+1} (2-c x)+3 \cosh ^{-1}(c x)\right)}{3 c (c x-1)^2}+\frac{3 b \left(\frac{\cosh ^{-1}(c x)}{1-c x}-\frac{1}{\sqrt{\frac{c x-1}{c x+1}}}\right)}{c}+\frac{3 b \left(\sqrt{\frac{c x-1}{c x+1}}-\frac{\cosh ^{-1}(c x)}{c x+1}\right)}{c}}{16 d^3}","\frac{3 x \left(a+b \cosh ^{-1}(c x)\right)}{8 d^3 \left(1-c^2 x^2\right)}+\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{4 d^3 \left(1-c^2 x^2\right)^2}+\frac{3 \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 c d^3}+\frac{3 b \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{8 c d^3}-\frac{3 b \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{8 c d^3}-\frac{3 b}{8 c d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}}",1,"((4*a*x)/(-1 + c^2*x^2)^2 - (6*a*x)/(-1 + c^2*x^2) + (b*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x) - 3*ArcCosh[c*x]))/(3*c*(1 + c*x)^2) + (b*((2 - c*x)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 3*ArcCosh[c*x]))/(3*c*(-1 + c*x)^2) + (3*b*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)))/c + (3*b*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)))/c - (3*a*Log[1 - c*x])/c + (3*a*Log[1 + c*x])/c - (3*b*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]]) - 4*PolyLog[2, -E^ArcCosh[c*x]]))/(2*c) + (3*b*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]) - 4*PolyLog[2, E^ArcCosh[c*x]]))/(2*c))/(16*d^3)","A",0
51,1,210,171,1.5379167,"\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^3),x]","-\frac{\frac{6 a}{c^2 x^2-1}-\frac{3 a}{\left(c^2 x^2-1\right)^2}+6 a \log \left(1-c^2 x^2\right)-12 a \log (x)+b \left(\frac{6 \cosh ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \cosh ^{-1}(c x)}{\left(c^2 x^2-1\right)^2}+6 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-6 \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)-\frac{c x \left(\frac{c x-1}{c x+1}\right)^{3/2}}{(c x-1)^3}+\frac{8 c x \sqrt{\frac{c x-1}{c x+1}}}{c x-1}+12 \cosh ^{-1}(c x) \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)-12 \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)}{12 d^3}","\frac{a+b \cosh ^{-1}(c x)}{2 d^3 \left(1-c^2 x^2\right)}+\frac{a+b \cosh ^{-1}(c x)}{4 d^3 \left(1-c^2 x^2\right)^2}+\frac{2 \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^3}+\frac{b \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{2 d^3}-\frac{b \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{2 d^3}-\frac{2 b c x}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}",1,"-1/12*((-3*a)/(-1 + c^2*x^2)^2 + (6*a)/(-1 + c^2*x^2) - 12*a*Log[x] + 6*a*Log[1 - c^2*x^2] + b*((8*c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(-1 + c*x) - (c*x*((-1 + c*x)/(1 + c*x))^(3/2))/(-1 + c*x)^3 - (3*ArcCosh[c*x])/(-1 + c^2*x^2)^2 + (6*ArcCosh[c*x])/(-1 + c^2*x^2) + 12*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 12*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] + 6*PolyLog[2, -E^(-2*ArcCosh[c*x])] - 6*PolyLog[2, E^(-2*ArcCosh[c*x])]))/d^3","A",0
52,1,362,230,1.781056,"\int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^3),x]","\frac{-\frac{84 a c^2 x}{c^2 x^2-1}+\frac{24 a c^2 x}{\left(c^2 x^2-1\right)^2}-90 a c \log (1-c x)+90 a c \log (c x+1)-\frac{96 a}{x}+\frac{96 b c \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-45 b c \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(e^{\cosh ^{-1}(c x)}+1\right)\right)-4 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)\right)+45 b c \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(1-e^{\cosh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)\right)-\frac{2 b c \left((c x-2) \sqrt{c x-1} \sqrt{c x+1}-3 \cosh ^{-1}(c x)\right)}{(c x-1)^2}+\frac{2 b c \left(\sqrt{c x-1} \sqrt{c x+1} (c x+2)-3 \cosh ^{-1}(c x)\right)}{(c x+1)^2}+42 b c \left(\frac{\cosh ^{-1}(c x)}{1-c x}-\frac{1}{\sqrt{\frac{c x-1}{c x+1}}}\right)+42 b c \left(\sqrt{\frac{c x-1}{c x+1}}-\frac{\cosh ^{-1}(c x)}{c x+1}\right)-\frac{96 b \cosh ^{-1}(c x)}{x}}{96 d^3}","\frac{15 c^2 x \left(a+b \cosh ^{-1}(c x)\right)}{8 d^3 \left(1-c^2 x^2\right)}+\frac{5 c^2 x \left(a+b \cosh ^{-1}(c x)\right)}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{a+b \cosh ^{-1}(c x)}{d^3 x \left(1-c^2 x^2\right)^2}+\frac{15 c \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 d^3}+\frac{15 b c \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{8 d^3}-\frac{15 b c \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{8 d^3}-\frac{7 b c}{8 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{b c \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{d^3}",1,"((-96*a)/x + (24*a*c^2*x)/(-1 + c^2*x^2)^2 - (84*a*c^2*x)/(-1 + c^2*x^2) - (2*b*c*((-2 + c*x)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*ArcCosh[c*x]))/(-1 + c*x)^2 + (2*b*c*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x) - 3*ArcCosh[c*x]))/(1 + c*x)^2 - (96*b*ArcCosh[c*x])/x + 42*b*c*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) + 42*b*c*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)) + (96*b*c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - 90*a*c*Log[1 - c*x] + 90*a*c*Log[1 + c*x] - 45*b*c*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]]) - 4*PolyLog[2, -E^ArcCosh[c*x]]) + 45*b*c*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]) - 4*PolyLog[2, E^ArcCosh[c*x]]))/(96*d^3)","A",0
53,1,273,250,3.0334183,"\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^3),x]","-\frac{\frac{12 a c^2}{c^2 x^2-1}-\frac{3 a c^2}{\left(c^2 x^2-1\right)^2}+18 a c^2 \log \left(1-c^2 x^2\right)-36 a c^2 \log (x)+\frac{6 a}{x^2}+b c^2 \left(\frac{12 \cosh ^{-1}(c x)}{c^2 x^2-1}-\frac{3 \cosh ^{-1}(c x)}{\left(c^2 x^2-1\right)^2}+\frac{6 \cosh ^{-1}(c x)}{c^2 x^2}+18 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-18 \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)+\frac{14 c x}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{c x}{\left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3}-\frac{6 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+36 \cosh ^{-1}(c x) \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)-36 \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)}{12 d^3}","\frac{3 c^2 \left(a+b \cosh ^{-1}(c x)\right)}{2 d^3 \left(1-c^2 x^2\right)}+\frac{3 c^2 \left(a+b \cosh ^{-1}(c x)\right)}{4 d^3 \left(1-c^2 x^2\right)^2}-\frac{a+b \cosh ^{-1}(c x)}{2 d^3 x^2 \left(1-c^2 x^2\right)^2}+\frac{6 c^2 \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^3}-\frac{2 b c^3 x}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c^3 x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{3 b c^2 \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{2 d^3}-\frac{3 b c^2 \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{2 d^3}+\frac{b c}{2 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}}",1,"-1/12*((6*a)/x^2 - (3*a*c^2)/(-1 + c^2*x^2)^2 + (12*a*c^2)/(-1 + c^2*x^2) - 36*a*c^2*Log[x] + 18*a*c^2*Log[1 - c^2*x^2] + b*c^2*(-((c*x)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3)) + (14*c*x)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) + (6*ArcCosh[c*x])/(c^2*x^2) - (3*ArcCosh[c*x])/(-1 + c^2*x^2)^2 + (12*ArcCosh[c*x])/(-1 + c^2*x^2) + 36*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 36*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] + 18*PolyLog[2, -E^(-2*ArcCosh[c*x])] - 18*PolyLog[2, E^(-2*ArcCosh[c*x])]))/d^3","A",0
54,1,471,310,1.932704,"\int \frac{a+b \cosh ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^3),x]","\frac{-105 a c^3 \log (1-c x)+105 a c^3 \log (c x+1)-\frac{144 a c^2}{x}-\frac{66 a c^4 x}{c^2 x^2-1}+\frac{12 a c^4 x}{\left(c^2 x^2-1\right)^2}-\frac{16 a}{x^3}-\frac{105}{2} b c^3 \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(e^{\cosh ^{-1}(c x)}+1\right)\right)-4 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)\right)+\frac{105}{2} b c^3 \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-4 \log \left(1-e^{\cosh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)\right)-\frac{b c^3 \left((c x-2) \sqrt{c x-1} \sqrt{c x+1}-3 \cosh ^{-1}(c x)\right)}{(c x-1)^2}+\frac{b c^3 \left(\sqrt{c x-1} \sqrt{c x+1} (c x+2)-3 \cosh ^{-1}(c x)\right)}{(c x+1)^2}+33 b c^3 \left(\frac{\cosh ^{-1}(c x)}{1-c x}-\frac{1}{\sqrt{\frac{c x-1}{c x+1}}}\right)+33 b c^3 \left(\sqrt{\frac{c x-1}{c x+1}}-\frac{\cosh ^{-1}(c x)}{c x+1}\right)+144 b c^2 \left(\frac{c \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{\cosh ^{-1}(c x)}{x}\right)+\frac{8 b \left(\frac{c x \left(c^2 x^2+c^2 x^2 \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)-1\right)}{\sqrt{c x-1} \sqrt{c x+1}}-2 \cosh ^{-1}(c x)\right)}{x^3}}{48 d^3}","\frac{35 c^3 \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 d^3}-\frac{7 c^2 \left(a+b \cosh ^{-1}(c x)\right)}{3 d^3 x \left(1-c^2 x^2\right)^2}-\frac{a+b \cosh ^{-1}(c x)}{3 d^3 x^3 \left(1-c^2 x^2\right)^2}+\frac{35 c^4 x \left(a+b \cosh ^{-1}(c x)\right)}{8 d^3 \left(1-c^2 x^2\right)}+\frac{35 c^4 x \left(a+b \cosh ^{-1}(c x)\right)}{12 d^3 \left(1-c^2 x^2\right)^2}+\frac{35 b c^3 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{8 d^3}-\frac{35 b c^3 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{8 d^3}-\frac{29 b c^3}{24 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{19 b c^3 \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{6 d^3}+\frac{b c}{6 d^3 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}",1,"((-16*a)/x^3 - (144*a*c^2)/x + (12*a*c^4*x)/(-1 + c^2*x^2)^2 - (66*a*c^4*x)/(-1 + c^2*x^2) - (b*c^3*((-2 + c*x)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*ArcCosh[c*x]))/(-1 + c*x)^2 + (b*c^3*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x) - 3*ArcCosh[c*x]))/(1 + c*x)^2 + 33*b*c^3*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) + 33*b*c^3*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x]/(1 + c*x)) + 144*b*c^2*(-(ArcCosh[c*x]/x) + (c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (8*b*(-2*ArcCosh[c*x] + (c*x*(-1 + c^2*x^2 + c^2*x^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/x^3 - 105*a*c^3*Log[1 - c*x] + 105*a*c^3*Log[1 + c*x] - (105*b*c^3*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 + E^ArcCosh[c*x]]) - 4*PolyLog[2, -E^ArcCosh[c*x]]))/2 + (105*b*c^3*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]) - 4*PolyLog[2, E^ArcCosh[c*x]]))/2)/(48*d^3)","A",0
55,1,77,53,0.0490317,"\int \frac{\cosh ^{-1}(a x)}{c-a^2 c x^2} \, dx","Integrate[ArcCosh[a*x]/(c - a^2*c*x^2),x]","\frac{\text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{a c}-\frac{\text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{a c}-\frac{\cosh ^{-1}(a x) \log \left(1-e^{\cosh ^{-1}(a x)}\right)}{a c}+\frac{\cosh ^{-1}(a x) \log \left(e^{\cosh ^{-1}(a x)}+1\right)}{a c}","\frac{\text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{a c}-\frac{\text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{a c}+\frac{2 \cosh ^{-1}(a x) \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{a c}",1,"-((ArcCosh[a*x]*Log[1 - E^ArcCosh[a*x]])/(a*c)) + (ArcCosh[a*x]*Log[1 + E^ArcCosh[a*x]])/(a*c) + PolyLog[2, -E^ArcCosh[a*x]]/(a*c) - PolyLog[2, E^ArcCosh[a*x]]/(a*c)","A",0
56,1,120,109,0.8859251,"\int \frac{\cosh ^{-1}(a x)}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[ArcCosh[a*x]/(c - a^2*c*x^2)^2,x]","\frac{-\frac{2 \left(\cosh ^{-1}(a x) \left(\left(a^2 x^2-1\right) \log \left(1-e^{\cosh ^{-1}(a x)}\right)+\left(1-a^2 x^2\right) \log \left(e^{\cosh ^{-1}(a x)}+1\right)+a x\right)+\sqrt{\frac{a x-1}{a x+1}} (a x+1)\right)}{a^2 x^2-1}+2 \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)-2 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{4 a c^2}","\frac{x \cosh ^{-1}(a x)}{2 c^2 \left(1-a^2 x^2\right)}+\frac{\text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{2 a c^2}-\frac{\text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{2 a c^2}-\frac{1}{2 a c^2 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x) \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{a c^2}",1,"((-2*(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + ArcCosh[a*x]*(a*x + (-1 + a^2*x^2)*Log[1 - E^ArcCosh[a*x]] + (1 - a^2*x^2)*Log[1 + E^ArcCosh[a*x]])))/(-1 + a^2*x^2) + 2*PolyLog[2, -E^ArcCosh[a*x]] - 2*PolyLog[2, E^ArcCosh[a*x]])/(4*a*c^2)","A",0
57,1,223,164,2.4861571,"\int \frac{\cosh ^{-1}(a x)}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[ArcCosh[a*x]/(c - a^2*c*x^2)^3,x]","\frac{36 \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)-36 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)-\frac{2 \sqrt{a x+1} (a x-2)}{(a x-1)^{3/2}}+\frac{2 \sqrt{a x-1} (a x+2)}{(a x+1)^{3/2}}+\frac{6 \cosh ^{-1}(a x)}{(a x-1)^2}-\frac{6 \cosh ^{-1}(a x)}{(a x+1)^2}+18 \left(\frac{\cosh ^{-1}(a x)}{1-a x}-\frac{1}{\sqrt{\frac{a x-1}{a x+1}}}\right)+18 \left(\sqrt{\frac{a x-1}{a x+1}}-\frac{\cosh ^{-1}(a x)}{a x+1}\right)+9 \cosh ^{-1}(a x) \left(\cosh ^{-1}(a x)-4 \log \left(1-e^{\cosh ^{-1}(a x)}\right)\right)-9 \cosh ^{-1}(a x) \left(\cosh ^{-1}(a x)-4 \log \left(e^{\cosh ^{-1}(a x)}+1\right)\right)}{96 a c^3}","\frac{3 x \cosh ^{-1}(a x)}{8 c^3 \left(1-a^2 x^2\right)}+\frac{x \cosh ^{-1}(a x)}{4 c^3 \left(1-a^2 x^2\right)^2}+\frac{3 \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{8 a c^3}-\frac{3 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{8 a c^3}-\frac{3}{8 a c^3 \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{12 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac{3 \cosh ^{-1}(a x) \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}",1,"((-2*(-2 + a*x)*Sqrt[1 + a*x])/(-1 + a*x)^(3/2) + (2*Sqrt[-1 + a*x]*(2 + a*x))/(1 + a*x)^(3/2) + (6*ArcCosh[a*x])/(-1 + a*x)^2 - (6*ArcCosh[a*x])/(1 + a*x)^2 + 18*(-(1/Sqrt[(-1 + a*x)/(1 + a*x)]) + ArcCosh[a*x]/(1 - a*x)) + 18*(Sqrt[(-1 + a*x)/(1 + a*x)] - ArcCosh[a*x]/(1 + a*x)) + 9*ArcCosh[a*x]*(ArcCosh[a*x] - 4*Log[1 - E^ArcCosh[a*x]]) - 9*ArcCosh[a*x]*(ArcCosh[a*x] - 4*Log[1 + E^ArcCosh[a*x]]) + 36*PolyLog[2, -E^ArcCosh[a*x]] - 36*PolyLog[2, E^ArcCosh[a*x]])/(96*a*c^3)","A",0
58,1,198,278,1.2184733,"\int x^4 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^4*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]),x]","\frac{-144 a \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+48 a c x \left(8 c^4 x^4-2 c^2 x^2-3\right) \sqrt{d-c^2 d x^2}+\frac{b \sqrt{d-c^2 d x^2} \left(-72 \cosh ^{-1}(c x)^2+18 \cosh \left(2 \cosh ^{-1}(c x)\right)-9 \cosh \left(4 \cosh ^{-1}(c x)\right)-2 \cosh \left(6 \cosh ^{-1}(c x)\right)+12 \cosh ^{-1}(c x) \left(-3 \sinh \left(2 \cosh ^{-1}(c x)\right)+3 \sinh \left(4 \cosh ^{-1}(c x)\right)+\sinh \left(6 \cosh ^{-1}(c x)\right)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}}{2304 c^5}","\frac{1}{6} x^5 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{24 c^2}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{32 b c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{16 c^4}-\frac{b c x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^4 \sqrt{d-c^2 d x^2}}{96 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^2 \sqrt{d-c^2 d x^2}}{32 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(48*a*c*x*Sqrt[d - c^2*d*x^2]*(-3 - 2*c^2*x^2 + 8*c^4*x^4) - 144*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (b*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/(2304*c^5)","A",0
59,1,151,201,1.0967515,"\int x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]),x]","-\frac{-16 a c x \left(2 c^2 x^2-1\right) \sqrt{d-c^2 d x^2}+16 a \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\frac{b \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}}{128 c^3}","-\frac{x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 c^2}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{16 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^2 \sqrt{d-c^2 d x^2}}{16 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/128*(-16*a*c*x*(-1 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2] + 16*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (b*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/c^3","A",0
60,1,144,124,0.6083915,"\int \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]),x]","\frac{1}{8} \left(4 a x \sqrt{d-c^2 d x^2}-\frac{4 a \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)}{c}-\frac{b \sqrt{d-c^2 d x^2} \left(2 \cosh ^{-1}(c x)^2+\cosh \left(2 \cosh ^{-1}(c x)\right)-2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)}{c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)","\frac{1}{2} x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{4 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{c x-1} \sqrt{c x+1}}",1,"(4*a*x*Sqrt[d - c^2*d*x^2] - (4*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/c - (b*Sqrt[d - c^2*d*x^2]*(2*ArcCosh[c*x]^2 + Cosh[2*ArcCosh[c*x]] - 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]]))/(c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/8","A",0
61,1,137,118,0.4668117,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^2,x]","-\frac{a \sqrt{d-c^2 d x^2}}{x}+a c \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\frac{1}{2} b c \sqrt{d-c^2 d x^2} \left(\frac{2 \log (c x)+\cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{2 \cosh ^{-1}(c x)}{c x}\right)","\frac{c \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}",1,"-((a*Sqrt[d - c^2*d*x^2])/x) + a*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (b*c*Sqrt[d - c^2*d*x^2]*((-2*ArcCosh[c*x])/(c*x) + (ArcCosh[c*x]^2 + 2*Log[c*x])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/2","A",0
62,1,88,119,0.1274981,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^4,x]","\frac{\sqrt{d-c^2 d x^2} \left(\frac{(c x-1)^{3/2} (c x+1)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 x^3}-\frac{1}{3} b c \left(c^2 \log (x)+\frac{1}{2 x^2}\right)\right)}{\sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 d x^3}-\frac{b c \sqrt{d-c^2 d x^2}}{6 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(Sqrt[d - c^2*d*x^2]*(((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(3*x^3) - (b*c*(1/(2*x^2) + c^2*Log[x]))/3))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
63,1,128,199,0.2177319,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x^6} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^6,x]","\frac{\sqrt{d-c^2 d x^2} \left(8 c^2 x^2 (c x-1)^{3/2} (c x+1)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+12 (c x-1)^{3/2} (c x+1)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)-b c x \left(8 c^4 x^4 \log (x)-2 c^2 x^2+3\right)\right)}{60 x^5 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{5 d x^5}-\frac{2 c^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{15 d x^3}-\frac{b c \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c^5 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(Sqrt[d - c^2*d*x^2]*(12*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + 8*c^2*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) - b*c*x*(3 - 2*c^2*x^2 + 8*c^4*x^4*Log[x])))/(60*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
64,1,146,279,0.2693134,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x^8} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^8,x]","\frac{\sqrt{d-c^2 d x^2} \left(16 c^2 x^2 (c x-1)^{3/2} (c x+1)^{3/2} \left(2 c^2 x^2+3\right) \left(a+b \cosh ^{-1}(c x)\right)+60 (c x-1)^{3/2} (c x+1)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)-b c x \left(32 c^6 x^6 \log (x)-8 c^4 x^4-3 c^2 x^2+10\right)\right)}{420 x^7 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 d x^7}-\frac{4 c^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{35 d x^5}-\frac{8 c^4 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{105 d x^3}-\frac{b c \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{8 b c^7 \log (x) \sqrt{d-c^2 d x^2}}{105 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^5 \sqrt{d-c^2 d x^2}}{105 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{140 x^4 \sqrt{c x-1} \sqrt{c x+1}}",1,"(Sqrt[d - c^2*d*x^2]*(60*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + 16*c^2*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(3 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]) - b*c*x*(10 - 3*c^2*x^2 - 8*c^4*x^4 + 32*c^6*x^6*Log[x])))/(420*x^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
65,1,152,272,0.3679641,"\int x^5 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(15 c^3 x^4 (c x-1)^{3/2} (c x+1)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{4 (c x-1)^{3/2} (c x+1)^{3/2} \left(3 c^2 x^2+2\right) \left(a+b \cosh ^{-1}(c x)\right)}{c}+b \left(-\frac{15}{7} c^6 x^7+\frac{3 c^4 x^5}{5}+\frac{4 c^2 x^3}{3}+8 x\right)\right)}{105 c^5 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 c^6 d^3}+\frac{2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{5 c^6 d^2}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^6 d}-\frac{b c x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^5 \sqrt{d-c^2 d x^2}}{175 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b x \sqrt{d-c^2 d x^2}}{105 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b x^3 \sqrt{d-c^2 d x^2}}{315 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(Sqrt[d - c^2*d*x^2]*(b*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (15*c^6*x^7)/7) + 15*c^3*x^4*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + (4*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(2 + 3*c^2*x^2)*(a + b*ArcCosh[c*x]))/c))/(105*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
66,1,128,195,0.187019,"\int x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(3 c^2 x^2 (c x-1)^{3/2} (c x+1)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+2 (c x-1)^{3/2} (c x+1)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{15} b c x \left(-9 c^4 x^4+5 c^2 x^2+30\right)\right)}{15 c^4 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{5 c^4 d^2}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^4 d}-\frac{b c x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^3 \sqrt{d-c^2 d x^2}}{45 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(Sqrt[d - c^2*d*x^2]*((b*c*x*(30 + 5*c^2*x^2 - 9*c^4*x^4))/15 + 2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + 3*c^2*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x])))/(15*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
67,1,98,118,0.1354347,"\int x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{d-c^2 d x^2} \left(3 a \left(c^2 x^2-1\right)^2+b c x \sqrt{c x-1} \sqrt{c x+1} \left(3-c^2 x^2\right)+3 b \left(c^2 x^2-1\right)^2 \cosh ^{-1}(c x)\right)}{9 c^2 \left(c^2 x^2-1\right)}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^2 d}+\frac{b x \sqrt{d-c^2 d x^2}}{3 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{c x-1} \sqrt{c x+1}}",1,"(Sqrt[d - c^2*d*x^2]*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - c^2*x^2) + 3*a*(-1 + c^2*x^2)^2 + 3*b*(-1 + c^2*x^2)^2*ArcCosh[c*x]))/(9*c^2*(-1 + c^2*x^2))","A",1
68,1,233,213,0.9074935,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x,x]","a \sqrt{d-c^2 d x^2}-a \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+a \sqrt{d} \log (x)+\frac{b \sqrt{d-c^2 d x^2} \left(i \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{i b \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}",1,"a*Sqrt[d - c^2*d*x^2] + a*Sqrt[d]*Log[x] - a*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
69,1,307,235,1.0972895,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^3,x]","\frac{1}{2} \left(-\frac{a \sqrt{d-c^2 d x^2}}{x^2}+a c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-a c^2 \sqrt{d} \log (x)+\frac{b d (c x+1) \left(i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+c x \sqrt{\frac{c x-1}{c x+1}}+c x \cosh ^{-1}(c x)-\cosh ^{-1}(c x)\right)}{x^2 \sqrt{d-c^2 d x^2}}\right)","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{d-c^2 d x^2}}{2 x \sqrt{c x-1} \sqrt{c x+1}}",1,"(-((a*Sqrt[d - c^2*d*x^2])/x^2) - a*c^2*Sqrt[d]*Log[x] + a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d*(1 + c*x)*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x] + c*x*ArcCosh[c*x] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, I/E^ArcCosh[c*x]]))/(x^2*Sqrt[d - c^2*d*x^2]))/2","A",0
70,1,290,315,1.0523586,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x^5} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^5,x]","\frac{1}{24} \left(-3 a c^4 \sqrt{d} \log (x)+\frac{3 a \left(c^2 x^2-2\right) \sqrt{d-c^2 d x^2}}{x^4}+3 a c^4 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+\frac{b \sqrt{d-c^2 d x^2} \left(-3 i c^4 x^4 \left(\text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)\right)-3 i c^4 x^4 \cosh ^{-1}(c x) \left(\log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)+3 c^3 x^3+3 c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-2 c x-6 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)\right)}{x^4 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)","\frac{c^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 x^2}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{4 x^4}+\frac{c^4 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{i b c^4 \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{8 x \sqrt{c x-1} \sqrt{c x+1}}",1,"((3*a*(-2 + c^2*x^2)*Sqrt[d - c^2*d*x^2])/x^4 - 3*a*c^4*Sqrt[d]*Log[x] + 3*a*c^4*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*Sqrt[d - c^2*d*x^2]*(-2*c*x + 3*c^3*x^3 - 6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + 3*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - (3*I)*c^4*x^4*ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]]) - (3*I)*c^4*x^4*(PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcCosh[c*x]])))/(x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/24","A",0
71,1,337,360,4.8165885,"\int x^4 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]","\frac{d \left(-1728 a \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-576 a c x \left(16 c^6 x^6-24 c^4 x^4+2 c^2 x^2+3\right) \sqrt{d-c^2 d x^2}+\frac{32 b \sqrt{d-c^2 d x^2} \left(-72 \cosh ^{-1}(c x)^2+18 \cosh \left(2 \cosh ^{-1}(c x)\right)-9 \cosh \left(4 \cosh ^{-1}(c x)\right)-2 \cosh \left(6 \cosh ^{-1}(c x)\right)+12 \cosh ^{-1}(c x) \left(-3 \sinh \left(2 \cosh ^{-1}(c x)\right)+3 \sinh \left(4 \cosh ^{-1}(c x)\right)+\sinh \left(6 \cosh ^{-1}(c x)\right)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b \sqrt{d-c^2 d x^2} \left(1440 \cosh ^{-1}(c x)^2-576 \cosh \left(2 \cosh ^{-1}(c x)\right)+144 \cosh \left(4 \cosh ^{-1}(c x)\right)+64 \cosh \left(6 \cosh ^{-1}(c x)\right)+9 \cosh \left(8 \cosh ^{-1}(c x)\right)-24 \cosh ^{-1}(c x) \left(-48 \sinh \left(2 \cosh ^{-1}(c x)\right)+24 \sinh \left(4 \cosh ^{-1}(c x)\right)+16 \sinh \left(6 \cosh ^{-1}(c x)\right)+3 \sinh \left(8 \cosh ^{-1}(c x)\right)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)}{73728 c^5}","\frac{1}{8} x^5 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{16} d x^5 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{d x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{64 c^2}-\frac{3 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{256 b c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{128 c^4}-\frac{b c d x^6 \sqrt{d-c^2 d x^2}}{32 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^4 \sqrt{d-c^2 d x^2}}{256 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b d x^2 \sqrt{d-c^2 d x^2}}{256 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d*(-576*a*c*x*Sqrt[d - c^2*d*x^2]*(3 + 2*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6) - 1728*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (32*b*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*Sqrt[d - c^2*d*x^2]*(1440*ArcCosh[c*x]^2 - 576*Cosh[2*ArcCosh[c*x]] + 144*Cosh[4*ArcCosh[c*x]] + 64*Cosh[6*ArcCosh[c*x]] + 9*Cosh[8*ArcCosh[c*x]] - 24*ArcCosh[c*x]*(-48*Sinh[2*ArcCosh[c*x]] + 24*Sinh[4*ArcCosh[c*x]] + 16*Sinh[6*ArcCosh[c*x]] + 3*Sinh[8*ArcCosh[c*x]])))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/(73728*c^5)","A",0
72,1,270,281,1.9854017,"\int x^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]","\frac{d \left(-144 a \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-48 a c x \left(8 c^4 x^4-14 c^2 x^2+3\right) \sqrt{d-c^2 d x^2}-\frac{18 b \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b \sqrt{d-c^2 d x^2} \left(72 \cosh ^{-1}(c x)^2-18 \cosh \left(2 \cosh ^{-1}(c x)\right)+9 \cosh \left(4 \cosh ^{-1}(c x)\right)+2 \cosh \left(6 \cosh ^{-1}(c x)\right)-12 \cosh ^{-1}(c x) \left(-3 \sinh \left(2 \cosh ^{-1}(c x)\right)+3 \sinh \left(4 \cosh ^{-1}(c x)\right)+\sinh \left(6 \cosh ^{-1}(c x)\right)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)}{2304 c^3}","-\frac{d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{16 c^2}+\frac{1}{6} x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{32 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^2 \sqrt{d-c^2 d x^2}}{32 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2}}{36 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d*(-48*a*c*x*Sqrt[d - c^2*d*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) - 144*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - (18*b*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*Sqrt[d - c^2*d*x^2]*(72*ArcCosh[c*x]^2 - 18*Cosh[2*ArcCosh[c*x]] + 9*Cosh[4*ArcCosh[c*x]] + 2*Cosh[6*ArcCosh[c*x]] - 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/(2304*c^3)","A",0
73,1,235,200,1.266054,"\int \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]","-\frac{3 a d^{3/2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)}{8 c}-\frac{1}{8} a d x \left(2 c^2 x^2-5\right) \sqrt{d-c^2 d x^2}-\frac{b d \sqrt{d-c^2 d x^2} \left(2 \cosh ^{-1}(c x)^2+\cosh \left(2 \cosh ^{-1}(c x)\right)-2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)}{8 c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b d \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)}{128 c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{1}{4} x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{3 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{16 b c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c d x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/8*(a*d*x*(-5 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (3*a*d^(3/2)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/(8*c) - (b*d*Sqrt[d - c^2*d*x^2]*(2*ArcCosh[c*x]^2 + Cosh[2*ArcCosh[c*x]] - 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]]))/(8*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(128*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
74,1,223,197,1.080036,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^2,x]","\frac{1}{8} \left(12 a c d^{3/2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-\frac{4 a d \left(c^2 x^2+2\right) \sqrt{d-c^2 d x^2}}{x}+4 b c d \sqrt{d-c^2 d x^2} \left(\frac{2 \log (c x)+\cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{2 \cosh ^{-1}(c x)}{c x}\right)+\frac{b c d \sqrt{d-c^2 d x^2} \left(2 \cosh ^{-1}(c x)^2+\cosh \left(2 \cosh ^{-1}(c x)\right)-2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)","-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{3 c d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{4 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x}+\frac{b c d \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{c x-1} \sqrt{c x+1}}",1,"((-4*a*d*(2 + c^2*x^2)*Sqrt[d - c^2*d*x^2])/x + 12*a*c*d^(3/2)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 4*b*c*d*Sqrt[d - c^2*d*x^2]*((-2*ArcCosh[c*x])/(c*x) + (ArcCosh[c*x]^2 + 2*Log[c*x])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (b*c*d*Sqrt[d - c^2*d*x^2]*(2*ArcCosh[c*x]^2 + Cosh[2*ArcCosh[c*x]] - 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/8","A",0
75,1,259,203,0.8376059,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^4,x]","\frac{d^2 \left(-2 a \sqrt{\frac{c x-1}{c x+1}} \left(4 c^4 x^4-5 c^2 x^2+1\right)+8 b c^3 x^3 (c x-1) \log (c x)+b c x (c x-1)\right)-6 a c^3 d^{3/2} x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+3 b c^3 d^2 x^3 (c x-1) \cosh ^{-1}(c x)^2-2 b d^2 \sqrt{\frac{c x-1}{c x+1}} \left(4 c^4 x^4-5 c^2 x^2+1\right) \cosh ^{-1}(c x)}{6 x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}}","\frac{c^2 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 x^3}-\frac{c^3 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d \sqrt{d-c^2 d x^2}}{6 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b c^3 d \log (x) \sqrt{d-c^2 d x^2}}{3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(-2*b*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 - 5*c^2*x^2 + 4*c^4*x^4)*ArcCosh[c*x] + 3*b*c^3*d^2*x^3*(-1 + c*x)*ArcCosh[c*x]^2 - 6*a*c^3*d^(3/2)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + d^2*(b*c*x*(-1 + c*x) - 2*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 - 5*c^2*x^2 + 4*c^4*x^4) + 8*b*c^3*x^3*(-1 + c*x)*Log[c*x]))/(6*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])","A",0
76,1,94,166,0.080936,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^6,x]","-\frac{d \sqrt{d-c^2 d x^2} \left(4 (c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+b c x \left(-4 c^4 x^4 \log (x)-4 c^2 x^2+1\right)\right)}{20 x^5 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{5 d x^5}-\frac{b c d \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^5 d \log (x) \sqrt{d-c^2 d x^2}}{5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d \sqrt{d-c^2 d x^2}}{5 x^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/20*(d*Sqrt[d - c^2*d*x^2]*(4*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + b*c*x*(1 - 4*c^2*x^2 - 4*c^4*x^4*Log[x])))/(x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
77,1,136,247,0.1479005,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^8,x]","-\frac{d \sqrt{d-c^2 d x^2} \left(12 c^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+30 (c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+b c x \left(-12 c^6 x^6 \log (x)+3 c^4 x^4-12 c^2 x^2+5\right)\right)}{210 x^7 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 d x^7}-\frac{2 c^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{35 d x^5}-\frac{b c d \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^7 d \log (x) \sqrt{d-c^2 d x^2}}{35 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{70 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d \sqrt{d-c^2 d x^2}}{35 x^4 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/210*(d*Sqrt[d - c^2*d*x^2]*(30*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 12*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + b*c*x*(5 - 12*c^2*x^2 + 3*c^4*x^4 - 12*c^6*x^6*Log[x])))/(x^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
78,1,154,328,0.3342874,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^{10}} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^10,x]","-\frac{d \sqrt{d-c^2 d x^2} \left(96 c^2 x^2 (c x-1)^{5/2} \left(2 c^2 x^2+5\right) (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+840 (c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+b c x \left(-192 c^8 x^8 \log (x)+48 c^6 x^6+18 c^4 x^4-200 c^2 x^2+105\right)\right)}{7560 x^9 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{9 d x^9}-\frac{4 c^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{63 d x^7}-\frac{8 c^4 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{315 d x^5}-\frac{b c d \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b c^9 d \log (x) \sqrt{d-c^2 d x^2}}{315 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c^7 d \sqrt{d-c^2 d x^2}}{315 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{420 x^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/7560*(d*Sqrt[d - c^2*d*x^2]*(840*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 96*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(5 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]) + b*c*x*(105 - 200*c^2*x^2 + 18*c^4*x^4 + 48*c^6*x^6 - 192*c^8*x^8*Log[x])))/(x^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
79,1,170,409,0.4532128,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^{12}} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^12,x]","-\frac{d \sqrt{d-c^2 d x^2} \left(12 c^2 x^2 (c x-1)^{5/2} \left(8 c^4 x^4+20 c^2 x^2+35\right) (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+630 (c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+b c x \left(-96 c^{10} x^{10} \log (x)+24 c^8 x^8+9 c^6 x^6+5 c^4 x^4-105 c^2 x^2+63\right)\right)}{6930 x^{11} \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{11 d x^{11}}-\frac{2 c^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{33 d x^9}-\frac{16 c^6 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{1155 d x^5}-\frac{8 c^4 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{231 d x^7}-\frac{b c d \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b c^{11} d \log (x) \sqrt{d-c^2 d x^2}}{1155 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b c^9 d \sqrt{d-c^2 d x^2}}{1155 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^7 d \sqrt{d-c^2 d x^2}}{770 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d \sqrt{d-c^2 d x^2}}{1386 x^6 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d \sqrt{d-c^2 d x^2}}{66 x^8 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/6930*(d*Sqrt[d - c^2*d*x^2]*(630*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 12*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(35 + 20*c^2*x^2 + 8*c^4*x^4)*(a + b*ArcCosh[c*x]) + b*c*x*(63 - 105*c^2*x^2 + 5*c^4*x^4 + 9*c^6*x^6 + 24*c^8*x^8 - 96*c^10*x^10*Log[x])))/(x^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
80,1,182,399,0.2339462,"\int x^7 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^7*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]","-\frac{d \sqrt{d-c^2 d x^2} \left(105 c^5 x^6 (c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{2 (c x-1)^{5/2} (c x+1)^{5/2} \left(35 c^4 x^4+20 c^2 x^2+8\right) \left(a+b \cosh ^{-1}(c x)\right)}{c}-b \left(\frac{105 c^{10} x^{11}}{11}-\frac{140 c^8 x^9}{9}+\frac{5 c^6 x^7}{7}+\frac{6 c^4 x^5}{5}+\frac{8 c^2 x^3}{3}+16 x\right)\right)}{1155 c^7 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\left(d-c^2 d x^2\right)^{11/2} \left(a+b \cosh ^{-1}(c x)\right)}{11 c^8 d^4}-\frac{\left(d-c^2 d x^2\right)^{9/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^8 d^3}+\frac{3 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 c^8 d^2}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{5 c^8 d}-\frac{4 b c d x^9 \sqrt{d-c^2 d x^2}}{297 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^7 \sqrt{d-c^2 d x^2}}{1617 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b d x \sqrt{d-c^2 d x^2}}{1155 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d x^3 \sqrt{d-c^2 d x^2}}{3465 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^{11} \sqrt{d-c^2 d x^2}}{121 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d x^5 \sqrt{d-c^2 d x^2}}{1925 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/1155*(d*Sqrt[d - c^2*d*x^2]*(-(b*(16*x + (8*c^2*x^3)/3 + (6*c^4*x^5)/5 + (5*c^6*x^7)/7 - (140*c^8*x^9)/9 + (105*c^10*x^11)/11)) + 105*c^5*x^6*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + (2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(8 + 20*c^2*x^2 + 35*c^4*x^4)*(a + b*ArcCosh[c*x]))/c))/(c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
81,1,164,321,0.1831769,"\int x^5 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]","-\frac{d \sqrt{d-c^2 d x^2} \left(35 c^3 x^4 (c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{4 (c x-1)^{5/2} (c x+1)^{5/2} \left(5 c^2 x^2+2\right) \left(a+b \cosh ^{-1}(c x)\right)}{c}-b \left(\frac{35 c^8 x^9}{9}-\frac{50 c^6 x^7}{7}+\frac{3 c^4 x^5}{5}+\frac{4 c^2 x^3}{3}+8 x\right)\right)}{315 c^5 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{9/2} \left(a+b \cosh ^{-1}(c x)\right)}{9 c^6 d^3}+\frac{2 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 c^6 d^2}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{5 c^6 d}-\frac{10 b c d x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^5 \sqrt{d-c^2 d x^2}}{525 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d x \sqrt{d-c^2 d x^2}}{315 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b d x^3 \sqrt{d-c^2 d x^2}}{945 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/315*(d*Sqrt[d - c^2*d*x^2]*(-(b*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (50*c^6*x^7)/7 + (35*c^8*x^9)/9)) + 35*c^3*x^4*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + (4*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(2 + 5*c^2*x^2)*(a + b*ArcCosh[c*x]))/c))/(c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
82,1,150,243,0.2379482,"\int x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]","-\frac{d \sqrt{d-c^2 d x^2} \left(5 c^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+2 (c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{5}{7} b c x \left(c^2 x^2-1\right)^3-\frac{19}{7} b c \left(\frac{c^4 x^5}{5}-\frac{2 c^2 x^3}{3}+x\right)\right)}{35 c^4 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 c^4 d^2}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{5 c^4 d}-\frac{8 b c d x^5 \sqrt{d-c^2 d x^2}}{175 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d x^3 \sqrt{d-c^2 d x^2}}{105 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/35*(d*Sqrt[d - c^2*d*x^2]*((-5*b*c*x*(-1 + c^2*x^2)^3)/7 - (19*b*c*(x - (2*c^2*x^3)/3 + (c^4*x^5)/5))/7 + 2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 5*c^2*x^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x])))/(c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
83,1,107,165,0.2169098,"\int x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]","-\frac{d \sqrt{d-c^2 d x^2} \left(15 a \left(c^2 x^2-1\right)^3+15 b \left(c^2 x^2-1\right)^3 \cosh ^{-1}(c x)+b c x \sqrt{c x-1} \sqrt{c x+1} \left(-3 c^4 x^4+10 c^2 x^2-15\right)\right)}{75 c^2 \left(c^2 x^2-1\right)}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{5 c^2 d}+\frac{b d x \sqrt{d-c^2 d x^2}}{5 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c d x^3 \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/75*(d*Sqrt[d - c^2*d*x^2]*(15*a*(-1 + c^2*x^2)^3 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-15 + 10*c^2*x^2 - 3*c^4*x^4) + 15*b*(-1 + c^2*x^2)^3*ArcCosh[c*x]))/(c^2*(-1 + c^2*x^2))","A",1
84,1,336,292,1.2497062,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x,x]","-a d^{3/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-\frac{1}{3} a d \left(c^2 x^2-4\right) \sqrt{d-c^2 d x^2}+a d^{3/2} \log (x)+\frac{b d \sqrt{d-c^2 d x^2} \left(i \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{b d \sqrt{d-c^2 d x^2} \left(9 c x+12 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left(3 \cosh ^{-1}(c x)\right)\right)}{36 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{1}{3} \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b c d x \sqrt{d-c^2 d x^2}}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/3*(a*d*(-4 + c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (b*d*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]))/(36*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + a*d^(3/2)*Log[x] - a*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
85,1,500,311,1.5781138,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^3,x]","\frac{1}{2} \left(3 a c^2 d^{3/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-3 a c^2 d^{3/2} \log (x)-\frac{a d \left(2 c^2 x^2+1\right) \sqrt{d-c^2 d x^2}}{x^2}+\frac{b d^2 (c x+1) \left(i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+c x \sqrt{\frac{c x-1}{c x+1}}+c x \cosh ^{-1}(c x)-\cosh ^{-1}(c x)\right)}{x^2 \sqrt{d-c^2 d x^2}}-\frac{2 b c^2 d \sqrt{d-c^2 d x^2} \left(i \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)","-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d \sqrt{d-c^2 d x^2}}{2 x \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}",1,"(-((a*d*(1 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2])/x^2) - 3*a*c^2*d^(3/2)*Log[x] + 3*a*c^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] - (2*b*c^2*d*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d^2*(1 + c*x)*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x] + c*x*ArcCosh[c*x] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, I/E^ArcCosh[c*x]]))/(x^2*Sqrt[d - c^2*d*x^2]))/2","A",0
86,1,574,321,1.2824538,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^5,x]","\frac{-15 a c^4 d^2 x^4 \sqrt{\frac{c x-1}{c x+1}}+21 a c^2 d^2 x^2 \sqrt{\frac{c x-1}{c x+1}}+9 a c^4 d^{3/2} x^4 \sqrt{\frac{c x-1}{c x+1}} \log (x) \sqrt{d-c^2 d x^2}-9 a c^4 d^{3/2} x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-6 a d^2 \sqrt{\frac{c x-1}{c x+1}}-9 i b c^5 d^2 x^5 \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)+9 i b c^5 d^2 x^5 \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)-9 i b c^4 d^2 x^4 (c x-1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)+9 i b c^4 d^2 x^4 (c x-1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-15 b c^4 d^2 x^4-15 b c^4 d^2 x^4 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+9 i b c^4 d^2 x^4 \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-9 i b c^4 d^2 x^4 \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+15 b c^3 d^2 x^3+2 b c^2 d^2 x^2+21 b c^2 d^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)-2 b c d^2 x-6 b d^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{24 x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}}","\frac{3 c^2 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 x^2}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{4 x^4}-\frac{3 c^4 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 i b c^4 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d \sqrt{d-c^2 d x^2}}{8 x \sqrt{c x-1} \sqrt{c x+1}}",1,"(-2*b*c*d^2*x + 2*b*c^2*d^2*x^2 + 15*b*c^3*d^2*x^3 - 15*b*c^4*d^2*x^4 - 6*a*d^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 21*a*c^2*d^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 15*a*c^4*d^2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 6*b*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + 21*b*c^2*d^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] - 15*b*c^4*d^2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + (9*I)*b*c^4*d^2*x^4*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (9*I)*b*c^5*d^2*x^5*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (9*I)*b*c^4*d^2*x^4*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + (9*I)*b*c^5*d^2*x^5*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + 9*a*c^4*d^(3/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*Log[x] - 9*a*c^4*d^(3/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] - (9*I)*b*c^4*d^2*x^4*(-1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (9*I)*b*c^4*d^2*x^4*(-1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]])/(24*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])","A",0
87,1,581,454,6.5900795,"\int x^4 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]","-\frac{3 a d^{5/2} \tan ^{-1}\left(\frac{c x \sqrt{-d \left(c^2 x^2-1\right)}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)}{256 c^5}+\sqrt{-d \left(c^2 x^2-1\right)} \left(\frac{1}{10} a c^4 d^2 x^9-\frac{3 a d^2 x}{256 c^4}-\frac{21}{80} a c^2 d^2 x^7-\frac{a d^2 x^3}{128 c^2}+\frac{31}{160} a d^2 x^5\right)+\frac{b d^2 \sqrt{-d (c x-1) (c x+1)} \left(18 \cosh \left(2 \cosh ^{-1}(c x)\right)-9 \cosh \left(4 \cosh ^{-1}(c x)\right)-2 \left(36 \cosh ^{-1}(c x)^2+\cosh \left(6 \cosh ^{-1}(c x)\right)+18 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)-18 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)-6 \cosh ^{-1}(c x) \sinh \left(6 \cosh ^{-1}(c x)\right)\right)\right)}{2304 c^5 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b d^2 \sqrt{-d (c x-1) (c x+1)} \left(1440 \cosh ^{-1}(c x)^2-576 \cosh \left(2 \cosh ^{-1}(c x)\right)+144 \cosh \left(4 \cosh ^{-1}(c x)\right)+64 \cosh \left(6 \cosh ^{-1}(c x)\right)+9 \cosh \left(8 \cosh ^{-1}(c x)\right)+1152 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)-576 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)-384 \cosh ^{-1}(c x) \sinh \left(6 \cosh ^{-1}(c x)\right)-72 \cosh ^{-1}(c x) \sinh \left(8 \cosh ^{-1}(c x)\right)\right)}{36864 c^5 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{b d^2 \sqrt{-d (c x-1) (c x+1)} \left(50400 \cosh ^{-1}(c x)^2-25200 \cosh \left(2 \cosh ^{-1}(c x)\right)+3600 \cosh \left(4 \cosh ^{-1}(c x)\right)+2600 \cosh \left(6 \cosh ^{-1}(c x)\right)+675 \cosh \left(8 \cosh ^{-1}(c x)\right)+72 \cosh \left(10 \cosh ^{-1}(c x)\right)+50400 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)-14400 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)-15600 \cosh ^{-1}(c x) \sinh \left(6 \cosh ^{-1}(c x)\right)-5400 \cosh ^{-1}(c x) \sinh \left(8 \cosh ^{-1}(c x)\right)-720 \cosh ^{-1}(c x) \sinh \left(10 \cosh ^{-1}(c x)\right)\right)}{3686400 c^5 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{1}{32} d^2 x^5 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{128 c^2}+\frac{1}{10} x^5 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{16} d x^5 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{512 b c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{256 c^4}-\frac{31 b c d^2 x^6 \sqrt{d-c^2 d x^2}}{960 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 x^4 \sqrt{d-c^2 d x^2}}{512 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^{10} \sqrt{d-c^2 d x^2}}{100 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b d^2 x^2 \sqrt{d-c^2 d x^2}}{512 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{21 b c^3 d^2 x^8 \sqrt{d-c^2 d x^2}}{640 \sqrt{c x-1} \sqrt{c x+1}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*((-3*a*d^2*x)/(256*c^4) - (a*d^2*x^3)/(128*c^2) + (31*a*d^2*x^5)/160 - (21*a*c^2*d^2*x^7)/80 + (a*c^4*d^2*x^9)/10) - (3*a*d^(5/2)*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(256*c^5) + (b*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*(36*ArcCosh[c*x]^2 + Cosh[6*ArcCosh[c*x]] + 18*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] - 18*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]] - 6*ArcCosh[c*x]*Sinh[6*ArcCosh[c*x]])))/(2304*c^5*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(1440*ArcCosh[c*x]^2 - 576*Cosh[2*ArcCosh[c*x]] + 144*Cosh[4*ArcCosh[c*x]] + 64*Cosh[6*ArcCosh[c*x]] + 9*Cosh[8*ArcCosh[c*x]] + 1152*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] - 576*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]] - 384*ArcCosh[c*x]*Sinh[6*ArcCosh[c*x]] - 72*ArcCosh[c*x]*Sinh[8*ArcCosh[c*x]]))/(36864*c^5*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(50400*ArcCosh[c*x]^2 - 25200*Cosh[2*ArcCosh[c*x]] + 3600*Cosh[4*ArcCosh[c*x]] + 2600*Cosh[6*ArcCosh[c*x]] + 675*Cosh[8*ArcCosh[c*x]] + 72*Cosh[10*ArcCosh[c*x]] + 50400*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] - 14400*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]] - 15600*ArcCosh[c*x]*Sinh[6*ArcCosh[c*x]] - 5400*ArcCosh[c*x]*Sinh[8*ArcCosh[c*x]] - 720*ArcCosh[c*x]*Sinh[10*ArcCosh[c*x]]))/(3686400*c^5*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
88,1,415,371,4.5465746,"\int x^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]","\frac{-2880 a d^{5/2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+192 a c d^2 x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(48 c^6 x^6-136 c^4 x^4+118 c^2 x^2-15\right) \sqrt{d-c^2 d x^2}-576 b d^2 \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)-64 b d^2 \sqrt{d-c^2 d x^2} \left(-72 \cosh ^{-1}(c x)^2+18 \cosh \left(2 \cosh ^{-1}(c x)\right)-9 \cosh \left(4 \cosh ^{-1}(c x)\right)-2 \cosh \left(6 \cosh ^{-1}(c x)\right)+12 \cosh ^{-1}(c x) \left(-3 \sinh \left(2 \cosh ^{-1}(c x)\right)+3 \sinh \left(4 \cosh ^{-1}(c x)\right)+\sinh \left(6 \cosh ^{-1}(c x)\right)\right)\right)+b d^2 \sqrt{d-c^2 d x^2} \left(-1440 \cosh ^{-1}(c x)^2+576 \cosh \left(2 \cosh ^{-1}(c x)\right)-144 \cosh \left(4 \cosh ^{-1}(c x)\right)-64 \cosh \left(6 \cosh ^{-1}(c x)\right)-9 \cosh \left(8 \cosh ^{-1}(c x)\right)+24 \cosh ^{-1}(c x) \left(-48 \sinh \left(2 \cosh ^{-1}(c x)\right)+24 \sinh \left(4 \cosh ^{-1}(c x)\right)+16 \sinh \left(6 \cosh ^{-1}(c x)\right)+3 \sinh \left(8 \cosh ^{-1}(c x)\right)\right)\right)}{73728 c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","-\frac{5 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{8} x^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{5}{48} d x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{256 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b d^2 x^2 \sqrt{d-c^2 d x^2}}{256 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{59 b c d^2 x^4 \sqrt{d-c^2 d x^2}}{768 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^8 \sqrt{d-c^2 d x^2}}{64 \sqrt{c x-1} \sqrt{c x+1}}+\frac{17 b c^3 d^2 x^6 \sqrt{d-c^2 d x^2}}{288 \sqrt{c x-1} \sqrt{c x+1}}",1,"(192*a*c*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(-15 + 118*c^2*x^2 - 136*c^4*x^4 + 48*c^6*x^6) - 2880*a*d^(5/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 576*b*d^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 64*b*d^2*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])) + b*d^2*Sqrt[d - c^2*d*x^2]*(-1440*ArcCosh[c*x]^2 + 576*Cosh[2*ArcCosh[c*x]] - 144*Cosh[4*ArcCosh[c*x]] - 64*Cosh[6*ArcCosh[c*x]] - 9*Cosh[8*ArcCosh[c*x]] + 24*ArcCosh[c*x]*(-48*Sinh[2*ArcCosh[c*x]] + 24*Sinh[4*ArcCosh[c*x]] + 16*Sinh[6*ArcCosh[c*x]] + 3*Sinh[8*ArcCosh[c*x]])))/(73728*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
89,1,347,293,2.4744037,"\int \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]","\frac{-720 a d^{5/2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+48 a c d^2 x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(8 c^4 x^4-26 c^2 x^2+33\right) \sqrt{d-c^2 d x^2}-288 b d^2 \sqrt{d-c^2 d x^2} \left(\cosh \left(2 \cosh ^{-1}(c x)\right)+2 \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-\sinh \left(2 \cosh ^{-1}(c x)\right)\right)\right)+36 b d^2 \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)+b d^2 \sqrt{d-c^2 d x^2} \left(-72 \cosh ^{-1}(c x)^2+18 \cosh \left(2 \cosh ^{-1}(c x)\right)-9 \cosh \left(4 \cosh ^{-1}(c x)\right)-2 \cosh \left(6 \cosh ^{-1}(c x)\right)+12 \cosh ^{-1}(c x) \left(-3 \sinh \left(2 \cosh ^{-1}(c x)\right)+3 \sinh \left(4 \cosh ^{-1}(c x)\right)+\sinh \left(6 \cosh ^{-1}(c x)\right)\right)\right)}{2304 c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{5}{16} d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{32 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{6} x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{5}{24} d x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{25 b c d^2 x^2 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{36 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d^2 x^4 \sqrt{d-c^2 d x^2}}{96 \sqrt{c x-1} \sqrt{c x+1}}",1,"(48*a*c*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4) - 720*a*d^(5/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 288*b*d^2*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + 36*b*d^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) + b*d^2*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(2304*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
90,1,305,284,1.8297313,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^2,x]","\frac{1}{128} d^2 \left(240 a c \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\frac{16 a \left(2 c^4 x^4-9 c^2 x^2-8\right) \sqrt{d-c^2 d x^2}}{x}+64 b c \sqrt{d-c^2 d x^2} \left(\frac{2 \log (c x)+\cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{2 \cosh ^{-1}(c x)}{c x}\right)+\frac{32 b c \sqrt{d-c^2 d x^2} \left(2 \cosh ^{-1}(c x)^2+\cosh \left(2 \cosh ^{-1}(c x)\right)-2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{b c \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)","-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{15 c d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{16 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{4} c^2 d x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x}+\frac{b c d^2 \log (x) \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^4 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}+\frac{9 b c^3 d^2 x^2 \sqrt{d-c^2 d x^2}}{16 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*((16*a*Sqrt[d - c^2*d*x^2]*(-8 - 9*c^2*x^2 + 2*c^4*x^4))/x + 240*a*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 64*b*c*Sqrt[d - c^2*d*x^2]*((-2*ArcCosh[c*x])/(c*x) + (ArcCosh[c*x]^2 + 2*Log[c*x])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (32*b*c*Sqrt[d - c^2*d*x^2]*(2*ArcCosh[c*x]^2 + Cosh[2*ArcCosh[c*x]] - 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b*c*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/128","A",0
91,1,319,293,1.4257536,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^4,x]","\frac{-4 d^3 \left(a \sqrt{\frac{c x-1}{c x+1}} \left(3 c^6 x^6+11 c^4 x^4-16 c^2 x^2+2\right)-14 b c^3 x^3 (c x-1) \log (c x)+b c x (1-c x)\right)-60 a c^3 d^{5/2} x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+30 b c^3 d^3 x^3 (c x-1) \cosh ^{-1}(c x)^2+3 b c^3 d^3 x^3 (c x-1) \cosh \left(2 \cosh ^{-1}(c x)\right)-2 b d^3 (c x-1) \cosh ^{-1}(c x) \left(3 c^3 x^3 \sinh \left(2 \cosh ^{-1}(c x)\right)+4 \sqrt{\frac{c x-1}{c x+1}} \left(7 c^3 x^3+7 c^2 x^2-c x-1\right)\right)}{24 x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}}","\frac{5 c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 x}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 x^3}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{4 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{6 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^2 \sqrt{d-c^2 d x^2}}{4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c^3 d^2 \log (x) \sqrt{d-c^2 d x^2}}{3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(30*b*c^3*d^3*x^3*(-1 + c*x)*ArcCosh[c*x]^2 - 60*a*c^3*d^(5/2)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 3*b*c^3*d^3*x^3*(-1 + c*x)*Cosh[2*ArcCosh[c*x]] - 4*d^3*(b*c*x*(1 - c*x) + a*Sqrt[(-1 + c*x)/(1 + c*x)]*(2 - 16*c^2*x^2 + 11*c^4*x^4 + 3*c^6*x^6) - 14*b*c^3*x^3*(-1 + c*x)*Log[c*x]) - 2*b*d^3*(-1 + c*x)*ArcCosh[c*x]*(4*Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 - c*x + 7*c^2*x^2 + 7*c^3*x^3) + 3*c^3*x^3*Sinh[2*ArcCosh[c*x]]))/(24*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])","A",0
92,1,400,293,3.592757,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^6,x]","\frac{d^2 \left(120 a c^5 \sqrt{d} x^5 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+8 a d \sqrt{\frac{c x-1}{c x+1}} \left(c^2 x^2-1\right) \left(23 c^4 x^4-11 c^2 x^2+3\right)-60 b c^4 d x^4 (1-c x) \left(2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left(2 \log (c x)+\cosh ^{-1}(c x)^2\right)\right)+40 b c^2 d x^2 (1-c x) \left(2 c^3 x^3 \log (c x)+c x-2 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \cosh ^{-1}(c x)\right)-b d (1-c x) \left(c x (10 \log (c x)+3)+20 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)+\log (c x) \cosh \left(5 \cosh ^{-1}(c x)\right)+(5 \log (c x)-1) \cosh \left(3 \cosh ^{-1}(c x)\right)-5 \cosh ^{-1}(c x) \sinh \left(3 \cosh ^{-1}(c x)\right)-\cosh ^{-1}(c x) \sinh \left(5 \cosh ^{-1}(c x)\right)\right)\right)}{120 x^5 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}}","-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{5 x^5}+\frac{c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 x^3}+\frac{c^5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{c^4 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{23 b c^5 d^2 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}}+\frac{11 b c^3 d^2 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*(8*a*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 + c^2*x^2)*(3 - 11*c^2*x^2 + 23*c^4*x^4) + 120*a*c^5*Sqrt[d]*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 40*b*c^2*d*x^2*(1 - c*x)*(c*x - 2*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] + 2*c^3*x^3*Log[c*x]) - 60*b*c^4*d*x^4*(1 - c*x)*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]^2 + 2*Log[c*x])) - b*d*(1 - c*x)*(20*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + Cosh[5*ArcCosh[c*x]]*Log[c*x] + Cosh[3*ArcCosh[c*x]]*(-1 + 5*Log[c*x]) + c*x*(3 + 10*Log[c*x]) - 5*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] - ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]])))/(120*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])","A",0
93,1,105,219,0.1000623,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^8,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(12 (c x-1)^{7/2} (c x+1)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)-b c x \left(12 c^6 x^6 \log (x)+18 c^4 x^4-9 c^2 x^2+2\right)\right)}{84 x^7 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 d x^7}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^7 d^2 \log (x) \sqrt{d-c^2 d x^2}}{7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}{14 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d^2 \sqrt{d-c^2 d x^2}}{28 x^4 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(12*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) - b*c*x*(2 - 9*c^2*x^2 + 18*c^4*x^4 + 12*c^6*x^6*Log[x])))/(84*x^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
94,1,147,314,0.1728876,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^{10}} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^10,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(48 c^2 x^2 (c x-1)^{7/2} (c x+1)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)+168 (c x-1)^{7/2} (c x+1)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)-b c x \left(48 c^8 x^8 \log (x)-12 c^6 x^6+90 c^4 x^4-76 c^2 x^2+21\right)\right)}{1512 x^9 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{9 d x^9}-\frac{2 c^2 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{63 d x^7}-\frac{b c d^2 \left(1-c^2 x^2\right)^4 \sqrt{d-c^2 d x^2}}{72 x^8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c^9 d^2 \log (x) \sqrt{d-c^2 d x^2}}{63 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{21 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^5 d^2 \sqrt{d-c^2 d x^2}}{42 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3 d^2 \sqrt{d-c^2 d x^2}}{189 x^6 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(168*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + 48*c^2*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) - b*c*x*(21 - 76*c^2*x^2 + 90*c^4*x^4 - 12*c^6*x^6 + 48*c^8*x^8*Log[x])))/(1512*x^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
95,1,165,385,0.2138642,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^{12}} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^12,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(480 c^2 x^2 (c x-1)^{7/2} \left(2 c^2 x^2+7\right) (c x+1)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)+7560 (c x-1)^{7/2} (c x+1)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)-b c x \left(960 c^{10} x^{10} \log (x)-240 c^8 x^8-90 c^6 x^6+2260 c^4 x^4-2415 c^2 x^2+756\right)\right)}{83160 x^{11} \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{11 d x^{11}}-\frac{4 c^2 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{99 d x^9}-\frac{8 c^4 \left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{693 d x^7}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{110 x^{10} \sqrt{c x-1} \sqrt{c x+1}}-\frac{8 b c^{11} d^2 \log (x) \sqrt{d-c^2 d x^2}}{693 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^9 d^2 \sqrt{d-c^2 d x^2}}{693 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^7 d^2 \sqrt{d-c^2 d x^2}}{924 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{113 b c^5 d^2 \sqrt{d-c^2 d x^2}}{4158 x^6 \sqrt{c x-1} \sqrt{c x+1}}+\frac{23 b c^3 d^2 \sqrt{d-c^2 d x^2}}{792 x^8 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(7560*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + 480*c^2*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(7 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]) - b*c*x*(756 - 2415*c^2*x^2 + 2260*c^4*x^4 - 90*c^6*x^6 - 240*c^8*x^8 + 960*c^10*x^10*Log[x])))/(83160*x^11*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
96,1,193,458,0.2638714,"\int x^7 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^7*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(231 c^5 x^6 (c x-1)^{7/2} (c x+1)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{2 (c x-1)^{7/2} (c x+1)^{7/2} \left(63 c^4 x^4+28 c^2 x^2+8\right) \left(a+b \cosh ^{-1}(c x)\right)}{c}+b \left(-\frac{231}{13} c^{12} x^{13}+\frac{567 c^{10} x^{11}}{11}-\frac{371 c^8 x^9}{9}+\frac{5 c^6 x^7}{7}+\frac{6 c^4 x^5}{5}+\frac{8 c^2 x^3}{3}+16 x\right)\right)}{3003 c^7 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\left(d-c^2 d x^2\right)^{13/2} \left(a+b \cosh ^{-1}(c x)\right)}{13 c^8 d^4}-\frac{3 \left(d-c^2 d x^2\right)^{11/2} \left(a+b \cosh ^{-1}(c x)\right)}{11 c^8 d^3}+\frac{\left(d-c^2 d x^2\right)^{9/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^8 d^2}-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 c^8 d}-\frac{53 b c d^2 x^9 \sqrt{d-c^2 d x^2}}{3861 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b d^2 x^7 \sqrt{d-c^2 d x^2}}{21021 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b d^2 x \sqrt{d-c^2 d x^2}}{3003 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^{13} \sqrt{d-c^2 d x^2}}{169 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d^2 x^3 \sqrt{d-c^2 d x^2}}{9009 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{27 b c^3 d^2 x^{11} \sqrt{d-c^2 d x^2}}{1573 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d^2 x^5 \sqrt{d-c^2 d x^2}}{5005 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(b*(16*x + (8*c^2*x^3)/3 + (6*c^4*x^5)/5 + (5*c^6*x^7)/7 - (371*c^8*x^9)/9 + (567*c^10*x^11)/11 - (231*c^12*x^13)/13) + 231*c^5*x^6*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + (2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(8 + 28*c^2*x^2 + 63*c^4*x^4)*(a + b*ArcCosh[c*x]))/c))/(3003*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
97,1,175,378,0.1997781,"\int x^5 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(63 c^3 x^4 (c x-1)^{7/2} (c x+1)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{4 (c x-1)^{7/2} (c x+1)^{7/2} \left(7 c^2 x^2+2\right) \left(a+b \cosh ^{-1}(c x)\right)}{c}+b \left(-\frac{63}{11} c^{10} x^{11}+\frac{161 c^8 x^9}{9}-\frac{113 c^6 x^7}{7}+\frac{3 c^4 x^5}{5}+\frac{4 c^2 x^3}{3}+8 x\right)\right)}{693 c^5 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\left(d-c^2 d x^2\right)^{11/2} \left(a+b \cosh ^{-1}(c x)\right)}{11 c^6 d^3}+\frac{2 \left(d-c^2 d x^2\right)^{9/2} \left(a+b \cosh ^{-1}(c x)\right)}{9 c^6 d^2}-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 c^6 d}-\frac{113 b c d^2 x^7 \sqrt{d-c^2 d x^2}}{4851 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 x^5 \sqrt{d-c^2 d x^2}}{1155 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d^2 x \sqrt{d-c^2 d x^2}}{693 c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^{11} \sqrt{d-c^2 d x^2}}{121 \sqrt{c x-1} \sqrt{c x+1}}+\frac{23 b c^3 d^2 x^9 \sqrt{d-c^2 d x^2}}{891 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b d^2 x^3 \sqrt{d-c^2 d x^2}}{2079 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(b*(8*x + (4*c^2*x^3)/3 + (3*c^4*x^5)/5 - (113*c^6*x^7)/7 + (161*c^8*x^9)/9 - (63*c^10*x^11)/11) + 63*c^3*x^4*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + (4*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(2 + 7*c^2*x^2)*(a + b*ArcCosh[c*x]))/c))/(693*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
98,1,160,298,0.1536533,"\int x^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(7 c^2 x^2 (c x-1)^{7/2} (c x+1)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)+2 (c x-1)^{7/2} (c x+1)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{7}{9} b c x \left(c^2 x^2-1\right)^4+\frac{25}{9} b c \left(-\frac{1}{7} c^6 x^7+\frac{3 c^4 x^5}{5}-c^2 x^3+x\right)\right)}{63 c^4 \sqrt{c x-1} \sqrt{c x+1}}","\frac{\left(d-c^2 d x^2\right)^{9/2} \left(a+b \cosh ^{-1}(c x)\right)}{9 c^4 d^2}-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 c^4 d}-\frac{b c d^2 x^5 \sqrt{d-c^2 d x^2}}{21 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 x^3 \sqrt{d-c^2 d x^2}}{189 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^9 \sqrt{d-c^2 d x^2}}{81 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d^2 x \sqrt{d-c^2 d x^2}}{63 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{19 b c^3 d^2 x^7 \sqrt{d-c^2 d x^2}}{441 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*((-7*b*c*x*(-1 + c^2*x^2)^4)/9 + (25*b*c*(x - c^2*x^3 + (3*c^4*x^5)/5 - (c^6*x^7)/7))/9 + 2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) + 7*c^2*x^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x])))/(63*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
99,1,117,218,0.2543854,"\int x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(35 a \left(c^2 x^2-1\right)^4+35 b \left(c^2 x^2-1\right)^4 \cosh ^{-1}(c x)+b c x \sqrt{c x-1} \sqrt{c x+1} \left(-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right)\right)}{245 c^2 \left(c^2 x^2-1\right)}","-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)}{7 c^2 d}+\frac{b d^2 x \sqrt{d-c^2 d x^2}}{7 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 x^3 \sqrt{d-c^2 d x^2}}{7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^7 \sqrt{d-c^2 d x^2}}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2}}{35 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(35*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(35 - 35*c^2*x^2 + 21*c^4*x^4 - 5*c^6*x^6) + 35*b*(-1 + c^2*x^2)^4*ArcCosh[c*x]))/(245*c^2*(-1 + c^2*x^2))","A",1
100,1,471,379,3.8538474,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x,x]","-a d^{5/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+\frac{1}{15} a d^2 \left(3 c^4 x^4-11 c^2 x^2+23\right) \sqrt{d-c^2 d x^2}+a d^{5/2} \log (x)+\frac{b d^2 \sqrt{d-c^2 d x^2} \left(i \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{b d^2 \sqrt{d-c^2 d x^2} \left(9 c x+12 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left(3 \cosh ^{-1}(c x)\right)\right)}{18 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{b d^2 \sqrt{d-c^2 d x^2} \left(25 \cosh \left(3 \cosh ^{-1}(c x)\right)+9 \left(\cosh \left(5 \cosh ^{-1}(c x)\right)-50 c x\right)+15 \cosh ^{-1}(c x) \left(30 \sqrt{\frac{c x-1}{c x+1}} (c x+1)-5 \sinh \left(3 \cosh ^{-1}(c x)\right)-3 \sinh \left(5 \cosh ^{-1}(c x)\right)\right)\right)}{3600 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{2 d^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{5} \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{i b d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{23 b c d^2 x \sqrt{d-c^2 d x^2}}{15 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 \sqrt{c x-1} \sqrt{c x+1}}+\frac{11 b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 \sqrt{c x-1} \sqrt{c x+1}}",1,"(a*d^2*Sqrt[d - c^2*d*x^2]*(23 - 11*c^2*x^2 + 3*c^4*x^4))/15 - (b*d^2*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]))/(18*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + a*d^(5/2)*Log[x] - a*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d^2*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b*d^2*Sqrt[d - c^2*d*x^2]*(25*Cosh[3*ArcCosh[c*x]] + 9*(-50*c*x + Cosh[5*ArcCosh[c*x]]) + 15*ArcCosh[c*x]*(30*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - 5*Sinh[3*ArcCosh[c*x]] - 3*Sinh[5*ArcCosh[c*x]])))/(3600*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
101,1,596,404,4.2877766,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^3,x]","\frac{1}{36} d^2 \left(90 a c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-90 a c^2 \sqrt{d} \log (x)+\frac{6 a \left(2 c^4 x^4-14 c^2 x^2-3\right) \sqrt{d-c^2 d x^2}}{x^2}-\frac{72 b c^2 \sqrt{d-c^2 d x^2} \left(i \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{18 b d (c x+1) \left(i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+c x \sqrt{\frac{c x-1}{c x+1}}+c x \cosh ^{-1}(c x)-\cosh ^{-1}(c x)\right)}{x^2 \sqrt{d-c^2 d x^2}}+\frac{b c^2 \sqrt{d-c^2 d x^2} \left(9 c x+12 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left(3 \cosh ^{-1}(c x)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)","-\frac{5}{2} c^2 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{5 c^2 d^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{6} c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{2 x^2}-\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 i b c^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{2 x \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{7 b c^3 d^2 x \sqrt{d-c^2 d x^2}}{3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*((6*a*Sqrt[d - c^2*d*x^2]*(-3 - 14*c^2*x^2 + 2*c^4*x^4))/x^2 + (b*c^2*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - 90*a*c^2*Sqrt[d]*Log[x] + 90*a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] - (72*b*c^2*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (18*b*d*(1 + c*x)*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x] + c*x*ArcCosh[c*x] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, I/E^ArcCosh[c*x]]))/(x^2*Sqrt[d - c^2*d*x^2])))/36","A",0
102,1,660,407,1.5008919,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{x^5} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^5,x]","\frac{-24 a c^6 d^3 x^6 \sqrt{\frac{c x-1}{c x+1}}-3 a c^4 d^3 x^4 \sqrt{\frac{c x-1}{c x+1}}+33 a c^2 d^3 x^2 \sqrt{\frac{c x-1}{c x+1}}+45 a c^4 d^{5/2} x^4 \sqrt{\frac{c x-1}{c x+1}} \log (x) \sqrt{d-c^2 d x^2}-45 a c^4 d^{5/2} x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-6 a d^3 \sqrt{\frac{c x-1}{c x+1}}+24 b c^6 d^3 x^6-24 b c^6 d^3 x^6 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)-24 b c^5 d^3 x^5-45 i b c^5 d^3 x^5 \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)+45 i b c^5 d^3 x^5 \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)-45 i b c^4 d^3 x^4 (c x-1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)+45 i b c^4 d^3 x^4 (c x-1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-27 b c^4 d^3 x^4-3 b c^4 d^3 x^4 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+45 i b c^4 d^3 x^4 \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-45 i b c^4 d^3 x^4 \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+27 b c^3 d^3 x^3+2 b c^2 d^3 x^2+33 b c^2 d^3 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)-2 b c d^3 x-6 b d^3 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{24 x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}}","\frac{5 c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{8 x^2}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}{4 x^4}+\frac{15}{8} c^4 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)-\frac{15 c^4 d^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{12 x^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{15 i b c^4 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{9 b c^3 d^2 \sqrt{d-c^2 d x^2}}{8 x \sqrt{c x-1} \sqrt{c x+1}}",1,"(-2*b*c*d^3*x + 2*b*c^2*d^3*x^2 + 27*b*c^3*d^3*x^3 - 27*b*c^4*d^3*x^4 - 24*b*c^5*d^3*x^5 + 24*b*c^6*d^3*x^6 - 6*a*d^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 33*a*c^2*d^3*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 3*a*c^4*d^3*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 24*a*c^6*d^3*x^6*Sqrt[(-1 + c*x)/(1 + c*x)] - 6*b*d^3*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + 33*b*c^2*d^3*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] - 3*b*c^4*d^3*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] - 24*b*c^6*d^3*x^6*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + (45*I)*b*c^4*d^3*x^4*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (45*I)*b*c^5*d^3*x^5*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (45*I)*b*c^4*d^3*x^4*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + (45*I)*b*c^5*d^3*x^5*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + 45*a*c^4*d^(5/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*Log[x] - 45*a*c^4*d^(5/2)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] - (45*I)*b*c^4*d^3*x^4*(-1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (45*I)*b*c^4*d^3*x^4*(-1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]])/(24*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])","A",0
103,1,54,66,0.115494,"\int \sqrt{1-x^2} \cosh ^{-1}(x) \, dx","Integrate[Sqrt[1 - x^2]*ArcCosh[x],x]","-\frac{\sqrt{-((x-1) (x+1))} \left(\cosh \left(2 \cosh ^{-1}(x)\right)+2 \cosh ^{-1}(x) \left(\cosh ^{-1}(x)-\sinh \left(2 \cosh ^{-1}(x)\right)\right)\right)}{8 \sqrt{\frac{x-1}{x+1}} (x+1)}","-\frac{\sqrt{1-x} x^2}{4 \sqrt{x-1}}+\frac{1}{2} \sqrt{1-x^2} x \cosh ^{-1}(x)-\frac{\sqrt{1-x} \cosh ^{-1}(x)^2}{4 \sqrt{x-1}}",1,"-1/8*(Sqrt[-((-1 + x)*(1 + x))]*(Cosh[2*ArcCosh[x]] + 2*ArcCosh[x]*(ArcCosh[x] - Sinh[2*ArcCosh[x]])))/(Sqrt[(-1 + x)/(1 + x)]*(1 + x))","A",0
104,1,140,236,0.297691,"\int \frac{x^5 \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^5*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{d-c^2 d x^2} \left(-15 a \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right)+b c x \sqrt{c x-1} \sqrt{c x+1} \left(9 c^4 x^4+20 c^2 x^2+120\right)-15 b \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right) \cosh ^{-1}(c x)\right)}{225 c^6 d (c x-1) (c x+1)}","-\frac{x^4 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{5 c^2 d}-\frac{8 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{15 c^6 d}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{15 c^4 d}-\frac{b x^5 \sqrt{c x-1} \sqrt{c x+1}}{25 c \sqrt{d-c^2 d x^2}}-\frac{8 b x \sqrt{c x-1} \sqrt{c x+1}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{4 b x^3 \sqrt{c x-1} \sqrt{c x+1}}{45 c^3 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 15*a*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6) - 15*b*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6)*ArcCosh[c*x]))/(225*c^6*d*(-1 + c*x)*(1 + c*x))","A",1
105,1,171,212,0.9329681,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{-\frac{16 a c x \left(2 c^2 x^2+3\right) \sqrt{d-c^2 d x^2}}{d}-\frac{48 a \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)}{\sqrt{d}}+\frac{b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(-16 \cosh \left(2 \cosh ^{-1}(c x)\right)-\cosh \left(4 \cosh ^{-1}(c x)\right)+4 \cosh ^{-1}(c x) \left(6 \cosh ^{-1}(c x)+8 \sinh \left(2 \cosh ^{-1}(c x)\right)+\sinh \left(4 \cosh ^{-1}(c x)\right)\right)\right)}{\sqrt{d-c^2 d x^2}}}{128 c^5}","-\frac{x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{4 c^2 d}+\frac{3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{16 b c^5 \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 c^4 d}-\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{16 c \sqrt{d-c^2 d x^2}}-\frac{3 b x^2 \sqrt{c x-1} \sqrt{c x+1}}{16 c^3 \sqrt{d-c^2 d x^2}}",1,"((-16*a*c*x*(3 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2])/d - (48*a*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqrt[d] + (b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-16*Cosh[2*ArcCosh[c*x]] - Cosh[4*ArcCosh[c*x]] + 4*ArcCosh[c*x]*(6*ArcCosh[c*x] + 8*Sinh[2*ArcCosh[c*x]] + Sinh[4*ArcCosh[c*x]])))/Sqrt[d - c^2*d*x^2])/(128*c^5)","A",1
106,1,113,156,0.2390027,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{d-c^2 d x^2} \left(-3 a \left(c^4 x^4+c^2 x^2-2\right)+b c x \sqrt{c x-1} \sqrt{c x+1} \left(c^2 x^2+6\right)-3 b \left(c^4 x^4+c^2 x^2-2\right) \cosh ^{-1}(c x)\right)}{9 c^4 d (c x-1) (c x+1)}","-\frac{x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^2 d}-\frac{2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^4 d}-\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^3 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(6 + c^2*x^2) - 3*a*(-2 + c^2*x^2 + c^4*x^4) - 3*b*(-2 + c^2*x^2 + c^4*x^4)*ArcCosh[c*x]))/(9*c^4*d*(-1 + c*x)*(1 + c*x))","A",1
107,1,141,132,0.6650155,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{-\frac{4 a c x \sqrt{d-c^2 d x^2}}{d}-\frac{4 a \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)}{\sqrt{d}}+\frac{b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(2 \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+\sinh \left(2 \cosh ^{-1}(c x)\right)\right)-\cosh \left(2 \cosh ^{-1}(c x)\right)\right)}{\sqrt{d-c^2 d x^2}}}{8 c^3}","-\frac{x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{2 c^2 d}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{4 c \sqrt{d-c^2 d x^2}}",1,"((-4*a*c*x*Sqrt[d - c^2*d*x^2])/d - (4*a*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqrt[d] + (b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] + Sinh[2*ArcCosh[c*x]])))/Sqrt[d - c^2*d*x^2])/(8*c^3)","A",1
108,1,85,72,0.1634911,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{d-c^2 d x^2} \left(-a c^2 x^2+a+\left(b-b c^2 x^2\right) \cosh ^{-1}(c x)+b c x \sqrt{c x-1} \sqrt{c x+1}\right)}{c^2 d (c x-1) (c x+1)}","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{c^2 d}-\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{c \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(a - a*c^2*x^2 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + (b - b*c^2*x^2)*ArcCosh[c*x]))/(c^2*d*(-1 + c*x)*(1 + c*x))","A",1
109,1,53,53,0.0369344,"\int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b c \sqrt{d-c^2 d x^2}}","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b c \sqrt{d-c^2 d x^2}}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2])","A",1
110,1,153,151,0.2885073,"\int \frac{a+b \cosh ^{-1}(c x)}{x \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x*Sqrt[d - c^2*d*x^2]),x]","-\frac{a \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)}{\sqrt{d}}+\frac{a \log (x)}{\sqrt{d}}-\frac{i b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+\cosh ^{-1}(c x) \left(\log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)\right)}{\sqrt{d-c^2 d x^2}}","\frac{2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}-\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}+\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}",1,"(a*Log[x])/Sqrt[d] - (a*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] - (I*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]]) + PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2]","A",0
111,1,71,71,0.0658407,"\int \frac{a+b \cosh ^{-1}(c x)}{x^2 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^2*Sqrt[d - c^2*d*x^2]),x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{x}-b c \log (x)\right)}{\sqrt{d-c^2 d x^2}}","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{d x}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \log (x)}{\sqrt{d-c^2 d x^2}}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/x - b*c*Log[x]))/Sqrt[d - c^2*d*x^2]","A",1
112,1,309,238,1.0884606,"\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^3*Sqrt[d - c^2*d*x^2]),x]","\frac{1}{2} \left(-\frac{a \sqrt{d-c^2 d x^2}}{d x^2}-\frac{a c^2 \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)}{\sqrt{d}}+\frac{a c^2 \log (x)}{\sqrt{d}}+\frac{b (c x+1) \left(-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)+i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)+i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+c x \sqrt{\frac{c x-1}{c x+1}}+c x \cosh ^{-1}(c x)-\cosh ^{-1}(c x)\right)}{x^2 \sqrt{d-c^2 d x^2}}\right)","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{2 d x^2}+\frac{c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}-\frac{i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{2 \sqrt{d-c^2 d x^2}}+\frac{i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 x \sqrt{d-c^2 d x^2}}",1,"(-((a*Sqrt[d - c^2*d*x^2])/(d*x^2)) + (a*c^2*Log[x])/Sqrt[d] - (a*c^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] + (b*(1 + c*x)*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x] + c*x*ArcCosh[c*x] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, (-I)/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, I/E^ArcCosh[c*x]]))/(x^2*Sqrt[d - c^2*d*x^2]))/2","A",0
113,1,174,155,0.3342233,"\int \frac{a+b \cosh ^{-1}(c x)}{x^4 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^4*Sqrt[d - c^2*d*x^2]),x]","-\frac{\sqrt{d-c^2 d x^2} \left(4 a c^2 x^2 \sqrt{c x-1} \sqrt{c x+1}+2 a \sqrt{c x-1} \sqrt{c x+1}+6 b c^3 x^3-4 b c^3 x^3 \log (c x-1)-4 b c^3 x^3 \log \left(\frac{1}{c x-1}+1\right)+2 b \sqrt{c x-1} \sqrt{c x+1} \left(2 c^2 x^2+1\right) \cosh ^{-1}(c x)+b c x\right)}{6 d x^3 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{2 c^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 d x}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 d x^3}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 x^2 \sqrt{d-c^2 d x^2}}-\frac{2 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log (x)}{3 \sqrt{d-c^2 d x^2}}",1,"-1/6*(Sqrt[d - c^2*d*x^2]*(b*c*x + 6*b*c^3*x^3 + 2*a*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 4*a*c^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 + 2*c^2*x^2)*ArcCosh[c*x] - 4*b*c^3*x^3*Log[-1 + c*x] - 4*b*c^3*x^3*Log[1 + (-1 + c*x)^(-1)]))/(d*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
114,1,145,233,0.1081078,"\int \frac{x^5 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{-3 a c^4 x^4-12 a c^2 x^2+24 a+b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}-3 b \left(c^4 x^4+4 c^2 x^2-8\right) \cosh ^{-1}(c x)+15 b c x \sqrt{c x-1} \sqrt{c x+1}+9 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{9 c^6 d \sqrt{d-c^2 d x^2}}","-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^6 d^3}+\frac{2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{c^6 d^2}+\frac{a+b \cosh ^{-1}(c x)}{c^6 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{c^6 d^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b x \sqrt{d-c^2 d x^2}}{3 c^5 d^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x^3 \sqrt{d-c^2 d x^2}}{9 c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(24*a - 12*a*c^2*x^2 - 3*a*c^4*x^4 + 15*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*b*(-8 + 4*c^2*x^2 + c^4*x^4)*ArcCosh[c*x] + 9*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(9*c^6*d*Sqrt[d - c^2*d*x^2])","A",1
115,1,192,226,1.47771,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{-4 a c d x \left(c^2 x^2-3\right)+12 a \sqrt{d} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+b d \left(8 c x \cosh ^{-1}(c x)-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(8 \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)+6 \cosh ^{-1}(c x)^2-\cosh \left(2 \cosh ^{-1}(c x)\right)+2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)\right)}{8 c^5 d^2 \sqrt{d-c^2 d x^2}}","\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{4 b c^5 d \sqrt{d-c^2 d x^2}}+\frac{3 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{2 c^4 d^2}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-c^2 x^2\right)}{2 c^5 d \sqrt{d-c^2 d x^2}}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{4 c^3 d \sqrt{d-c^2 d x^2}}",1,"(-4*a*c*d*x*(-3 + c^2*x^2) + 12*a*Sqrt[d]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b*d*(8*c*x*ArcCosh[c*x] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(6*ArcCosh[c*x]^2 - Cosh[2*ArcCosh[c*x]] + 8*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)] + 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]])))/(8*c^5*d^2*Sqrt[d - c^2*d*x^2])","A",0
116,1,97,150,0.0742996,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{-a c^2 x^2+2 a+b \left(2-c^2 x^2\right) \cosh ^{-1}(c x)+b c x \sqrt{c x-1} \sqrt{c x+1}+b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{c^4 d \sqrt{d-c^2 d x^2}}","\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{c^4 d^2}+\frac{a+b \cosh ^{-1}(c x)}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{c^4 d^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x \sqrt{d-c^2 d x^2}}{c^3 d^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(2*a - a*c^2*x^2 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*(2 - c^2*x^2)*ArcCosh[c*x] + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(c^4*d*Sqrt[d - c^2*d*x^2])","A",1
117,1,159,143,0.6763085,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{2 a \sqrt{d} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+2 a c d x+b d \left(2 c x \cosh ^{-1}(c x)-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(2 \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)+\cosh ^{-1}(c x)^2\right)\right)}{2 c^3 d^2 \sqrt{d-c^2 d x^2}}","\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b c^3 d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-c^2 x^2\right)}{2 c^3 d \sqrt{d-c^2 d x^2}}",1,"(2*a*c*d*x + 2*a*Sqrt[d]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b*d*(2*c*x*ArcCosh[c*x] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(ArcCosh[c*x]^2 + 2*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])))/(2*c^3*d^2*Sqrt[d - c^2*d*x^2])","A",0
118,1,90,76,0.2378125,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{c^2 d^2 \left(c^2 x^2-1\right)}-\frac{b \sqrt{-d \left(c^2 x^2-1\right)} \tanh ^{-1}(c x)}{c^2 d^2 \sqrt{c x-1} \sqrt{c x+1}}","\frac{a+b \cosh ^{-1}(c x)}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{c^2 d \sqrt{d-c^2 d x^2}}",1,"-((Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(c^2*d^2*(-1 + c^2*x^2))) - (b*Sqrt[-(d*(-1 + c^2*x^2))]*ArcTanh[c*x])/(c^2*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
119,1,72,84,0.0315082,"\int \frac{a+b \cosh ^{-1}(c x)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(3/2),x]","\frac{2 a c x-b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-c^2 x^2\right)+2 b c x \cosh ^{-1}(c x)}{2 c d \sqrt{d-c^2 d x^2}}","\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-c^2 x^2\right)}{2 c d \sqrt{d-c^2 d x^2}}",1,"(2*a*c*x + 2*b*c*x*ArcCosh[c*x] - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2*c*d*Sqrt[d - c^2*d*x^2])","A",1
120,1,301,229,2.5730264,"\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^(3/2)),x]","-\frac{\frac{a \sqrt{d-c^2 d x^2}}{c^2 x^2-1}+a \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+a \left(-\sqrt{d}\right) \log (x)+\frac{i b d \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+i \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)-i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)}{\sqrt{d-c^2 d x^2}}}{d^2}","\frac{a+b \cosh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}+\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}",1,"-(((a*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2) - a*Sqrt[d]*Log[x] + a*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (I*b*d*(I*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - I*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2])/d^2)","A",0
121,1,114,158,0.0893552,"\int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^(3/2)),x]","\frac{4 a c^2 x^2-2 a-b c x \sqrt{c x-1} \sqrt{c x+1} \log \left(1-c^2 x^2\right)+2 b \left(2 c^2 x^2-1\right) \cosh ^{-1}(c x)-2 b c x \sqrt{c x-1} \sqrt{c x+1} \log (x)}{2 d x \sqrt{d-c^2 d x^2}}","\frac{2 c^2 x \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{d x \sqrt{d-c^2 d x^2}}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \sqrt{d-c^2 d x^2} \log \left(1-c^2 x^2\right)}{2 d^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(-2*a + 4*a*c^2*x^2 + 2*b*(-1 + 2*c^2*x^2)*ArcCosh[c*x] - 2*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x] - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(2*d*x*Sqrt[d - c^2*d*x^2])","A",1
122,1,405,329,4.450497,"\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^(3/2)),x]","\frac{1}{2} \left(-\frac{3 a c^2 \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)}{d^{3/2}}+\frac{3 a c^2 \log (x)}{d^{3/2}}-\frac{a \left(3 c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}{d^2 x^2 \left(c^2 x^2-1\right)}-\frac{b c^2 \left(\left(\frac{1}{c^2 x^2}-1\right) \cosh ^{-1}(c x)+3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}-2 \cosh ^{-1}(c x) \cosh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)+3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+2 \cosh ^{-1}(c x) \sinh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)+2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)}{d \sqrt{d-c^2 d x^2}}\right)","\frac{3 c^2 \left(a+b \cosh ^{-1}(c x)\right)}{2 d \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{2 d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{2 d \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d x \sqrt{d-c^2 d x^2}}+\frac{b c^2 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{d \sqrt{d-c^2 d x^2}}",1,"(-((a*(-1 + 3*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(d^2*x^2*(-1 + c^2*x^2))) + (3*a*c^2*Log[x])/d^(3/2) - (3*a*c^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/d^(3/2) - (b*c^2*(-((Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x)) + (-1 + 1/(c^2*x^2))*ArcCosh[c*x] - 2*ArcCosh[c*x]*Cosh[ArcCosh[c*x]/2]^2 + (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^2))/(d*Sqrt[d - c^2*d*x^2]))/2","A",0
123,1,161,250,0.1246149,"\int \frac{a+b \cosh ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(3/2)),x]","\frac{16 a c^4 x^4-8 a c^2 x^2-2 a-10 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1} \log (x)+2 b \left(8 c^4 x^4-4 c^2 x^2-1\right) \cosh ^{-1}(c x)-3 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1} \log \left(1-c^2 x^2\right)+b c x \sqrt{c x-1} \sqrt{c x+1}}{6 d x^3 \sqrt{d-c^2 d x^2}}","-\frac{4 c^2 \left(a+b \cosh ^{-1}(c x)\right)}{3 d x \sqrt{d-c^2 d x^2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left(a+b \cosh ^{-1}(c x)\right)}{3 d \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{d-c^2 d x^2}}{6 d^2 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 \sqrt{d-c^2 d x^2} \log \left(1-c^2 x^2\right)}{2 d^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(-2*a - 8*a*c^2*x^2 + 16*a*c^4*x^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*(-1 - 4*c^2*x^2 + 8*c^4*x^4)*ArcCosh[c*x] - 10*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[x] - 3*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Log[1 - c^2*x^2])/(6*d*x^3*Sqrt[d - c^2*d*x^2])","A",1
124,1,167,243,0.1688702,"\int \frac{x^5 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{6 a c^4 x^4-24 a c^2 x^2+16 a-6 b c^3 x^3 \sqrt{c x-1} \sqrt{c x+1}-11 b \sqrt{c x-1} \sqrt{c x+1} \left(c^2 x^2-1\right) \tanh ^{-1}(c x)+2 b \left(3 c^4 x^4-12 c^2 x^2+8\right) \cosh ^{-1}(c x)+5 b c x \sqrt{c x-1} \sqrt{c x+1}}{6 c^6 d^2 \left(c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{c^6 d^3}-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 c^6 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{11 b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^6 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \sqrt{d-c^2 d x^2}}{c^5 d^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x \sqrt{d-c^2 d x^2}}{6 c^5 d^3 \sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)}",1,"(16*a - 24*a*c^2*x^2 + 6*a*c^4*x^4 + 5*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 6*b*c^3*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*(8 - 12*c^2*x^2 + 3*c^4*x^4)*ArcCosh[c*x] - 11*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2)*ArcTanh[c*x])/(6*c^6*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",1
125,1,225,224,0.7598835,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{\frac{2 a c x \left(4 c^2 x^2-3\right) \sqrt{d-c^2 d x^2}}{\left(c^2 x^2-1\right)^2}-6 a \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\frac{b d \left(-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)+2 c x \cosh ^{-1}(c x)}{c^2 x^2-1}-8 c x \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(8 \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)+3 \cosh ^{-1}(c x)^2\right)\right)}{\sqrt{d-c^2 d x^2}}}{6 c^5 d^3}","\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-c^2 x^2\right)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{6 c^5 d \left(d-c^2 d x^2\right)^{3/2}}",1,"((2*a*c*x*(-3 + 4*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2)^2 - 6*a*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (b*d*(-8*c*x*ArcCosh[c*x] - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + 2*c*x*ArcCosh[c*x])/(-1 + c^2*x^2) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(3*ArcCosh[c*x]^2 + 8*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])))/Sqrt[d - c^2*d*x^2])/(6*c^5*d^3)","A",0
126,1,122,158,0.1308058,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{-6 a c^2 x^2+4 a+b \left(4-6 c^2 x^2\right) \cosh ^{-1}(c x)-5 b \sqrt{c x-1} \sqrt{c x+1} \left(c^2 x^2-1\right) \tanh ^{-1}(c x)-b c x \sqrt{c x-1} \sqrt{c x+1}}{6 c^4 d^2 \left(c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}","-\frac{a+b \cosh ^{-1}(c x)}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 c^4 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{5 b \sqrt{d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^4 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \sqrt{d-c^2 d x^2}}{6 c^3 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}",1,"(4*a - 6*a*c^2*x^2 - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*(4 - 6*c^2*x^2)*ArcCosh[c*x] - 5*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2)*ArcTanh[c*x])/(6*c^4*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",1
127,1,101,133,0.2087291,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(\frac{b \left(\frac{1}{1-c^2 x^2}+\log \left(1-c^2 x^2\right)\right)}{c^3}-\frac{2 x^3 \left(a+b \cosh ^{-1}(c x)\right)}{(c x-1)^{3/2} (c x+1)^{3/2}}\right)}{6 d^2 \sqrt{d-c^2 d x^2}}","\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-c^2 x^2\right)}{6 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{6 c^3 d \left(d-c^2 d x^2\right)^{3/2}}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((-2*x^3*(a + b*ArcCosh[c*x]))/((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (b*((1 - c^2*x^2)^(-1) + Log[1 - c^2*x^2]))/c^3))/(6*d^2*Sqrt[d - c^2*d*x^2])","A",1
128,1,119,127,0.2589886,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{\sqrt{d-c^2 d x^2} \left(2 a+b c x \sqrt{c x-1} \sqrt{c x+1}+2 b \cosh ^{-1}(c x)\right)}{6 c^2 d^3 \left(c^2 x^2-1\right)^2}-\frac{b \sqrt{-d \left(c^2 x^2-1\right)} \tanh ^{-1}(c x)}{6 c^2 d^3 \sqrt{c x-1} \sqrt{c x+1}}","\frac{a+b \cosh ^{-1}(c x)}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{6 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{6 c d \left(d-c^2 d x^2\right)^{3/2}}",1,"(Sqrt[d - c^2*d*x^2]*(2*a + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*ArcCosh[c*x]))/(6*c^2*d^3*(-1 + c^2*x^2)^2) - (b*Sqrt[-(d*(-1 + c^2*x^2))]*ArcTanh[c*x])/(6*c^2*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
129,1,132,162,0.0879299,"\int \frac{a+b \cosh ^{-1}(c x)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d - c^2*d*x^2)^(5/2),x]","\frac{4 a c^3 x^3-6 a c x-2 b \sqrt{c x-1} \sqrt{c x+1} \left(c^2 x^2-1\right) \log \left(1-c^2 x^2\right)+2 b c x \left(2 c^2 x^2-3\right) \cosh ^{-1}(c x)-b \sqrt{c x-1} \sqrt{c x+1}}{6 c d^2 \left(c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}","\frac{2 x \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-c^2 x^2\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1}}{6 c d \left(d-c^2 d x^2\right)^{3/2}}",1,"(-6*a*c*x + 4*a*c^3*x^3 - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*b*c*x*(-3 + 2*c^2*x^2)*ArcCosh[c*x] - 2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1 + c^2*x^2)*Log[1 - c^2*x^2])/(6*c*d^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^2])","A",1
130,1,377,317,7.0848876,"\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x*(d - c^2*d*x^2)^(5/2)),x]","-\frac{a \log \left(\sqrt{d} \sqrt{-d \left(c^2 x^2-1\right)}+d\right)}{d^{5/2}}+\sqrt{-d \left(c^2 x^2-1\right)} \left(\frac{a}{3 d^3 \left(c^2 x^2-1\right)^2}-\frac{a}{d^3 \left(c^2 x^2-1\right)}\right)+\frac{a \log (x)}{d^{5/2}}+\frac{b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(-24 i \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)+24 i \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-24 i \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)+24 i \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)-\frac{8 \cosh ^{-1}(c x) \sinh ^4\left(\frac{1}{2} \cosh ^{-1}(c x)\right)}{\left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3}-14 \cosh ^{-1}(c x) \tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)+14 \cosh ^{-1}(c x) \coth \left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\frac{1}{2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \text{csch}^4\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\text{csch}^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\text{sech}^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-28 \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)}{24 d^2 \sqrt{-d (c x-1) (c x+1)}}","\frac{a+b \cosh ^{-1}(c x)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{a+b \cosh ^{-1}(c x)}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{b c x \sqrt{c x-1} \sqrt{c x+1}}{6 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{7 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*(a/(3*d^3*(-1 + c^2*x^2)^2) - a/(d^3*(-1 + c^2*x^2))) + (a*Log[x])/d^(5/2) - (a*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/d^(5/2) + (b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(14*ArcCosh[c*x]*Coth[ArcCosh[c*x]/2] - Csch[ArcCosh[c*x]/2]^2 - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Csch[ArcCosh[c*x]/2]^4)/2 - (24*I)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (24*I)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - 28*Log[Tanh[ArcCosh[c*x]/2]] - (24*I)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (24*I)*PolyLog[2, I/E^ArcCosh[c*x]] - Sech[ArcCosh[c*x]/2]^2 - (8*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^4)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3) - 14*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]))/(24*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])","A",0
131,1,147,248,0.3775091,"\int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^2*(d - c^2*d*x^2)^(5/2)),x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(\frac{4 c^2 x \left(2 c^2 x^2-3\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{a+b \cosh ^{-1}(c x)}{x (c x-1)^{3/2} (c x+1)^{3/2}}-\frac{1}{6} b c \left(\frac{1}{c^2 x^2-1}+5 \log \left(1-c^2 x^2\right)+6 \log (x)\right)\right)}{d^2 \sqrt{d-c^2 d x^2}}","\frac{8 c^2 x \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left(a+b \cosh ^{-1}(c x)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{d x \left(d-c^2 d x^2\right)^{3/2}}-\frac{b c \sqrt{d-c^2 d x^2}}{6 d^3 \sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)}+\frac{b c \log (x) \sqrt{d-c^2 d x^2}}{d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c \sqrt{d-c^2 d x^2} \log \left(1-c^2 x^2\right)}{6 d^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/(x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (4*c^2*x*(-3 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]))/(3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - (b*c*((-1 + c^2*x^2)^(-1) + 6*Log[x] + 5*Log[1 - c^2*x^2]))/6))/(d^2*Sqrt[d - c^2*d*x^2])","A",1
132,1,500,479,7.2861186,"\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)),x]","-\frac{5 a c^2 \log \left(\sqrt{d} \sqrt{-d \left(c^2 x^2-1\right)}+d\right)}{2 d^{5/2}}+\frac{5 a c^2 \log (x)}{2 d^{5/2}}+\sqrt{-d \left(c^2 x^2-1\right)} \left(-\frac{2 a c^2}{d^3 \left(c^2 x^2-1\right)}+\frac{a c^2}{3 d^3 \left(c^2 x^2-1\right)^2}-\frac{a}{2 d^3 x^2}\right)+\frac{b c^2 \left(\frac{6 (c x-1) (c x+1) \cosh ^{-1}(c x)}{c^2 x^2}-30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)+30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+\frac{6 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+26 \cosh ^{-1}(c x) \cosh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)+30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)-26 \cosh ^{-1}(c x) \sinh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\cosh ^{-1}(c x) \tanh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\cosh ^{-1}(c x) \coth ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\coth \left(\frac{1}{2} \cosh ^{-1}(c x)\right)-26 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)}{12 d^2 \sqrt{-d (c x-1) (c x+1)}}","\frac{5 c^2 \left(a+b \cosh ^{-1}(c x)\right)}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left(a+b \cosh ^{-1}(c x)\right)}{6 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2 \left(d-c^2 d x^2\right)^{3/2}}-\frac{5 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{3 b c \sqrt{c x-1} \sqrt{c x+1}}{4 d^2 x \sqrt{d-c^2 d x^2}}-\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{4 d^2 x \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{13 b c^2 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}(c x)}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b c^3 x \sqrt{c x-1} \sqrt{c x+1}}{12 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a/(d^3*x^2) + (a*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a*c^2)/(d^3*(-1 + c^2*x^2))) + (5*a*c^2*Log[x])/(2*d^(5/2)) - (5*a*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (b*c^2*((6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) + (6*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x])/(c^2*x^2) + 26*ArcCosh[c*x]*Cosh[ArcCosh[c*x]/2]^2 - Coth[ArcCosh[c*x]/2] - ArcCosh[c*x]*Coth[ArcCosh[c*x]/2]^2 - (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - 26*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] - (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] - 26*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^2 - Tanh[ArcCosh[c*x]/2] - ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]^2))/(12*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])","A",0
133,1,169,338,0.4387065,"\int \frac{a+b \cosh ^{-1}(c x)}{x^4 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^(5/2)),x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(\frac{2 c^2 \left(8 c^4 x^4-12 c^2 x^2+3\right) \left(a+b \cosh ^{-1}(c x)\right)}{x (c x-1)^{3/2} (c x+1)^{3/2}}+\frac{a+b \cosh ^{-1}(c x)}{x^3 (c x-1)^{3/2} (c x+1)^{3/2}}-b c \left(\frac{1}{2 x^2 \left(c^2 x^2-1\right)}+4 c^2 \log \left(1-c^2 x^2\right)+8 c^2 \log (x)\right)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}","-\frac{2 c^2 \left(a+b \cosh ^{-1}(c x)\right)}{d x \left(d-c^2 d x^2\right)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3 \left(d-c^2 d x^2\right)^{3/2}}+\frac{16 c^4 x \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left(a+b \cosh ^{-1}(c x)\right)}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{b c \sqrt{d-c^2 d x^2}}{6 d^3 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3 \sqrt{d-c^2 d x^2}}{6 d^3 \sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)}+\frac{8 b c^3 \log (x) \sqrt{d-c^2 d x^2}}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b c^3 \sqrt{d-c^2 d x^2} \log \left(1-c^2 x^2\right)}{3 d^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/(x^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (2*c^2*(3 - 12*c^2*x^2 + 8*c^4*x^4)*(a + b*ArcCosh[c*x]))/(x*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) - b*c*(1/(2*x^2*(-1 + c^2*x^2)) + 8*c^2*Log[x] + 4*c^2*Log[1 - c^2*x^2])))/(3*d^2*Sqrt[d - c^2*d*x^2])","A",1
134,1,116,246,0.0897989,"\int \frac{\cosh ^{-1}(a x)}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcCosh[a*x]/(c - a^2*c*x^2)^(7/2),x]","\frac{\sqrt{a x-1} \sqrt{a x+1} \left(-8 a^2 x^2-16 \left(a^2 x^2-1\right)^2 \log \left(1-a^2 x^2\right)+11\right)+4 a x \left(8 a^4 x^4-20 a^2 x^2+15\right) \cosh ^{-1}(a x)}{60 a c^3 \left(a^2 x^2-1\right)^2 \sqrt{c-a^2 c x^2}}","\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{15 a c^3 \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1}}{20 a c^3 \left(1-a^2 x^2\right)^2 \sqrt{c-a^2 c x^2}}-\frac{4 \sqrt{a x-1} \sqrt{a x+1} \log \left(1-a^2 x^2\right)}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left(c-a^2 c x^2\right)^{3/2}}+\frac{x \cosh ^{-1}(a x)}{5 c \left(c-a^2 c x^2\right)^{5/2}}",1,"(4*a*x*(15 - 20*a^2*x^2 + 8*a^4*x^4)*ArcCosh[a*x] + Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(11 - 8*a^2*x^2 - 16*(-1 + a^2*x^2)^2*Log[1 - a^2*x^2]))/(60*a*c^3*(-1 + a^2*x^2)^2*Sqrt[c - a^2*c*x^2])","A",1
135,1,93,145,0.2594753,"\int \frac{x^4 \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^4*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(-16 \cosh \left(2 \cosh ^{-1}(a x)\right)-\cosh \left(4 \cosh ^{-1}(a x)\right)+4 \cosh ^{-1}(a x) \left(6 \cosh ^{-1}(a x)+8 \sinh \left(2 \cosh ^{-1}(a x)\right)+\sinh \left(4 \cosh ^{-1}(a x)\right)\right)\right)}{128 a^5 \sqrt{-((a x-1) (a x+1))}}","\frac{3 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{16 a^5 \sqrt{1-a x}}-\frac{3 x^2 \sqrt{a x-1}}{16 a^3 \sqrt{1-a x}}-\frac{x^3 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{4 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{8 a^4}-\frac{x^4 \sqrt{a x-1}}{16 a \sqrt{1-a x}}",1,"(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(-16*Cosh[2*ArcCosh[a*x]] - Cosh[4*ArcCosh[a*x]] + 4*ArcCosh[a*x]*(6*ArcCosh[a*x] + 8*Sinh[2*ArcCosh[a*x]] + Sinh[4*ArcCosh[a*x]])))/(128*a^5*Sqrt[-((-1 + a*x)*(1 + a*x))])","A",1
136,1,74,110,0.1280436,"\int \frac{x^3 \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]","-\frac{a x \sqrt{a x-1} \sqrt{a x+1} \left(a^2 x^2+6\right)-3 \left(a^4 x^4+a^2 x^2-2\right) \cosh ^{-1}(a x)}{9 a^4 \sqrt{1-a^2 x^2}}","-\frac{2 x \sqrt{a x-1}}{3 a^3 \sqrt{1-a x}}-\frac{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{3 a^4}-\frac{x^3 \sqrt{a x-1}}{9 a \sqrt{1-a x}}",1,"-1/9*(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(6 + a^2*x^2) - 3*(-2 + a^2*x^2 + a^4*x^4)*ArcCosh[a*x])/(a^4*Sqrt[1 - a^2*x^2])","A",1
137,1,75,88,0.1553327,"\int \frac{x^2 \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]","-\frac{\sqrt{-((a x-1) (a x+1))} \left(2 \cosh ^{-1}(a x) \left(\cosh ^{-1}(a x)+\sinh \left(2 \cosh ^{-1}(a x)\right)\right)-\cosh \left(2 \cosh ^{-1}(a x)\right)\right)}{8 a^3 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^2}{4 a^3 \sqrt{1-a x}}-\frac{x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{2 a^2}-\frac{x^2 \sqrt{a x-1}}{4 a \sqrt{1-a x}}",1,"-1/8*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(-Cosh[2*ArcCosh[a*x]] + 2*ArcCosh[a*x]*(ArcCosh[a*x] + Sinh[2*ArcCosh[a*x]])))/(a^3*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",1
138,1,55,49,0.0860743,"\int \frac{x \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{\left(a^2 x^2-1\right) \cosh ^{-1}(a x)-a x \sqrt{a x-1} \sqrt{a x+1}}{a^2 \sqrt{1-a^2 x^2}}","-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{a^2}-\frac{x \sqrt{a x-1}}{a \sqrt{1-a x}}",1,"(-(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (-1 + a^2*x^2)*ArcCosh[a*x])/(a^2*Sqrt[1 - a^2*x^2])","A",1
139,1,45,32,0.0206685,"\int \frac{\cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 a \sqrt{1-a^2 x^2}}","\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^2}{2 a \sqrt{1-a x}}",1,"(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/(2*a*Sqrt[1 - a^2*x^2])","A",1
140,1,113,103,0.1449014,"\int \frac{\cosh ^{-1}(a x)}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]/(x*Sqrt[1 - a^2*x^2]),x]","\frac{i \sqrt{-((a x-1) (a x+1))} \left(\text{Li}_2\left(-i e^{-\cosh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\cosh ^{-1}(a x)}\right)+\cosh ^{-1}(a x) \left(\log \left(1-i e^{-\cosh ^{-1}(a x)}\right)-\log \left(1+i e^{-\cosh ^{-1}(a x)}\right)\right)\right)}{\sqrt{\frac{a x-1}{a x+1}} (a x+1)}","-\frac{i \sqrt{a x-1} \text{Li}_2\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{i \sqrt{a x-1} \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{2 \sqrt{a x-1} \cosh ^{-1}(a x) \tan ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}",1,"(I*Sqrt[-((-1 + a*x)*(1 + a*x))]*(ArcCosh[a*x]*(Log[1 - I/E^ArcCosh[a*x]] - Log[1 + I/E^ArcCosh[a*x]]) + PolyLog[2, (-I)/E^ArcCosh[a*x]] - PolyLog[2, I/E^ArcCosh[a*x]]))/(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",0
141,1,57,48,0.0302946,"\int \frac{\cosh ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]/(x^2*Sqrt[1 - a^2*x^2]),x]","\frac{\left(a^2 x^2-1\right) \cosh ^{-1}(a x)-a x \sqrt{a x-1} \sqrt{a x+1} \log (x)}{x \sqrt{1-a^2 x^2}}","-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{x}-\frac{a \sqrt{a x-1} \log (x)}{\sqrt{1-a x}}",1,"((-1 + a^2*x^2)*ArcCosh[a*x] - a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Log[x])/(x*Sqrt[1 - a^2*x^2])","A",1
142,1,234,167,0.2881695,"\int \frac{\cosh ^{-1}(a x)}{x^3 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]/(x^3*Sqrt[1 - a^2*x^2]),x]","\frac{(a x+1) \left(-i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \text{Li}_2\left(-i e^{-\cosh ^{-1}(a x)}\right)+i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(a x)}\right)-i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x) \log \left(1-i e^{-\cosh ^{-1}(a x)}\right)+i a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x) \log \left(1+i e^{-\cosh ^{-1}(a x)}\right)+a x \sqrt{\frac{a x-1}{a x+1}}+a x \cosh ^{-1}(a x)-\cosh ^{-1}(a x)\right)}{2 x^2 \sqrt{1-a^2 x^2}}","-\frac{i a^2 \sqrt{a x-1} \text{Li}_2\left(-i e^{\cosh ^{-1}(a x)}\right)}{2 \sqrt{1-a x}}+\frac{i a^2 \sqrt{a x-1} \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)}{2 \sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)}{2 x^2}+\frac{a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \tan ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{a \sqrt{a x-1}}{2 x \sqrt{1-a x}}",1,"((1 + a*x)*(a*x*Sqrt[(-1 + a*x)/(1 + a*x)] - ArcCosh[a*x] + a*x*ArcCosh[a*x] - I*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 - I/E^ArcCosh[a*x]] + I*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] - I*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[2, (-I)/E^ArcCosh[a*x]] + I*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[2, I/E^ArcCosh[a*x]]))/(2*x^2*Sqrt[1 - a^2*x^2])","A",0
143,1,100,98,0.1124317,"\int \frac{(f x)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((f*x)^(3/2)*(a + b*ArcCosh[c*x]))/Sqrt[1 - c^2*x^2],x]","\frac{2}{35} x (f x)^{3/2} \left(\frac{2 b c x \sqrt{c x-1} \sqrt{c x+1} \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right)}{\sqrt{1-c^2 x^2}}+7 \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)\right)","\frac{4 b c \sqrt{c x-1} (f x)^{7/2} \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right)}{35 f^2 \sqrt{1-c x}}+\frac{2 (f x)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{5 f}",1,"(2*x*(f*x)^(3/2)*(7*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2] + (2*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2])/Sqrt[1 - c^2*x^2]))/35","A",1
144,1,115,141,0.0347198,"\int \frac{(f x)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f*x)^(3/2)*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{2 x (f x)^{3/2} \left(2 b c x \sqrt{c x-1} \sqrt{c x+1} \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right)+7 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)\right)}{35 \sqrt{d-c^2 d x^2}}","\frac{4 b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{7/2} \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};c^2 x^2\right)}{35 f^2 \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{1-c^2 x^2} (f x)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{5 f \sqrt{d-c^2 d x^2}}",1,"(2*x*(f*x)^(3/2)*(7*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, 5/4, 9/4, c^2*x^2] + 2*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, c^2*x^2]))/(35*Sqrt[d - c^2*d*x^2])","A",1
145,1,387,429,1.1784458,"\int (f x)^m \left(d-c^2 d x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)^3*(a + b*ArcCosh[c*x]),x]","d^3 x (f x)^m \left(-\frac{c^6 x^6 \left(a+b \cosh ^{-1}(c x)\right)}{m+7}+\frac{3 c^4 x^4 \left(a+b \cosh ^{-1}(c x)\right)}{m+5}-\frac{3 c^2 x^2 \left(a+b \cosh ^{-1}(c x)\right)}{m+3}+\frac{a+b \cosh ^{-1}(c x)}{m+1}-\frac{b c x \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{\left(m^2+3 m+2\right) \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^7 x^7 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m}{2}+4;\frac{m}{2}+5;c^2 x^2\right)}{(m+7) (m+8) \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c^5 x^5 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+6}{2};\frac{m+8}{2};c^2 x^2\right)}{(m+5) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 x^3 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};c^2 x^2\right)}{\left(m^2+7 m+12\right) \sqrt{c x-1} \sqrt{c x+1}}\right)","-\frac{c^6 d^3 (f x)^{m+7} \left(a+b \cosh ^{-1}(c x)\right)}{f^7 (m+7)}+\frac{3 c^4 d^3 (f x)^{m+5} \left(a+b \cosh ^{-1}(c x)\right)}{f^5 (m+5)}-\frac{3 c^2 d^3 (f x)^{m+3} \left(a+b \cosh ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{d^3 (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+1)}-\frac{3 b c d^3 \left(35 m^3+455 m^2+1813 m+2161\right) \sqrt{1-c^2 x^2} (f x)^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^3 \left(m^4+27 m^3+284 m^2+1329 m+2271\right) \left(1-c^2 x^2\right) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 (m+7)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^3 \left(1-c^2 x^2\right) (f x)^{m+6}}{f^6 (m+7)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d^3 (m+9) (2 m+13) \left(1-c^2 x^2\right) (f x)^{m+4}}{f^4 (m+5)^2 (m+7)^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"d^3*x*(f*x)^m*((a + b*ArcCosh[c*x])/(1 + m) - (3*c^2*x^2*(a + b*ArcCosh[c*x]))/(3 + m) + (3*c^4*x^4*(a + b*ArcCosh[c*x]))/(5 + m) - (c^6*x^6*(a + b*ArcCosh[c*x]))/(7 + m) + (b*c^7*x^7*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, 4 + m/2, 5 + m/2, c^2*x^2])/((7 + m)*(8 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((12 + 7*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c^5*x^5*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (6 + m)/2, (8 + m)/2, c^2*x^2])/((5 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))","A",1
146,1,290,307,0.4505419,"\int (f x)^m \left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x]),x]","d^2 x (f x)^m \left(\frac{c^4 x^4 \left(a+b \cosh ^{-1}(c x)\right)}{m+5}-\frac{2 c^2 x^2 \left(a+b \cosh ^{-1}(c x)\right)}{m+3}+\frac{a+b \cosh ^{-1}(c x)}{m+1}-\frac{b c x \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{\left(m^2+3 m+2\right) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 x^5 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+6}{2};\frac{m+8}{2};c^2 x^2\right)}{(m+5) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 x^3 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};c^2 x^2\right)}{\left(m^2+7 m+12\right) \sqrt{c x-1} \sqrt{c x+1}}\right)","\frac{c^4 d^2 (f x)^{m+5} \left(a+b \cosh ^{-1}(c x)\right)}{f^5 (m+5)}-\frac{2 c^2 d^2 (f x)^{m+3} \left(a+b \cosh ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{d^2 (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+1)}-\frac{b c d^2 \left(15 m^2+100 m+149\right) \sqrt{1-c^2 x^2} (f x)^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{f^2 (m+1) (m+2) (m+3)^2 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \left(m^2+13 m+38\right) \left(1-c^2 x^2\right) (f x)^{m+2}}{f^2 (m+3)^2 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d^2 \left(1-c^2 x^2\right) (f x)^{m+4}}{f^4 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"d^2*x*(f*x)^m*((a + b*ArcCosh[c*x])/(1 + m) - (2*c^2*x^2*(a + b*ArcCosh[c*x]))/(3 + m) + (c^4*x^4*(a + b*ArcCosh[c*x]))/(5 + m) - (b*c*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^3*x^3*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((12 + 7*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*x^5*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (6 + m)/2, (8 + m)/2, c^2*x^2])/((5 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))","A",1
147,1,191,184,0.2378641,"\int (f x)^m \left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x]),x]","d x (f x)^m \left(-\frac{c^2 x^2 \left(a+b \cosh ^{-1}(c x)\right)}{m+3}+\frac{a+b \cosh ^{-1}(c x)}{m+1}-\frac{b c x \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{\left(m^2+3 m+2\right) \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 x^3 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};c^2 x^2\right)}{\left(m^2+7 m+12\right) \sqrt{c x-1} \sqrt{c x+1}}\right)","-\frac{c^2 d (f x)^{m+3} \left(a+b \cosh ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{d (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+1)}-\frac{b c d (3 m+7) \sqrt{1-c^2 x^2} (f x)^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{f^2 (m+1) (m+2) (m+3)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2}}{f^2 (m+3)^2}",1,"d*x*(f*x)^m*((a + b*ArcCosh[c*x])/(1 + m) - (c^2*x^2*(a + b*ArcCosh[c*x]))/(3 + m) - (b*c*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*x^3*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((12 + 7*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))","A",1
148,0,0,30,4.1013304,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2),x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{d-c^2 d x^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{d-c^2 d x^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2), x]","A",-1
149,0,0,161,11.2097998,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^2,x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^2} \, dx","\frac{(1-m) \text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{d-c^2 d x^2},x\right)}{2 d}+\frac{(f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{2 d^2 f \left(1-c^2 x^2\right)}-\frac{b c \sqrt{1-c^2 x^2} (f x)^{m+2} \, _2F_1\left(\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{2 d^2 f^2 (m+2) \sqrt{c x-1} \sqrt{c x+1}}",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^2, x]","A",-1
150,0,0,294,15.0213306,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3,x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^3} \, dx","\frac{(1-m) (3-m) \text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{d-c^2 d x^2},x\right)}{8 d^2}+\frac{(3-m) (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{8 d^3 f \left(1-c^2 x^2\right)}+\frac{(f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{4 d^3 f \left(1-c^2 x^2\right)^2}-\frac{b c (3-m) \sqrt{1-c^2 x^2} (f x)^{m+2} \, _2F_1\left(\frac{3}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{8 d^3 f^2 (m+2) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{1-c^2 x^2} (f x)^{m+2} \, _2F_1\left(\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{4 d^3 f^2 (m+2) \sqrt{c x-1} \sqrt{c x+1}}",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^3, x]","A",-1
151,1,350,723,1.3102886,"\int (f x)^m \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]","\frac{d^2 x \sqrt{d-c^2 d x^2} (f x)^m \left(\frac{15 \left(-\frac{(m+2) \left(\frac{b c x \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{m+2}+\frac{\sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}\right)}{m+1}+(m+2) \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)-b c x\right)}{(m+2)^2 (m+4)}-\frac{5 (c x-1)^{3/2} (c x+1)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}{m+4}+(c x-1)^{5/2} (c x+1)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)+\frac{5 b c x \left(\frac{c^2 x^2}{m+4}-\frac{1}{m+2}\right)}{m+4}-b c x \left(\frac{c^4 x^4}{m+6}-\frac{2 c^2 x^2}{m+4}+\frac{1}{m+2}\right)\right)}{(m+6) \sqrt{c x-1} \sqrt{c x+1}}","-\frac{15 b c d^2 \sqrt{d-c^2 d x^2} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{f^2 (m+1) (m+2)^2 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{15 d^2 \sqrt{d-c^2 d x^2} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{f (m+4) (m+6) \left(m^2+3 m+2\right) \sqrt{1-c x} \sqrt{c x+1}}+\frac{15 d^2 \sqrt{d-c^2 d x^2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+6) \left(m^2+6 m+8\right)}+\frac{\left(d-c^2 d x^2\right)^{5/2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+6)}+\frac{5 d \left(d-c^2 d x^2\right)^{3/2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+4) (m+6)}-\frac{b c d^2 \sqrt{d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c d^2 \sqrt{d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2) (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{15 b c d^2 \sqrt{d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 d^2 \sqrt{d-c^2 d x^2} (f x)^{m+6}}{f^6 (m+6)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d^2 \sqrt{d-c^2 d x^2} (f x)^{m+4}}{f^4 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 d^2 \sqrt{d-c^2 d x^2} (f x)^{m+4}}{f^4 (m+4)^2 (m+6) \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*x*(f*x)^m*Sqrt[d - c^2*d*x^2]*((5*b*c*x*(-(2 + m)^(-1) + (c^2*x^2)/(4 + m)))/(4 + m) - b*c*x*((2 + m)^(-1) - (2*c^2*x^2)/(4 + m) + (c^4*x^4)/(6 + m)) - (5*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(4 + m) + (-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + (15*(-(b*c*x) + (2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]) - ((2 + m)*((Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/(2 + m)))/(1 + m)))/((2 + m)^2*(4 + m))))/((6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",0
152,1,274,455,0.7882724,"\int (f x)^m \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]","-\frac{d x \sqrt{d-c^2 d x^2} (f x)^m \left(\frac{3 b c x \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{(m+1) (m+2)^2}+\frac{3 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{(m+1) (m+2) \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{m+2}+(c x-1)^{3/2} (c x+1)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)+b c x \left(\frac{1}{m+2}-\frac{c^2 x^2}{m+4}\right)+\frac{3 b c x}{(m+2)^2}\right)}{(m+4) \sqrt{c x-1} \sqrt{c x+1}}","-\frac{3 b c d \sqrt{d-c^2 d x^2} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{f^2 (m+1) (m+2)^2 (m+4) \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 d \sqrt{d-c^2 d x^2} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{f (m+4) \left(m^2+3 m+2\right) \sqrt{1-c x} \sqrt{c x+1}}+\frac{3 d \sqrt{d-c^2 d x^2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f \left(m^2+6 m+8\right)}+\frac{\left(d-c^2 d x^2\right)^{3/2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+4)}-\frac{b c d \sqrt{d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2) (m+4) \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c d \sqrt{d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d \sqrt{d-c^2 d x^2} (f x)^{m+4}}{f^4 (m+4)^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"-((d*x*(f*x)^m*Sqrt[d - c^2*d*x^2]*((3*b*c*x)/(2 + m)^2 + b*c*x*((2 + m)^(-1) - (c^2*x^2)/(4 + m)) - (3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2 + m) + (-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + (3*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((1 + m)*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/((1 + m)*(2 + m)^2)))/((4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))","A",0
153,1,223,278,0.2942678,"\int (f x)^m \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]),x]","\frac{x \sqrt{d-c^2 d x^2} (f x)^m \left(-b c x \sqrt{c x-1} \sqrt{c x+1} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)-(m+2) \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)+(m+1) \left(a (m+2) \left(c^2 x^2-1\right)+b (m+2) \left(c^2 x^2-1\right) \cosh ^{-1}(c x)-b c x \sqrt{c x-1} \sqrt{c x+1}\right)\right)}{(m+1) (m+2)^2 (c x-1) (c x+1)}","-\frac{b c \sqrt{d-c^2 d x^2} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{f^2 (m+1) (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{d-c^2 d x^2} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{f \left(m^2+3 m+2\right) \sqrt{1-c x} \sqrt{c x+1}}+\frac{\sqrt{d-c^2 d x^2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+2)}-\frac{b c \sqrt{d-c^2 d x^2} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(x*(f*x)^m*Sqrt[d - c^2*d*x^2]*((1 + m)*(-(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + a*(2 + m)*(-1 + c^2*x^2) + b*(2 + m)*(-1 + c^2*x^2)*ArcCosh[c*x]) - (2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/((1 + m)*(2 + m)^2*(-1 + c*x)*(1 + c*x))","A",1
154,1,147,176,0.0774788,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]","\frac{x (f x)^m \left(b c x \sqrt{c x-1} \sqrt{c x+1} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)+(m+2) \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)\right)}{(m+1) (m+2) \sqrt{d-c^2 d x^2}}","\frac{b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{f^2 (m+1) (m+2) \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{f (m+1) \sqrt{d-c^2 d x^2}}",1,"(x*(f*x)^m*((2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/((1 + m)*(2 + m)*Sqrt[d - c^2*d*x^2])","A",1
155,1,216,300,0.2504857,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(3/2),x]","\frac{x (f x)^m \left(-b c m x \sqrt{c x-1} \sqrt{c x+1} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)-m (m+2) \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)+(m+1) \left((m+2) \left(a+b \cosh ^{-1}(c x)\right)+b c x \sqrt{c x-1} \sqrt{c x+1} \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)\right)\right)}{d (m+1) (m+2) \sqrt{d-c^2 d x^2}}","-\frac{b c m \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{d f^2 (m+1) (m+2) \sqrt{d-c^2 d x^2}}-\frac{m \sqrt{1-c^2 x^2} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{d f (m+1) \sqrt{d-c^2 d x^2}}+\frac{(f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{d f \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{d f^2 (m+2) \sqrt{d-c^2 d x^2}}",1,"(x*(f*x)^m*(-(m*(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2]) + (1 + m)*((2 + m)*(a + b*ArcCosh[c*x]) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, c^2*x^2]) - b*c*m*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/(d*(1 + m)*(2 + m)*Sqrt[d - c^2*d*x^2])","A",1
156,1,319,450,0.7091579,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d - c^2*d*x^2)^(5/2),x]","\frac{x \sqrt{c x-1} \sqrt{c x+1} (f x)^m \left(\frac{(m-2) \left(b c m x \sqrt{c x-1} \sqrt{c x+1} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)+m (m+2) \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)-(m+1) \left((m+2) \left(a+b \cosh ^{-1}(c x)\right)+b c x \sqrt{c x-1} \sqrt{c x+1} \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)\right)\right)}{(m+1) (m+2) \sqrt{c x-1} \sqrt{c x+1}}-\frac{a+b \cosh ^{-1}(c x)}{(c x-1)^{3/2} (c x+1)^{3/2}}+\frac{b c x \, _2F_1\left(2,\frac{m}{2}+1;\frac{m}{2}+2;c^2 x^2\right)}{m+2}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}","-\frac{b c (2-m) m \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{3 d^2 f^2 (m+1) (m+2) \sqrt{d-c^2 d x^2}}-\frac{(2-m) m \sqrt{1-c^2 x^2} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 f (m+1) \sqrt{d-c^2 d x^2}}+\frac{(2-m) (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 f \sqrt{d-c^2 d x^2}}+\frac{(f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 d f \left(d-c^2 d x^2\right)^{3/2}}+\frac{b c (2-m) \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{3 d^2 f^2 (m+2) \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _2F_1\left(2,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{3 d^2 f^2 (m+2) \sqrt{d-c^2 d x^2}}",1,"(x*(f*x)^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/((-1 + c*x)^(3/2)*(1 + c*x)^(3/2))) + (b*c*x*Hypergeometric2F1[2, 1 + m/2, 2 + m/2, c^2*x^2])/(2 + m) + ((-2 + m)*(m*(2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - (1 + m)*((2 + m)*(a + b*ArcCosh[c*x]) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, c^2*x^2]) + b*c*m*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/((1 + m)*(2 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/(3*d^2*Sqrt[d - c^2*d*x^2])","A",1
157,1,387,817,2.3493572,"\int (f x)^m (\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)*(a + b*ArcCosh[c*x]),x]","\frac{\text{d1}^2 \text{d2}^2 x \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^m \left(\frac{5 \left(\frac{3 \left(-b c x \sqrt{c x-1} \sqrt{c x+1} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)-(m+2) \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)+(m+1) \left(a (m+2) \left(c^2 x^2-1\right)+b (m+2) \left(c^2 x^2-1\right) \cosh ^{-1}(c x)-b c x \sqrt{c x-1} \sqrt{c x+1}\right)\right)}{(m+1) (m+2)^2 (c x-1) (c x+1)}-(c x-1) (c x+1) \left(a+b \cosh ^{-1}(c x)\right)+\frac{b c x \left(\frac{c^2 x^2}{m+4}-\frac{1}{m+2}\right)}{\sqrt{c x-1} \sqrt{c x+1}}\right)}{m+4}+\left(c^2 x^2-1\right)^2 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b c x \left(\frac{c^4 x^4}{m+6}-\frac{2 c^2 x^2}{m+4}+\frac{1}{m+2}\right)}{\sqrt{c x-1} \sqrt{c x+1}}\right)}{m+6}","\frac{(c x \text{d1}+\text{d1})^{5/2} (\text{d2}-c \text{d2} x)^{5/2} \left(a+b \cosh ^{-1}(c x)\right) (f x)^{m+1}}{f (m+6)}+\frac{5 \text{d1} \text{d2} (c x \text{d1}+\text{d1})^{3/2} (\text{d2}-c \text{d2} x)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) (f x)^{m+1}}{f (m+4) (m+6)}+\frac{15 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} \left(a+b \cosh ^{-1}(c x)\right) (f x)^{m+1}}{f (m+6) \left(m^2+6 m+8\right)}+\frac{15 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} \left(a+b \cosh ^{-1}(c x)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) (f x)^{m+1}}{f (m+4) (m+6) \left(m^2+3 m+2\right) \sqrt{1-c x} \sqrt{c x+1}}-\frac{15 b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right) (f x)^{m+2}}{f^2 (m+1) (m+2)^2 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2) (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{15 b c \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+4}}{f^4 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b c^3 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+4}}{f^4 (m+4)^2 (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^5 \text{d1}^2 \text{d2}^2 \sqrt{c x \text{d1}+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+6}}{f^6 (m+6)^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d1^2*d2^2*x*(f*x)^m*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(-((b*c*x*((2 + m)^(-1) - (2*c^2*x^2)/(4 + m) + (c^4*x^4)/(6 + m)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (-1 + c^2*x^2)^2*(a + b*ArcCosh[c*x]) + (5*((b*c*x*(-(2 + m)^(-1) + (c^2*x^2)/(4 + m)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (-1 + c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]) + (3*((1 + m)*(-(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + a*(2 + m)*(-1 + c^2*x^2) + b*(2 + m)*(-1 + c^2*x^2)*ArcCosh[c*x]) - (2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/((1 + m)*(2 + m)^2*(-1 + c*x)*(1 + c*x))))/(4 + m)))/(6 + m)","A",1
158,1,288,503,1.046829,"\int (f x)^m (\text{d1}+c \text{d1} x)^{3/2} (\text{d2}-c \text{d2} x)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)*(a + b*ArcCosh[c*x]),x]","\frac{\text{d1} \text{d2} x \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^m \left(-\frac{3 b c x \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{(m+1) (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{(m+1) (m+2) (c x-1) (c x+1)}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{m+2}-(c x-1) (c x+1) \left(a+b \cosh ^{-1}(c x)\right)+\frac{b c x \left(\frac{c^2 x^2}{m+4}-\frac{1}{m+2}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c x}{(m+2)^2 \sqrt{c x-1} \sqrt{c x+1}}\right)}{m+4}","-\frac{3 b c \text{d1} \text{d2} \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{f^2 (m+1) (m+2)^2 (m+4) \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 \text{d1} \text{d2} \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{f (m+4) \left(m^2+3 m+2\right) \sqrt{1-c x} \sqrt{c x+1}}+\frac{3 \text{d1} \text{d2} \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f \left(m^2+6 m+8\right)}+\frac{(c \text{d1} x+\text{d1})^{3/2} (\text{d2}-c \text{d2} x)^{3/2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+4)}+\frac{b c^3 \text{d1} \text{d2} \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+4}}{f^4 (m+4)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \text{d1} \text{d2} \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2) (m+4) \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c \text{d1} \text{d2} \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2)^2 (m+4) \sqrt{c x-1} \sqrt{c x+1}}",1,"(d1*d2*x*(f*x)^m*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*((-3*b*c*x)/((2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*x*(-(2 + m)^(-1) + (c^2*x^2)/(4 + m)))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*(a + b*ArcCosh[c*x]))/(2 + m) - (-1 + c*x)*(1 + c*x)*(a + b*ArcCosh[c*x]) - (3*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/((1 + m)*(2 + m)*(-1 + c*x)*(1 + c*x)) - (3*b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2])/((1 + m)*(2 + m)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/(4 + m)","A",1
159,1,229,302,0.2212079,"\int (f x)^m \sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*ArcCosh[c*x]),x]","\frac{x \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^m \left(-b c x \sqrt{c x-1} \sqrt{c x+1} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)-(m+2) \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)+(m+1) \left(a (m+2) \left(c^2 x^2-1\right)+b (m+2) \left(c^2 x^2-1\right) \cosh ^{-1}(c x)-b c x \sqrt{c x-1} \sqrt{c x+1}\right)\right)}{(m+1) (m+2)^2 (c x-1) (c x+1)}","-\frac{b c \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{f^2 (m+1) (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{f \left(m^2+3 m+2\right) \sqrt{1-c x} \sqrt{c x+1}}+\frac{\sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+2)}-\frac{b c \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x} (f x)^{m+2}}{f^2 (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(x*(f*x)^m*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*((1 + m)*(-(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + a*(2 + m)*(-1 + c^2*x^2) + b*(2 + m)*(-1 + c^2*x^2)*ArcCosh[c*x]) - (2 + m)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2] - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, c^2*x^2]))/((1 + m)*(2 + m)^2*(-1 + c*x)*(1 + c*x))","A",1
160,1,264,188,6.2658936,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{\text{d1}+c \text{d1} x} \sqrt{\text{d2}-c \text{d2} x}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]),x]","\frac{2^{-m-3} \sqrt{c \text{d1} x+\text{d1}} \left(\frac{c x}{c x+1}\right)^{1-m} (f x)^m \left(b m \left(\frac{c x}{c x+1}\right)^m \sinh \left(2 \cosh ^{-1}(c x)\right) \left(\sqrt{\pi } c (m+1) x \sqrt{\frac{c x-1}{c x+1}} \Gamma (m+1) \, _3\tilde{F}_2\left(1,\frac{m+2}{2},\frac{m+2}{2};\frac{m+3}{2},\frac{m+4}{2};c^2 x^2\right)-2^{m+2} (c x-1) \cosh ^{-1}(c x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+3}{2};c^2 x^2\right)\right)+a 2^{m+3} (m+1) (c x-1) F_1\left(-m;-m,\frac{1}{2};1-m;\frac{1}{c x+1},\frac{2}{c x+1}\right)\right)}{c^2 \text{d1} m (m+1) x \sqrt{\frac{c x-1}{c x+1}} \sqrt{\text{d2}-c \text{d2} x}}","\frac{b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{f^2 (m+1) (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{\sqrt{1-c^2 x^2} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{f (m+1) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}",1,"(2^(-3 - m)*(f*x)^m*((c*x)/(1 + c*x))^(1 - m)*Sqrt[d1 + c*d1*x]*(2^(3 + m)*a*(1 + m)*(-1 + c*x)*AppellF1[-m, -m, 1/2, 1 - m, (1 + c*x)^(-1), 2/(1 + c*x)] + b*m*((c*x)/(1 + c*x))^m*(-(2^(2 + m)*(-1 + c*x)*ArcCosh[c*x]*Hypergeometric2F1[1, (2 + m)/2, (3 + m)/2, c^2*x^2]) + c*(1 + m)*Sqrt[Pi]*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Gamma[1 + m]*HypergeometricPFQRegularized[{1, (2 + m)/2, (2 + m)/2}, {(3 + m)/2, (4 + m)/2}, c^2*x^2])*Sinh[2*ArcCosh[c*x]]))/(c^2*d1*m*(1 + m)*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d2 - c*d2*x])","C",0
161,0,0,336,2.5614428,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{(\text{d1}+c \text{d1} x)^{3/2} (\text{d2}-c \text{d2} x)^{3/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/((d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)),x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{(\text{d1}+c \text{d1} x)^{3/2} (\text{d2}-c \text{d2} x)^{3/2}} \, dx","-\frac{b c m \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{\text{d1} \text{d2} f^2 (m+1) (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}-\frac{m \sqrt{1-c^2 x^2} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{\text{d1} \text{d2} f (m+1) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{(f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{\text{d1} \text{d2} f \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{\text{d1} \text{d2} f^2 (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}",1,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/((d1 + c*d1*x)^(3/2)*(d2 - c*d2*x)^(3/2)), x]","F",-1
162,0,0,504,2.6987622,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{(\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/((d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)),x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{(\text{d1}+c \text{d1} x)^{5/2} (\text{d2}-c \text{d2} x)^{5/2}} \, dx","-\frac{b c (2-m) m \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;c^2 x^2\right)}{3 \text{d1}^2 \text{d2}^2 f^2 (m+1) (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}-\frac{(2-m) m \sqrt{1-c^2 x^2} (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 \text{d1}^2 \text{d2}^2 f (m+1) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{(2-m) (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 \text{d1}^2 \text{d2}^2 f \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{(f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 \text{d1} \text{d2} f (c \text{d1} x+\text{d1})^{3/2} (\text{d2}-c \text{d2} x)^{3/2}}+\frac{b c (2-m) \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{3 \text{d1}^2 \text{d2}^2 f^2 (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2} \, _2F_1\left(2,\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{3 \text{d1}^2 \text{d2}^2 f^2 (m+2) \sqrt{c \text{d1} x+\text{d1}} \sqrt{\text{d2}-c \text{d2} x}}",1,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/((d1 + c*d1*x)^(5/2)*(d2 - c*d2*x)^(5/2)), x]","F",-1
163,1,124,128,0.0826062,"\int \frac{(f x)^m \cosh ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx","Integrate[((f*x)^m*ArcCosh[a*x])/Sqrt[1 - a^2*x^2],x]","\frac{x (f x)^m \left(\frac{a x \sqrt{a x-1} \sqrt{a x+1} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;a^2 x^2\right)}{(m+2) \sqrt{1-a^2 x^2}}+\cosh ^{-1}(a x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right)\right)}{m+1}","\frac{a \sqrt{a x-1} (f x)^{m+2} \, _3F_2\left(1,\frac{m}{2}+1,\frac{m}{2}+1;\frac{m}{2}+\frac{3}{2},\frac{m}{2}+2;a^2 x^2\right)}{f^2 (m+1) (m+2) \sqrt{1-a x}}+\frac{\cosh ^{-1}(a x) (f x)^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};a^2 x^2\right)}{f (m+1)}",1,"(x*(f*x)^m*(ArcCosh[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, a^2*x^2] + (a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, a^2*x^2])/((2 + m)*Sqrt[1 - a^2*x^2])))/(1 + m)","A",1
164,1,125,266,0.2781892,"\int \left(c-a^2 c x^2\right)^3 \cosh ^{-1}(a x)^2 \, dx","Integrate[(c - a^2*c*x^2)^3*ArcCosh[a*x]^2,x]","\frac{c^3 \left(-2250 a^7 x^7+14742 a^5 x^5-52990 a^3 x^3-11025 a x \left(5 a^6 x^6-21 a^4 x^4+35 a^2 x^2-35\right) \cosh ^{-1}(a x)^2+210 \sqrt{a x-1} \sqrt{a x+1} \left(75 a^6 x^6-351 a^4 x^4+757 a^2 x^2-2161\right) \cosh ^{-1}(a x)+453810 a x\right)}{385875 a}","-\frac{2}{343} a^6 c^3 x^7+\frac{234 a^4 c^3 x^5}{6125}-\frac{1514 a^2 c^3 x^3}{11025}+\frac{1}{7} c^3 x \left(1-a^2 x^2\right)^3 \cosh ^{-1}(a x)^2+\frac{6}{35} c^3 x \left(1-a^2 x^2\right)^2 \cosh ^{-1}(a x)^2+\frac{8}{35} c^3 x \left(1-a^2 x^2\right) \cosh ^{-1}(a x)^2+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^2+\frac{2 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)}{49 a}-\frac{12 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{175 a}+\frac{16 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{105 a}-\frac{32 c^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{35 a}+\frac{4322 c^3 x}{3675}",1,"(c^3*(453810*a*x - 52990*a^3*x^3 + 14742*a^5*x^5 - 2250*a^7*x^7 + 210*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-2161 + 757*a^2*x^2 - 351*a^4*x^4 + 75*a^6*x^6)*ArcCosh[a*x] - 11025*a*x*(-35 + 35*a^2*x^2 - 21*a^4*x^4 + 5*a^6*x^6)*ArcCosh[a*x]^2))/(385875*a)","A",1
165,1,101,195,0.2250169,"\int \left(c-a^2 c x^2\right)^2 \cosh ^{-1}(a x)^2 \, dx","Integrate[(c - a^2*c*x^2)^2*ArcCosh[a*x]^2,x]","\frac{c^2 \left(54 a^5 x^5-380 a^3 x^3+225 a x \left(3 a^4 x^4-10 a^2 x^2+15\right) \cosh ^{-1}(a x)^2-30 \sqrt{a x-1} \sqrt{a x+1} \left(9 a^4 x^4-38 a^2 x^2+149\right) \cosh ^{-1}(a x)+4470 a x\right)}{3375 a}","\frac{2}{125} a^4 c^2 x^5-\frac{76}{675} a^2 c^2 x^3+\frac{1}{5} c^2 x \left(1-a^2 x^2\right)^2 \cosh ^{-1}(a x)^2+\frac{4}{15} c^2 x \left(1-a^2 x^2\right) \cosh ^{-1}(a x)^2+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^2-\frac{2 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}{25 a}+\frac{8 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{45 a}-\frac{16 c^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{15 a}+\frac{298 c^2 x}{225}",1,"(c^2*(4470*a*x - 380*a^3*x^3 + 54*a^5*x^5 - 30*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(149 - 38*a^2*x^2 + 9*a^4*x^4)*ArcCosh[a*x] + 225*a*x*(15 - 10*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*x]^2))/(3375*a)","A",1
166,1,73,112,0.117166,"\int \left(c-a^2 c x^2\right) \cosh ^{-1}(a x)^2 \, dx","Integrate[(c - a^2*c*x^2)*ArcCosh[a*x]^2,x]","\frac{c \left(-2 a^3 x^3-9 a x \left(a^2 x^2-3\right) \cosh ^{-1}(a x)^2+6 \sqrt{a x-1} \sqrt{a x+1} \left(a^2 x^2-7\right) \cosh ^{-1}(a x)+42 a x\right)}{27 a}","-\frac{2}{27} a^2 c x^3+\frac{1}{3} c x \left(1-a^2 x^2\right) \cosh ^{-1}(a x)^2+\frac{2}{3} c x \cosh ^{-1}(a x)^2+\frac{2 c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}{9 a}-\frac{4 c \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{3 a}+\frac{14 c x}{9}",1,"(c*(42*a*x - 2*a^3*x^3 + 6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-7 + a^2*x^2)*ArcCosh[a*x] - 9*a*x*(-3 + a^2*x^2)*ArcCosh[a*x]^2))/(27*a)","A",1
167,1,95,98,0.077695,"\int \frac{\cosh ^{-1}(a x)^2}{c-a^2 c x^2} \, dx","Integrate[ArcCosh[a*x]^2/(c - a^2*c*x^2),x]","\frac{2 \cosh ^{-1}(a x) \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)-2 \cosh ^{-1}(a x) \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)-2 \text{Li}_3\left(-e^{\cosh ^{-1}(a x)}\right)+2 \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)+\cosh ^{-1}(a x)^2 \left(-\log \left(1-e^{\cosh ^{-1}(a x)}\right)\right)+\cosh ^{-1}(a x)^2 \log \left(e^{\cosh ^{-1}(a x)}+1\right)}{a c}","\frac{2 \cosh ^{-1}(a x) \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{a c}-\frac{2 \cosh ^{-1}(a x) \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{a c}-\frac{2 \text{Li}_3\left(-e^{\cosh ^{-1}(a x)}\right)}{a c}+\frac{2 \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)}{a c}+\frac{2 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{a c}",1,"(-(ArcCosh[a*x]^2*Log[1 - E^ArcCosh[a*x]]) + ArcCosh[a*x]^2*Log[1 + E^ArcCosh[a*x]] + 2*ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]] - 2*ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]] - 2*PolyLog[3, -E^ArcCosh[a*x]] + 2*PolyLog[3, E^ArcCosh[a*x]])/(a*c)","A",0
168,1,191,163,0.939842,"\int \frac{\cosh ^{-1}(a x)^2}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[ArcCosh[a*x]^2/(c - a^2*c*x^2)^2,x]","\frac{-8 \cosh ^{-1}(a x) \text{Li}_2\left(-e^{-\cosh ^{-1}(a x)}\right)+8 \cosh ^{-1}(a x) \text{Li}_2\left(e^{-\cosh ^{-1}(a x)}\right)-8 \text{Li}_3\left(-e^{-\cosh ^{-1}(a x)}\right)+8 \text{Li}_3\left(e^{-\cosh ^{-1}(a x)}\right)-4 \cosh ^{-1}(a x)^2 \log \left(1-e^{-\cosh ^{-1}(a x)}\right)+4 \cosh ^{-1}(a x)^2 \log \left(e^{-\cosh ^{-1}(a x)}+1\right)+4 \cosh ^{-1}(a x) \tanh \left(\frac{1}{2} \cosh ^{-1}(a x)\right)-4 \cosh ^{-1}(a x) \coth \left(\frac{1}{2} \cosh ^{-1}(a x)\right)+\cosh ^{-1}(a x)^2 \left(-\text{csch}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)\right)-\cosh ^{-1}(a x)^2 \text{sech}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)+8 \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(a x)\right)\right)}{8 a c^2}","\frac{x \cosh ^{-1}(a x)^2}{2 c^2 \left(1-a^2 x^2\right)}+\frac{\cosh ^{-1}(a x) \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{a c^2}-\frac{\cosh ^{-1}(a x) \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{a c^2}-\frac{\text{Li}_3\left(-e^{\cosh ^{-1}(a x)}\right)}{a c^2}+\frac{\text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)}{a c^2}-\frac{\cosh ^{-1}(a x)}{a c^2 \sqrt{a x-1} \sqrt{a x+1}}-\frac{\tanh ^{-1}(a x)}{a c^2}+\frac{\cosh ^{-1}(a x)^2 \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{a c^2}",1,"(-4*ArcCosh[a*x]*Coth[ArcCosh[a*x]/2] - ArcCosh[a*x]^2*Csch[ArcCosh[a*x]/2]^2 - 4*ArcCosh[a*x]^2*Log[1 - E^(-ArcCosh[a*x])] + 4*ArcCosh[a*x]^2*Log[1 + E^(-ArcCosh[a*x])] + 8*Log[Tanh[ArcCosh[a*x]/2]] - 8*ArcCosh[a*x]*PolyLog[2, -E^(-ArcCosh[a*x])] + 8*ArcCosh[a*x]*PolyLog[2, E^(-ArcCosh[a*x])] - 8*PolyLog[3, -E^(-ArcCosh[a*x])] + 8*PolyLog[3, E^(-ArcCosh[a*x])] - ArcCosh[a*x]^2*Sech[ArcCosh[a*x]/2]^2 + 4*ArcCosh[a*x]*Tanh[ArcCosh[a*x]/2])/(8*a*c^2)","A",0
169,1,319,258,4.8101076,"\int \frac{\cosh ^{-1}(a x)^2}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[ArcCosh[a*x]^2/(c - a^2*c*x^2)^3,x]","-\frac{72 \left(2 \cosh ^{-1}(a x) \text{Li}_2\left(-e^{-\cosh ^{-1}(a x)}\right)-2 \cosh ^{-1}(a x) \text{Li}_2\left(e^{-\cosh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-e^{-\cosh ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{-\cosh ^{-1}(a x)}\right)+\cosh ^{-1}(a x)^2 \log \left(1-e^{-\cosh ^{-1}(a x)}\right)-\cosh ^{-1}(a x)^2 \log \left(e^{-\cosh ^{-1}(a x)}+1\right)\right)-\frac{32 \cosh ^{-1}(a x) \sinh ^4\left(\frac{1}{2} \cosh ^{-1}(a x)\right)}{\left(\frac{a x-1}{a x+1}\right)^{3/2} (a x+1)^3}-80 \cosh ^{-1}(a x) \tanh \left(\frac{1}{2} \cosh ^{-1}(a x)\right)+80 \cosh ^{-1}(a x) \coth \left(\frac{1}{2} \cosh ^{-1}(a x)\right)-3 \cosh ^{-1}(a x)^2 \text{csch}^4\left(\frac{1}{2} \cosh ^{-1}(a x)\right)-2 \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x) \text{csch}^4\left(\frac{1}{2} \cosh ^{-1}(a x)\right)+2 \left(9 \cosh ^{-1}(a x)^2-2\right) \text{csch}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)+3 \cosh ^{-1}(a x)^2 \text{sech}^4\left(\frac{1}{2} \cosh ^{-1}(a x)\right)+2 \left(9 \cosh ^{-1}(a x)^2-2\right) \text{sech}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)-160 \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(a x)\right)\right)}{192 a c^3}","-\frac{x}{12 c^3 \left(1-a^2 x^2\right)}+\frac{3 x \cosh ^{-1}(a x)^2}{8 c^3 \left(1-a^2 x^2\right)}+\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left(1-a^2 x^2\right)^2}+\frac{3 \cosh ^{-1}(a x) \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}-\frac{3 \cosh ^{-1}(a x) \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}-\frac{3 \text{Li}_3\left(-e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}+\frac{3 \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}-\frac{3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x)}{6 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}-\frac{5 \tanh ^{-1}(a x)}{6 a c^3}+\frac{3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}",1,"-1/192*(80*ArcCosh[a*x]*Coth[ArcCosh[a*x]/2] + 2*(-2 + 9*ArcCosh[a*x]^2)*Csch[ArcCosh[a*x]/2]^2 - 2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]*Csch[ArcCosh[a*x]/2]^4 - 3*ArcCosh[a*x]^2*Csch[ArcCosh[a*x]/2]^4 - 160*Log[Tanh[ArcCosh[a*x]/2]] + 72*(ArcCosh[a*x]^2*Log[1 - E^(-ArcCosh[a*x])] - ArcCosh[a*x]^2*Log[1 + E^(-ArcCosh[a*x])] + 2*ArcCosh[a*x]*PolyLog[2, -E^(-ArcCosh[a*x])] - 2*ArcCosh[a*x]*PolyLog[2, E^(-ArcCosh[a*x])] + 2*PolyLog[3, -E^(-ArcCosh[a*x])] - 2*PolyLog[3, E^(-ArcCosh[a*x])]) + 2*(-2 + 9*ArcCosh[a*x]^2)*Sech[ArcCosh[a*x]/2]^2 + 3*ArcCosh[a*x]^2*Sech[ArcCosh[a*x]/2]^4 - (32*ArcCosh[a*x]*Sinh[ArcCosh[a*x]/2]^4)/(((-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x)^3) - 80*ArcCosh[a*x]*Tanh[ArcCosh[a*x]/2])/(a*c^3)","A",0
170,1,237,371,0.4389356,"\int x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{d-c^2 d x^2} \left(225 a^2 \left(c^2 x^2-1\right)^2 \left(3 c^2 x^2+2\right)-30 a b c x \sqrt{c x-1} \sqrt{c x+1} \left(9 c^4 x^4-5 c^2 x^2-30\right)+30 b \cosh ^{-1}(c x) \left(15 a \left(3 c^2 x^2+2\right) \left(c^2 x^2-1\right)^2+b c x \sqrt{c x-1} \sqrt{c x+1} \left(-9 c^4 x^4+5 c^2 x^2+30\right)\right)+225 b^2 \left(c^2 x^2-1\right)^2 \left(3 c^2 x^2+2\right) \cosh ^{-1}(c x)^2+2 b^2 \left(27 c^6 x^6-16 c^4 x^4-439 c^2 x^2+428\right)\right)}{3375 c^4 \left(c^2 x^2-1\right)}","-\frac{x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{15 c^2}-\frac{2 b c x^5 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{25 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{5} x^4 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{2 b x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{45 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{15 c^4}+\frac{4 a b x \sqrt{d-c^2 d x^2}}{15 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{22 b^2 x^2 \sqrt{d-c^2 d x^2}}{3375 c^2}+\frac{2}{125} b^2 x^4 \sqrt{d-c^2 d x^2}-\frac{856 b^2 \sqrt{d-c^2 d x^2}}{3375 c^4}+\frac{4 b^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{15 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(Sqrt[d - c^2*d*x^2]*(225*a^2*(-1 + c^2*x^2)^2*(2 + 3*c^2*x^2) - 30*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-30 - 5*c^2*x^2 + 9*c^4*x^4) + 2*b^2*(428 - 439*c^2*x^2 - 16*c^4*x^4 + 27*c^6*x^6) + 30*b*(15*a*(-1 + c^2*x^2)^2*(2 + 3*c^2*x^2) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(30 + 5*c^2*x^2 - 9*c^4*x^4))*ArcCosh[c*x] + 225*b^2*(-1 + c^2*x^2)^2*(2 + 3*c^2*x^2)*ArcCosh[c*x]^2))/(3375*c^4*(-1 + c^2*x^2))","A",1
171,1,241,319,2.1058026,"\int x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2,x]","-\frac{-96 a^2 c x \left(2 c^2 x^2-1\right) \sqrt{d-c^2 d x^2}+96 a^2 \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\frac{12 a b \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b^2 \sqrt{d-c^2 d x^2} \left(32 \cosh ^{-1}(c x)^3+12 \cosh \left(4 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-3 \left(8 \cosh ^{-1}(c x)^2+1\right) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}}{768 c^3}","\frac{b x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{8 c^2}-\frac{b c x^4 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{4} x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{24 b c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b^2 x \sqrt{d-c^2 d x^2}}{64 c^2}+\frac{1}{32} b^2 x^3 \sqrt{d-c^2 d x^2}-\frac{b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/768*(-96*a^2*c*x*(-1 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2] + 96*a^2*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (12*a*b*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b^2*Sqrt[d - c^2*d*x^2]*(32*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*ArcCosh[c*x]^2)*Sinh[4*ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/c^3","A",0
172,1,181,186,0.3671035,"\int x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{d-c^2 d x^2} \left(9 a^2 \left(c^2 x^2-1\right)^2-6 a b c x \sqrt{c x-1} \sqrt{c x+1} \left(c^2 x^2-3\right)+6 b \cosh ^{-1}(c x) \left(3 a \left(c^2 x^2-1\right)^2+b c x \sqrt{c x-1} \sqrt{c x+1} \left(3-c^2 x^2\right)\right)+9 b^2 \left(c^2 x^2-1\right)^2 \cosh ^{-1}(c x)^2+2 b^2 \left(c^4 x^4-8 c^2 x^2+7\right)\right)}{27 c^2 \left(c^2 x^2-1\right)}","\frac{2 b x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^2 d}-\frac{2 b c x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2}{27} b^2 x^2 \sqrt{d-c^2 d x^2}-\frac{14 b^2 \sqrt{d-c^2 d x^2}}{27 c^2}",1,"(Sqrt[d - c^2*d*x^2]*(-6*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-3 + c^2*x^2) + 9*a^2*(-1 + c^2*x^2)^2 + 2*b^2*(7 - 8*c^2*x^2 + c^4*x^4) + 6*b*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 - c^2*x^2) + 3*a*(-1 + c^2*x^2)^2)*ArcCosh[c*x] + 9*b^2*(-1 + c^2*x^2)^2*ArcCosh[c*x]^2))/(27*c^2*(-1 + c^2*x^2))","A",1
173,1,235,204,1.0850406,"\int \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2,x]","\frac{1}{24} \left(12 a^2 x \sqrt{d-c^2 d x^2}-\frac{12 a^2 \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)}{c}-\frac{6 a b \sqrt{d-c^2 d x^2} \left(2 \cosh ^{-1}(c x)^2+\cosh \left(2 \cosh ^{-1}(c x)\right)-2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)}{c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b^2 \sqrt{d-c^2 d x^2} \left(-4 \cosh ^{-1}(c x)^3-6 \cosh \left(2 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)+\left(6 \cosh ^{-1}(c x)^2+3\right) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)}{c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{6 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{2} x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{b c x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{4} b^2 x \sqrt{d-c^2 d x^2}+\frac{b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 c \sqrt{c x-1} \sqrt{c x+1}}",1,"(12*a^2*x*Sqrt[d - c^2*d*x^2] - (12*a^2*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/c - (6*a*b*Sqrt[d - c^2*d*x^2]*(2*ArcCosh[c*x]^2 + Cosh[2*ArcCosh[c*x]] - 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]]))/(c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b^2*Sqrt[d - c^2*d*x^2]*(-4*ArcCosh[c*x]^3 - 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] + (3 + 6*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]))/(c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))/24","A",0
174,1,449,402,1.3421575,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x,x]","a^2 \sqrt{d-c^2 d x^2}-a^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+a^2 \sqrt{d} \log (c x)+\frac{2 a b \sqrt{d-c^2 d x^2} \left(i \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+b^2 \sqrt{d-c^2 d x^2} \left(\frac{i \left(2 \cosh ^{-1}(c x) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 \cosh ^{-1}(c x) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+2 \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)+\cosh ^{-1}(c x)^2 \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\cosh ^{-1}(c x)^2 \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\cosh ^{-1}(c x)^2+\frac{2 c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{1-c x}+2\right)","\frac{2 i b \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 a b c x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}+\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 i b^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+2 b^2 \sqrt{d-c^2 d x^2}-\frac{2 b^2 c x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}",1,"a^2*Sqrt[d - c^2*d*x^2] + a^2*Sqrt[d]*Log[c*x] - a^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + b^2*Sqrt[d - c^2*d*x^2]*(2 + (2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x])/(1 - c*x) + ArcCosh[c*x]^2 + (I*(ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] - ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] + 2*PolyLog[3, (-I)/E^ArcCosh[c*x]] - 2*PolyLog[3, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))","A",0
175,1,270,234,1.7072327,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x^2,x]","-\frac{a^2 \sqrt{d-c^2 d x^2}}{x}+a^2 c \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+a b c \sqrt{d-c^2 d x^2} \left(\frac{2 \log (c x)+\cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{2 \cosh ^{-1}(c x)}{c x}\right)+\frac{1}{3} b^2 c \sqrt{d-c^2 d x^2} \left(\frac{3 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{1-c x}+\cosh ^{-1}(c x) \left(\frac{\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+3\right)+6 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{3 \cosh ^{-1}(c x)}{c x}\right)\right)","\frac{c \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{3 b \sqrt{c x-1} \sqrt{c x+1}}+\frac{c \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x}+\frac{2 b c \sqrt{d-c^2 d x^2} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b^2 c \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}",1,"-((a^2*Sqrt[d - c^2*d*x^2])/x) + a^2*c*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + a*b*c*Sqrt[d - c^2*d*x^2]*((-2*ArcCosh[c*x])/(c*x) + (ArcCosh[c*x]^2 + 2*Log[c*x])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (b^2*c*Sqrt[d - c^2*d*x^2]*(ArcCosh[c*x]*((-3*ArcCosh[c*x])/(c*x) + (ArcCosh[c*x]*(3 + ArcCosh[c*x]) + 6*Log[1 + E^(-2*ArcCosh[c*x])])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (3*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(1 - c*x)))/3","A",0
176,1,547,427,85.4674519,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x^3,x]","\frac{1}{2} a \left(-\frac{a \sqrt{d-c^2 d x^2}}{x^2}+a c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-a c^2 \sqrt{d} \log (x)+\frac{2 b d (c x+1) \left(i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+c x \sqrt{\frac{c x-1}{c x+1}}+c x \cosh ^{-1}(c x)-\cosh ^{-1}(c x)\right)}{x^2 \sqrt{d-c^2 d x^2}}\right)+\frac{1}{2} b^2 c^2 \sqrt{d-c^2 d x^2} \left(-\frac{\cosh ^{-1}(c x)^2}{c^2 x^2}+\frac{2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{c x-c^2 x^2}-\frac{i \left(2 \cosh ^{-1}(c x) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 \cosh ^{-1}(c x) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+2 \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)+\cosh ^{-1}(c x)^2 \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\cosh ^{-1}(c x)^2 \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)","-\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{i b c^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 x^2}+\frac{c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{i b^2 c^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b^2 c^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}",1,"(a*(-((a*Sqrt[d - c^2*d*x^2])/x^2) - a*c^2*Sqrt[d]*Log[x] + a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*b*d*(1 + c*x)*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x] + c*x*ArcCosh[c*x] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, I/E^ArcCosh[c*x]]))/(x^2*Sqrt[d - c^2*d*x^2])))/2 + (b^2*c^2*Sqrt[d - c^2*d*x^2]*((2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x])/(c*x - c^2*x^2) - ArcCosh[c*x]^2/(c^2*x^2) - (I*((4*I)*ArcTan[Tanh[ArcCosh[c*x]/2]] + ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] - ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] + 2*PolyLog[3, (-I)/E^ArcCosh[c*x]] - 2*PolyLog[3, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/2","A",0
177,1,304,336,1.0076567,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x^4,x]","-\frac{d (c x+1) \left(a^2 c^3 x^3-a^2 c^2 x^2-a^2 c x+a^2-2 a b c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} \log (c x)-b \cosh ^{-1}(c x) \left(-2 a (c x-1)^2 (c x+1)+2 b c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)+b c x \sqrt{\frac{c x-1}{c x+1}}\right)-a b c x \sqrt{\frac{c x-1}{c x+1}}+b^2 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+b^2 c^3 x^3-b^2 c^2 x^2-b^2 \left(c^3 x^3 \left(\sqrt{\frac{c x-1}{c x+1}}-1\right)+c^2 x^2+c x-1\right) \cosh ^{-1}(c x)^2\right)}{3 x^3 \sqrt{d-c^2 d x^2}}","-\frac{b c \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d x^3}-\frac{c^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c^3 \sqrt{d-c^2 d x^2} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^2 \sqrt{d-c^2 d x^2}}{3 x}+\frac{b^2 c^3 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b^2 c^3 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{3 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/3*(d*(1 + c*x)*(a^2 - a^2*c*x - a^2*c^2*x^2 - b^2*c^2*x^2 + a^2*c^3*x^3 + b^2*c^3*x^3 - a*b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - b^2*(-1 + c*x + c^2*x^2 + c^3*x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)]))*ArcCosh[c*x]^2 - b*ArcCosh[c*x]*(b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*a*(-1 + c*x)^2*(1 + c*x) + 2*b*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Log[1 + E^(-2*ArcCosh[c*x])]) - 2*a*b*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Log[c*x] + b^2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, -E^(-2*ArcCosh[c*x])]))/(x^3*Sqrt[d - c^2*d*x^2])","A",0
178,1,262,495,0.5845516,"\int x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]","-\frac{d \sqrt{d-c^2 d x^2} \left(11025 a^2 \left(5 c^2 x^2+2\right) \left(c^2 x^2-1\right)^3-210 a b c x \sqrt{c x-1} \sqrt{c x+1} \left(75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right)-210 b \cosh ^{-1}(c x) \left(b c x \sqrt{c x-1} \sqrt{c x+1} \left(75 c^6 x^6-168 c^4 x^4+35 c^2 x^2+210\right)-105 a \left(c^2 x^2-1\right)^3 \left(5 c^2 x^2+2\right)\right)+11025 b^2 \left(5 c^2 x^2+2\right) \left(c^2 x^2-1\right)^3 \cosh ^{-1}(c x)^2+2 b^2 \left(1125 c^8 x^8-3303 c^6 x^6+499 c^4 x^4+20371 c^2 x^2-18692\right)\right)}{385875 c^4 \left(c^2 x^2-1\right)}","-\frac{d x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{35 c^2}-\frac{16 b c d x^5 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{175 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{7} x^4 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{3}{35} d x^4 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{2 b d x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{105 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{35 c^4}+\frac{4 a b d x \sqrt{d-c^2 d x^2}}{35 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d x^7 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3358 b^2 d x^2 \sqrt{d-c^2 d x^2}}{385875 c^2}-\frac{2}{343} b^2 c^2 d x^6 \sqrt{d-c^2 d x^2}+\frac{484 b^2 d x^4 \sqrt{d-c^2 d x^2}}{42875}-\frac{37384 b^2 d \sqrt{d-c^2 d x^2}}{385875 c^4}+\frac{4 b^2 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{35 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/385875*(d*Sqrt[d - c^2*d*x^2]*(11025*a^2*(-1 + c^2*x^2)^3*(2 + 5*c^2*x^2) - 210*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6) + 2*b^2*(-18692 + 20371*c^2*x^2 + 499*c^4*x^4 - 3303*c^6*x^6 + 1125*c^8*x^8) - 210*b*(-105*a*(-1 + c^2*x^2)^3*(2 + 5*c^2*x^2) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(210 + 35*c^2*x^2 - 168*c^4*x^4 + 75*c^6*x^6))*ArcCosh[c*x] + 11025*b^2*(-1 + c^2*x^2)^3*(2 + 5*c^2*x^2)*ArcCosh[c*x]^2))/(c^4*(-1 + c^2*x^2))","A",1
179,1,485,441,4.370507,"\int x^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]","\frac{-864 a^2 d^{3/2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-288 a^2 c d x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(8 c^4 x^4-14 c^2 x^2+3\right) \sqrt{d-c^2 d x^2}-216 a b d \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)-12 a b d \sqrt{d-c^2 d x^2} \left(-72 \cosh ^{-1}(c x)^2+18 \cosh \left(2 \cosh ^{-1}(c x)\right)-9 \cosh \left(4 \cosh ^{-1}(c x)\right)-2 \cosh \left(6 \cosh ^{-1}(c x)\right)+12 \cosh ^{-1}(c x) \left(-3 \sinh \left(2 \cosh ^{-1}(c x)\right)+3 \sinh \left(4 \cosh ^{-1}(c x)\right)+\sinh \left(6 \cosh ^{-1}(c x)\right)\right)\right)-18 b^2 d \sqrt{d-c^2 d x^2} \left(32 \cosh ^{-1}(c x)^3+12 \cosh \left(4 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-3 \left(8 \cosh ^{-1}(c x)^2+1\right) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)+b^2 d \sqrt{d-c^2 d x^2} \left(288 \cosh ^{-1}(c x)^3+12 \left(-18 \cosh \left(2 \cosh ^{-1}(c x)\right)+9 \cosh \left(4 \cosh ^{-1}(c x)\right)+2 \cosh \left(6 \cosh ^{-1}(c x)\right)\right) \cosh ^{-1}(c x)-72 \cosh ^{-1}(c x)^2 \left(-3 \sinh \left(2 \cosh ^{-1}(c x)\right)+3 \sinh \left(4 \cosh ^{-1}(c x)\right)+\sinh \left(6 \cosh ^{-1}(c x)\right)\right)+108 \sinh \left(2 \cosh ^{-1}(c x)\right)-27 \sinh \left(4 \cosh ^{-1}(c x)\right)-4 \sinh \left(6 \cosh ^{-1}(c x)\right)\right)}{13824 c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{b d x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{16 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{16 c^2}-\frac{7 b c d x^4 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{48 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{6} x^3 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{1}{8} d x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{48 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 d x^6 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{18 \sqrt{c x-1} \sqrt{c x+1}}+\frac{7 b^2 d x \sqrt{d-c^2 d x^2}}{1152 c^2}-\frac{1}{108} b^2 c^2 d x^5 \sqrt{d-c^2 d x^2}+\frac{43 b^2 d x^3 \sqrt{d-c^2 d x^2}}{1728}+\frac{7 b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(-288*a^2*c*d*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(3 - 14*c^2*x^2 + 8*c^4*x^4) - 864*a^2*d^(3/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 216*a*b*d*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - 18*b^2*d*Sqrt[d - c^2*d*x^2]*(32*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*ArcCosh[c*x]^2)*Sinh[4*ArcCosh[c*x]]) - 12*a*b*d*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])) + b^2*d*Sqrt[d - c^2*d*x^2]*(288*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*(-18*Cosh[2*ArcCosh[c*x]] + 9*Cosh[4*ArcCosh[c*x]] + 2*Cosh[6*ArcCosh[c*x]]) + 108*Sinh[2*ArcCosh[c*x]] - 27*Sinh[4*ArcCosh[c*x]] - 4*Sinh[6*ArcCosh[c*x]] - 72*ArcCosh[c*x]^2*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(13824*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
180,1,208,348,0.5276681,"\int x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]","-\frac{d \sqrt{d-c^2 d x^2} \left(225 a^2 \left(c^2 x^2-1\right)^3-30 a b c x \sqrt{c x-1} \sqrt{c x+1} \left(3 c^4 x^4-10 c^2 x^2+15\right)-30 b \cosh ^{-1}(c x) \left(b c x \sqrt{c x-1} \sqrt{c x+1} \left(3 c^4 x^4-10 c^2 x^2+15\right)-15 a \left(c^2 x^2-1\right)^3\right)+225 b^2 \left(c^2 x^2-1\right)^3 \cosh ^{-1}(c x)^2+2 b^2 \left(9 c^6 x^6-47 c^4 x^4+187 c^2 x^2-149\right)\right)}{1125 c^2 \left(c^2 x^2-1\right)}","\frac{2 b d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{5 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{5 c^2 d}-\frac{4 b c d x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{15 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d x^5 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{25 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b^2 d \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{125 c^2 (1-c x) (c x+1)}-\frac{8 b^2 d \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{225 c^2 (1-c x) (c x+1)}-\frac{16 b^2 d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{75 c^2 (1-c x) (c x+1)}",1,"-1/1125*(d*Sqrt[d - c^2*d*x^2]*(225*a^2*(-1 + c^2*x^2)^3 - 30*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15 - 10*c^2*x^2 + 3*c^4*x^4) + 2*b^2*(-149 + 187*c^2*x^2 - 47*c^4*x^4 + 9*c^6*x^6) - 30*b*(-15*a*(-1 + c^2*x^2)^3 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15 - 10*c^2*x^2 + 3*c^4*x^4))*ArcCosh[c*x] + 225*b^2*(-1 + c^2*x^2)^3*ArcCosh[c*x]^2))/(c^2*(-1 + c^2*x^2))","A",1
181,1,374,336,2.9539267,"\int \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]","\frac{-288 a^2 d^{3/2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-96 a^2 c d x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(2 c^2 x^2-5\right) \sqrt{d-c^2 d x^2}-192 a b d \sqrt{d-c^2 d x^2} \left(\cosh \left(2 \cosh ^{-1}(c x)\right)+2 \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-\sinh \left(2 \cosh ^{-1}(c x)\right)\right)\right)+12 a b d \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)-32 b^2 d \sqrt{d-c^2 d x^2} \left(4 \cosh ^{-1}(c x)^3+6 \cosh \left(2 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-3 \left(2 \cosh ^{-1}(c x)^2+1\right) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)+b^2 d \sqrt{d-c^2 d x^2} \left(32 \cosh ^{-1}(c x)^3+12 \cosh \left(4 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-3 \left(8 \cosh ^{-1}(c x)^2+1\right) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)}{768 c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","-\frac{d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{8 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{4} x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{3}{8} d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{b d \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c d x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{15}{64} b^2 d x \sqrt{d-c^2 d x^2}+\frac{1}{32} b^2 d x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}+\frac{9 b^2 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 c \sqrt{c x-1} \sqrt{c x+1}}",1,"(-96*a^2*c*d*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-5 + 2*c^2*x^2)*Sqrt[d - c^2*d*x^2] - 288*a^2*d^(3/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 192*a*b*d*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) - 32*b^2*d*Sqrt[d - c^2*d*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*(1 + 2*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]) + 12*a*b*d*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) + b^2*d*Sqrt[d - c^2*d*x^2]*(32*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*ArcCosh[c*x]^2)*Sinh[4*ArcCosh[c*x]]))/(768*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
182,1,650,573,2.9130774,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x,x]","-a^2 d^{3/2} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)-\frac{1}{3} a^2 d \left(c^2 x^2-4\right) \sqrt{d-c^2 d x^2}+a^2 d^{3/2} \log (c x)+\frac{2 a b d \sqrt{d-c^2 d x^2} \left(i \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-i \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-c x+c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{a b d \sqrt{d-c^2 d x^2} \left(9 c x+12 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \cosh ^{-1}(c x)-\cosh \left(3 \cosh ^{-1}(c x)\right)\right)}{18 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}+b^2 d \sqrt{d-c^2 d x^2} \left(\frac{i \left(2 \cosh ^{-1}(c x) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 \cosh ^{-1}(c x) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+2 \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)+\cosh ^{-1}(c x)^2 \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\cosh ^{-1}(c x)^2 \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\cosh ^{-1}(c x)^2+\frac{2 c x \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{1-c x}+2\right)-\frac{1}{54} b^2 d \sqrt{d-c^2 d x^2} \left(9 \left(\cosh \left(2 \cosh ^{-1}(c x)\right)-1\right) \cosh ^{-1}(c x)^2+\frac{3 \sqrt{\frac{c x-1}{c x+1}} \left(9 c x-\cosh \left(3 \cosh ^{-1}(c x)\right)\right) \cosh ^{-1}(c x)}{c x-1}+2 \left(\cosh \left(2 \cosh ^{-1}(c x)\right)-13\right)\right)","\frac{2 i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b d \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 a b c d x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{3} \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2+d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c^3 d x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 i b^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2}{27} b^2 c^2 d x^2 \sqrt{d-c^2 d x^2}+\frac{68}{27} b^2 d \sqrt{d-c^2 d x^2}-\frac{2 b^2 c d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}",1,"-1/3*(a^2*d*(-4 + c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (b^2*d*Sqrt[d - c^2*d*x^2]*(2*(-13 + Cosh[2*ArcCosh[c*x]]) + 9*ArcCosh[c*x]^2*(-1 + Cosh[2*ArcCosh[c*x]]) + (3*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*(9*c*x - Cosh[3*ArcCosh[c*x]]))/(-1 + c*x)))/54 - (a*b*d*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]))/(18*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + a^2*d^(3/2)*Log[c*x] - a^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (2*a*b*d*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + b^2*d*Sqrt[d - c^2*d*x^2]*(2 + (2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x])/(1 - c*x) + ArcCosh[c*x]^2 + (I*(ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] - ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] + 2*PolyLog[3, (-I)/E^ArcCosh[c*x]] - 2*PolyLog[3, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)))","A",0
183,1,433,453,4.1069868,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^2,x]","\frac{36 a^2 c d^{3/2} x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-12 a^2 d \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(c^2 x^2+2\right) \sqrt{d-c^2 d x^2}-24 a b d \sqrt{d-c^2 d x^2} \left(2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left(2 \log (c x)+\cosh ^{-1}(c x)^2\right)\right)+6 a b c d x \sqrt{d-c^2 d x^2} \left(\cosh \left(2 \cosh ^{-1}(c x)\right)+2 \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-\sinh \left(2 \cosh ^{-1}(c x)\right)\right)\right)-8 b^2 d \sqrt{d-c^2 d x^2} \left(3 c x \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+\cosh ^{-1}(c x) \left(3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+3\right)+6 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)\right)\right)+b^2 c d x \sqrt{d-c^2 d x^2} \left(4 \cosh ^{-1}(c x)^3+6 \cosh \left(2 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-3 \left(2 \cosh ^{-1}(c x)^2+1\right) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)}{24 x \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{c d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{2 b \sqrt{c x-1} \sqrt{c x+1}}+\frac{c d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x}+\frac{2 b c d \sqrt{d-c^2 d x^2} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b^2 c d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt{c x-1} \sqrt{c x+1}}",1,"(-12*a^2*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(2 + c^2*x^2)*Sqrt[d - c^2*d*x^2] + 36*a^2*c*d^(3/2)*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 24*a*b*d*Sqrt[d - c^2*d*x^2]*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]^2 + 2*Log[c*x])) - 8*b^2*d*Sqrt[d - c^2*d*x^2]*(ArcCosh[c*x]*(3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]*(3 + ArcCosh[c*x]) + 6*Log[1 + E^(-2*ArcCosh[c*x])])) + 3*c*x*PolyLog[2, -E^(-2*ArcCosh[c*x])]) + 6*a*b*c*d*x*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + b^2*c*d*x*Sqrt[d - c^2*d*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*(1 + 2*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]))/(24*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
184,1,1129,630,176.1172009,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^3,x]","\frac{1}{2} d \sqrt{d-c^2 d x^2} \left(\frac{4 x^2 \cosh ^{-1}(c x) c^4}{(c x-1)^{3/2} \sqrt{c x+1}}-\frac{2 x \cosh ^{-1}(c x)^2 c^3}{c x-1}-\frac{4 x \cosh ^{-1}(c x) c^3}{(c x-1)^{3/2} \sqrt{c x+1}}-\frac{2 x \tan ^{-1}\left(\frac{1}{\sqrt{c^2 x^2-1}}\right) c^3}{(c x-1) \sqrt{c^2 x^2-1}}-\frac{4 x c^3}{c x-1}+\frac{2 \cosh ^{-1}(c x)^2 c^2}{c x-1}-\frac{2 \cosh ^{-1}(c x) c^2}{(c x-1)^{3/2} \sqrt{c x+1}}+\frac{2 \tan ^{-1}\left(\frac{1}{\sqrt{c^2 x^2-1}}\right) c^2}{(c x-1) \sqrt{c^2 x^2-1}}-\frac{3 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)^2 \log \left(1-i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}+\frac{3 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)^2 \log \left(1+i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}-\frac{6 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}+\frac{6 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}-\frac{6 i \sqrt{\frac{c x-1}{c x+1}} \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}+\frac{6 i \sqrt{\frac{c x-1}{c x+1}} \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}+\frac{4 c^2}{c x-1}+\frac{\cosh ^{-1}(c x)^2 c}{x-c x^2}+\frac{2 \cosh ^{-1}(c x) c}{x (c x-1)^{3/2} \sqrt{c x+1}}+\frac{\cosh ^{-1}(c x)^2}{x^2 (c x-1)}\right) b^2-2 a c^2 d \sqrt{-d (c x-1) (c x+1)} \left(-\frac{c x}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\cosh ^{-1}(c x)+\frac{i \cosh ^{-1}(c x) \left(\log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{i \left(\text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right) b+\frac{i a c^2 d^2 \left(-\frac{i (c x-1) \cosh ^{-1}(c x) (c x+1)}{c^2 x^2}+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right) (c x+1)-\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right) (c x+1)+\sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right) (c x+1)-\sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right) (c x+1)-\frac{i \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}\right) b}{\sqrt{-d (c x-1) (c x+1)}}-\frac{3}{2} a^2 c^2 d^{3/2} \log (x)+\frac{3}{2} a^2 c^2 d^{3/2} \log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right)+\left(-c^2 d a^2-\frac{d a^2}{2 x^2}\right) \sqrt{-d \left(c^2 x^2-1\right)}","-\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{b c d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{x \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 i b^2 c^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 i b^2 c^2 d \sqrt{d-c^2 d x^2} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-2 b^2 c^2 d \sqrt{d-c^2 d x^2}+\frac{b^2 c^2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}",1,"(-(a^2*c^2*d) - (a^2*d)/(2*x^2))*Sqrt[-(d*(-1 + c^2*x^2))] - (3*a^2*c^2*d^(3/2)*Log[x])/2 + (3*a^2*c^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/2 - 2*a*b*c^2*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-((c*x)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + ArcCosh[c*x] + (I*ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (I*(PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (I*a*b*c^2*d^2*(((-I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) - (I*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x])/(c^2*x^2) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[-(d*(-1 + c*x)*(1 + c*x))] + (b^2*d*Sqrt[d - c^2*d*x^2]*((4*c^2)/(-1 + c*x) - (4*c^3*x)/(-1 + c*x) - (2*c^2*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (2*c*ArcCosh[c*x])/(x*(-1 + c*x)^(3/2)*Sqrt[1 + c*x]) - (4*c^3*x*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (4*c^4*x^2*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (2*c^2*ArcCosh[c*x]^2)/(-1 + c*x) + ArcCosh[c*x]^2/(x^2*(-1 + c*x)) - (2*c^3*x*ArcCosh[c*x]^2)/(-1 + c*x) + (c*ArcCosh[c*x]^2)/(x - c*x^2) + (2*c^2*ArcTan[1/Sqrt[-1 + c^2*x^2]])/((-1 + c*x)*Sqrt[-1 + c^2*x^2]) - (2*c^3*x*ArcTan[1/Sqrt[-1 + c^2*x^2]])/((-1 + c*x)*Sqrt[-1 + c^2*x^2]) - ((3*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]])/(-1 + c*x) + ((3*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]])/(-1 + c*x) - ((6*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]])/(-1 + c*x) + ((6*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]])/(-1 + c*x) - ((6*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[3, (-I)/E^ArcCosh[c*x]])/(-1 + c*x) + ((6*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[3, I/E^ArcCosh[c*x]])/(-1 + c*x)))/2","A",0
185,1,583,426,2.1204512,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x^4,x]","\frac{-4 a^2 c^4 d^2 x^4 \sqrt{\frac{c x-1}{c x+1}}+5 a^2 c^2 d^2 x^2 \sqrt{\frac{c x-1}{c x+1}}-3 a^2 c^3 d^{3/2} x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-a^2 d^2 \sqrt{\frac{c x-1}{c x+1}}+8 a b c^4 d^2 x^4 \log (c x)-8 a b c^3 d^2 x^3 \log (c x)+a b c^2 d^2 x^2-b d^2 (c x-1) \cosh ^{-1}(c x)^2 \left(b \left(4 c^3 x^3 \left(\sqrt{\frac{c x-1}{c x+1}}-1\right)+4 c^2 x^2 \sqrt{\frac{c x-1}{c x+1}}-c x \sqrt{\frac{c x-1}{c x+1}}-\sqrt{\frac{c x-1}{c x+1}}\right)-3 a c^3 x^3\right)+b d^2 (c x-1) \cosh ^{-1}(c x) \left(2 a \sqrt{\frac{c x-1}{c x+1}} \left(-4 c^3 x^3-4 c^2 x^2+c x+1\right)+8 b c^3 x^3 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)+b c x\right)-a b c d^2 x-b^2 c^4 d^2 x^4 \sqrt{\frac{c x-1}{c x+1}}-4 b^2 c^3 d^2 x^3 (c x-1) \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+b^2 c^3 d^2 x^3 (c x-1) \cosh ^{-1}(c x)^3+b^2 c^2 d^2 x^2 \sqrt{\frac{c x-1}{c x+1}}}{3 x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}}","\frac{c^2 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x}-\frac{b c d \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 x^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 x^3}-\frac{c^3 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{3 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 c^3 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{8 b c^3 d \sqrt{d-c^2 d x^2} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^2 d \sqrt{d-c^2 d x^2}}{3 x}+\frac{4 b^2 c^3 d \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b^2 c^3 d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(-(a*b*c*d^2*x) + a*b*c^2*d^2*x^2 - a^2*d^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 5*a^2*c^2*d^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + b^2*c^2*d^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*a^2*c^4*d^2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - b^2*c^4*d^2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - b*d^2*(-1 + c*x)*(-3*a*c^3*x^3 + b*(-Sqrt[(-1 + c*x)/(1 + c*x)] - c*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*c^3*x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)])))*ArcCosh[c*x]^2 + b^2*c^3*d^2*x^3*(-1 + c*x)*ArcCosh[c*x]^3 - 3*a^2*c^3*d^(3/2)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b*d^2*(-1 + c*x)*ArcCosh[c*x]*(b*c*x + 2*a*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x - 4*c^2*x^2 - 4*c^3*x^3) + 8*b*c^3*x^3*Log[1 + E^(-2*ArcCosh[c*x])]) - 8*a*b*c^3*d^2*x^3*Log[c*x] + 8*a*b*c^4*d^2*x^4*Log[c*x] - 4*b^2*c^3*d^2*x^3*(-1 + c*x)*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])","A",0
186,1,288,880,0.754466,"\int x^3 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(3969 a^2 \left(7 c^2 x^2+2\right) \left(c^2 x^2-1\right)^4-126 a b c x \sqrt{c x-1} \sqrt{c x+1} \left(49 c^8 x^8-171 c^6 x^6+189 c^4 x^4-21 c^2 x^2-126\right)+126 b \cosh ^{-1}(c x) \left(63 a \left(7 c^2 x^2+2\right) \left(c^2 x^2-1\right)^4+b c x \sqrt{c x-1} \sqrt{c x+1} \left(-49 c^8 x^8+171 c^6 x^6-189 c^4 x^4+21 c^2 x^2+126\right)\right)+3969 b^2 \left(7 c^2 x^2+2\right) \left(c^2 x^2-1\right)^4 \cosh ^{-1}(c x)^2+2 b^2 \left(343 c^{10} x^{10}-1490 c^8 x^8+2152 c^6 x^6-106 c^4 x^4-7039 c^2 x^2+6140\right)\right)}{250047 c^4 \left(c^2 x^2-1\right)}","-\frac{2 b c^5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^9}{81 \sqrt{c x-1} \sqrt{c x+1}}+\frac{38 b c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^7}{441 \sqrt{c x-1} \sqrt{c x+1}}-\frac{10 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^6}{3087}-\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^5}{21 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{9} \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2 x^4+\frac{5}{63} d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 x^4+\frac{1}{21} d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 x^4+\frac{484 b^2 d^2 \sqrt{d-c^2 d x^2} x^4}{77175}+\frac{2 b d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^3}{189 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 x^2}{63 c^2}+\frac{3358 b^2 d^2 \sqrt{d-c^2 d x^2} x^2}{694575 c^2}+\frac{4 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) x}{63 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 a b d^2 \sqrt{d-c^2 d x^2} x}{63 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{63 c^4}+\frac{2 b^2 d^2 \left(1-c^2 x^2\right)^5 \sqrt{d-c^2 d x^2}}{729 c^4 (1-c x) (c x+1)}-\frac{20 b^2 d^2 \left(1-c^2 x^2\right)^4 \sqrt{d-c^2 d x^2}}{3969 c^4 (1-c x) (c x+1)}+\frac{2 b^2 d^2 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{4725 c^4 (1-c x) (c x+1)}-\frac{37384 b^2 d^2 \sqrt{d-c^2 d x^2}}{694575 c^4}+\frac{8 b^2 d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{8505 c^4 (1-c x) (c x+1)}+\frac{16 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{2835 c^4 (1-c x) (c x+1)}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(3969*a^2*(-1 + c^2*x^2)^4*(2 + 7*c^2*x^2) - 126*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-126 - 21*c^2*x^2 + 189*c^4*x^4 - 171*c^6*x^6 + 49*c^8*x^8) + 2*b^2*(6140 - 7039*c^2*x^2 - 106*c^4*x^4 + 2152*c^6*x^6 - 1490*c^8*x^8 + 343*c^10*x^10) + 126*b*(63*a*(-1 + c^2*x^2)^4*(2 + 7*c^2*x^2) + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(126 + 21*c^2*x^2 - 189*c^4*x^4 + 171*c^6*x^6 - 49*c^8*x^8))*ArcCosh[c*x] + 3969*b^2*(-1 + c^2*x^2)^4*(2 + 7*c^2*x^2)*ArcCosh[c*x]^2))/(250047*c^4*(-1 + c^2*x^2))","A",1
187,1,910,841,5.9098175,"\int x^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]","-\frac{d^2 \left(-110592 a^2 c^8 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^8-110592 a^2 c^7 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^7+313344 a^2 c^6 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^6+313344 a^2 c^5 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^5-271872 a^2 c^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^4-271872 a^2 c^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^3+34560 a^2 c^2 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x^2+34560 a^2 c \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right) x+34560 a^2 c \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} x+11520 b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^3+34560 a^2 \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+13824 a b \sqrt{d-c^2 d x^2} \cosh \left(2 \cosh ^{-1}(c x)\right)+3456 a b \sqrt{d-c^2 d x^2} \cosh \left(4 \cosh ^{-1}(c x)\right)-1536 a b \sqrt{d-c^2 d x^2} \cosh \left(6 \cosh ^{-1}(c x)\right)+216 a b \sqrt{d-c^2 d x^2} \cosh \left(8 \cosh ^{-1}(c x)\right)-6912 b^2 \sqrt{d-c^2 d x^2} \sinh \left(2 \cosh ^{-1}(c x)\right)-864 b^2 \sqrt{d-c^2 d x^2} \sinh \left(4 \cosh ^{-1}(c x)\right)+256 b^2 \sqrt{d-c^2 d x^2} \sinh \left(6 \cosh ^{-1}(c x)\right)-27 b^2 \sqrt{d-c^2 d x^2} \sinh \left(8 \cosh ^{-1}(c x)\right)+24 b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \left(576 b \cosh \left(2 \cosh ^{-1}(c x)\right)+144 b \cosh \left(4 \cosh ^{-1}(c x)\right)-64 b \cosh \left(6 \cosh ^{-1}(c x)\right)+9 b \cosh \left(8 \cosh ^{-1}(c x)\right)-1152 a \sinh \left(2 \cosh ^{-1}(c x)\right)-576 a \sinh \left(4 \cosh ^{-1}(c x)\right)+384 a \sinh \left(6 \cosh ^{-1}(c x)\right)-72 a \sinh \left(8 \cosh ^{-1}(c x)\right)\right)-288 b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2 \left(-120 a+48 b \sinh \left(2 \cosh ^{-1}(c x)\right)+24 b \sinh \left(4 \cosh ^{-1}(c x)\right)-16 b \sinh \left(6 \cosh ^{-1}(c x)\right)+3 b \sinh \left(8 \cosh ^{-1}(c x)\right)\right)\right)}{884736 c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","-\frac{b c^5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^8}{32 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^4 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} x^7}{256 (1-c x) (c x+1)}+\frac{17 b c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^6}{144 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{864} b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^5-\frac{43 b^2 c^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} x^5}{4608 (1-c x) (c x+1)}-\frac{59 b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^4}{384 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{8} \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2 x^3+\frac{5}{48} d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 x^3+\frac{5}{64} d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 x^3+\frac{215 b^2 d^2 \sqrt{d-c^2 d x^2} x^3}{13824}+\frac{73 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} x^3}{18432 (1-c x) (c x+1)}+\frac{5 b d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^2}{128 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 x}{128 c^2}+\frac{35 b^2 d^2 \sqrt{d-c^2 d x^2} x}{9216 c^2}+\frac{73 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} x}{12288 c^2 (1-c x) (c x+1)}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{384 b c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{35 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{9216 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{73 b^2 d^2 \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{12288 c^3 (1-c x) (c x+1)}",1,"-1/884736*(d^2*(34560*a^2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 34560*a^2*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 271872*a^2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 271872*a^2*c^4*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 313344*a^2*c^5*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 313344*a^2*c^6*x^6*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 110592*a^2*c^7*x^7*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 110592*a^2*c^8*x^8*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 11520*b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^3 + 34560*a^2*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 34560*a^2*c*Sqrt[d]*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 13824*a*b*Sqrt[d - c^2*d*x^2]*Cosh[2*ArcCosh[c*x]] + 3456*a*b*Sqrt[d - c^2*d*x^2]*Cosh[4*ArcCosh[c*x]] - 1536*a*b*Sqrt[d - c^2*d*x^2]*Cosh[6*ArcCosh[c*x]] + 216*a*b*Sqrt[d - c^2*d*x^2]*Cosh[8*ArcCosh[c*x]] - 6912*b^2*Sqrt[d - c^2*d*x^2]*Sinh[2*ArcCosh[c*x]] - 864*b^2*Sqrt[d - c^2*d*x^2]*Sinh[4*ArcCosh[c*x]] + 256*b^2*Sqrt[d - c^2*d*x^2]*Sinh[6*ArcCosh[c*x]] - 27*b^2*Sqrt[d - c^2*d*x^2]*Sinh[8*ArcCosh[c*x]] + 24*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*(576*b*Cosh[2*ArcCosh[c*x]] + 144*b*Cosh[4*ArcCosh[c*x]] - 64*b*Cosh[6*ArcCosh[c*x]] + 9*b*Cosh[8*ArcCosh[c*x]] - 1152*a*Sinh[2*ArcCosh[c*x]] - 576*a*Sinh[4*ArcCosh[c*x]] + 384*a*Sinh[6*ArcCosh[c*x]] - 72*a*Sinh[8*ArcCosh[c*x]]) - 288*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^2*(-120*a + 48*b*Sinh[2*ArcCosh[c*x]] + 24*b*Sinh[4*ArcCosh[c*x]] - 16*b*Sinh[6*ArcCosh[c*x]] + 3*b*Sinh[8*ArcCosh[c*x]])))/(c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
188,1,234,470,0.6147294,"\int x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]","\frac{d^2 \sqrt{d-c^2 d x^2} \left(3675 a^2 \left(c^2 x^2-1\right)^4-210 a b c x \sqrt{c x-1} \sqrt{c x+1} \left(5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right)+210 b \cosh ^{-1}(c x) \left(35 a \left(c^2 x^2-1\right)^4+b c x \sqrt{c x-1} \sqrt{c x+1} \left(-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right)\right)+3675 b^2 \left(c^2 x^2-1\right)^4 \cosh ^{-1}(c x)^2+2 b^2 \left(75 c^8 x^8-426 c^6 x^6+1108 c^4 x^4-2918 c^2 x^2+2161\right)\right)}{25725 c^2 \left(c^2 x^2-1\right)}","\frac{2 b d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{7 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left(d-c^2 d x^2\right)^{7/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{7 c^2 d}-\frac{2 b c^5 d^2 x^7 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{49 \sqrt{c x-1} \sqrt{c x+1}}+\frac{6 b c^3 d^2 x^5 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{35 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b^2 d^2 \left(1-c^2 x^2\right)^4 \sqrt{d-c^2 d x^2}}{343 c^2 (1-c x) (c x+1)}-\frac{12 b^2 d^2 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{1225 c^2 (1-c x) (c x+1)}-\frac{16 b^2 d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{735 c^2 (1-c x) (c x+1)}-\frac{32 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{245 c^2 (1-c x) (c x+1)}",1,"(d^2*Sqrt[d - c^2*d*x^2]*(3675*a^2*(-1 + c^2*x^2)^4 - 210*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + 2*b^2*(2161 - 2918*c^2*x^2 + 1108*c^4*x^4 - 426*c^6*x^6 + 75*c^8*x^8) + 210*b*(35*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(35 - 35*c^2*x^2 + 21*c^4*x^4 - 5*c^6*x^6))*ArcCosh[c*x] + 3675*b^2*(-1 + c^2*x^2)^4*ArcCosh[c*x]^2))/(25725*c^2*(-1 + c^2*x^2))","A",1
189,1,740,486,3.4881327,"\int \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]","\frac{d^2 \left(9504 a^2 c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}+9504 a^2 c x \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}-4320 a^2 c \sqrt{d} x \sqrt{\frac{c x-1}{c x+1}} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-4320 a^2 \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+2304 a^2 c^6 x^6 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}+2304 a^2 c^5 x^5 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}-7488 a^2 c^4 x^4 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}-7488 a^2 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}-3240 a b \sqrt{d-c^2 d x^2} \cosh \left(2 \cosh ^{-1}(c x)\right)+324 a b \sqrt{d-c^2 d x^2} \cosh \left(4 \cosh ^{-1}(c x)\right)-24 a b \sqrt{d-c^2 d x^2} \cosh \left(6 \cosh ^{-1}(c x)\right)-12 b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \left(-540 a \sinh \left(2 \cosh ^{-1}(c x)\right)+108 a \sinh \left(4 \cosh ^{-1}(c x)\right)-12 a \sinh \left(6 \cosh ^{-1}(c x)\right)+270 b \cosh \left(2 \cosh ^{-1}(c x)\right)-27 b \cosh \left(4 \cosh ^{-1}(c x)\right)+2 b \cosh \left(6 \cosh ^{-1}(c x)\right)\right)+72 b \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^2 \left(-60 a+45 b \sinh \left(2 \cosh ^{-1}(c x)\right)-9 b \sinh \left(4 \cosh ^{-1}(c x)\right)+b \sinh \left(6 \cosh ^{-1}(c x)\right)\right)-1440 b^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)^3+1620 b^2 \sqrt{d-c^2 d x^2} \sinh \left(2 \cosh ^{-1}(c x)\right)-81 b^2 \sqrt{d-c^2 d x^2} \sinh \left(4 \cosh ^{-1}(c x)\right)+4 b^2 \sqrt{d-c^2 d x^2} \sinh \left(6 \cosh ^{-1}(c x)\right)\right)}{13824 c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{b d^2 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{18 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{48 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{48 b c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{16} d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{5 b c d^2 x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{16 \sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{6} x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{5}{24} d x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{1}{108} b^2 d^2 x (1-c x)^2 (c x+1)^2 \sqrt{d-c^2 d x^2}+\frac{245 b^2 d^2 x \sqrt{d-c^2 d x^2}}{1152}+\frac{65 b^2 d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}{1728}+\frac{115 b^2 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{1152 c \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*(9504*a^2*c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 9504*a^2*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 7488*a^2*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 7488*a^2*c^4*x^4*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 2304*a^2*c^5*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] + 2304*a^2*c^6*x^6*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2] - 1440*b^2*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^3 - 4320*a^2*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 4320*a^2*c*Sqrt[d]*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 3240*a*b*Sqrt[d - c^2*d*x^2]*Cosh[2*ArcCosh[c*x]] + 324*a*b*Sqrt[d - c^2*d*x^2]*Cosh[4*ArcCosh[c*x]] - 24*a*b*Sqrt[d - c^2*d*x^2]*Cosh[6*ArcCosh[c*x]] + 1620*b^2*Sqrt[d - c^2*d*x^2]*Sinh[2*ArcCosh[c*x]] - 81*b^2*Sqrt[d - c^2*d*x^2]*Sinh[4*ArcCosh[c*x]] + 4*b^2*Sqrt[d - c^2*d*x^2]*Sinh[6*ArcCosh[c*x]] - 12*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*(270*b*Cosh[2*ArcCosh[c*x]] - 27*b*Cosh[4*ArcCosh[c*x]] + 2*b*Cosh[6*ArcCosh[c*x]] - 540*a*Sinh[2*ArcCosh[c*x]] + 108*a*Sinh[4*ArcCosh[c*x]] - 12*a*Sinh[6*ArcCosh[c*x]]) + 72*b*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^2*(-60*a + 45*b*Sinh[2*ArcCosh[c*x]] - 9*b*Sinh[4*ArcCosh[c*x]] + b*Sinh[6*ArcCosh[c*x]])))/(13824*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
190,1,1031,836,7.2454654,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x,x]","a^2 \log (c x) d^{5/2}-a^2 \log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right) d^{5/2}+\frac{a b \sqrt{-d (c x-1) (c x+1)} \left(-12 \left(\frac{c x-1}{c x+1}\right)^{3/2} \cosh ^{-1}(c x) (c x+1)^3-9 c x+\cosh \left(3 \cosh ^{-1}(c x)\right)\right) d^2}{9 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{1}{27} b^2 \sqrt{-d (c x-1) (c x+1)} \left(-9 \cosh ^{-1}(c x)^2-\frac{3 \cosh \left(3 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{27 c x \cosh ^{-1}(c x)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\left(9 \cosh ^{-1}(c x)^2+2\right) \cosh \left(2 \cosh ^{-1}(c x)\right)-26\right) d^2+2 a b \sqrt{-d (c x-1) (c x+1)} \left(-\frac{c x}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\cosh ^{-1}(c x)+\frac{i \cosh ^{-1}(c x) \left(\log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{i \left(\text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right) d^2+b^2 \sqrt{-d (c x-1) (c x+1)} \left(\cosh ^{-1}(c x)^2-\frac{2 c x \cosh ^{-1}(c x)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{i \left(\log \left(1-i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)^2-\log \left(1+i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)^2+2 \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)-2 \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)+2 \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+2\right) d^2-\frac{a b \sqrt{-d (c x-1) (c x+1)} \left(-450 c x+450 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)+25 \cosh \left(3 \cosh ^{-1}(c x)\right)+9 \cosh \left(5 \cosh ^{-1}(c x)\right)-75 \cosh ^{-1}(c x) \sinh \left(3 \cosh ^{-1}(c x)\right)-45 \cosh ^{-1}(c x) \sinh \left(5 \cosh ^{-1}(c x)\right)\right) d^2}{1800 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{b^2 \sqrt{-d (c x-1) (c x+1)} \left(6750 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2-1125 \sinh \left(3 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)^2-675 \sinh \left(5 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)^2-13500 c x \cosh ^{-1}(c x)+750 \cosh \left(3 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)+270 \cosh \left(5 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)+13500 \sqrt{\frac{c x-1}{c x+1}} (c x+1)-250 \sinh \left(3 \cosh ^{-1}(c x)\right)-54 \sinh \left(5 \cosh ^{-1}(c x)\right)\right) d^2}{54000 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\sqrt{-d \left(c^2 x^2-1\right)} \left(\frac{1}{5} a^2 c^4 d^2 x^4-\frac{11}{15} a^2 c^2 d^2 x^2+\frac{23 a^2 d^2}{15}\right)","-\frac{2 b c^5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^5}{25 \sqrt{c x-1} \sqrt{c x+1}}+\frac{22 b c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x^3}{45 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2}{27} b^2 c^2 d^2 \sqrt{d-c^2 d x^2} x^2-\frac{2 b^2 c d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) x}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{16 b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) x}{15 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 a b c d^2 \sqrt{d-c^2 d x^2} x}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{1}{5} \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{1}{3} d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2+d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b^2 d^2 \left(1-c^2 x^2\right)^3 \sqrt{d-c^2 d x^2}}{125 (1-c x) (c x+1)}+\frac{68}{27} b^2 d^2 \sqrt{d-c^2 d x^2}+\frac{8 b^2 d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2}}{225 (1-c x) (c x+1)}+\frac{16 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}{75 (1-c x) (c x+1)}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*((23*a^2*d^2)/15 - (11*a^2*c^2*d^2*x^2)/15 + (a^2*c^4*d^2*x^4)/5) + (a*b*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-9*c*x - 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] + Cosh[3*ArcCosh[c*x]]))/(9*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b^2*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-26 + (27*c*x*ArcCosh[c*x])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - 9*ArcCosh[c*x]^2 + (2 + 9*ArcCosh[c*x]^2)*Cosh[2*ArcCosh[c*x]] - (3*ArcCosh[c*x]*Cosh[3*ArcCosh[c*x]])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/27 + a^2*d^(5/2)*Log[c*x] - a^2*d^(5/2)*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]] + 2*a*b*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-((c*x)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + ArcCosh[c*x] + (I*ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (I*(PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + b^2*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(2 - (2*c*x*ArcCosh[c*x])/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + ArcCosh[c*x]^2 + (I*(ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] - ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] + 2*PolyLog[3, (-I)/E^ArcCosh[c*x]] - 2*PolyLog[3, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) - (a*b*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-450*c*x + 450*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + 25*Cosh[3*ArcCosh[c*x]] + 9*Cosh[5*ArcCosh[c*x]] - 75*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] - 45*ArcCosh[c*x]*Sinh[5*ArcCosh[c*x]]))/(1800*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b^2*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(13500*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - 13500*c*x*ArcCosh[c*x] + 6750*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2 + 750*ArcCosh[c*x]*Cosh[3*ArcCosh[c*x]] + 270*ArcCosh[c*x]*Cosh[5*ArcCosh[c*x]] - 250*Sinh[3*ArcCosh[c*x]] - 1125*ArcCosh[c*x]^2*Sinh[3*ArcCosh[c*x]] - 54*Sinh[5*ArcCosh[c*x]] - 675*ArcCosh[c*x]^2*Sinh[5*ArcCosh[c*x]]))/(54000*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
191,1,554,607,5.9365785,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^2,x]","\frac{d^2 \left(1440 a^2 c \sqrt{d} x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+96 a^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(2 c^4 x^4-9 c^2 x^2-8\right) \sqrt{d-c^2 d x^2}-768 a b \sqrt{d-c^2 d x^2} \left(2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left(2 \log (c x)+\cosh ^{-1}(c x)^2\right)\right)+384 a b c x \sqrt{d-c^2 d x^2} \left(\cosh \left(2 \cosh ^{-1}(c x)\right)+2 \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-\sinh \left(2 \cosh ^{-1}(c x)\right)\right)\right)-12 a b c x \sqrt{d-c^2 d x^2} \left(8 \cosh ^{-1}(c x)^2+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)-256 b^2 \sqrt{d-c^2 d x^2} \left(3 c x \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+\cosh ^{-1}(c x) \left(3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+3\right)+6 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)\right)\right)+64 b^2 c x \sqrt{d-c^2 d x^2} \left(4 \cosh ^{-1}(c x)^3+6 \cosh \left(2 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-3 \left(2 \cosh ^{-1}(c x)^2+1\right) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)-b^2 c x \sqrt{d-c^2 d x^2} \left(32 \cosh ^{-1}(c x)^3+12 \cosh \left(4 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-3 \left(8 \cosh ^{-1}(c x)^2+1\right) \sinh \left(4 \cosh ^{-1}(c x)\right)\right)\right)}{768 x \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","-\frac{15}{8} c^2 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2+\frac{5 c d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{8 b \sqrt{c x-1} \sqrt{c x+1}}+\frac{c d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{4} c^2 d x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x}+\frac{15 b c^3 d^2 x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{8 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b^2 c d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{31}{64} b^2 c^2 d^2 x \sqrt{d-c^2 d x^2}-\frac{1}{32} b^2 c^2 d^2 x (1-c x) (c x+1) \sqrt{d-c^2 d x^2}-\frac{89 b^2 c d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{64 \sqrt{c x-1} \sqrt{c x+1}}",1,"(d^2*(96*a^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(-8 - 9*c^2*x^2 + 2*c^4*x^4) + 1440*a^2*c*Sqrt[d]*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 768*a*b*Sqrt[d - c^2*d*x^2]*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]^2 + 2*Log[c*x])) - 256*b^2*Sqrt[d - c^2*d*x^2]*(ArcCosh[c*x]*(3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]*(3 + ArcCosh[c*x]) + 6*Log[1 + E^(-2*ArcCosh[c*x])])) + 3*c*x*PolyLog[2, -E^(-2*ArcCosh[c*x])]) + 384*a*b*c*x*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + 64*b^2*c*x*Sqrt[d - c^2*d*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*(1 + 2*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]) - 12*a*b*c*x*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) - b^2*c*x*Sqrt[d - c^2*d*x^2]*(32*ArcCosh[c*x]^3 + 12*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 3*(1 + 8*ArcCosh[c*x]^2)*Sinh[4*ArcCosh[c*x]])))/(768*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
192,1,1384,890,90.3063052,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^3,x]","\frac{i a b c^2 \left(-\frac{i (c x-1) \cosh ^{-1}(c x) (c x+1)}{c^2 x^2}+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right) (c x+1)-\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right) (c x+1)+\sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right) (c x+1)-\sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right) (c x+1)-\frac{i \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}\right) d^3}{\sqrt{-d (c x-1) (c x+1)}}-\frac{5}{2} a^2 c^2 \log (x) d^{5/2}+\frac{5}{2} a^2 c^2 \log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right) d^{5/2}-\frac{a b c^2 \sqrt{-d (c x-1) (c x+1)} \left(-12 \left(\frac{c x-1}{c x+1}\right)^{3/2} \cosh ^{-1}(c x) (c x+1)^3-9 c x+\cosh \left(3 \cosh ^{-1}(c x)\right)\right) d^2}{18 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}-4 a b c^2 \sqrt{-d (c x-1) (c x+1)} \left(-\frac{c x}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\cosh ^{-1}(c x)+\frac{i \cosh ^{-1}(c x) \left(\log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{i \left(\text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right) d^2+\frac{1}{54} b^2 \sqrt{d-c^2 d x^2} \left(-\frac{12 x^4 \cosh ^{-1}(c x) c^6}{(c x-1)^{3/2} \sqrt{c x+1}}+\frac{18 x^3 \cosh ^{-1}(c x)^2 c^5}{c x-1}+\frac{12 x^3 \cosh ^{-1}(c x) c^5}{(c x-1)^{3/2} \sqrt{c x+1}}+\frac{4 x^3 c^5}{c x-1}-\frac{18 x^2 \cosh ^{-1}(c x)^2 c^4}{c x-1}+\frac{252 x^2 \cosh ^{-1}(c x) c^4}{(c x-1)^{3/2} \sqrt{c x+1}}-\frac{4 x^2 c^4}{c x-1}-\frac{126 x \cosh ^{-1}(c x)^2 c^3}{c x-1}-\frac{252 x \cosh ^{-1}(c x) c^3}{(c x-1)^{3/2} \sqrt{c x+1}}-\frac{54 x \tan ^{-1}\left(\frac{1}{\sqrt{c^2 x^2-1}}\right) c^3}{(c x-1) \sqrt{c^2 x^2-1}}-\frac{244 x c^3}{c x-1}+\frac{126 \cosh ^{-1}(c x)^2 c^2}{c x-1}-\frac{54 \cosh ^{-1}(c x) c^2}{(c x-1)^{3/2} \sqrt{c x+1}}+\frac{54 \tan ^{-1}\left(\frac{1}{\sqrt{c^2 x^2-1}}\right) c^2}{(c x-1) \sqrt{c^2 x^2-1}}-\frac{135 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)^2 \log \left(1-i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}+\frac{135 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)^2 \log \left(1+i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}-\frac{270 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}+\frac{270 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}-\frac{270 i \sqrt{\frac{c x-1}{c x+1}} \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}+\frac{270 i \sqrt{\frac{c x-1}{c x+1}} \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right) c^2}{c x-1}+\frac{244 c^2}{c x-1}+\frac{27 \cosh ^{-1}(c x)^2 c}{x-c x^2}+\frac{54 \cosh ^{-1}(c x) c}{x (c x-1)^{3/2} \sqrt{c x+1}}+\frac{27 \cosh ^{-1}(c x)^2}{x^2 (c x-1)}\right) d^2+\sqrt{-d \left(c^2 x^2-1\right)} \left(\frac{1}{3} a^2 d^2 x^2 c^4-\frac{7}{3} a^2 d^2 c^2-\frac{a^2 d^2}{2 x^2}\right)","-\frac{2 b d^2 x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) c^5}{9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{27} b^2 d^2 x^2 \sqrt{d-c^2 d x^2} c^4+\frac{5 b^2 d^2 x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) c^3}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) c^3}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 a b d^2 x \sqrt{d-c^2 d x^2} c^3}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{5}{6} d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 c^2-\frac{5}{2} d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 c^2+\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) c^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b^2 d^2 \sqrt{c^2 x^2-1} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right) c^2}{(1-c x) (c x+1)}-\frac{5 i b d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) c^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{5 i b d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) c^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{5 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right) c^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{5 i b^2 d^2 \sqrt{d-c^2 d x^2} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right) c^2}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{170}{27} b^2 d^2 \sqrt{d-c^2 d x^2} c^2+\frac{b^2 d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} c^2}{9 (1-c x) (c x+1)}+\frac{5 b^2 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} c^2}{3 (1-c x) (c x+1)}-\frac{b d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) c}{x \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 x^2}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*((-7*a^2*c^2*d^2)/3 - (a^2*d^2)/(2*x^2) + (a^2*c^4*d^2*x^2)/3) - (a*b*c^2*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-9*c*x - 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] + Cosh[3*ArcCosh[c*x]]))/(18*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (5*a^2*c^2*d^(5/2)*Log[x])/2 + (5*a^2*c^2*d^(5/2)*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/2 - 4*a*b*c^2*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-((c*x)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + ArcCosh[c*x] + (I*ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (I*(PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (I*a*b*c^2*d^3*(((-I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) - (I*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x])/(c^2*x^2) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[-(d*(-1 + c*x)*(1 + c*x))] + (b^2*d^2*Sqrt[d - c^2*d*x^2]*((244*c^2)/(-1 + c*x) - (244*c^3*x)/(-1 + c*x) - (4*c^4*x^2)/(-1 + c*x) + (4*c^5*x^3)/(-1 + c*x) - (54*c^2*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (54*c*ArcCosh[c*x])/(x*(-1 + c*x)^(3/2)*Sqrt[1 + c*x]) - (252*c^3*x*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (252*c^4*x^2*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (12*c^5*x^3*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) - (12*c^6*x^4*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (126*c^2*ArcCosh[c*x]^2)/(-1 + c*x) + (27*ArcCosh[c*x]^2)/(x^2*(-1 + c*x)) - (126*c^3*x*ArcCosh[c*x]^2)/(-1 + c*x) - (18*c^4*x^2*ArcCosh[c*x]^2)/(-1 + c*x) + (18*c^5*x^3*ArcCosh[c*x]^2)/(-1 + c*x) + (27*c*ArcCosh[c*x]^2)/(x - c*x^2) + (54*c^2*ArcTan[1/Sqrt[-1 + c^2*x^2]])/((-1 + c*x)*Sqrt[-1 + c^2*x^2]) - (54*c^3*x*ArcTan[1/Sqrt[-1 + c^2*x^2]])/((-1 + c*x)*Sqrt[-1 + c^2*x^2]) - ((135*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]])/(-1 + c*x) + ((135*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]])/(-1 + c*x) - ((270*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]])/(-1 + c*x) + ((270*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]])/(-1 + c*x) - ((270*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[3, (-I)/E^ArcCosh[c*x]])/(-1 + c*x) + ((270*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[3, I/E^ArcCosh[c*x]])/(-1 + c*x)))/54","A",0
193,1,803,638,3.1656503,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/x^4,x]","\frac{-12 a^2 c^6 d^3 \sqrt{\frac{c x-1}{c x+1}} x^6+6 a b c^4 d^3 \cosh \left(2 \cosh ^{-1}(c x)\right) x^4+112 a b c^4 d^3 \log (c x) x^4-3 b^2 c^4 d^3 \sinh \left(2 \cosh ^{-1}(c x)\right) x^4-44 a^2 c^4 d^3 \sqrt{\frac{c x-1}{c x+1}} x^4-8 b^2 c^4 d^3 \sqrt{\frac{c x-1}{c x+1}} x^4+20 b^2 c^3 d^3 (c x-1) \cosh ^{-1}(c x)^3 x^3-60 a^2 c^3 d^{5/2} \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right) x^3-6 a b c^3 d^3 \cosh \left(2 \cosh ^{-1}(c x)\right) x^3-112 a b c^3 d^3 \log (c x) x^3-56 b^2 c^3 d^3 (c x-1) \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right) x^3+3 b^2 c^3 d^3 \sinh \left(2 \cosh ^{-1}(c x)\right) x^3+8 a b c^2 d^3 x^2+64 a^2 c^2 d^3 \sqrt{\frac{c x-1}{c x+1}} x^2+8 b^2 c^2 d^3 \sqrt{\frac{c x-1}{c x+1}} x^2-8 a b c d^3 x+2 b d^3 (c x-1) \cosh ^{-1}(c x) \left(3 b c^3 \cosh \left(2 \cosh ^{-1}(c x)\right) x^3+56 b c^3 \log \left(1+e^{-2 \cosh ^{-1}(c x)}\right) x^3-6 a c^3 \sinh \left(2 \cosh ^{-1}(c x)\right) x^3-56 a c^3 \sqrt{\frac{c x-1}{c x+1}} x^3-56 a c^2 \sqrt{\frac{c x-1}{c x+1}} x^2+4 b c x+8 a c \sqrt{\frac{c x-1}{c x+1}} x+8 a \sqrt{\frac{c x-1}{c x+1}}\right)-2 b d^3 (c x-1) \cosh ^{-1}(c x)^2 \left(-30 a c^3 x^3+3 b c^3 \sinh \left(2 \cosh ^{-1}(c x)\right) x^3+4 b \left(7 c^3 \left(\sqrt{\frac{c x-1}{c x+1}}-1\right) x^3+7 c^2 \sqrt{\frac{c x-1}{c x+1}} x^2-c \sqrt{\frac{c x-1}{c x+1}} x-\sqrt{\frac{c x-1}{c x+1}}\right)\right)-8 a^2 d^3 \sqrt{\frac{c x-1}{c x+1}}}{24 x^3 \sqrt{\frac{c x-1}{c x+1}} \sqrt{d-c^2 d x^2}}","-\frac{b c d^2 \left(1-c^2 x^2\right)^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 c^2 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 x}-\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 x^3}-\frac{5 b c^5 d^2 x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5}{2} c^4 d^2 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{5 c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^3}{6 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{7 b c^3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)}{3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{14 b c^3 d^2 \sqrt{d-c^2 d x^2} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b^2 c^2 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}{3 x}+\frac{7}{12} b^2 c^4 d^2 x \sqrt{d-c^2 d x^2}+\frac{7 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{23 b^2 c^3 d^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{12 \sqrt{c x-1} \sqrt{c x+1}}",1,"(-8*a*b*c*d^3*x + 8*a*b*c^2*d^3*x^2 - 8*a^2*d^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 64*a^2*c^2*d^3*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 8*b^2*c^2*d^3*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 44*a^2*c^4*d^3*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 8*b^2*c^4*d^3*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 12*a^2*c^6*d^3*x^6*Sqrt[(-1 + c*x)/(1 + c*x)] + 20*b^2*c^3*d^3*x^3*(-1 + c*x)*ArcCosh[c*x]^3 - 60*a^2*c^3*d^(5/2)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 6*a*b*c^3*d^3*x^3*Cosh[2*ArcCosh[c*x]] + 6*a*b*c^4*d^3*x^4*Cosh[2*ArcCosh[c*x]] - 112*a*b*c^3*d^3*x^3*Log[c*x] + 112*a*b*c^4*d^3*x^4*Log[c*x] - 56*b^2*c^3*d^3*x^3*(-1 + c*x)*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 3*b^2*c^3*d^3*x^3*Sinh[2*ArcCosh[c*x]] - 3*b^2*c^4*d^3*x^4*Sinh[2*ArcCosh[c*x]] + 2*b*d^3*(-1 + c*x)*ArcCosh[c*x]*(4*b*c*x + 8*a*Sqrt[(-1 + c*x)/(1 + c*x)] + 8*a*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 56*a*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] - 56*a*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 3*b*c^3*x^3*Cosh[2*ArcCosh[c*x]] + 56*b*c^3*x^3*Log[1 + E^(-2*ArcCosh[c*x])] - 6*a*c^3*x^3*Sinh[2*ArcCosh[c*x]]) - 2*b*d^3*(-1 + c*x)*ArcCosh[c*x]^2*(-30*a*c^3*x^3 + 4*b*(-Sqrt[(-1 + c*x)/(1 + c*x)] - c*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 7*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 7*c^3*x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)])) + 3*b*c^3*x^3*Sinh[2*ArcCosh[c*x]]))/(24*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])","A",0
194,1,255,421,0.553879,"\int \frac{x^5 \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^5*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{d-c^2 d x^2} \left(-225 a^2 \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right)+30 a b c x \sqrt{c x-1} \sqrt{c x+1} \left(9 c^4 x^4+20 c^2 x^2+120\right)+30 b \cosh ^{-1}(c x) \left(b c x \sqrt{c x-1} \sqrt{c x+1} \left(9 c^4 x^4+20 c^2 x^2+120\right)-15 a \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right)\right)-2 b^2 \left(27 c^6 x^6+109 c^4 x^4+1936 c^2 x^2-2072\right)-225 b^2 \left(3 c^6 x^6+c^4 x^4+4 c^2 x^2-8\right) \cosh ^{-1}(c x)^2\right)}{3375 c^6 d (c x-1) (c x+1)}","-\frac{2 b x^5 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{25 c \sqrt{d-c^2 d x^2}}-\frac{x^4 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{5 c^2 d}-\frac{8 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{15 c^6 d}-\frac{16 a b x \sqrt{c x-1} \sqrt{c x+1}}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{15 c^4 d}-\frac{8 b x^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{45 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^4 (1-c x) (c x+1)}{125 c^2 \sqrt{d-c^2 d x^2}}-\frac{4144 b^2 (1-c x) (c x+1)}{3375 c^6 \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{15 c^5 \sqrt{d-c^2 d x^2}}-\frac{272 b^2 x^2 (1-c x) (c x+1)}{3375 c^4 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(30*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 225*a^2*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6) - 2*b^2*(-2072 + 1936*c^2*x^2 + 109*c^4*x^4 + 27*c^6*x^6) + 30*b*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(120 + 20*c^2*x^2 + 9*c^4*x^4) - 15*a*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6))*ArcCosh[c*x] - 225*b^2*(-8 + 4*c^2*x^2 + c^4*x^4 + 3*c^6*x^6)*ArcCosh[c*x]^2))/(3375*c^6*d*(-1 + c*x)*(1 + c*x))","A",1
195,1,295,355,1.5404512,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{32 a^2 c \sqrt{d} x \left(c^2 x^2-1\right) \left(2 c^2 x^2+3\right)-96 a^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)-4 a b \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(16 \cosh \left(2 \cosh ^{-1}(c x)\right)+\cosh \left(4 \cosh ^{-1}(c x)\right)-4 \cosh ^{-1}(c x) \left(6 \cosh ^{-1}(c x)+8 \sinh \left(2 \cosh ^{-1}(c x)\right)+\sinh \left(4 \cosh ^{-1}(c x)\right)\right)\right)+b^2 \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(32 \cosh ^{-1}(c x)^3-4 \left(16 \cosh \left(2 \cosh ^{-1}(c x)\right)+\cosh \left(4 \cosh ^{-1}(c x)\right)\right) \cosh ^{-1}(c x)+8 \cosh ^{-1}(c x)^2 \left(8 \sinh \left(2 \cosh ^{-1}(c x)\right)+\sinh \left(4 \cosh ^{-1}(c x)\right)\right)+32 \sinh \left(2 \cosh ^{-1}(c x)\right)+\sinh \left(4 \cosh ^{-1}(c x)\right)\right)}{256 c^5 \sqrt{d} \sqrt{d-c^2 d x^2}}","-\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{8 c \sqrt{d-c^2 d x^2}}-\frac{x^3 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{4 c^2 d}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^3}{8 b c^5 \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{8 c^4 d}-\frac{3 b x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{8 c^3 \sqrt{d-c^2 d x^2}}-\frac{b^2 x^3 (1-c x) (c x+1)}{32 c^2 \sqrt{d-c^2 d x^2}}+\frac{15 b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{64 c^5 \sqrt{d-c^2 d x^2}}-\frac{15 b^2 x (1-c x) (c x+1)}{64 c^4 \sqrt{d-c^2 d x^2}}",1,"(32*a^2*c*Sqrt[d]*x*(-1 + c^2*x^2)*(3 + 2*c^2*x^2) - 96*a^2*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + b^2*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(32*ArcCosh[c*x]^3 - 4*ArcCosh[c*x]*(16*Cosh[2*ArcCosh[c*x]] + Cosh[4*ArcCosh[c*x]]) + 32*Sinh[2*ArcCosh[c*x]] + Sinh[4*ArcCosh[c*x]] + 8*ArcCosh[c*x]^2*(8*Sinh[2*ArcCosh[c*x]] + Sinh[4*ArcCosh[c*x]])) - 4*a*b*Sqrt[d]*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(16*Cosh[2*ArcCosh[c*x]] + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*(6*ArcCosh[c*x] + 8*Sinh[2*ArcCosh[c*x]] + Sinh[4*ArcCosh[c*x]])))/(256*c^5*Sqrt[d]*Sqrt[d - c^2*d*x^2])","A",0
196,1,201,292,0.4540805,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{d-c^2 d x^2} \left(-9 a^2 \left(c^4 x^4+c^2 x^2-2\right)+6 a b c x \sqrt{c x-1} \sqrt{c x+1} \left(c^2 x^2+6\right)+6 b \cosh ^{-1}(c x) \left(b c x \sqrt{c x-1} \sqrt{c x+1} \left(c^2 x^2+6\right)-3 a \left(c^4 x^4+c^2 x^2-2\right)\right)-2 b^2 \left(c^4 x^4+19 c^2 x^2-20\right)-9 b^2 \left(c^4 x^4+c^2 x^2-2\right) \cosh ^{-1}(c x)^2\right)}{27 c^4 d (c x-1) (c x+1)}","-\frac{x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^2 d}-\frac{2 b x^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^4 d}-\frac{4 a b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^3 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x^2 (1-c x) (c x+1)}{27 c^2 \sqrt{d-c^2 d x^2}}-\frac{40 b^2 (1-c x) (c x+1)}{27 c^4 \sqrt{d-c^2 d x^2}}-\frac{4 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^3 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(6*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(6 + c^2*x^2) - 9*a^2*(-2 + c^2*x^2 + c^4*x^4) - 2*b^2*(-20 + 19*c^2*x^2 + c^4*x^4) + 6*b*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(6 + c^2*x^2) - 3*a*(-2 + c^2*x^2 + c^4*x^4))*ArcCosh[c*x] - 9*b^2*(-2 + c^2*x^2 + c^4*x^4)*ArcCosh[c*x]^2))/(27*c^4*d*(-1 + c*x)*(1 + c*x))","A",1
197,1,228,226,0.8832265,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{-\frac{12 a^2 c x \sqrt{d-c^2 d x^2}}{d}-\frac{12 a^2 \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)}{\sqrt{d}}+\frac{6 a b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(2 \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+\sinh \left(2 \cosh ^{-1}(c x)\right)\right)-\cosh \left(2 \cosh ^{-1}(c x)\right)\right)}{\sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(4 \cosh ^{-1}(c x)^3-6 \cosh \left(2 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)+\left(6 \cosh ^{-1}(c x)^2+3\right) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)}{\sqrt{d-c^2 d x^2}}}{24 c^3}","-\frac{x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 c^2 d}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{2 c \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^3}{6 b c^3 \sqrt{d-c^2 d x^2}}-\frac{b^2 x (1-c x) (c x+1)}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{4 c^3 \sqrt{d-c^2 d x^2}}",1,"((-12*a^2*c*x*Sqrt[d - c^2*d*x^2])/d - (12*a^2*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqrt[d] + (b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(4*ArcCosh[c*x]^3 - 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] + (3 + 6*ArcCosh[c*x]^2)*Sinh[2*ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2] + (6*a*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] + Sinh[2*ArcCosh[c*x]])))/Sqrt[d - c^2*d*x^2])/(24*c^3)","A",0
198,1,149,155,0.3933292,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{d-c^2 d x^2} \left(a^2 \left(1-c^2 x^2\right)+2 b \cosh ^{-1}(c x) \left(-a c^2 x^2+a+b c x \sqrt{c x-1} \sqrt{c x+1}\right)+2 a b c x \sqrt{c x-1} \sqrt{c x+1}-2 b^2 \left(c^2 x^2-1\right)+b^2 \left(1-c^2 x^2\right) \cosh ^{-1}(c x)^2\right)}{c^2 d (c x-1) (c x+1)}","-\frac{2 a b x \sqrt{c x-1} \sqrt{c x+1}}{c \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{c^2 d}-\frac{2 b^2 (1-c x) (c x+1)}{c^2 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c \sqrt{d-c^2 d x^2}}",1,"(Sqrt[d - c^2*d*x^2]*(2*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + a^2*(1 - c^2*x^2) - 2*b^2*(-1 + c^2*x^2) + 2*b*(a - a*c^2*x^2 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])*ArcCosh[c*x] + b^2*(1 - c^2*x^2)*ArcCosh[c*x]^2))/(c^2*d*(-1 + c*x)*(1 + c*x))","A",1
199,1,53,53,0.0476923,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^3}{3 b c \sqrt{d-c^2 d x^2}}","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^3}{3 b c \sqrt{d-c^2 d x^2}}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(3*b*c*Sqrt[d - c^2*d*x^2])","A",1
200,1,315,273,0.6262184,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x*Sqrt[d - c^2*d*x^2]),x]","-\frac{a^2 \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)}{\sqrt{d}}+\frac{a^2 \log (c x)}{\sqrt{d}}-\frac{2 i a b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+\cosh ^{-1}(c x) \left(\log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)\right)}{\sqrt{d-c^2 d x^2}}+\frac{i b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(-2 \cosh ^{-1}(c x) \left(\text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)\right)-2 \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)+2 \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)-\left(\cosh ^{-1}(c x)^2 \left(\log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\log \left(1+i e^{-\cosh ^{-1}(c x)}\right)\right)\right)\right)}{\sqrt{d-c^2 d x^2}}","-\frac{2 i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}+\frac{2 i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}}+\frac{2 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}-\frac{2 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}",1,"(a^2*Log[c*x])/Sqrt[d] - (a^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] - ((2*I)*a*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]]) + PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2] + (I*b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-(ArcCosh[c*x]^2*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x]])) - 2*ArcCosh[c*x]*(PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcCosh[c*x]]) - 2*PolyLog[3, (-I)/E^ArcCosh[c*x]] + 2*PolyLog[3, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2]","A",0
201,1,179,186,0.8642257,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x^2 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x^2*Sqrt[d - c^2*d*x^2]),x]","\frac{\frac{a^2 \left(c^2 x^2-1\right)}{x}-\frac{2 a b \left(c^2 x^2-1\right) \left(\frac{c x \log (c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\cosh ^{-1}(c x)\right)}{x}+b^2 c \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+\cosh ^{-1}(c x) \left(\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{c x}-\cosh ^{-1}(c x)-2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)\right)}{\sqrt{d-c^2 d x^2}}","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{d x}-\frac{c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}}-\frac{2 b c \sqrt{c x-1} \sqrt{c x+1} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}+\frac{b^2 c \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}",1,"((a^2*(-1 + c^2*x^2))/x - (2*a*b*(-1 + c^2*x^2)*(-ArcCosh[c*x] + (c*x*Log[c*x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/x + b^2*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(ArcCosh[c*x]*(-ArcCosh[c*x] + (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/(c*x) - 2*Log[1 + E^(-2*ArcCosh[c*x])]) + PolyLog[2, -E^(-2*ArcCosh[c*x])]))/Sqrt[d - c^2*d*x^2]","A",0
202,1,551,430,84.8304156,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x^3 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x^3*Sqrt[d - c^2*d*x^2]),x]","\frac{1}{2} a \left(-\frac{a \sqrt{d-c^2 d x^2}}{d x^2}-\frac{a c^2 \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)}{\sqrt{d}}+\frac{a c^2 \log (x)}{\sqrt{d}}+\frac{2 b (c x+1) \left(-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)+i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)+i c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+c x \sqrt{\frac{c x-1}{c x+1}}+c x \cosh ^{-1}(c x)-\cosh ^{-1}(c x)\right)}{x^2 \sqrt{d-c^2 d x^2}}\right)-\frac{b^2 c^2 \sqrt{d-c^2 d x^2} \left(\frac{\cosh ^{-1}(c x)^2}{c^2 x^2}+\frac{2 \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)}{c^2 x^2-c x}-\frac{i \left(2 \cosh ^{-1}(c x) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 \cosh ^{-1}(c x) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+2 \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)+\cosh ^{-1}(c x)^2 \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\cosh ^{-1}(c x)^2 \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)-4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right)}{2 d}","-\frac{i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}+\frac{i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{\sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{x \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 d x^2}+\frac{c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}}+\frac{i b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}-\frac{i b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{\sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{\sqrt{d-c^2 d x^2}}",1,"(a*(-((a*Sqrt[d - c^2*d*x^2])/(d*x^2)) + (a*c^2*Log[x])/Sqrt[d] - (a*c^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] + (2*b*(1 + c*x)*(c*x*Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCosh[c*x] + c*x*ArcCosh[c*x] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, (-I)/E^ArcCosh[c*x]] + I*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[2, I/E^ArcCosh[c*x]]))/(x^2*Sqrt[d - c^2*d*x^2])))/2 - (b^2*c^2*Sqrt[d - c^2*d*x^2]*((2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x])/(-(c*x) + c^2*x^2) + ArcCosh[c*x]^2/(c^2*x^2) - (I*((-4*I)*ArcTan[Tanh[ArcCosh[c*x]/2]] + ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] - ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] + 2*PolyLog[3, (-I)/E^ArcCosh[c*x]] - 2*PolyLog[3, I/E^ArcCosh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/(2*d)","A",0
203,1,346,328,1.6409058,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x^4 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x^4*Sqrt[d - c^2*d*x^2]),x]","\frac{(c x-1) \sqrt{c x+1} \left(a^2 \sqrt{c x-1} \sqrt{c x+1} \left(2 c^2 x^2+1\right)-4 a b c^3 x^3 \log (c x)+a b c x-b^2 c^2 x^2 \sqrt{c x-1} \sqrt{c x+1}\right)+b \sqrt{c x-1} (c x+1) \cosh ^{-1}(c x) \left(2 a \left(2 c^3 x^3-2 c^2 x^2+c x-1\right)-4 b c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)+b c x \sqrt{\frac{c x-1}{c x+1}}\right)+\frac{2 b^2 c^3 x^3 (c x-1)^{3/2} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{\sqrt{\frac{c x-1}{c x+1}}}-b^2 \sqrt{c x-1} (c x+1) \left(2 c^3 x^3 \left(\sqrt{\frac{c x-1}{c x+1}}-1\right)+2 c^2 x^2-c x+1\right) \cosh ^{-1}(c x)^2}{3 x^3 \sqrt{c x-1} \sqrt{d-c^2 d x^2}}","-\frac{2 c^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d x}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 x^2 \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d x^3}-\frac{2 c^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 \sqrt{d-c^2 d x^2}}-\frac{4 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2 (1-c x) (c x+1)}{3 x \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{3 \sqrt{d-c^2 d x^2}}",1,"(-(b^2*Sqrt[-1 + c*x]*(1 + c*x)*(1 - c*x + 2*c^2*x^2 + 2*c^3*x^3*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)]))*ArcCosh[c*x]^2) + b*Sqrt[-1 + c*x]*(1 + c*x)*ArcCosh[c*x]*(b*c*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 2*a*(-1 + c*x - 2*c^2*x^2 + 2*c^3*x^3) - 4*b*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*Log[1 + E^(-2*ArcCosh[c*x])]) + (-1 + c*x)*Sqrt[1 + c*x]*(a*b*c*x - b^2*c^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + a^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 + 2*c^2*x^2) - 4*a*b*c^3*x^3*Log[c*x]) + (2*b^2*c^3*x^3*(-1 + c*x)^(3/2)*PolyLog[2, -E^(-2*ArcCosh[c*x])])/Sqrt[(-1 + c*x)/(1 + c*x)])/(3*x^3*Sqrt[-1 + c*x]*Sqrt[d - c^2*d*x^2])","A",0
204,1,358,556,3.8386752,"\int \frac{x^5 \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{-36 a^2 \left(c^4 x^4+4 c^2 x^2-8\right)+3 a b \left(-60 \cosh \left(2 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-3 \cosh \left(4 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)+135 \cosh ^{-1}(c x)+62 \sinh \left(2 \cosh ^{-1}(c x)\right)+\sinh \left(4 \cosh ^{-1}(c x)\right)-72 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)-b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(216 \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)-216 \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)+378 \sqrt{\frac{c x-1}{c x+1}} (c x+1)+189 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2-378 c x \cosh ^{-1}(c x)-6 \cosh \left(3 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)+216 \cosh ^{-1}(c x) \log \left(1-e^{-\cosh ^{-1}(c x)}\right)-216 \cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right)+9 \cosh ^{-1}(c x)^2 \sinh \left(3 \cosh ^{-1}(c x)\right)+2 \sinh \left(3 \cosh ^{-1}(c x)\right)+54 \cosh ^{-1}(c x)^2 \tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)-54 \cosh ^{-1}(c x)^2 \coth \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)}{108 c^6 d \sqrt{d-c^2 d x^2}}","\frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^6 d^2}+\frac{4 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{16 a b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^5 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^4 d^2}+\frac{2 b x^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{c^6 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{94 b^2 (1-c x) (c x+1)}{27 c^6 d \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x^2 (1-c x) (c x+1)}{27 c^4 d \sqrt{d-c^2 d x^2}}",1,"(-36*a^2*(-8 + 4*c^2*x^2 + c^4*x^4) + 3*a*b*(135*ArcCosh[c*x] - 60*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 72*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + 62*Sinh[2*ArcCosh[c*x]] + Sinh[4*ArcCosh[c*x]]) - b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(378*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - 378*c*x*ArcCosh[c*x] + 189*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2 - 6*ArcCosh[c*x]*Cosh[3*ArcCosh[c*x]] - 54*ArcCosh[c*x]^2*Coth[ArcCosh[c*x]/2] + 216*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] - 216*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 216*PolyLog[2, -E^(-ArcCosh[c*x])] - 216*PolyLog[2, E^(-ArcCosh[c*x])] + 2*Sinh[3*ArcCosh[c*x]] + 9*ArcCosh[c*x]^2*Sinh[3*ArcCosh[c*x]] + 54*ArcCosh[c*x]^2*Tanh[ArcCosh[c*x]/2]))/(108*c^6*d*Sqrt[d - c^2*d*x^2])","A",0
205,1,343,440,2.0036124,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{-4 a^2 c \sqrt{d} x \left(c^2 x^2-3\right)+12 a^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+2 a b \sqrt{d} \left(8 c x \cosh ^{-1}(c x)-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(8 \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)+6 \cosh ^{-1}(c x)^2-\cosh \left(2 \cosh ^{-1}(c x)\right)+2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)\right)\right)+b^2 \sqrt{d} \left(8 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)+8 c x \cosh ^{-1}(c x)^2-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(4 \cosh ^{-1}(c x)^3-2 \cosh ^{-1}(c x) \left(\cosh \left(2 \cosh ^{-1}(c x)\right)-8 \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)\right)+2 \cosh ^{-1}(c x)^2 \left(\sinh \left(2 \cosh ^{-1}(c x)\right)+4\right)+\sinh \left(2 \cosh ^{-1}(c x)\right)\right)\right)}{8 c^5 d^{3/2} \sqrt{d-c^2 d x^2}}","\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^3}{2 b c^5 d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{c^5 d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{3 x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{2 c^4 d^2}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{2 c^3 d \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{c^5 d \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{4 c^5 d \sqrt{d-c^2 d x^2}}+\frac{b^2 x (1-c x) (c x+1)}{4 c^4 d \sqrt{d-c^2 d x^2}}",1,"(-4*a^2*c*Sqrt[d]*x*(-3 + c^2*x^2) + 12*a^2*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 2*a*b*Sqrt[d]*(8*c*x*ArcCosh[c*x] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(6*ArcCosh[c*x]^2 - Cosh[2*ArcCosh[c*x]] + 8*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)] + 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]])) + b^2*Sqrt[d]*(8*c*x*ArcCosh[c*x]^2 + 8*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(4*ArcCosh[c*x]^3 - 2*ArcCosh[c*x]*(Cosh[2*ArcCosh[c*x]] - 8*Log[1 - E^(-2*ArcCosh[c*x])]) + Sinh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]^2*(4 + Sinh[2*ArcCosh[c*x]]))))/(8*c^5*d^(3/2)*Sqrt[d - c^2*d*x^2])","A",0
206,1,302,413,1.6435413,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{-2 a^2 \left(c^2 x^2-2\right)+2 a b \left(-\cosh ^{-1}(c x) \left(\cosh \left(2 \cosh ^{-1}(c x)\right)-3\right)+\sinh \left(2 \cosh ^{-1}(c x)\right)-2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)+b^2 \left(-4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)-\cosh \left(2 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)^2+3 \cosh ^{-1}(c x)^2-2 \cosh \left(2 \cosh ^{-1}(c x)\right)-4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-e^{-\cosh ^{-1}(c x)}\right)+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right)+2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)+2\right)}{2 c^4 d \sqrt{d-c^2 d x^2}}","\frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{c^4 d^2}+\frac{4 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 a b x \sqrt{c x-1} \sqrt{c x+1}}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 (1-c x) (c x+1)}{c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{c^3 d \sqrt{d-c^2 d x^2}}",1,"(-2*a^2*(-2 + c^2*x^2) + 2*a*b*(-(ArcCosh[c*x]*(-3 + Cosh[2*ArcCosh[c*x]])) - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + Sinh[2*ArcCosh[c*x]]) + b^2*(2 + 3*ArcCosh[c*x]^2 - 2*Cosh[2*ArcCosh[c*x]] - ArcCosh[c*x]^2*Cosh[2*ArcCosh[c*x]] - 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] - 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-ArcCosh[c*x])] + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-ArcCosh[c*x])] + 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]]))/(2*c^4*d*Sqrt[d - c^2*d*x^2])","A",0
207,1,270,257,2.0470195,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{3 a^2 \sqrt{d} \sqrt{d-c^2 d x^2} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+3 a^2 c d x+3 a b d \left(2 c x \cosh ^{-1}(c x)-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(2 \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)+\cosh ^{-1}(c x)^2\right)\right)-b^2 d \left(\cosh ^{-1}(c x) \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+3\right)+6 \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)\right)-3 c x \cosh ^{-1}(c x)\right)-3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)\right)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}","\frac{x \left(a+b \cosh ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^3}{3 b c^3 d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^3 d \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{c^3 d \sqrt{d-c^2 d x^2}}",1,"(3*a^2*c*d*x + 3*a^2*Sqrt[d]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 3*a*b*d*(2*c*x*ArcCosh[c*x] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(ArcCosh[c*x]^2 + 2*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])) - b^2*d*(ArcCosh[c*x]*(-3*c*x*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(ArcCosh[c*x]*(3 + ArcCosh[c*x]) + 6*Log[1 - E^(-2*ArcCosh[c*x])])) - 3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*c^3*d^2*Sqrt[d - c^2*d*x^2])","A",0
208,1,210,196,1.0052309,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\frac{a^2+2 a b \cosh ^{-1}(c x)-2 a b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)-2 b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)+2 b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)+b^2 \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\log \left(1-e^{-\cosh ^{-1}(c x)}\right)-\log \left(e^{-\cosh ^{-1}(c x)}+1\right)\right)\right)}{c^2 d \sqrt{d-c^2 d x^2}}","\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{c^2 d \sqrt{d-c^2 d x^2}}",1,"(a^2 + 2*a*b*ArcCosh[c*x] + b^2*ArcCosh[c*x]*(ArcCosh[c*x] - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(Log[1 - E^(-ArcCosh[c*x])] - Log[1 + E^(-ArcCosh[c*x])])) - 2*a*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] - 2*b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-ArcCosh[c*x])] + 2*b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-ArcCosh[c*x])])/(c^2*d*Sqrt[d - c^2*d*x^2])","A",0
209,1,126,198,0.4590873,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(3/2),x]","\frac{\frac{\sqrt{c x-1} \sqrt{c x+1} \left(\left(a+b \cosh ^{-1}(c x)\right) \left(a+b \cosh ^{-1}(c x)-2 b \log \left(1-e^{\cosh ^{-1}(c x)}\right)-2 b \log \left(e^{\cosh ^{-1}(c x)}+1\right)\right)-2 b^2 \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)-2 b^2 \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)\right)}{c}+x \left(a+b \cosh ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}","\frac{x \left(a+b \cosh ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{c d \sqrt{d-c^2 d x^2}}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{c d \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{c d \sqrt{d-c^2 d x^2}}",1,"(x*(a + b*ArcCosh[c*x])^2 + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])*(a + b*ArcCosh[c*x] - 2*b*Log[1 - E^ArcCosh[c*x]] - 2*b*Log[1 + E^ArcCosh[c*x]]) - 2*b^2*PolyLog[2, -E^ArcCosh[c*x]] - 2*b^2*PolyLog[2, E^ArcCosh[c*x]]))/c)/(d*Sqrt[d - c^2*d*x^2])","A",0
210,1,577,471,3.6221827,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x*(d - c^2*d*x^2)^(3/2)),x]","-\frac{\frac{a^2 \sqrt{d-c^2 d x^2}}{c^2 x^2-1}+a^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{d-c^2 d x^2}+d\right)+a^2 \left(-\sqrt{d}\right) \log (c x)+\frac{2 i a b d \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+i \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)-i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)}{\sqrt{d-c^2 d x^2}}+\frac{b^2 d \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(2 i \cosh ^{-1}(c x) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 i \cosh ^{-1}(c x) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+2 \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)-2 \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)+2 i \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)-2 i \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)+\frac{\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)^2}{1-c x}+i \cosh ^{-1}(c x)^2 \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-i \cosh ^{-1}(c x)^2 \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+2 \cosh ^{-1}(c x) \log \left(1-e^{-\cosh ^{-1}(c x)}\right)-2 \cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right)\right)}{\sqrt{d-c^2 d x^2}}}{d^2}","-\frac{2 i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{2 i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}+\frac{4 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}",1,"-(((a^2*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2) - a^2*Sqrt[d]*Log[c*x] + a^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + ((2*I)*a*b*d*(I*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - I*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2] + (b^2*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*((Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2)/(1 - c*x) + 2*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + I*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] - I*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 2*PolyLog[2, -E^(-ArcCosh[c*x])] + (2*I)*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - (2*I)*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] - 2*PolyLog[2, E^(-ArcCosh[c*x])] + (2*I)*PolyLog[3, (-I)/E^ArcCosh[c*x]] - (2*I)*PolyLog[3, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2])/d^2)","A",0
211,1,315,341,1.6909231,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x^2 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x^2*(d - c^2*d*x^2)^(3/2)),x]","\frac{a^2 \left(2 c^2 x^2-1\right)+2 a b \left(c^2 x^2 \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left(\log (c x)+\log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)\right)\right)\right)+b^2 \left(\cosh ^{-1}(c x) \left(c^2 x^2 \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-2 c x \left(\cosh ^{-1}(c x)+\log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)+\log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)\right)\right)+c x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+c x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)\right)}{d x \sqrt{d-c^2 d x^2}}","\frac{2 c^2 x \left(a+b \cosh ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}+\frac{2 c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{d x \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}",1,"(a^2*(-1 + 2*c^2*x^2) + 2*a*b*(c^2*x^2*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(Log[c*x] + Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]))) + b^2*(ArcCosh[c*x]*(c^2*x^2*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - 2*c*x*(ArcCosh[c*x] + Log[1 - E^(-2*ArcCosh[c*x])] + Log[1 + E^(-2*ArcCosh[c*x])]))) + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-2*ArcCosh[c*x])] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(d*x*Sqrt[d - c^2*d*x^2])","A",0
212,1,979,650,92.2826142,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x^3 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x^3*(d - c^2*d*x^2)^(3/2)),x]","\frac{b^2 \sqrt{d-c^2 d x^2} \left(-2 \cosh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)^2+2 \sinh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)^2+\left(\frac{1}{c^2 x^2}-1\right) \cosh ^{-1}(c x)^2+3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)^2-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)^2-\frac{2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{c x}+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(1-e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)-4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(1+e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)+6 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)-6 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)-4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)+6 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)-6 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)\right) c^2}{2 d^2 \left(c^2 x^2-1\right)}+\frac{a \left(3 a \sqrt{d} \log (x) c^2-3 a \sqrt{d} \log \left(d+\sqrt{d-c^2 d x^2} \sqrt{d}\right) c^2-\frac{2 b d \left(-2 \cosh ^{-1}(c x) \cosh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)+2 \cosh ^{-1}(c x) \sinh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+\left(\frac{1}{c^2 x^2}-1\right) \cosh ^{-1}(c x)+3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)+3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)\right) c^2}{\sqrt{d-c^2 d x^2}}-\frac{a \left(3 c^2 x^2-1\right) \sqrt{d-c^2 d x^2}}{x^2 \left(c^2 x^2-1\right)}\right)}{2 d^2}","-\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{2 d \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{d x \sqrt{d-c^2 d x^2}}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)^2}{d \sqrt{d-c^2 d x^2}}+\frac{4 b c^2 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{3 i b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{d \sqrt{d-c^2 d x^2}}",1,"(b^2*c^2*Sqrt[d - c^2*d*x^2]*((-2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/(c*x) + (-1 + 1/(c^2*x^2))*ArcCosh[c*x]^2 + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[Tanh[ArcCosh[c*x]/2]] - 2*ArcCosh[c*x]^2*Cosh[ArcCosh[c*x]/2]^2 + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] - 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-ArcCosh[c*x])] + (6*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - (6*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] - 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-ArcCosh[c*x])] + (6*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[3, (-I)/E^ArcCosh[c*x]] - (6*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[3, I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]^2*Sinh[ArcCosh[c*x]/2]^2))/(2*d^2*(-1 + c^2*x^2)) + (a*(-((a*(-1 + 3*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(x^2*(-1 + c^2*x^2))) + 3*a*c^2*Sqrt[d]*Log[x] - 3*a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] - (2*b*c^2*d*(-((Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x)) + (-1 + 1/(c^2*x^2))*ArcCosh[c*x] - 2*ArcCosh[c*x]*Cosh[ArcCosh[c*x]/2]^2 + (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^2))/Sqrt[d - c^2*d*x^2]))/(2*d^2)","A",0
213,1,529,496,2.4249727,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x^4 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x^4*(d - c^2*d*x^2)^(3/2)),x]","\frac{a^2 \left(8 c^4 x^4-4 c^2 x^2-1\right)+a b \left(6 c^4 x^4 \cosh ^{-1}(c x)+2 c^2 x^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(5 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left(5 \log (c x)+3 \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)\right)\right)+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(c x+2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)\right)\right)+b^2 \left(-c^4 x^4+3 c^4 x^4 \cosh ^{-1}(c x)^2+5 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+3 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)-8 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2-6 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)-10 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)+c^2 x^2+5 c^2 x^2 (c x-1) (c x+1) \cosh ^{-1}(c x)^2+c x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)+(c x-1) (c x+1) \cosh ^{-1}(c x)^2\right)}{3 d x^3 \sqrt{d-c^2 d x^2}}","-\frac{4 c^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d x \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 d x^2 \sqrt{d-c^2 d x^2}}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{3 d x^3 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d \sqrt{d-c^2 d x^2}}+\frac{8 c^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d \sqrt{d-c^2 d x^2}}-\frac{16 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 d \sqrt{d-c^2 d x^2}}-\frac{20 b c^3 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 d \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2 (1-c x) (c x+1)}{3 d x \sqrt{d-c^2 d x^2}}-\frac{5 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{3 d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{d \sqrt{d-c^2 d x^2}}",1,"(a^2*(-1 - 4*c^2*x^2 + 8*c^4*x^4) + a*b*(6*c^4*x^4*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(c*x + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]) + 2*c^2*x^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(5*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(5*Log[c*x] + 3*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]))) + b^2*(c^2*x^2 - c^4*x^4 + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + 3*c^4*x^4*ArcCosh[c*x]^2 + (-1 + c*x)*(1 + c*x)*ArcCosh[c*x]^2 + 5*c^2*x^2*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x]^2 - 8*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2 - 6*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 10*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] + 5*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 3*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*d*x^3*Sqrt[d - c^2*d*x^2])","A",0
214,1,437,568,5.5482276,"\int \frac{x^5 \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","-\frac{8 a^2 \left(3 c^4 x^4-12 c^2 x^2+8\right)+2 a b \left(-36 \cosh \left(2 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)+3 \cosh \left(4 \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)+25 \cosh ^{-1}(c x)+4 \sinh \left(2 \cosh ^{-1}(c x)\right)-3 \sinh \left(4 \cosh ^{-1}(c x)\right)-33 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)+11 \sinh \left(3 \cosh ^{-1}(c x)\right) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)+b^2 \left(88 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)-88 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)+25 \cosh ^{-1}(c x)^2-4 \left(9 \cosh ^{-1}(c x)^2+7\right) \cosh \left(2 \cosh ^{-1}(c x)\right)+3 \left(\cosh ^{-1}(c x)^2+2\right) \cosh \left(4 \cosh ^{-1}(c x)\right)-66 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-e^{-\cosh ^{-1}(c x)}\right)+66 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right)+8 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)-6 \cosh ^{-1}(c x) \sinh \left(4 \cosh ^{-1}(c x)\right)+22 \cosh ^{-1}(c x) \log \left(1-e^{-\cosh ^{-1}(c x)}\right) \sinh \left(3 \cosh ^{-1}(c x)\right)-22 \cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right) \sinh \left(3 \cosh ^{-1}(c x)\right)+22\right)}{24 c^6 d \left(d-c^2 d x^2\right)^{3/2}}","\frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{8 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^6 d^3}-\frac{22 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 a b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b x \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}-\frac{11 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{7 b^2 (1-c x) (c x+1)}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 x^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}",1,"-1/24*(8*a^2*(8 - 12*c^2*x^2 + 3*c^4*x^4) + 2*a*b*(25*ArcCosh[c*x] - 36*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] + 3*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] - 33*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + 4*Sinh[2*ArcCosh[c*x]] + 11*Log[Tanh[ArcCosh[c*x]/2]]*Sinh[3*ArcCosh[c*x]] - 3*Sinh[4*ArcCosh[c*x]]) + b^2*(22 + 25*ArcCosh[c*x]^2 - 4*(7 + 9*ArcCosh[c*x]^2)*Cosh[2*ArcCosh[c*x]] + 3*(2 + ArcCosh[c*x]^2)*Cosh[4*ArcCosh[c*x]] - 66*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + 66*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 88*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, -E^(-ArcCosh[c*x])] - 88*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, E^(-ArcCosh[c*x])] + 8*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] + 22*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - 22*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - 6*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(c^6*d*(d - c^2*d*x^2)^(3/2))","A",0
215,1,382,482,2.5606085,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{\frac{a^2 c x \left(4 c^2 x^2-3\right) \sqrt{d-c^2 d x^2}}{\left(c^2 x^2-1\right)^2}-3 a^2 \sqrt{d} \tan ^{-1}\left(\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left(c^2 x^2-1\right)}\right)+\frac{a b d \left(-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)+2 c x \cosh ^{-1}(c x)}{c^2 x^2-1}-8 c x \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(8 \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)+3 \cosh ^{-1}(c x)^2\right)\right)}{\sqrt{d-c^2 d x^2}}+\frac{b^2 d \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(-\frac{c x \left(c^2 x^2+\left(4 c^2 x^2-3\right) \cosh ^{-1}(c x)^2-1\right)}{\left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3}+\cosh ^{-1}(c x) \left(\frac{1}{1-c^2 x^2}+\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+4\right)+8 \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)\right)}{\sqrt{d-c^2 d x^2}}}{3 c^5 d^3}","\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^3}{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{4 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{x \left(a+b \cosh ^{-1}(c x)\right)^2}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{4 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 (1-c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}",1,"((a^2*c*x*(-3 + 4*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2)^2 - 3*a^2*Sqrt[d]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (a*b*d*(-8*c*x*ArcCosh[c*x] - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + 2*c*x*ArcCosh[c*x])/(-1 + c^2*x^2) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(3*ArcCosh[c*x]^2 + 8*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])))/Sqrt[d - c^2*d*x^2] + (b^2*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-((c*x*(-1 + c^2*x^2 + (-3 + 4*c^2*x^2)*ArcCosh[c*x]^2))/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3)) + ArcCosh[c*x]*((1 - c^2*x^2)^(-1) + ArcCosh[c*x]*(4 + ArcCosh[c*x]) + 8*Log[1 - E^(-2*ArcCosh[c*x])]) - 4*PolyLog[2, E^(-2*ArcCosh[c*x])]))/Sqrt[d - c^2*d*x^2])/(3*c^5*d^3)","A",0
216,1,341,336,4.5351791,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{4 a^2 \left(3 c^2 x^2-2\right)-a b \left(\cosh ^{-1}(c x) \left(4-12 \cosh \left(2 \cosh ^{-1}(c x)\right)\right)-2 \sinh \left(2 \cosh ^{-1}(c x)\right)+5 \left(\sinh \left(3 \cosh ^{-1}(c x)\right)-3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)\right) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)-b^2 \left(20 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)-20 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)+2 \cosh ^{-1}(c x)^2-2 \left(3 \cosh ^{-1}(c x)^2+1\right) \cosh \left(2 \cosh ^{-1}(c x)\right)-15 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \left(\log \left(1-e^{-\cosh ^{-1}(c x)}\right)-\log \left(e^{-\cosh ^{-1}(c x)}+1\right)\right)-2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)+5 \cosh ^{-1}(c x) \log \left(1-e^{-\cosh ^{-1}(c x)}\right) \sinh \left(3 \cosh ^{-1}(c x)\right)-5 \cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right) \sinh \left(3 \cosh ^{-1}(c x)\right)+2\right)}{12 c^4 d \left(d-c^2 d x^2\right)^{3/2}}","\frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{10 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}-\frac{5 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}",1,"(4*a^2*(-2 + 3*c^2*x^2) - b^2*(2 + 2*ArcCosh[c*x]^2 - 2*(1 + 3*ArcCosh[c*x]^2)*Cosh[2*ArcCosh[c*x]] - 15*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*(Log[1 - E^(-ArcCosh[c*x])] - Log[1 + E^(-ArcCosh[c*x])]) + 20*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, -E^(-ArcCosh[c*x])] - 20*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, E^(-ArcCosh[c*x])] - 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] + 5*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - 5*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]]) - a*b*(ArcCosh[c*x]*(4 - 12*Cosh[2*ArcCosh[c*x]]) - 2*Sinh[2*ArcCosh[c*x]] + 5*Log[Tanh[ArcCosh[c*x]/2]]*(-3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + Sinh[3*ArcCosh[c*x]])))/(12*c^4*d*(d - c^2*d*x^2)^(3/2))","A",0
217,1,264,389,1.7828875,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{\frac{a^2 c^3 x^3}{1-c^2 x^2}+a b \left(\frac{\sqrt{\frac{c x-1}{c x+1}} \left(2 \left(c^2 x^2-1\right) \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)-1\right)}{c x-1}+\frac{2 c^3 x^3 \cosh ^{-1}(c x)}{1-c^2 x^2}\right)+b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(-\frac{c x \left(c^2 x^2+c^2 x^2 \cosh ^{-1}(c x)^2-1\right)}{\left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3}+\cosh ^{-1}(c x) \left(\frac{1}{1-c^2 x^2}+\cosh ^{-1}(c x)+2 \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)\right)-\text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)\right)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}","\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 c d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 (1-c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}",1,"((a^2*c^3*x^3)/(1 - c^2*x^2) + a*b*((2*c^3*x^3*ArcCosh[c*x])/(1 - c^2*x^2) + (Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 + 2*(-1 + c^2*x^2)*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]))/(-1 + c*x)) + b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-((c*x*(-1 + c^2*x^2 + c^2*x^2*ArcCosh[c*x]^2))/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3)) + ArcCosh[c*x]*((1 - c^2*x^2)^(-1) + ArcCosh[c*x] + 2*Log[1 - E^(-2*ArcCosh[c*x])]) - PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*c^3*d^2*Sqrt[d - c^2*d*x^2])","A",0
218,1,332,298,2.5410175,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\frac{4 a^2+a b \left(8 \cosh ^{-1}(c x)+2 \sinh \left(2 \cosh ^{-1}(c x)\right)+\left(\sinh \left(3 \cosh ^{-1}(c x)\right)-3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)\right) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)\right)+b^2 \left(4 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)-4 \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3 \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)+4 \cosh ^{-1}(c x)^2+2 \cosh \left(2 \cosh ^{-1}(c x)\right)-3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-e^{-\cosh ^{-1}(c x)}\right)+3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right)+2 \cosh ^{-1}(c x) \sinh \left(2 \cosh ^{-1}(c x)\right)+\cosh ^{-1}(c x) \log \left(1-e^{-\cosh ^{-1}(c x)}\right) \sinh \left(3 \cosh ^{-1}(c x)\right)-\cosh ^{-1}(c x) \log \left(e^{-\cosh ^{-1}(c x)}+1\right) \sinh \left(3 \cosh ^{-1}(c x)\right)-2\right)}{12 c^2 d \left(d-c^2 d x^2\right)^{3/2}}","\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 c d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{3 c^2 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}",1,"(4*a^2 + b^2*(-2 + 4*ArcCosh[c*x]^2 + 2*Cosh[2*ArcCosh[c*x]] - 3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + 3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 4*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, -E^(-ArcCosh[c*x])] - 4*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, E^(-ArcCosh[c*x])] + 2*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] + ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]]) + a*b*(8*ArcCosh[c*x] + 2*Sinh[2*ArcCosh[c*x]] + Log[Tanh[ArcCosh[c*x]/2]]*(-3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + Sinh[3*ArcCosh[c*x]])))/(12*c^2*d*(d - c^2*d*x^2)^(3/2))","A",0
219,1,289,331,1.5216859,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(5/2),x]","\frac{\frac{a^2 c x \left(2 c^2 x^2-3\right)}{c^2 x^2-1}+a b \left(\frac{\sqrt{\frac{c x-1}{c x+1}} \left(\left(4-4 c^2 x^2\right) \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)-1\right)}{c x-1}+2 c x \left(\frac{1}{1-c^2 x^2}+2\right) \cosh ^{-1}(c x)\right)+b^2 \left(-\frac{\cosh ^{-1}(c x) \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)+c x \cosh ^{-1}(c x)\right)}{c^2 x^2-1}+2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)+c x \left(2 \cosh ^{-1}(c x)^2-1\right)-2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+2 \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)\right)\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}","\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 c d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{2 x \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{4 b \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}+\frac{x \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{3 c d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}",1,"((a^2*c*x*(-3 + 2*c^2*x^2))/(-1 + c^2*x^2) + a*b*(2*c*x*(2 + (1 - c^2*x^2)^(-1))*ArcCosh[c*x] + (Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 + (4 - 4*c^2*x^2)*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]))/(-1 + c*x)) + b^2*(-((ArcCosh[c*x]*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + c*x*ArcCosh[c*x]))/(-1 + c^2*x^2)) + c*x*(-1 + 2*ArcCosh[c*x]^2) - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 - E^(-2*ArcCosh[c*x])]) + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*c*d^2*Sqrt[d - c^2*d*x^2])","A",0
220,1,806,597,10.4496089,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x*(d - c^2*d*x^2)^(5/2)),x]","\frac{\log (c x) a^2}{d^{5/2}}-\frac{\log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right) a^2}{d^{5/2}}+\frac{b \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(-\frac{1}{2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \text{csch}^4\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\text{csch}^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\frac{8 \cosh ^{-1}(c x) \sinh ^4\left(\frac{1}{2} \cosh ^{-1}(c x)\right)}{\left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3}-\text{sech}^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)+14 \cosh ^{-1}(c x) \coth \left(\frac{1}{2} \cosh ^{-1}(c x)\right)-24 i \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)+24 i \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)-28 \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)-24 i \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)+24 i \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-14 \cosh ^{-1}(c x) \tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right) a}{12 d^2 \sqrt{-d (c x-1) (c x+1)}}+\sqrt{-d \left(c^2 x^2-1\right)} \left(\frac{a^2}{3 d^3 \left(c^2 x^2-1\right)^2}-\frac{a^2}{d^3 \left(c^2 x^2-1\right)}\right)+\frac{b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(-\frac{1}{2} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2 \text{csch}^4\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-2 \cosh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\frac{8 \cosh ^{-1}(c x)^2 \sinh ^4\left(\frac{1}{2} \cosh ^{-1}(c x)\right)}{\left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3}-2 \cosh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)+14 \cosh ^{-1}(c x)^2 \coth \left(\frac{1}{2} \cosh ^{-1}(c x)\right)-4 \coth \left(\frac{1}{2} \cosh ^{-1}(c x)\right)-56 \cosh ^{-1}(c x) \log \left(1-e^{-\cosh ^{-1}(c x)}\right)-24 i \cosh ^{-1}(c x)^2 \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)+24 i \cosh ^{-1}(c x)^2 \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)+56 \cosh ^{-1}(c x) \log \left(1+e^{-\cosh ^{-1}(c x)}\right)-56 \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)-48 i \cosh ^{-1}(c x) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)+48 i \cosh ^{-1}(c x) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)+56 \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)-48 i \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)+48 i \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)-14 \cosh ^{-1}(c x)^2 \tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)+4 \tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)}{24 d^2 \sqrt{-d (c x-1) (c x+1)}}","-\frac{2 i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 i b \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{b c x \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{14 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{7 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{7 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{2 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2}{3 d^2 \sqrt{d-c^2 d x^2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*(a^2/(3*d^3*(-1 + c^2*x^2)^2) - a^2/(d^3*(-1 + c^2*x^2))) + (a^2*Log[c*x])/d^(5/2) - (a^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/d^(5/2) + (a*b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(14*ArcCosh[c*x]*Coth[ArcCosh[c*x]/2] - Csch[ArcCosh[c*x]/2]^2 - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Csch[ArcCosh[c*x]/2]^4)/2 - (24*I)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (24*I)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - 28*Log[Tanh[ArcCosh[c*x]/2]] - (24*I)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (24*I)*PolyLog[2, I/E^ArcCosh[c*x]] - Sech[ArcCosh[c*x]/2]^2 - (8*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^4)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3) - 14*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]))/(12*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) + (b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-4*Coth[ArcCosh[c*x]/2] + 14*ArcCosh[c*x]^2*Coth[ArcCosh[c*x]/2] - 2*ArcCosh[c*x]*Csch[ArcCosh[c*x]/2]^2 - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2*Csch[ArcCosh[c*x]/2]^4)/2 - 56*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] - (24*I)*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] + (24*I)*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] + 56*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] - 56*PolyLog[2, -E^(-ArcCosh[c*x])] - (48*I)*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (48*I)*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] + 56*PolyLog[2, E^(-ArcCosh[c*x])] - (48*I)*PolyLog[3, (-I)/E^ArcCosh[c*x]] + (48*I)*PolyLog[3, I/E^ArcCosh[c*x]] - 2*ArcCosh[c*x]*Sech[ArcCosh[c*x]/2]^2 - (8*ArcCosh[c*x]^2*Sinh[ArcCosh[c*x]/2]^4)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3) + 4*Tanh[ArcCosh[c*x]/2] - 14*ArcCosh[c*x]^2*Tanh[ArcCosh[c*x]/2]))/(24*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])","A",0
221,1,457,476,3.0812784,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x^2 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x^2*(d - c^2*d*x^2)^(5/2)),x]","\frac{c \left(\frac{a^2 \left(8 c^4 x^4-12 c^2 x^2+3\right)}{c x \left(c^2 x^2-1\right)}+a b \left(-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)+2 c x \cosh ^{-1}(c x)}{c^2 x^2-1}+10 c x \cosh ^{-1}(c x)-2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(3 \log (c x)+5 \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)-\frac{3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{c x}\right)\right)+b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\frac{\cosh ^{-1}(c x)}{1-c^2 x^2}+3 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+5 \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)+\frac{c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{c x}+\frac{5 c x \cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{c x \cosh ^{-1}(c x)^2}{\left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3}-8 \cosh ^{-1}(c x)^2-10 \cosh ^{-1}(c x) \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)-6 \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}","\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{8 c^2 x \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 b c \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2 \sqrt{d-c^2 d x^2}}+\frac{4 c^2 x \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{d x \left(d-c^2 d x^2\right)^{3/2}}-\frac{b^2 c \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 b^2 c \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 x}{3 d^2 \sqrt{d-c^2 d x^2}}",1,"(c*((a^2*(3 - 12*c^2*x^2 + 8*c^4*x^4))/(c*x*(-1 + c^2*x^2)) + a*b*(10*c*x*ArcCosh[c*x] - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + 2*c*x*ArcCosh[c*x])/(-1 + c^2*x^2) - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*((-3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/(c*x) + 3*Log[c*x] + 5*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])) + b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*((c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) + ArcCosh[c*x]/(1 - c^2*x^2) - 8*ArcCosh[c*x]^2 - (c*x*ArcCosh[c*x]^2)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3) + (5*c*x*ArcCosh[c*x]^2)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2)/(c*x) - 10*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 6*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] + 3*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 5*PolyLog[2, E^(-2*ArcCosh[c*x])])))/(3*d^2*Sqrt[d - c^2*d*x^2])","A",0
222,1,1181,796,99.1556547,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x^3 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]","\frac{5 a^2 \log (x) c^2}{2 d^{5/2}}-\frac{5 a^2 \log \left(d+\sqrt{-d \left(c^2 x^2-1\right)} \sqrt{d}\right) c^2}{2 d^{5/2}}+\frac{a b \left(26 \cosh ^{-1}(c x) \cosh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\cosh ^{-1}(c x) \coth ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-26 \cosh ^{-1}(c x) \sinh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-\cosh ^{-1}(c x) \tanh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)+\frac{6 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+\frac{6 (c x-1) (c x+1) \cosh ^{-1}(c x)}{c^2 x^2}-\coth \left(\frac{1}{2} \cosh ^{-1}(c x)\right)-30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right)+30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right)-26 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)-30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right)+30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right)-\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right) c^2}{6 d^2 \sqrt{-d (c x-1) (c x+1)}}-\frac{b^2 \sqrt{d-c^2 d x^2} \left(26 \cosh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)^2-\coth ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)^2-26 \sinh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)^2-\tanh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)^2+6 \left(1-\frac{1}{c^2 x^2}\right) \cosh ^{-1}(c x)^2-30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(1-i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)^2+30 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(1+i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)^2+\frac{12 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{c x}-2 \coth \left(\frac{1}{2} \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-52 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(1-e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)+52 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(1+e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)-60 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)+60 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(i e^{-\cosh ^{-1}(c x)}\right) \cosh ^{-1}(c x)-2 \tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right) \cosh ^{-1}(c x)-4 \cosh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)+4 \sinh ^2\left(\frac{1}{2} \cosh ^{-1}(c x)\right)-24 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left(\tanh \left(\frac{1}{2} \cosh ^{-1}(c x)\right)\right)-52 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(-e^{-\cosh ^{-1}(c x)}\right)+52 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_2\left(e^{-\cosh ^{-1}(c x)}\right)-60 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_3\left(-i e^{-\cosh ^{-1}(c x)}\right)+60 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{Li}_3\left(i e^{-\cosh ^{-1}(c x)}\right)\right) c^2}{12 d^3 \left(c^2 x^2-1\right)}+\sqrt{-d \left(c^2 x^2-1\right)} \left(-\frac{2 c^2 a^2}{d^3 \left(c^2 x^2-1\right)}-\frac{a^2}{2 d^3 x^2}+\frac{c^2 a^2}{3 d^3 \left(c^2 x^2-1\right)^2}\right)","-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right) c^3}{3 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}+\frac{5 \left(a+b \cosh ^{-1}(c x)\right)^2 c^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 \left(a+b \cosh ^{-1}(c x)\right)^2 c^2}{6 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{5 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2 \tan ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right) c^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{26 b \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right) \tanh ^{-1}\left(e^{\cosh ^{-1}(c x)}\right) c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{\cosh ^{-1}(c x)}\right) c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(-i e^{\cosh ^{-1}(c x)}\right) c^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(i e^{\cosh ^{-1}(c x)}\right) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{\cosh ^{-1}(c x)}\right) c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(-i e^{\cosh ^{-1}(c x)}\right) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_3\left(i e^{\cosh ^{-1}(c x)}\right) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right) c}{d^2 x \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{2 d x^2 \left(d-c^2 d x^2\right)^{3/2}}",1,"Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a^2/(d^3*x^2) + (a^2*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a^2*c^2)/(d^3*(-1 + c^2*x^2))) + (5*a^2*c^2*Log[x])/(2*d^(5/2)) - (5*a^2*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (a*b*c^2*((6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) + (6*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x])/(c^2*x^2) + 26*ArcCosh[c*x]*Cosh[ArcCosh[c*x]/2]^2 - Coth[ArcCosh[c*x]/2] - ArcCosh[c*x]*Coth[ArcCosh[c*x]/2]^2 - (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - 26*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] - (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] - 26*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^2 - Tanh[ArcCosh[c*x]/2] - ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]^2))/(6*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) - (b^2*c^2*Sqrt[d - c^2*d*x^2]*((12*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/(c*x) + 6*(1 - 1/(c^2*x^2))*ArcCosh[c*x]^2 - 24*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[Tanh[ArcCosh[c*x]/2]] - 4*Cosh[ArcCosh[c*x]/2]^2 + 26*ArcCosh[c*x]^2*Cosh[ArcCosh[c*x]/2]^2 - 2*ArcCosh[c*x]*Coth[ArcCosh[c*x]/2] - ArcCosh[c*x]^2*Coth[ArcCosh[c*x]/2]^2 - 52*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] - (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] + 52*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] - 52*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-ArcCosh[c*x])] - (60*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (60*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] + 52*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-ArcCosh[c*x])] - (60*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[3, (-I)/E^ArcCosh[c*x]] + (60*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[3, I/E^ArcCosh[c*x]] + 4*Sinh[ArcCosh[c*x]/2]^2 - 26*ArcCosh[c*x]^2*Sinh[ArcCosh[c*x]/2]^2 - 2*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2] - ArcCosh[c*x]^2*Tanh[ArcCosh[c*x]/2]^2))/(12*d^3*(-1 + c^2*x^2))","A",0
223,1,534,562,3.6774777,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{x^4 \left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]","\frac{\frac{a^2 \left(16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right)}{x^3 \left(c^2 x^2-1\right)}+a b c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\frac{1}{1-c^2 x^2}+\frac{1}{c^2 x^2}+\frac{2 \left(16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right) \left(\frac{c x-1}{c x+1}\right)^{3/2} \cosh ^{-1}(c x)}{c^3 x^3 (c x-1)^3}-16 \log (c x)-16 \log \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)\right)+b^2 c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{c^3 x^3}+\frac{\cosh ^{-1}(c x)}{1-c^2 x^2}+\frac{\cosh ^{-1}(c x)}{c^2 x^2}+8 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+8 \text{Li}_2\left(e^{-2 \cosh ^{-1}(c x)}\right)-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+\frac{c x \sqrt{\frac{c x-1}{c x+1}}}{1-c x}+\frac{8 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{c x}+\frac{8 c x \cosh ^{-1}(c x)^2}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}-\frac{c x \cosh ^{-1}(c x)^2}{\left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3}-16 \cosh ^{-1}(c x)^2-16 \cosh ^{-1}(c x) \log \left(1-e^{-2 \cosh ^{-1}(c x)}\right)-16 \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}","\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 x^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2}}-\frac{2 c^2 \left(a+b \cosh ^{-1}(c x)\right)^2}{d x \left(d-c^2 d x^2\right)^{3/2}}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{3 d x^3 \left(d-c^2 d x^2\right)^{3/2}}+\frac{16 c^4 x \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{8 c^4 x \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d \left(d-c^2 d x^2\right)^{3/2}}+\frac{16 c^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{c x-1} \sqrt{c x+1} \log \left(1-e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{32 b c^3 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left(e^{2 \cosh ^{-1}(c x)}\right) \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 c^2}{3 d^2 x \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(-e^{2 \cosh ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(c x)}\right)}{3 d^2 \sqrt{d-c^2 d x^2}}",1,"((a^2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6))/(x^3*(-1 + c^2*x^2)) + a*b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(1/(c^2*x^2) + (1 - c^2*x^2)^(-1) + (2*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6)*ArcCosh[c*x])/(c^3*x^3*(-1 + c*x)^3) - 16*Log[c*x] - 16*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]) + b^2*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*((c*x*Sqrt[(-1 + c*x)/(1 + c*x)])/(1 - c*x) - (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) + ArcCosh[c*x]/(c^2*x^2) + ArcCosh[c*x]/(1 - c^2*x^2) - 16*ArcCosh[c*x]^2 - (c*x*ArcCosh[c*x]^2)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3) + (8*c*x*ArcCosh[c*x]^2)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2)/(c^3*x^3) + (8*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2)/(c*x) - 16*ArcCosh[c*x]*Log[1 - E^(-2*ArcCosh[c*x])] - 16*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] + 8*PolyLog[2, -E^(-2*ArcCosh[c*x])] + 8*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*d^2*Sqrt[d - c^2*d*x^2])","A",0
224,1,220,429,1.3821777,"\int \frac{\cosh ^{-1}(a x)^2}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcCosh[a*x]^2/(c - a^2*c*x^2)^(7/2),x]","-\frac{a x \left(\frac{1}{1-a^2 x^2}+10\right)+2 \left(a x \left(\frac{4}{a^2 x^2-1}-\frac{3}{\left(a^2 x^2-1\right)^2}+8 \sqrt{\frac{a x-1}{a x+1}}-8\right)+8 \sqrt{\frac{a x-1}{a x+1}}\right) \cosh ^{-1}(a x)^2+\frac{\left(\frac{a x-1}{a x+1}\right)^{3/2} \cosh ^{-1}(a x) \left(8 a^2 x^2+32 \left(a^2 x^2-1\right)^2 \log \left(1-e^{-2 \cosh ^{-1}(a x)}\right)-11\right)}{(a x-1)^3}-16 \sqrt{\frac{a x-1}{a x+1}} (a x+1) \text{Li}_2\left(e^{-2 \cosh ^{-1}(a x)}\right)}{30 a c^3 \sqrt{c-a^2 c x^2}}","-\frac{8 \sqrt{a x-1} \sqrt{a x+1} \text{Li}_2\left(e^{2 \cosh ^{-1}(a x)}\right)}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}-\frac{x}{30 c^3 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{15 a c^3 \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}{10 a c^3 \left(1-a^2 x^2\right)^2 \sqrt{c-a^2 c x^2}}-\frac{16 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left(1-e^{2 \cosh ^{-1}(a x)}\right)}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^2}{15 c^2 \left(c-a^2 c x^2\right)^{3/2}}+\frac{x \cosh ^{-1}(a x)^2}{5 c \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/30*(a*x*(10 + (1 - a^2*x^2)^(-1)) + 2*(8*Sqrt[(-1 + a*x)/(1 + a*x)] + a*x*(-8 + 8*Sqrt[(-1 + a*x)/(1 + a*x)] - 3/(-1 + a^2*x^2)^2 + 4/(-1 + a^2*x^2)))*ArcCosh[a*x]^2 + (((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x]*(-11 + 8*a^2*x^2 + 32*(-1 + a^2*x^2)^2*Log[1 - E^(-2*ArcCosh[a*x])]))/(-1 + a*x)^3 - 16*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*PolyLog[2, E^(-2*ArcCosh[a*x])])/(a*c^3*Sqrt[c - a^2*c*x^2])","A",0
225,1,116,243,0.2645224,"\int \frac{x^4 \cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^4*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(32 \cosh ^{-1}(a x)^3-4 \left(16 \cosh \left(2 \cosh ^{-1}(a x)\right)+\cosh \left(4 \cosh ^{-1}(a x)\right)\right) \cosh ^{-1}(a x)+8 \cosh ^{-1}(a x)^2 \left(8 \sinh \left(2 \cosh ^{-1}(a x)\right)+\sinh \left(4 \cosh ^{-1}(a x)\right)\right)+32 \sinh \left(2 \cosh ^{-1}(a x)\right)+\sinh \left(4 \cosh ^{-1}(a x)\right)\right)}{256 a^5 \sqrt{1-a^2 x^2}}","\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^3}{8 a^5 \sqrt{1-a x}}+\frac{15 \sqrt{a x-1} \cosh ^{-1}(a x)}{64 a^5 \sqrt{1-a x}}-\frac{15 x \sqrt{1-a x} \sqrt{a x+1}}{64 a^4}-\frac{3 x^2 \sqrt{a x-1} \cosh ^{-1}(a x)}{8 a^3 \sqrt{1-a x}}-\frac{x^3 \sqrt{1-a x} \sqrt{a x+1}}{32 a^2}-\frac{x^3 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{4 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{8 a^4}-\frac{x^4 \sqrt{a x-1} \cosh ^{-1}(a x)}{8 a \sqrt{1-a x}}",1,"(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(32*ArcCosh[a*x]^3 - 4*ArcCosh[a*x]*(16*Cosh[2*ArcCosh[a*x]] + Cosh[4*ArcCosh[a*x]]) + 32*Sinh[2*ArcCosh[a*x]] + Sinh[4*ArcCosh[a*x]] + 8*ArcCosh[a*x]^2*(8*Sinh[2*ArcCosh[a*x]] + Sinh[4*ArcCosh[a*x]])))/(256*a^5*Sqrt[1 - a^2*x^2])","A",1
226,1,123,177,0.1484627,"\int \frac{x^3 \cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\left(-\frac{40}{27 a^4}-\frac{2 x^2}{27 a^2}\right) \sqrt{1-a^2 x^2}-\frac{\sqrt{1-a^2 x^2} \left(a^2 x^2+2\right) \cosh ^{-1}(a x)^2}{3 a^4}+\frac{2 x \sqrt{1-a^2 x^2} \left(a^2 x^2+6\right) \cosh ^{-1}(a x)}{9 a^3 \sqrt{a x-1} \sqrt{a x+1}}","-\frac{40 \sqrt{1-a x} \sqrt{a x+1}}{27 a^4}-\frac{4 x \sqrt{a x-1} \cosh ^{-1}(a x)}{3 a^3 \sqrt{1-a x}}-\frac{2 x^2 \sqrt{1-a x} \sqrt{a x+1}}{27 a^2}-\frac{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{3 a^2}-\frac{2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{3 a^4}-\frac{2 x^3 \sqrt{a x-1} \cosh ^{-1}(a x)}{9 a \sqrt{1-a x}}",1,"(-40/(27*a^4) - (2*x^2)/(27*a^2))*Sqrt[1 - a^2*x^2] + (2*x*Sqrt[1 - a^2*x^2]*(6 + a^2*x^2)*ArcCosh[a*x])/(9*a^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcCosh[a*x]^2)/(3*a^4)","A",1
227,1,87,151,0.1756177,"\int \frac{x^2 \cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]","-\frac{\sqrt{-((a x-1) (a x+1))} \left(4 \cosh ^{-1}(a x)^3-6 \cosh \left(2 \cosh ^{-1}(a x)\right) \cosh ^{-1}(a x)+\left(6 \cosh ^{-1}(a x)^2+3\right) \sinh \left(2 \cosh ^{-1}(a x)\right)\right)}{24 a^3 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt{1-a x}}+\frac{\sqrt{a x-1} \cosh ^{-1}(a x)}{4 a^3 \sqrt{1-a x}}-\frac{x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 a^2}-\frac{x \sqrt{1-a x} \sqrt{a x+1}}{4 a^2}-\frac{x^2 \sqrt{a x-1} \cosh ^{-1}(a x)}{2 a \sqrt{1-a x}}",1,"-1/24*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(4*ArcCosh[a*x]^3 - 6*ArcCosh[a*x]*Cosh[2*ArcCosh[a*x]] + (3 + 6*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]]))/(a^3*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",1
228,1,54,79,0.0978099,"\int \frac{x \cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{1-a^2 x^2} \left(-\cosh ^{-1}(a x)^2+\frac{2 a x \cosh ^{-1}(a x)}{\sqrt{a x-1} \sqrt{a x+1}}-2\right)}{a^2}","-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{a^2}-\frac{2 \sqrt{1-a x} \sqrt{a x+1}}{a^2}-\frac{2 x \sqrt{a x-1} \cosh ^{-1}(a x)}{a \sqrt{1-a x}}",1,"(Sqrt[1 - a^2*x^2]*(-2 + (2*a*x*ArcCosh[a*x])/(Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - ArcCosh[a*x]^2))/a^2","A",1
229,1,45,32,0.0226836,"\int \frac{\cosh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]^2/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{3 a \sqrt{1-a^2 x^2}}","\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^3}{3 a \sqrt{1-a x}}",1,"(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3)/(3*a*Sqrt[1 - a^2*x^2])","A",1
230,1,151,183,0.1893004,"\int \frac{\cosh ^{-1}(a x)^2}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]^2/(x*Sqrt[1 - a^2*x^2]),x]","\frac{i \sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(-2 \cosh ^{-1}(a x) \left(\text{Li}_2\left(-i e^{-\cosh ^{-1}(a x)}\right)-\text{Li}_2\left(i e^{-\cosh ^{-1}(a x)}\right)\right)-2 \text{Li}_3\left(-i e^{-\cosh ^{-1}(a x)}\right)+2 \text{Li}_3\left(i e^{-\cosh ^{-1}(a x)}\right)-\left(\cosh ^{-1}(a x)^2 \left(\log \left(1-i e^{-\cosh ^{-1}(a x)}\right)-\log \left(1+i e^{-\cosh ^{-1}(a x)}\right)\right)\right)\right)}{\sqrt{1-a^2 x^2}}","-\frac{2 i \sqrt{a x-1} \cosh ^{-1}(a x) \text{Li}_2\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{2 i \sqrt{a x-1} \cosh ^{-1}(a x) \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{2 i \sqrt{a x-1} \text{Li}_3\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{2 i \sqrt{a x-1} \text{Li}_3\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{2 \sqrt{a x-1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}",1,"(I*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(-(ArcCosh[a*x]^2*(Log[1 - I/E^ArcCosh[a*x]] - Log[1 + I/E^ArcCosh[a*x]])) - 2*ArcCosh[a*x]*(PolyLog[2, (-I)/E^ArcCosh[a*x]] - PolyLog[2, I/E^ArcCosh[a*x]]) - 2*PolyLog[3, (-I)/E^ArcCosh[a*x]] + 2*PolyLog[3, I/E^ArcCosh[a*x]]))/Sqrt[1 - a^2*x^2]","A",0
231,1,111,124,0.4773789,"\int \frac{\cosh ^{-1}(a x)^2}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]^2/(x^2*Sqrt[1 - a^2*x^2]),x]","\frac{a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(\text{Li}_2\left(-e^{-2 \cosh ^{-1}(a x)}\right)+\cosh ^{-1}(a x) \left(\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a x}-\cosh ^{-1}(a x)-2 \log \left(e^{-2 \cosh ^{-1}(a x)}+1\right)\right)\right)}{\sqrt{-((a x-1) (a x+1))}}","-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{x}-\frac{a \sqrt{a x-1} \text{Li}_2\left(-e^{2 \cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{a \sqrt{a x-1} \cosh ^{-1}(a x)^2}{\sqrt{1-a x}}-\frac{2 a \sqrt{a x-1} \cosh ^{-1}(a x) \log \left(e^{2 \cosh ^{-1}(a x)}+1\right)}{\sqrt{1-a x}}",1,"(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(ArcCosh[a*x]*(-ArcCosh[a*x] + (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x])/(a*x) - 2*Log[1 + E^(-2*ArcCosh[a*x])]) + PolyLog[2, -E^(-2*ArcCosh[a*x])]))/Sqrt[-((-1 + a*x)*(1 + a*x))]","A",0
232,1,233,296,1.0092712,"\int \frac{\cosh ^{-1}(a x)^2}{x^3 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]^2/(x^3*Sqrt[1 - a^2*x^2]),x]","\frac{i a^2 \sqrt{-((a x-1) (a x+1))} \left(\frac{i \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^2}{a^2 x^2}+2 \cosh ^{-1}(a x) \text{Li}_2\left(-i e^{-\cosh ^{-1}(a x)}\right)-2 \cosh ^{-1}(a x) \text{Li}_2\left(i e^{-\cosh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-i e^{-\cosh ^{-1}(a x)}\right)-2 \text{Li}_3\left(i e^{-\cosh ^{-1}(a x)}\right)+\frac{2 i \cosh ^{-1}(a x)}{a x}+\cosh ^{-1}(a x)^2 \log \left(1-i e^{-\cosh ^{-1}(a x)}\right)-\cosh ^{-1}(a x)^2 \log \left(1+i e^{-\cosh ^{-1}(a x)}\right)-4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \cosh ^{-1}(a x)\right)\right)\right)}{2 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","-\frac{i a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \text{Li}_2\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{i a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{i a^2 \sqrt{a x-1} \text{Li}_3\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{i a^2 \sqrt{a x-1} \text{Li}_3\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 x^2}-\frac{a^2 \sqrt{a x-1} \tan ^{-1}\left(\sqrt{a x-1} \sqrt{a x+1}\right)}{\sqrt{1-a x}}+\frac{a^2 \sqrt{a x-1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{a \sqrt{a x-1} \cosh ^{-1}(a x)}{x \sqrt{1-a x}}",1,"((I/2)*a^2*Sqrt[-((-1 + a*x)*(1 + a*x))]*(((2*I)*ArcCosh[a*x])/(a*x) + (I*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^2)/(a^2*x^2) - (4*I)*ArcTan[Tanh[ArcCosh[a*x]/2]] + ArcCosh[a*x]^2*Log[1 - I/E^ArcCosh[a*x]] - ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] + 2*ArcCosh[a*x]*PolyLog[2, (-I)/E^ArcCosh[a*x]] - 2*ArcCosh[a*x]*PolyLog[2, I/E^ArcCosh[a*x]] + 2*PolyLog[3, (-I)/E^ArcCosh[a*x]] - 2*PolyLog[3, I/E^ArcCosh[a*x]]))/(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",0
233,0,0,1154,1.5908296,"\int (f x)^m \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2,x]","\int (f x)^m \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","\frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2 (f x)^{m+1}}{f (m+6)}+\frac{5 d \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 (f x)^{m+1}}{f (m+4) (m+6)}+\frac{15 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 (f x)^{m+1}}{f (m+6) \left(m^2+6 m+8\right)}-\frac{2 b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) (f x)^{m+2}}{f^2 (m+2) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{10 b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) (f x)^{m+2}}{f^2 (m+2) (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{30 b c d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) (f x)^{m+2}}{f^2 (m+2)^2 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{10 b^2 c^2 d^2 (3 m+10) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right) (f x)^{m+3}}{f^3 (m+2) (m+3) (m+4)^3 (m+6) (1-c x) (c x+1)}-\frac{2 b^2 c^2 d^2 \left(15 m^2+130 m+264\right) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right) (f x)^{m+3}}{f^3 (m+2) (m+3) (m+4)^2 (m+6)^3 (1-c x) (c x+1)}-\frac{30 b^2 c^2 d^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right) (f x)^{m+3}}{f^3 (m+2)^2 (m+3) (m+4) (m+6) (1-c x) (c x+1)}-\frac{2 b^2 c^2 d^2 \left(m^2+15 m+52\right) \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} (f x)^{m+3}}{f^3 (m+4)^2 (m+6)^3 (1-c x) (c x+1)}-\frac{10 b^2 c^2 d^2 \sqrt{d-c^2 d x^2} (f x)^{m+3}}{f^3 (m+4)^3 (m+6)}+\frac{4 b c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) (f x)^{m+4}}{f^4 (m+4) (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{10 b c^3 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) (f x)^{m+4}}{f^4 (m+4)^2 (m+6) \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b^2 c^4 d^2 \left(1-c^2 x^2\right) \sqrt{d-c^2 d x^2} (f x)^{m+5}}{f^5 (m+6)^3 (1-c x) (c x+1)}-\frac{2 b c^5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right) (f x)^{m+6}}{f^6 (m+6)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{15 d^3 \text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}},x\right)}{(m+6) \left(m^2+6 m+8\right)}",0,"Integrate[(f*x)^m*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2, x]","A",-1
234,0,0,584,0.5269277,"\int (f x)^m \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2,x]","\int (f x)^m \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","\frac{3 d^2 \text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}},x\right)}{m^2+6 m+8}-\frac{2 b c d \sqrt{d-c^2 d x^2} (f x)^{m+2} \left(a+b \cosh ^{-1}(c x)\right)}{f^2 (m+2) (m+4) \sqrt{c x-1} \sqrt{c x+1}}-\frac{6 b c d \sqrt{d-c^2 d x^2} (f x)^{m+2} \left(a+b \cosh ^{-1}(c x)\right)}{f^2 (m+2)^2 (m+4) \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 d \sqrt{d-c^2 d x^2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{f \left(m^2+6 m+8\right)}+\frac{\left(d-c^2 d x^2\right)^{3/2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{f (m+4)}+\frac{2 b c^3 d \sqrt{d-c^2 d x^2} (f x)^{m+4} \left(a+b \cosh ^{-1}(c x)\right)}{f^4 (m+4)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b^2 c^2 d (3 m+10) \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} (f x)^{m+3} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{f^3 (m+2) (m+3) (m+4)^3 (1-c x) (c x+1)}-\frac{6 b^2 c^2 d \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} (f x)^{m+3} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{f^3 (m+2)^2 (m+3) (m+4) (1-c x) (c x+1)}-\frac{2 b^2 c^2 d \sqrt{d-c^2 d x^2} (f x)^{m+3}}{f^3 (m+4)^3}",0,"Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2, x]","A",-1
235,0,0,240,0.3422895,"\int (f x)^m \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[(f*x)^m*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2,x]","\int (f x)^m \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","\frac{d \text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}},x\right)}{m+2}-\frac{2 b c \sqrt{d-c^2 d x^2} (f x)^{m+2} \left(a+b \cosh ^{-1}(c x)\right)}{f^2 (m+2)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{d-c^2 d x^2} (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)^2}{f (m+2)}-\frac{2 b^2 c^2 \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} (f x)^{m+3} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};c^2 x^2\right)}{f^3 (m+2)^2 (m+3) (1-c x) (c x+1)}",0,"Integrate[(f*x)^m*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2, x]","A",-1
236,0,0,34,3.5368957,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d-c^2 d x^2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2], x]","A",-1
237,0,0,34,4.5881369,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{3/2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(3/2), x]","A",-1
238,0,0,34,4.7772266,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d-c^2 d x^2\right)^{5/2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^2)/(d - c^2*d*x^2)^(5/2), x]","A",-1
239,0,0,29,0.7554891,"\int \frac{(f x)^m \cosh ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((f*x)^m*ArcCosh[c*x]^2)/Sqrt[1 - c^2*x^2],x]","\int \frac{(f x)^m \cosh ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}} \, dx","\text{Int}\left(\frac{\cosh ^{-1}(c x)^2 (f x)^m}{\sqrt{1-c^2 x^2}},x\right)",0,"Integrate[((f*x)^m*ArcCosh[c*x]^2)/Sqrt[1 - c^2*x^2], x]","A",-1
240,1,179,505,0.4171194,"\int \left(c-a^2 c x^2\right)^3 \cosh ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)^3*ArcCosh[a*x]^3,x]","\frac{c^3 \left(2 \sqrt{a x-1} \sqrt{a x+1} \left(16875 a^6 x^6-134541 a^4 x^4+747937 a^2 x^2-22329151\right)-385875 a x \left(5 a^6 x^6-21 a^4 x^4+35 a^2 x^2-35\right) \cosh ^{-1}(a x)^3+11025 \sqrt{a x-1} \sqrt{a x+1} \left(75 a^6 x^6-351 a^4 x^4+757 a^2 x^2-2161\right) \cosh ^{-1}(a x)^2-210 a x \left(1125 a^6 x^6-7371 a^4 x^4+26495 a^2 x^2-226905\right) \cosh ^{-1}(a x)\right)}{13505625 a}","-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)+\frac{702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac{1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac{6 c^3 \left(1-a^2 x^2\right)^4}{2401 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2664 c^3 \left(1-a^2 x^2\right)^3}{214375 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1184 c^3 \left(1-a^2 x^2\right)^2}{42875 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{7104 c^3 \left(1-a^2 x^2\right)}{42875 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{7} c^3 x \left(1-a^2 x^2\right)^3 \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left(1-a^2 x^2\right)^2 \cosh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left(1-a^2 x^2\right) \cosh ^{-1}(a x)^3+\frac{16}{315} a c^3 x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{976 c^3 \sqrt{a x-1} \sqrt{a x+1}}{315 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac{4322 c^3 x \cosh ^{-1}(a x)}{1225}+\frac{3 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)^2}{49 a}-\frac{18 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{8 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac{48 c^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{35 a}",1,"(c^3*(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-22329151 + 747937*a^2*x^2 - 134541*a^4*x^4 + 16875*a^6*x^6) - 210*a*x*(-226905 + 26495*a^2*x^2 - 7371*a^4*x^4 + 1125*a^6*x^6)*ArcCosh[a*x] + 11025*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-2161 + 757*a^2*x^2 - 351*a^4*x^4 + 75*a^6*x^6)*ArcCosh[a*x]^2 - 385875*a*x*(-35 + 35*a^2*x^2 - 21*a^4*x^4 + 5*a^6*x^6)*ArcCosh[a*x]^3))/(13505625*a)","A",1
241,1,147,388,0.2192188,"\int \left(c-a^2 c x^2\right)^2 \cosh ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)^2*ArcCosh[a*x]^3,x]","\frac{c^2 \left(-2 \sqrt{a x-1} \sqrt{a x+1} \left(81 a^4 x^4-842 a^2 x^2+31841\right)+1125 a x \left(3 a^4 x^4-10 a^2 x^2+15\right) \cosh ^{-1}(a x)^3-225 \sqrt{a x-1} \sqrt{a x+1} \left(9 a^4 x^4-38 a^2 x^2+149\right) \cosh ^{-1}(a x)^2+30 a x \left(27 a^4 x^4-190 a^2 x^2+2235\right) \cosh ^{-1}(a x)\right)}{16875 a}","\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6 c^2 \left(1-a^2 x^2\right)^3}{625 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{8 c^2 \left(1-a^2 x^2\right)^2}{375 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{16 c^2 \left(1-a^2 x^2\right)}{125 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{5} c^2 x \left(1-a^2 x^2\right)^2 \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left(1-a^2 x^2\right) \cosh ^{-1}(a x)^3+\frac{8}{135} a c^2 x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{488 c^2 \sqrt{a x-1} \sqrt{a x+1}}{135 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{3 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{4 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{8 c^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{5 a}",1,"(c^2*(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(31841 - 842*a^2*x^2 + 81*a^4*x^4) + 30*a*x*(2235 - 190*a^2*x^2 + 27*a^4*x^4)*ArcCosh[a*x] - 225*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(149 - 38*a^2*x^2 + 9*a^4*x^4)*ArcCosh[a*x]^2 + 1125*a*x*(15 - 10*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*x]^3))/(16875*a)","A",1
242,1,109,175,0.1187301,"\int \left(c-a^2 c x^2\right) \cosh ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)*ArcCosh[a*x]^3,x]","\frac{c \left(2 \sqrt{a x-1} \sqrt{a x+1} \left(a^2 x^2-61\right)-9 a x \left(a^2 x^2-3\right) \cosh ^{-1}(a x)^3+9 \sqrt{a x-1} \sqrt{a x+1} \left(a^2 x^2-7\right) \cosh ^{-1}(a x)^2-6 a x \left(a^2 x^2-21\right) \cosh ^{-1}(a x)\right)}{27 a}","-\frac{2}{9} a^2 c x^3 \cosh ^{-1}(a x)+\frac{1}{3} c x \left(1-a^2 x^2\right) \cosh ^{-1}(a x)^3+\frac{2}{27} a c x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{122 c \sqrt{a x-1} \sqrt{a x+1}}{27 a}+\frac{2}{3} c x \cosh ^{-1}(a x)^3+\frac{14}{3} c x \cosh ^{-1}(a x)+\frac{c (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{3 a}-\frac{2 c \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{a}",1,"(c*(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-61 + a^2*x^2) - 6*a*x*(-21 + a^2*x^2)*ArcCosh[a*x] + 9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-7 + a^2*x^2)*ArcCosh[a*x]^2 - 9*a*x*(-3 + a^2*x^2)*ArcCosh[a*x]^3))/(27*a)","A",1
243,1,129,144,0.1081515,"\int \frac{\cosh ^{-1}(a x)^3}{c-a^2 c x^2} \, dx","Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2),x]","\frac{3 \cosh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)-3 \cosh ^{-1}(a x)^2 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)-6 \cosh ^{-1}(a x) \text{Li}_3\left(-e^{\cosh ^{-1}(a x)}\right)+6 \cosh ^{-1}(a x) \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)+6 \text{Li}_4\left(-e^{\cosh ^{-1}(a x)}\right)-6 \text{Li}_4\left(e^{\cosh ^{-1}(a x)}\right)+\cosh ^{-1}(a x)^3 \left(-\log \left(1-e^{\cosh ^{-1}(a x)}\right)\right)+\cosh ^{-1}(a x)^3 \log \left(e^{\cosh ^{-1}(a x)}+1\right)}{a c}","\frac{3 \cosh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{a c}-\frac{3 \cosh ^{-1}(a x)^2 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{a c}-\frac{6 \cosh ^{-1}(a x) \text{Li}_3\left(-e^{\cosh ^{-1}(a x)}\right)}{a c}+\frac{6 \cosh ^{-1}(a x) \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)}{a c}+\frac{6 \text{Li}_4\left(-e^{\cosh ^{-1}(a x)}\right)}{a c}-\frac{6 \text{Li}_4\left(e^{\cosh ^{-1}(a x)}\right)}{a c}+\frac{2 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{a c}",1,"(-(ArcCosh[a*x]^3*Log[1 - E^ArcCosh[a*x]]) + ArcCosh[a*x]^3*Log[1 + E^ArcCosh[a*x]] + 3*ArcCosh[a*x]^2*PolyLog[2, -E^ArcCosh[a*x]] - 3*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]] - 6*ArcCosh[a*x]*PolyLog[3, -E^ArcCosh[a*x]] + 6*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]] + 6*PolyLog[4, -E^ArcCosh[a*x]] - 6*PolyLog[4, E^ArcCosh[a*x]])/(a*c)","A",0
244,1,276,260,2.1637289,"\int \frac{\cosh ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^2} \, dx","Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2)^2,x]","\frac{-24 \cosh ^{-1}(a x)^2 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)-48 \cosh ^{-1}(a x) \text{Li}_3\left(-e^{-\cosh ^{-1}(a x)}\right)+48 \cosh ^{-1}(a x) \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)-24 \left(\cosh ^{-1}(a x)^2-2\right) \text{Li}_2\left(-e^{-\cosh ^{-1}(a x)}\right)-48 \text{Li}_2\left(e^{-\cosh ^{-1}(a x)}\right)-48 \text{Li}_4\left(-e^{-\cosh ^{-1}(a x)}\right)-48 \text{Li}_4\left(e^{\cosh ^{-1}(a x)}\right)+2 \cosh ^{-1}(a x)^4+8 \cosh ^{-1}(a x)^3 \log \left(e^{-\cosh ^{-1}(a x)}+1\right)-8 \cosh ^{-1}(a x)^3 \log \left(1-e^{\cosh ^{-1}(a x)}\right)+48 \cosh ^{-1}(a x) \log \left(1-e^{-\cosh ^{-1}(a x)}\right)-48 \cosh ^{-1}(a x) \log \left(e^{-\cosh ^{-1}(a x)}+1\right)+12 \cosh ^{-1}(a x)^2 \tanh \left(\frac{1}{2} \cosh ^{-1}(a x)\right)-12 \cosh ^{-1}(a x)^2 \coth \left(\frac{1}{2} \cosh ^{-1}(a x)\right)-2 \cosh ^{-1}(a x)^3 \text{csch}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)-2 \cosh ^{-1}(a x)^3 \text{sech}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)-\pi ^4}{16 a c^2}","\frac{x \cosh ^{-1}(a x)^3}{2 c^2 \left(1-a^2 x^2\right)}+\frac{3 \cosh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{2 a c^2}-\frac{3 \cosh ^{-1}(a x)^2 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{2 a c^2}-\frac{3 \cosh ^{-1}(a x) \text{Li}_3\left(-e^{\cosh ^{-1}(a x)}\right)}{a c^2}+\frac{3 \cosh ^{-1}(a x) \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)}{a c^2}-\frac{3 \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{a c^2}+\frac{3 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{a c^2}+\frac{3 \text{Li}_4\left(-e^{\cosh ^{-1}(a x)}\right)}{a c^2}-\frac{3 \text{Li}_4\left(e^{\cosh ^{-1}(a x)}\right)}{a c^2}-\frac{3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x)^3 \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{a c^2}-\frac{6 \cosh ^{-1}(a x) \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{a c^2}",1,"(-Pi^4 + 2*ArcCosh[a*x]^4 - 12*ArcCosh[a*x]^2*Coth[ArcCosh[a*x]/2] - 2*ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^2 + 48*ArcCosh[a*x]*Log[1 - E^(-ArcCosh[a*x])] - 48*ArcCosh[a*x]*Log[1 + E^(-ArcCosh[a*x])] + 8*ArcCosh[a*x]^3*Log[1 + E^(-ArcCosh[a*x])] - 8*ArcCosh[a*x]^3*Log[1 - E^ArcCosh[a*x]] - 24*(-2 + ArcCosh[a*x]^2)*PolyLog[2, -E^(-ArcCosh[a*x])] - 48*PolyLog[2, E^(-ArcCosh[a*x])] - 24*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]] - 48*ArcCosh[a*x]*PolyLog[3, -E^(-ArcCosh[a*x])] + 48*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]] - 48*PolyLog[4, -E^(-ArcCosh[a*x])] - 48*PolyLog[4, E^ArcCosh[a*x]] - 2*ArcCosh[a*x]^3*Sech[ArcCosh[a*x]/2]^2 + 12*ArcCosh[a*x]^2*Tanh[ArcCosh[a*x]/2])/(16*a*c^2)","A",0
245,1,455,387,8.1920464,"\int \frac{\cosh ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^3} \, dx","Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2)^3,x]","-\frac{72 \cosh ^{-1}(a x)^2 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)+144 \cosh ^{-1}(a x) \text{Li}_3\left(-e^{-\cosh ^{-1}(a x)}\right)-144 \cosh ^{-1}(a x) \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)+8 \left(9 \cosh ^{-1}(a x)^2-20\right) \text{Li}_2\left(-e^{-\cosh ^{-1}(a x)}\right)+160 \text{Li}_2\left(e^{-\cosh ^{-1}(a x)}\right)+144 \text{Li}_4\left(-e^{-\cosh ^{-1}(a x)}\right)+144 \text{Li}_4\left(e^{\cosh ^{-1}(a x)}\right)-6 \cosh ^{-1}(a x)^4-24 \cosh ^{-1}(a x)^3 \log \left(e^{-\cosh ^{-1}(a x)}+1\right)+24 \cosh ^{-1}(a x)^3 \log \left(1-e^{\cosh ^{-1}(a x)}\right)-160 \cosh ^{-1}(a x) \log \left(1-e^{-\cosh ^{-1}(a x)}\right)+160 \cosh ^{-1}(a x) \log \left(e^{-\cosh ^{-1}(a x)}+1\right)-\frac{16 \cosh ^{-1}(a x)^2 \sinh ^4\left(\frac{1}{2} \cosh ^{-1}(a x)\right)}{\left(\frac{a x-1}{a x+1}\right)^{3/2} (a x+1)^3}-40 \cosh ^{-1}(a x)^2 \tanh \left(\frac{1}{2} \cosh ^{-1}(a x)\right)+8 \tanh \left(\frac{1}{2} \cosh ^{-1}(a x)\right)+40 \cosh ^{-1}(a x)^2 \coth \left(\frac{1}{2} \cosh ^{-1}(a x)\right)-8 \coth \left(\frac{1}{2} \cosh ^{-1}(a x)\right)-\cosh ^{-1}(a x)^3 \text{csch}^4\left(\frac{1}{2} \cosh ^{-1}(a x)\right)-\sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^2 \text{csch}^4\left(\frac{1}{2} \cosh ^{-1}(a x)\right)+6 \cosh ^{-1}(a x)^3 \text{csch}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)-4 \cosh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)+\cosh ^{-1}(a x)^3 \text{sech}^4\left(\frac{1}{2} \cosh ^{-1}(a x)\right)+6 \cosh ^{-1}(a x)^3 \text{sech}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)-4 \cosh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \cosh ^{-1}(a x)\right)+3 \pi ^4}{64 a c^3}","\frac{3 x \cosh ^{-1}(a x)^3}{8 c^3 \left(1-a^2 x^2\right)}+\frac{x \cosh ^{-1}(a x)^3}{4 c^3 \left(1-a^2 x^2\right)^2}-\frac{x \cosh ^{-1}(a x)}{4 c^3 \left(1-a^2 x^2\right)}+\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{8 a c^3}-\frac{9 \cosh ^{-1}(a x)^2 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{8 a c^3}-\frac{9 \cosh ^{-1}(a x) \text{Li}_3\left(-e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}+\frac{9 \cosh ^{-1}(a x) \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}-\frac{5 \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)}{2 a c^3}+\frac{5 \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)}{2 a c^3}+\frac{9 \text{Li}_4\left(-e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}-\frac{9 \text{Li}_4\left(e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}+\frac{1}{4 a c^3 \sqrt{a x-1} \sqrt{a x+1}}-\frac{9 \cosh ^{-1}(a x)^2}{8 a c^3 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x)^2}{4 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}+\frac{3 \cosh ^{-1}(a x)^3 \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{4 a c^3}-\frac{5 \cosh ^{-1}(a x) \tanh ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{a c^3}",1,"-1/64*(3*Pi^4 - 6*ArcCosh[a*x]^4 - 8*Coth[ArcCosh[a*x]/2] + 40*ArcCosh[a*x]^2*Coth[ArcCosh[a*x]/2] - 4*ArcCosh[a*x]*Csch[ArcCosh[a*x]/2]^2 + 6*ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^2 - Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^2*Csch[ArcCosh[a*x]/2]^4 - ArcCosh[a*x]^3*Csch[ArcCosh[a*x]/2]^4 - 160*ArcCosh[a*x]*Log[1 - E^(-ArcCosh[a*x])] + 160*ArcCosh[a*x]*Log[1 + E^(-ArcCosh[a*x])] - 24*ArcCosh[a*x]^3*Log[1 + E^(-ArcCosh[a*x])] + 24*ArcCosh[a*x]^3*Log[1 - E^ArcCosh[a*x]] + 8*(-20 + 9*ArcCosh[a*x]^2)*PolyLog[2, -E^(-ArcCosh[a*x])] + 160*PolyLog[2, E^(-ArcCosh[a*x])] + 72*ArcCosh[a*x]^2*PolyLog[2, E^ArcCosh[a*x]] + 144*ArcCosh[a*x]*PolyLog[3, -E^(-ArcCosh[a*x])] - 144*ArcCosh[a*x]*PolyLog[3, E^ArcCosh[a*x]] + 144*PolyLog[4, -E^(-ArcCosh[a*x])] + 144*PolyLog[4, E^ArcCosh[a*x]] - 4*ArcCosh[a*x]*Sech[ArcCosh[a*x]/2]^2 + 6*ArcCosh[a*x]^3*Sech[ArcCosh[a*x]/2]^2 + ArcCosh[a*x]^3*Sech[ArcCosh[a*x]/2]^4 - (16*ArcCosh[a*x]^2*Sinh[ArcCosh[a*x]/2]^4)/(((-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x)^3) + 8*Tanh[ArcCosh[a*x]/2] - 40*ArcCosh[a*x]^2*Tanh[ArcCosh[a*x]/2])/(a*c^3)","A",0
246,1,189,605,1.2025363,"\int \left(c-a^2 c x^2\right)^{5/2} \cosh ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^3,x]","\frac{c^2 \sqrt{c-a^2 c x^2} \left(-4320 \cosh ^{-1}(a x)^4-72 \left(270 \cosh \left(2 \cosh ^{-1}(a x)\right)-27 \cosh \left(4 \cosh ^{-1}(a x)\right)+2 \cosh \left(6 \cosh ^{-1}(a x)\right)\right) \cosh ^{-1}(a x)^2-9720 \cosh \left(2 \cosh ^{-1}(a x)\right)+243 \cosh \left(4 \cosh ^{-1}(a x)\right)-8 \cosh \left(6 \cosh ^{-1}(a x)\right)+288 \cosh ^{-1}(a x)^3 \left(45 \sinh \left(2 \cosh ^{-1}(a x)\right)-9 \sinh \left(4 \cosh ^{-1}(a x)\right)+\sinh \left(6 \cosh ^{-1}(a x)\right)\right)+12 \cosh ^{-1}(a x) \left(1620 \sinh \left(2 \cosh ^{-1}(a x)\right)-81 \sinh \left(4 \cosh ^{-1}(a x)\right)+4 \sinh \left(6 \cosh ^{-1}(a x)\right)\right)\right)}{55296 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","-\frac{865 a c^2 x^2 \sqrt{c-a^2 c x^2}}{2304 \sqrt{a x-1} \sqrt{a x+1}}+\frac{c^2 \left(1-a^2 x^2\right)^3 \sqrt{c-a^2 c x^2}}{216 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 a c^2 x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt{a x-1} \sqrt{a x+1}}+\frac{5}{16} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{245}{384} c^2 x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{1}{36} c^2 x (1-a x)^2 (a x+1)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{65}{576} c^2 x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{5 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{c^2 \left(1-a^2 x^2\right)^3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{5 c^2 \left(1-a^2 x^2\right)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{115 c^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{768 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{6} x \left(c-a^2 c x^2\right)^{5/2} \cosh ^{-1}(a x)^3+\frac{5}{24} c x \left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^3+\frac{65 a^3 c^2 x^4 \sqrt{c-a^2 c x^2}}{2304 \sqrt{a x-1} \sqrt{a x+1}}",1,"(c^2*Sqrt[c - a^2*c*x^2]*(-4320*ArcCosh[a*x]^4 - 9720*Cosh[2*ArcCosh[a*x]] + 243*Cosh[4*ArcCosh[a*x]] - 8*Cosh[6*ArcCosh[a*x]] - 72*ArcCosh[a*x]^2*(270*Cosh[2*ArcCosh[a*x]] - 27*Cosh[4*ArcCosh[a*x]] + 2*Cosh[6*ArcCosh[a*x]]) + 288*ArcCosh[a*x]^3*(45*Sinh[2*ArcCosh[a*x]] - 9*Sinh[4*ArcCosh[a*x]] + Sinh[6*ArcCosh[a*x]]) + 12*ArcCosh[a*x]*(1620*Sinh[2*ArcCosh[a*x]] - 81*Sinh[4*ArcCosh[a*x]] + 4*Sinh[6*ArcCosh[a*x]])))/(55296*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",0
247,1,148,402,0.4445926,"\int \left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^3 \, dx","Integrate[(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^3,x]","-\frac{c \sqrt{c-a^2 c x^2} \left(96 \cosh ^{-1}(a x)^4-24 \left(\cosh \left(4 \cosh ^{-1}(a x)\right)-16 \cosh \left(2 \cosh ^{-1}(a x)\right)\right) \cosh ^{-1}(a x)^2-3 \left(\cosh \left(4 \cosh ^{-1}(a x)\right)-64 \cosh \left(2 \cosh ^{-1}(a x)\right)\right)+32 \cosh ^{-1}(a x)^3 \left(\sinh \left(4 \cosh ^{-1}(a x)\right)-8 \sinh \left(2 \cosh ^{-1}(a x)\right)\right)+12 \cosh ^{-1}(a x) \left(\sinh \left(4 \cosh ^{-1}(a x)\right)-32 \sinh \left(2 \cosh ^{-1}(a x)\right)\right)\right)}{1024 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","-\frac{51 a c x^2 \sqrt{c-a^2 c x^2}}{128 \sqrt{a x-1} \sqrt{a x+1}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{4} x \left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^3+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{45}{64} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{3}{32} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 c \left(1-a^2 x^2\right)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{27 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{128 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 a^3 c x^4 \sqrt{c-a^2 c x^2}}{128 \sqrt{a x-1} \sqrt{a x+1}}",1,"-1/1024*(c*Sqrt[c - a^2*c*x^2]*(96*ArcCosh[a*x]^4 - 3*(-64*Cosh[2*ArcCosh[a*x]] + Cosh[4*ArcCosh[a*x]]) - 24*ArcCosh[a*x]^2*(-16*Cosh[2*ArcCosh[a*x]] + Cosh[4*ArcCosh[a*x]]) + 12*ArcCosh[a*x]*(-32*Sinh[2*ArcCosh[a*x]] + Sinh[4*ArcCosh[a*x]]) + 32*ArcCosh[a*x]^3*(-8*Sinh[2*ArcCosh[a*x]] + Sinh[4*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",0
248,1,98,231,0.2074967,"\int \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3 \, dx","Integrate[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3,x]","-\frac{\sqrt{-c (a x-1) (a x+1)} \left(2 \cosh ^{-1}(a x)^4+\left(6 \cosh ^{-1}(a x)^2+3\right) \cosh \left(2 \cosh ^{-1}(a x)\right)-2 \left(2 \cosh ^{-1}(a x)^2+3\right) \cosh ^{-1}(a x) \sinh \left(2 \cosh ^{-1}(a x)\right)\right)}{16 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","-\frac{3 a x^2 \sqrt{c-a^2 c x^2}}{8 \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{8 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{4 \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{8 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3}{4} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)",1,"-1/16*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(2*ArcCosh[a*x]^4 + (3 + 6*ArcCosh[a*x]^2)*Cosh[2*ArcCosh[a*x]] - 2*ArcCosh[a*x]*(3 + 2*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]]))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",1
249,1,46,46,0.0303908,"\int \frac{\cosh ^{-1}(a x)^3}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[ArcCosh[a*x]^3/Sqrt[c - a^2*c*x^2],x]","\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}}","\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^4}{4 a \sqrt{c-a^2 c x^2}}",1,"(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^4)/(4*a*Sqrt[c - a^2*c*x^2])","A",1
250,1,145,241,0.2557659,"\int \frac{\cosh ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(3/2),x]","\frac{\frac{\sqrt{a x-1} \sqrt{a x+1} \left(-6 \cosh ^{-1}(a x) \text{Li}_2\left(-e^{\cosh ^{-1}(a x)}\right)-6 \cosh ^{-1}(a x) \text{Li}_2\left(e^{\cosh ^{-1}(a x)}\right)+6 \text{Li}_3\left(-e^{\cosh ^{-1}(a x)}\right)+6 \text{Li}_3\left(e^{\cosh ^{-1}(a x)}\right)+\cosh ^{-1}(a x)^3-3 \cosh ^{-1}(a x)^2 \log \left(1-e^{\cosh ^{-1}(a x)}\right)-3 \cosh ^{-1}(a x)^2 \log \left(e^{\cosh ^{-1}(a x)}+1\right)\right)}{a}+x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}","-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{Li}_2\left(e^{2 \cosh ^{-1}(a x)}\right)}{a c \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \text{Li}_3\left(e^{2 \cosh ^{-1}(a x)}\right)}{2 a c \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left(1-e^{2 \cosh ^{-1}(a x)}\right)}{a c \sqrt{c-a^2 c x^2}}",1,"(x*ArcCosh[a*x]^3 + (Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(ArcCosh[a*x]^3 - 3*ArcCosh[a*x]^2*Log[1 - E^ArcCosh[a*x]] - 3*ArcCosh[a*x]^2*Log[1 + E^ArcCosh[a*x]] - 6*ArcCosh[a*x]*PolyLog[2, -E^ArcCosh[a*x]] - 6*ArcCosh[a*x]*PolyLog[2, E^ArcCosh[a*x]] + 6*PolyLog[3, -E^ArcCosh[a*x]] + 6*PolyLog[3, E^ArcCosh[a*x]]))/a)/(c*Sqrt[c - a^2*c*x^2])","A",0
251,1,258,413,0.9970223,"\int \frac{\cosh ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(5/2),x]","\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(\frac{6 \cosh ^{-1}(a x)^2}{1-a^2 x^2}-24 \cosh ^{-1}(a x) \text{Li}_2\left(e^{2 \cosh ^{-1}(a x)}\right)+12 \text{Li}_3\left(e^{2 \cosh ^{-1}(a x)}\right)+12 \log \left(\sqrt{\frac{a x-1}{a x+1}} (a x+1)\right)-\frac{4 a x \left(\frac{a x-1}{a x+1}\right)^{3/2} \cosh ^{-1}(a x)^3}{(a x-1)^3}+\frac{8 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^3}{a x-1}+8 \cosh ^{-1}(a x)^3-\frac{12 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)}{a x-1}-24 \cosh ^{-1}(a x)^2 \log \left(1-e^{2 \cosh ^{-1}(a x)}\right)-i \pi ^3\right)}{12 a c^2 \sqrt{c-a^2 c x^2}}","-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{Li}_2\left(e^{2 \cosh ^{-1}(a x)}\right)}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \text{Li}_3\left(e^{2 \cosh ^{-1}(a x)}\right)}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \log \left(1-a^2 x^2\right)}{2 a c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \cosh ^{-1}(a x)^3}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{3 a c^2 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{2 a c^2 \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left(1-e^{2 \cosh ^{-1}(a x)}\right)}{a c^2 \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{3 c \left(c-a^2 c x^2\right)^{3/2}}",1,"(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*((-I)*Pi^3 - (12*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x])/(-1 + a*x) + (6*ArcCosh[a*x]^2)/(1 - a^2*x^2) + 8*ArcCosh[a*x]^3 + (8*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3)/(-1 + a*x) - (4*a*x*((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x]^3)/(-1 + a*x)^3 - 24*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])] + 12*Log[Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)] - 24*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x])] + 12*PolyLog[3, E^(2*ArcCosh[a*x])]))/(12*a*c^2*Sqrt[c - a^2*c*x^2])","C",0
252,1,363,607,1.8533655,"\int \frac{\cosh ^{-1}(a x)^3}{\left(c-a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcCosh[a*x]^3/(c - a^2*c*x^2)^(7/2),x]","-\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(\frac{3}{1-a^2 x^2}+\frac{24 \cosh ^{-1}(a x)^2}{a^2 x^2-1}-\frac{9 \cosh ^{-1}(a x)^2}{\left(a^2 x^2-1\right)^2}+96 \cosh ^{-1}(a x) \text{Li}_2\left(e^{2 \cosh ^{-1}(a x)}\right)-48 \text{Li}_3\left(e^{2 \cosh ^{-1}(a x)}\right)-60 \log \left(\sqrt{\frac{a x-1}{a x+1}} (a x+1)\right)+\frac{16 a x \left(\frac{a x-1}{a x+1}\right)^{3/2} \cosh ^{-1}(a x)^3}{(a x-1)^3}-\frac{32 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^3}{a x-1}-\frac{12 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^3}{(a x-1)^3 (a x+1)^2}-32 \cosh ^{-1}(a x)^3-\frac{6 a x \left(\frac{a x-1}{a x+1}\right)^{3/2} \cosh ^{-1}(a x)}{(a x-1)^3}+\frac{60 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)}{a x-1}+96 \cosh ^{-1}(a x)^2 \log \left(1-e^{2 \cosh ^{-1}(a x)}\right)+4 i \pi ^3\right)}{60 a c^3 \sqrt{c-a^2 c x^2}}","-\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{Li}_2\left(e^{2 \cosh ^{-1}(a x)}\right)}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{a x-1} \sqrt{a x+1} \text{Li}_3\left(e^{2 \cosh ^{-1}(a x)}\right)}{5 a c^3 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{a x-1} \sqrt{a x+1}}{20 a c^3 \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \log \left(1-a^2 x^2\right)}{2 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \cosh ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{5 a c^3 \left(1-a^2 x^2\right) \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{20 a c^3 \left(1-a^2 x^2\right)^2 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}-\frac{x \cosh ^{-1}(a x)}{10 c^3 (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}-\frac{8 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left(1-e^{2 \cosh ^{-1}(a x)}\right)}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \cosh ^{-1}(a x)^3}{15 c^2 \left(c-a^2 c x^2\right)^{3/2}}+\frac{x \cosh ^{-1}(a x)^3}{5 c \left(c-a^2 c x^2\right)^{5/2}}",1,"-1/60*(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*((4*I)*Pi^3 + 3/(1 - a^2*x^2) + (60*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x])/(-1 + a*x) - (6*a*x*((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x])/(-1 + a*x)^3 - (9*ArcCosh[a*x]^2)/(-1 + a^2*x^2)^2 + (24*ArcCosh[a*x]^2)/(-1 + a^2*x^2) - 32*ArcCosh[a*x]^3 - (32*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3)/(-1 + a*x) + (16*a*x*((-1 + a*x)/(1 + a*x))^(3/2)*ArcCosh[a*x]^3)/(-1 + a*x)^3 - (12*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3)/((-1 + a*x)^3*(1 + a*x)^2) + 96*ArcCosh[a*x]^2*Log[1 - E^(2*ArcCosh[a*x])] - 60*Log[Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)] + 96*ArcCosh[a*x]*PolyLog[2, E^(2*ArcCosh[a*x])] - 48*PolyLog[3, E^(2*ArcCosh[a*x])]))/(a*c^3*Sqrt[c - a^2*c*x^2])","C",0
253,1,136,315,0.4511439,"\int \frac{x^4 \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^4*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(-192 \left(2 \cosh ^{-1}(a x)^2+1\right) \cosh \left(2 \cosh ^{-1}(a x)\right)-3 \left(8 \cosh ^{-1}(a x)^2+1\right) \cosh \left(4 \cosh ^{-1}(a x)\right)+4 \cosh ^{-1}(a x) \left(24 \cosh ^{-1}(a x)^3+32 \left(2 \cosh ^{-1}(a x)^2+3\right) \sinh \left(2 \cosh ^{-1}(a x)\right)+\left(8 \cosh ^{-1}(a x)^2+3\right) \sinh \left(4 \cosh ^{-1}(a x)\right)\right)\right)}{1024 a^5 \sqrt{-((a x-1) (a x+1))}}","\frac{3 \sqrt{a x-1} \cosh ^{-1}(a x)^4}{32 a^5 \sqrt{1-a x}}+\frac{45 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{128 a^5 \sqrt{1-a x}}-\frac{45 x \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{64 a^4}-\frac{45 x^2 \sqrt{a x-1}}{128 a^3 \sqrt{1-a x}}-\frac{9 x^2 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{16 a^3 \sqrt{1-a x}}-\frac{3 x^3 \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{32 a^2}-\frac{x^3 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{4 a^2}-\frac{3 x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{8 a^4}-\frac{3 x^4 \sqrt{a x-1}}{128 a \sqrt{1-a x}}-\frac{3 x^4 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{16 a \sqrt{1-a x}}",1,"(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(-192*(1 + 2*ArcCosh[a*x]^2)*Cosh[2*ArcCosh[a*x]] - 3*(1 + 8*ArcCosh[a*x]^2)*Cosh[4*ArcCosh[a*x]] + 4*ArcCosh[a*x]*(24*ArcCosh[a*x]^3 + 32*(3 + 2*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]] + (3 + 8*ArcCosh[a*x]^2)*Sinh[4*ArcCosh[a*x]])))/(1024*a^5*Sqrt[-((-1 + a*x)*(1 + a*x))])","A",1
254,1,140,243,0.1695709,"\int \frac{x^3 \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^3*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{1-a^2 x^2} \left(2 a x \left(a^2 x^2+60\right)-9 \sqrt{a x-1} \sqrt{a x+1} \left(a^2 x^2+2\right) \cosh ^{-1}(a x)^3+9 a x \left(a^2 x^2+6\right) \cosh ^{-1}(a x)^2-6 \sqrt{a x-1} \sqrt{a x+1} \left(a^2 x^2+20\right) \cosh ^{-1}(a x)\right)}{27 a^4 \sqrt{a x-1} \sqrt{a x+1}}","-\frac{40 \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a^4}-\frac{40 x \sqrt{a x-1}}{9 a^3 \sqrt{1-a x}}-\frac{2 x \sqrt{a x-1} \cosh ^{-1}(a x)^2}{a^3 \sqrt{1-a x}}-\frac{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{3 a^2}-\frac{2 x^2 \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{9 a^2}-\frac{2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{3 a^4}-\frac{2 x^3 \sqrt{a x-1}}{27 a \sqrt{1-a x}}-\frac{x^3 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{3 a \sqrt{1-a x}}",1,"(Sqrt[1 - a^2*x^2]*(2*a*x*(60 + a^2*x^2) - 6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(20 + a^2*x^2)*ArcCosh[a*x] + 9*a*x*(6 + a^2*x^2)*ArcCosh[a*x]^2 - 9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(2 + a^2*x^2)*ArcCosh[a*x]^3))/(27*a^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])","A",1
255,1,98,188,0.2241494,"\int \frac{x^2 \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x^2*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]","-\frac{\sqrt{-((a x-1) (a x+1))} \left(2 \cosh ^{-1}(a x) \left(\cosh ^{-1}(a x)^3+\left(2 \cosh ^{-1}(a x)^2+3\right) \sinh \left(2 \cosh ^{-1}(a x)\right)\right)-3 \left(2 \cosh ^{-1}(a x)^2+1\right) \cosh \left(2 \cosh ^{-1}(a x)\right)\right)}{16 a^3 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^4}{8 a^3 \sqrt{1-a x}}+\frac{3 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{8 a^3 \sqrt{1-a x}}-\frac{x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{2 a^2}-\frac{3 x \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{4 a^2}-\frac{3 x^2 \sqrt{a x-1}}{8 a \sqrt{1-a x}}-\frac{3 x^2 \sqrt{a x-1} \cosh ^{-1}(a x)^2}{4 a \sqrt{1-a x}}",1,"-1/16*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(-3*(1 + 2*ArcCosh[a*x]^2)*Cosh[2*ArcCosh[a*x]] + 2*ArcCosh[a*x]*(ArcCosh[a*x]^3 + (3 + 2*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]])))/(a^3*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",1
256,1,101,110,0.0971031,"\int \frac{x \cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[(x*ArcCosh[a*x]^3)/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{1-a^2 x^2} \left(6 a x-\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3+3 a x \cosh ^{-1}(a x)^2-6 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)\right)}{a^2 \sqrt{a x-1} \sqrt{a x+1}}","-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{a^2}-\frac{6 \sqrt{1-a x} \sqrt{a x+1} \cosh ^{-1}(a x)}{a^2}-\frac{6 x \sqrt{a x-1}}{a \sqrt{1-a x}}-\frac{3 x \sqrt{a x-1} \cosh ^{-1}(a x)^2}{a \sqrt{1-a x}}",1,"(Sqrt[1 - a^2*x^2]*(6*a*x - 6*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x] + 3*a*x*ArcCosh[a*x]^2 - Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^3))/(a^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])","A",1
257,1,45,32,0.0221138,"\int \frac{\cosh ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]^3/Sqrt[1 - a^2*x^2],x]","\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^4}{4 a \sqrt{1-a^2 x^2}}","\frac{\sqrt{a x-1} \cosh ^{-1}(a x)^4}{4 a \sqrt{1-a x}}",1,"(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^4)/(4*a*Sqrt[1 - a^2*x^2])","A",1
258,1,488,265,0.6557593,"\int \frac{\cosh ^{-1}(a x)^3}{x \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]^3/(x*Sqrt[1 - a^2*x^2]),x]","\frac{i \sqrt{-((a x-1) (a x+1))} \left(192 \cosh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\cosh ^{-1}(a x)}\right)+192 i \pi  \cosh ^{-1}(a x) \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)+384 \cosh ^{-1}(a x) \text{Li}_3\left(-i e^{-\cosh ^{-1}(a x)}\right)-384 \cosh ^{-1}(a x) \text{Li}_3\left(-i e^{\cosh ^{-1}(a x)}\right)-48 \left(\pi -2 i \cosh ^{-1}(a x)\right)^2 \text{Li}_2\left(-i e^{-\cosh ^{-1}(a x)}\right)-48 \pi ^2 \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)+192 i \pi  \text{Li}_3\left(-i e^{-\cosh ^{-1}(a x)}\right)-192 i \pi  \text{Li}_3\left(i e^{\cosh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{-\cosh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{\cosh ^{-1}(a x)}\right)-16 \cosh ^{-1}(a x)^4-32 i \pi  \cosh ^{-1}(a x)^3+24 \pi ^2 \cosh ^{-1}(a x)^2+8 i \pi ^3 \cosh ^{-1}(a x)-64 \cosh ^{-1}(a x)^3 \log \left(1+i e^{-\cosh ^{-1}(a x)}\right)+64 \cosh ^{-1}(a x)^3 \log \left(1+i e^{\cosh ^{-1}(a x)}\right)-96 i \pi  \cosh ^{-1}(a x)^2 \log \left(1+i e^{-\cosh ^{-1}(a x)}\right)+96 i \pi  \cosh ^{-1}(a x)^2 \log \left(1-i e^{\cosh ^{-1}(a x)}\right)+48 \pi ^2 \cosh ^{-1}(a x) \log \left(1+i e^{-\cosh ^{-1}(a x)}\right)-48 \pi ^2 \cosh ^{-1}(a x) \log \left(1-i e^{\cosh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(1+i e^{-\cosh ^{-1}(a x)}\right)-8 i \pi ^3 \log \left(1+i e^{\cosh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(\pi +2 i \cosh ^{-1}(a x)\right)\right)\right)+7 \pi ^4\right)}{64 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","-\frac{3 i \sqrt{a x-1} \cosh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{3 i \sqrt{a x-1} \cosh ^{-1}(a x)^2 \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{6 i \sqrt{a x-1} \cosh ^{-1}(a x) \text{Li}_3\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{6 i \sqrt{a x-1} \cosh ^{-1}(a x) \text{Li}_3\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{6 i \sqrt{a x-1} \text{Li}_4\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{6 i \sqrt{a x-1} \text{Li}_4\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{2 \sqrt{a x-1} \cosh ^{-1}(a x)^3 \tan ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}",1,"((I/64)*Sqrt[-((-1 + a*x)*(1 + a*x))]*(7*Pi^4 + (8*I)*Pi^3*ArcCosh[a*x] + 24*Pi^2*ArcCosh[a*x]^2 - (32*I)*Pi*ArcCosh[a*x]^3 - 16*ArcCosh[a*x]^4 + (8*I)*Pi^3*Log[1 + I/E^ArcCosh[a*x]] + 48*Pi^2*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] - (96*I)*Pi*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] - 64*ArcCosh[a*x]^3*Log[1 + I/E^ArcCosh[a*x]] - 48*Pi^2*ArcCosh[a*x]*Log[1 - I*E^ArcCosh[a*x]] + (96*I)*Pi*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh[a*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcCosh[a*x]] + 64*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcCosh[a*x])/4]] - 48*(Pi - (2*I)*ArcCosh[a*x])^2*PolyLog[2, (-I)/E^ArcCosh[a*x]] + 192*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcCosh[a*x]] + 384*ArcCosh[a*x]*PolyLog[3, (-I)/E^ArcCosh[a*x]] - 384*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcCosh[a*x]] + 384*PolyLog[4, (-I)/E^ArcCosh[a*x]] + 384*PolyLog[4, (-I)*E^ArcCosh[a*x]]))/(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",0
259,1,137,166,0.5412946,"\int \frac{\cosh ^{-1}(a x)^3}{x^2 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]),x]","\frac{a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(6 \cosh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \cosh ^{-1}(a x)}\right)+3 \text{Li}_3\left(-e^{-2 \cosh ^{-1}(a x)}\right)+2 \cosh ^{-1}(a x)^2 \left(\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a x}-\cosh ^{-1}(a x)-3 \log \left(e^{-2 \cosh ^{-1}(a x)}+1\right)\right)\right)}{2 \sqrt{-((a x-1) (a x+1))}}","-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{x}-\frac{3 a \sqrt{a x-1} \cosh ^{-1}(a x) \text{Li}_2\left(-e^{2 \cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{3 a \sqrt{a x-1} \text{Li}_3\left(-e^{2 \cosh ^{-1}(a x)}\right)}{2 \sqrt{1-a x}}+\frac{a \sqrt{a x-1} \cosh ^{-1}(a x)^3}{\sqrt{1-a x}}-\frac{3 a \sqrt{a x-1} \cosh ^{-1}(a x)^2 \log \left(e^{2 \cosh ^{-1}(a x)}+1\right)}{\sqrt{1-a x}}",1,"(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(2*ArcCosh[a*x]^2*(-ArcCosh[a*x] + (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x])/(a*x) - 3*Log[1 + E^(-2*ArcCosh[a*x])]) + 6*ArcCosh[a*x]*PolyLog[2, -E^(-2*ArcCosh[a*x])] + 3*PolyLog[3, -E^(-2*ArcCosh[a*x])]))/(2*Sqrt[-((-1 + a*x)*(1 + a*x))])","A",0
260,1,1051,460,6.2646475,"\int \frac{\cosh ^{-1}(a x)^3}{x^3 \sqrt{1-a^2 x^2}} \, dx","Integrate[ArcCosh[a*x]^3/(x^3*Sqrt[1 - a^2*x^2]),x]","-\frac{i a^2 (a x+1) \left(-16 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^4+\frac{64 i (a x-1) \cosh ^{-1}(a x)^3}{a^2 x^2}-64 \sqrt{\frac{a x-1}{a x+1}} \log \left(1+i e^{-\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)^3+64 \sqrt{\frac{a x-1}{a x+1}} \log \left(1+i e^{\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)^3-32 i \pi  \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^3-96 i \pi  \sqrt{\frac{a x-1}{a x+1}} \log \left(1+i e^{-\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)^2+96 i \pi  \sqrt{\frac{a x-1}{a x+1}} \log \left(1-i e^{\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)^2+192 \sqrt{\frac{a x-1}{a x+1}} \text{Li}_2\left(-i e^{\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)^2+\frac{192 i \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^2}{a x}+24 \pi ^2 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)^2-384 \sqrt{\frac{a x-1}{a x+1}} \log \left(1-i e^{-\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)+48 \pi ^2 \sqrt{\frac{a x-1}{a x+1}} \log \left(1+i e^{-\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)+384 \sqrt{\frac{a x-1}{a x+1}} \log \left(1+i e^{-\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)-48 \pi ^2 \sqrt{\frac{a x-1}{a x+1}} \log \left(1-i e^{\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)+192 i \pi  \sqrt{\frac{a x-1}{a x+1}} \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)+384 \sqrt{\frac{a x-1}{a x+1}} \text{Li}_3\left(-i e^{-\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)-384 \sqrt{\frac{a x-1}{a x+1}} \text{Li}_3\left(-i e^{\cosh ^{-1}(a x)}\right) \cosh ^{-1}(a x)+8 i \pi ^3 \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)+8 i \pi ^3 \sqrt{\frac{a x-1}{a x+1}} \log \left(1+i e^{-\cosh ^{-1}(a x)}\right)-8 i \pi ^3 \sqrt{\frac{a x-1}{a x+1}} \log \left(1+i e^{\cosh ^{-1}(a x)}\right)+8 i \pi ^3 \sqrt{\frac{a x-1}{a x+1}} \log \left(\tan \left(\frac{1}{4} \left(2 i \cosh ^{-1}(a x)+\pi \right)\right)\right)-48 \sqrt{\frac{a x-1}{a x+1}} \left(-4 \cosh ^{-1}(a x)^2-4 i \pi  \cosh ^{-1}(a x)+\pi ^2+8\right) \text{Li}_2\left(-i e^{-\cosh ^{-1}(a x)}\right)+384 \sqrt{\frac{a x-1}{a x+1}} \text{Li}_2\left(i e^{-\cosh ^{-1}(a x)}\right)-48 \pi ^2 \sqrt{\frac{a x-1}{a x+1}} \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)+192 i \pi  \sqrt{\frac{a x-1}{a x+1}} \text{Li}_3\left(-i e^{-\cosh ^{-1}(a x)}\right)-192 i \pi  \sqrt{\frac{a x-1}{a x+1}} \text{Li}_3\left(i e^{\cosh ^{-1}(a x)}\right)+384 \sqrt{\frac{a x-1}{a x+1}} \text{Li}_4\left(-i e^{-\cosh ^{-1}(a x)}\right)+384 \sqrt{\frac{a x-1}{a x+1}} \text{Li}_4\left(-i e^{\cosh ^{-1}(a x)}\right)+7 \pi ^4 \sqrt{\frac{a x-1}{a x+1}}\right)}{128 \sqrt{1-a^2 x^2}}","-\frac{3 i a^2 \sqrt{a x-1} \cosh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\cosh ^{-1}(a x)}\right)}{2 \sqrt{1-a x}}+\frac{3 i a^2 \sqrt{a x-1} \cosh ^{-1}(a x)^2 \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)}{2 \sqrt{1-a x}}+\frac{3 i a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \text{Li}_3\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{3 i a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \text{Li}_3\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{3 i a^2 \sqrt{a x-1} \text{Li}_2\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{3 i a^2 \sqrt{a x-1} \text{Li}_2\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{3 i a^2 \sqrt{a x-1} \text{Li}_4\left(-i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{3 i a^2 \sqrt{a x-1} \text{Li}_4\left(i e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3}{2 x^2}+\frac{a^2 \sqrt{a x-1} \cosh ^{-1}(a x)^3 \tan ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}-\frac{6 a^2 \sqrt{a x-1} \cosh ^{-1}(a x) \tan ^{-1}\left(e^{\cosh ^{-1}(a x)}\right)}{\sqrt{1-a x}}+\frac{3 a \sqrt{a x-1} \cosh ^{-1}(a x)^2}{2 x \sqrt{1-a x}}",1,"((-1/128*I)*a^2*(1 + a*x)*(7*Pi^4*Sqrt[(-1 + a*x)/(1 + a*x)] + (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] + 24*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2 + ((192*I)*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2)/(a*x) + ((64*I)*(-1 + a*x)*ArcCosh[a*x]^3)/(a^2*x^2) - (32*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3 - 16*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^4 - 384*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 - I/E^ArcCosh[a*x]] + (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[1 + I/E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] + 48*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 + I/E^ArcCosh[a*x]] - (96*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*Log[1 + I/E^ArcCosh[a*x]] - 64*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3*Log[1 + I/E^ArcCosh[a*x]] - 48*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*Log[1 - I*E^ArcCosh[a*x]] + (96*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*Log[1 - I*E^ArcCosh[a*x]] - (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[1 + I*E^ArcCosh[a*x]] + 64*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^3*Log[1 + I*E^ArcCosh[a*x]] + (8*I)*Pi^3*Sqrt[(-1 + a*x)/(1 + a*x)]*Log[Tan[(Pi + (2*I)*ArcCosh[a*x])/4]] - 48*Sqrt[(-1 + a*x)/(1 + a*x)]*(8 + Pi^2 - (4*I)*Pi*ArcCosh[a*x] - 4*ArcCosh[a*x]^2)*PolyLog[2, (-I)/E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[2, I/E^ArcCosh[a*x]] + 192*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]^2*PolyLog[2, (-I)*E^ArcCosh[a*x]] - 48*Pi^2*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]] + (192*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[3, (-I)/E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*PolyLog[3, (-I)/E^ArcCosh[a*x]] - 384*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]] - (192*I)*Pi*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[3, I*E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[4, (-I)/E^ArcCosh[a*x]] + 384*Sqrt[(-1 + a*x)/(1 + a*x)]*PolyLog[4, (-I)*E^ArcCosh[a*x]]))/Sqrt[1 - a^2*x^2]","B",0
261,0,0,33,3.6833601,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^3}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^3)/Sqrt[1 - c^2*x^2],x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^3}{\sqrt{1-c^2 x^2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^3}{\sqrt{1-c^2 x^2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^3)/Sqrt[1 - c^2*x^2], x]","A",-1
262,1,45,67,0.2796453,"\int \frac{\left(c-a^2 c x^2\right)^3}{\cosh ^{-1}(a x)} \, dx","Integrate[(c - a^2*c*x^2)^3/ArcCosh[a*x],x]","\frac{c^3 \left(35 \text{Shi}\left(\cosh ^{-1}(a x)\right)-21 \text{Shi}\left(3 \cosh ^{-1}(a x)\right)+7 \text{Shi}\left(5 \cosh ^{-1}(a x)\right)-\text{Shi}\left(7 \cosh ^{-1}(a x)\right)\right)}{64 a}","\frac{35 c^3 \text{Shi}\left(\cosh ^{-1}(a x)\right)}{64 a}-\frac{21 c^3 \text{Shi}\left(3 \cosh ^{-1}(a x)\right)}{64 a}+\frac{7 c^3 \text{Shi}\left(5 \cosh ^{-1}(a x)\right)}{64 a}-\frac{c^3 \text{Shi}\left(7 \cosh ^{-1}(a x)\right)}{64 a}",1,"(c^3*(35*SinhIntegral[ArcCosh[a*x]] - 21*SinhIntegral[3*ArcCosh[a*x]] + 7*SinhIntegral[5*ArcCosh[a*x]] - SinhIntegral[7*ArcCosh[a*x]]))/(64*a)","A",1
263,1,34,50,0.1718913,"\int \frac{\left(c-a^2 c x^2\right)^2}{\cosh ^{-1}(a x)} \, dx","Integrate[(c - a^2*c*x^2)^2/ArcCosh[a*x],x]","\frac{c^2 \left(10 \text{Shi}\left(\cosh ^{-1}(a x)\right)-5 \text{Shi}\left(3 \cosh ^{-1}(a x)\right)+\text{Shi}\left(5 \cosh ^{-1}(a x)\right)\right)}{16 a}","\frac{5 c^2 \text{Shi}\left(\cosh ^{-1}(a x)\right)}{8 a}-\frac{5 c^2 \text{Shi}\left(3 \cosh ^{-1}(a x)\right)}{16 a}+\frac{c^2 \text{Shi}\left(5 \cosh ^{-1}(a x)\right)}{16 a}",1,"(c^2*(10*SinhIntegral[ArcCosh[a*x]] - 5*SinhIntegral[3*ArcCosh[a*x]] + SinhIntegral[5*ArcCosh[a*x]]))/(16*a)","A",1
264,1,25,29,0.1207052,"\int \frac{c-a^2 c x^2}{\cosh ^{-1}(a x)} \, dx","Integrate[(c - a^2*c*x^2)/ArcCosh[a*x],x]","\frac{c \left(3 \text{Shi}\left(\cosh ^{-1}(a x)\right)-\text{Shi}\left(3 \cosh ^{-1}(a x)\right)\right)}{4 a}","\frac{3 c \text{Shi}\left(\cosh ^{-1}(a x)\right)}{4 a}-\frac{c \text{Shi}\left(3 \cosh ^{-1}(a x)\right)}{4 a}",1,"(c*(3*SinhIntegral[ArcCosh[a*x]] - SinhIntegral[3*ArcCosh[a*x]]))/(4*a)","A",1
265,0,0,23,1.5773803,"\int \frac{1}{\left(c-a^2 c x^2\right) \cosh ^{-1}(a x)} \, dx","Integrate[1/((c - a^2*c*x^2)*ArcCosh[a*x]),x]","\int \frac{1}{\left(c-a^2 c x^2\right) \cosh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{\left(c-a^2 c x^2\right) \cosh ^{-1}(a x)},x\right)",0,"Integrate[1/((c - a^2*c*x^2)*ArcCosh[a*x]), x]","A",-1
266,0,0,23,6.139094,"\int \frac{1}{\left(c-a^2 c x^2\right)^2 \cosh ^{-1}(a x)} \, dx","Integrate[1/((c - a^2*c*x^2)^2*ArcCosh[a*x]),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^2 \cosh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{\left(c-a^2 c x^2\right)^2 \cosh ^{-1}(a x)},x\right)",0,"Integrate[1/((c - a^2*c*x^2)^2*ArcCosh[a*x]), x]","A",-1
267,1,188,339,0.50663,"\int \frac{x^4 \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x^4*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{1-c^2 x^2} \left(-\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+2 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 \log \left(a+b \cosh ^{-1}(c x)\right)\right)}{32 c^5 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","-\frac{\sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^5 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^5 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^5 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^5 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^5 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^5 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \log \left(a+b \cosh ^{-1}(c x)\right)}{16 b c^5 \sqrt{c x-1}}",1,"(Sqrt[1 - c^2*x^2]*(-(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])]) + 2*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])] + Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcCosh[c*x])] - 2*Log[a + b*ArcCosh[c*x]] + Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 2*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] - Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])]))/(32*c^5*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",0
268,1,171,297,0.4574829,"\int \frac{x^3 \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x^3*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{1-c^2 x^2} \left(-2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)}{16 c^4 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","-\frac{\sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b c^4 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^4 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^4 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b c^4 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^4 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^4 \sqrt{c x-1}}",1,"(Sqrt[1 - c^2*x^2]*(-2*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] + Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] + 2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])]))/(16*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",0
269,1,103,139,0.3115861,"\int \frac{x^2 \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]),x]","-\frac{\sqrt{-((c x-1) (c x+1))} \left(-\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\log \left(a+b \cosh ^{-1}(c x)\right)\right)}{8 b c^3 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{\sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \log \left(a+b \cosh ^{-1}(c x)\right)}{8 b c^3 \sqrt{c x-1}}",1,"-1/8*(Sqrt[-((-1 + c*x)*(1 + c*x))]*(-(Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])]) + Log[a + b*ArcCosh[c*x]] + Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(b*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
270,1,127,197,0.3145329,"\int \frac{x \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{1-c^2 x^2} \left(-\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)}{4 c^2 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","-\frac{\sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b c^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b c^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b c^2 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b c^2 \sqrt{c x-1}}",1,"(Sqrt[1 - c^2*x^2]*(-(Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]]) + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])]))/(4*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",0
271,1,105,139,0.2044525,"\int \frac{\sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{-((c x-1) (c x+1))} \left(\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\log \left(a+b \cosh ^{-1}(c x)\right)\right)}{2 b c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{\sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \log \left(a+b \cosh ^{-1}(c x)\right)}{2 b c \sqrt{c x-1}}",1,"(Sqrt[-((-1 + c*x)*(1 + c*x))]*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] - Log[a + b*ArcCosh[c*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])]))/(2*b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
272,0,0,117,1.2485756,"\int \frac{\sqrt{1-c^2 x^2}}{x \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcCosh[c*x])),x]","\int \frac{\sqrt{1-c^2 x^2}}{x \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)-\frac{\sqrt{c x-1} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b \sqrt{1-c x}}+\frac{\sqrt{c x-1} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b \sqrt{1-c x}}",0,"Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcCosh[c*x])), x]","A",-1
273,0,0,66,1.088443,"\int \frac{\sqrt{1-c^2 x^2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcCosh[c*x])),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)-\frac{c \sqrt{c x-1} \log \left(a+b \cosh ^{-1}(c x)\right)}{b \sqrt{1-c x}}",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcCosh[c*x])), x]","A",-1
274,0,0,31,1.4199642,"\int \frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])), x]","A",-1
275,0,0,31,0.8958862,"\int \frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])), x]","A",-1
276,1,215,397,0.9412791,"\int \frac{x^3 \left(1-c^2 x^2\right)^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x^3*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{1-c^2 x^2} \left(-3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)}{64 c^4 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","-\frac{3 \sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{64 b c^4 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^4 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^4 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^4 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{64 b c^4 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^4 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^4 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^4 \sqrt{c x-1}}",1,"(Sqrt[1 - c^2*x^2]*(-3*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] + Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] - Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcCosh[c*x])] + 3*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])]))/(64*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",0
277,1,188,339,0.7409178,"\int \frac{x^2 \left(1-c^2 x^2\right)^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x^2*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]),x]","-\frac{\sqrt{1-c^2 x^2} \left(-\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+2 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+2 \log \left(a+b \cosh ^{-1}(c x)\right)\right)}{32 c^3 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","\frac{\sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \log \left(a+b \cosh ^{-1}(c x)\right)}{16 b c^3 \sqrt{c x-1}}",1,"-1/32*(Sqrt[1 - c^2*x^2]*(-(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])]) - 2*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])] + Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcCosh[c*x])] + 2*Log[a + b*ArcCosh[c*x]] + Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 2*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] - Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])]))/(c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",0
278,1,172,297,0.6890935,"\int \frac{x \left(1-c^2 x^2\right)^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{1-c^2 x^2} \left(-2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)}{16 c^2 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","-\frac{\sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b c^2 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^2 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b c^2 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^2 \sqrt{c x-1}}",1,"(Sqrt[1 - c^2*x^2]*(-2*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] - Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] + 2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])]))/(16*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",0
279,1,147,239,0.4579851,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(a + b*ArcCosh[c*x]),x]","-\frac{\sqrt{1-c^2 x^2} \left(-4 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+4 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 \log \left(a+b \cosh ^{-1}(c x)\right)\right)}{8 b c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{\sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 b c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b c \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 b c \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \log \left(a+b \cosh ^{-1}(c x)\right)}{8 b c \sqrt{c x-1}}",1,"-1/8*(Sqrt[1 - c^2*x^2]*(-4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] + Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])] + 3*Log[a + b*ArcCosh[c*x]] + 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
280,0,0,216,1.276639,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcCosh[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)-\frac{5 \sqrt{c x-1} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b \sqrt{1-c x}}+\frac{\sqrt{c x-1} \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b \sqrt{1-c x}}+\frac{5 \sqrt{c x-1} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b \sqrt{1-c x}}-\frac{\sqrt{c x-1} \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b \sqrt{1-c x}}",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcCosh[c*x])), x]","A",-1
281,0,0,164,1.4234181,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcCosh[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)+\frac{c \sqrt{c x-1} \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b \sqrt{1-c x}}-\frac{c \sqrt{c x-1} \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b \sqrt{1-c x}}-\frac{3 c \sqrt{c x-1} \log \left(a+b \cosh ^{-1}(c x)\right)}{2 b \sqrt{1-c x}}",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcCosh[c*x])), x]","A",-1
282,0,0,31,1.4467603,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])), x]","A",-1
283,0,0,31,0.9103266,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])), x]","A",-1
284,1,216,397,1.3327014,"\int \frac{x^3 \left(1-c^2 x^2\right)^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x^3*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{1-c^2 x^2} \left(-6 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+8 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-3 \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{9 a}{b}\right) \text{Chi}\left(9 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+6 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-8 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{9 a}{b}\right) \text{Shi}\left(9 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)}{256 c^4 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","-\frac{3 \sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{128 b c^4 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^4 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{256 b c^4 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{9 a}{b}\right) \text{Chi}\left(\frac{9 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{256 b c^4 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{128 b c^4 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^4 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{256 b c^4 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{9 a}{b}\right) \text{Shi}\left(\frac{9 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{256 b c^4 \sqrt{c x-1}}",1,"(Sqrt[1 - c^2*x^2]*(-6*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + 8*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] - 3*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcCosh[c*x])] + Cosh[(9*a)/b]*CoshIntegral[9*(a/b + ArcCosh[c*x])] + 6*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 8*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 3*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])] - Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcCosh[c*x])]))/(256*c^4*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",0
285,1,233,439,1.215448,"\int \frac{x^2 \left(1-c^2 x^2\right)^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{1-c^2 x^2} \left(4 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+4 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-4 \cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{8 a}{b}\right) \text{Chi}\left(8 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-4 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-4 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+4 \sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{8 a}{b}\right) \text{Shi}\left(8 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-5 \log \left(a+b \cosh ^{-1}(c x)\right)\right)}{128 c^3 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","\frac{\sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{8 a}{b}\right) \text{Chi}\left(\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{128 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{8 a}{b}\right) \text{Shi}\left(\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{128 b c^3 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \log \left(a+b \cosh ^{-1}(c x)\right)}{128 b c^3 \sqrt{c x-1}}",1,"(Sqrt[1 - c^2*x^2]*(4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] + 4*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])] - 4*Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcCosh[c*x])] + Cosh[(8*a)/b]*CoshIntegral[8*(a/b + ArcCosh[c*x])] - 5*Log[a + b*ArcCosh[c*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 4*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] + 4*Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] - Sinh[(8*a)/b]*SinhIntegral[8*(a/b + ArcCosh[c*x])]))/(128*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",0
286,1,216,397,1.0873656,"\int \frac{x \left(1-c^2 x^2\right)^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{1-c^2 x^2} \left(-5 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+9 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-5 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+5 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-9 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+5 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)}{64 c^2 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","-\frac{5 \sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{64 b c^2 \sqrt{c x-1}}+\frac{9 \sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^2 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^2 \sqrt{c x-1}}+\frac{5 \sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{64 b c^2 \sqrt{c x-1}}-\frac{9 \sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^2 \sqrt{c x-1}}+\frac{5 \sqrt{1-c x} \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^2 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b c^2 \sqrt{c x-1}}",1,"(Sqrt[1 - c^2*x^2]*(-5*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + 9*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] - 5*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] + Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcCosh[c*x])] + 5*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 9*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 5*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] - Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])]))/(64*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",0
287,1,191,339,0.7544236,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(a + b*ArcCosh[c*x]),x]","\frac{\sqrt{1-c^2 x^2} \left(15 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-6 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-15 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+6 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-10 \log \left(a+b \cosh ^{-1}(c x)\right)\right)}{32 b c \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{15 \sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c \sqrt{c x-1}}-\frac{15 \sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 b c \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \log \left(a+b \cosh ^{-1}(c x)\right)}{16 b c \sqrt{c x-1}}",1,"(Sqrt[1 - c^2*x^2]*(15*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] - 6*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcCosh[c*x])] + Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcCosh[c*x])] - 10*Log[a + b*ArcCosh[c*x]] - 15*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 6*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] - Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])]))/(32*b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))","A",0
288,0,0,310,1.314545,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)-\frac{11 \sqrt{c x-1} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b \sqrt{1-c x}}+\frac{7 \sqrt{c x-1} \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b \sqrt{1-c x}}-\frac{\sqrt{c x-1} \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b \sqrt{1-c x}}+\frac{11 \sqrt{c x-1} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b \sqrt{1-c x}}-\frac{7 \sqrt{c x-1} \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b \sqrt{1-c x}}+\frac{\sqrt{c x-1} \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b \sqrt{1-c x}}",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])), x]","A",-1
289,0,0,255,1.3356478,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcCosh[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)+\frac{c \sqrt{c x-1} \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b \sqrt{1-c x}}-\frac{c \sqrt{c x-1} \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 b \sqrt{1-c x}}-\frac{c \sqrt{c x-1} \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b \sqrt{1-c x}}+\frac{c \sqrt{c x-1} \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 b \sqrt{1-c x}}-\frac{15 c \sqrt{c x-1} \log \left(a+b \cosh ^{-1}(c x)\right)}{8 b \sqrt{1-c x}}",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcCosh[c*x])), x]","A",-1
290,0,0,31,1.4247274,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcCosh[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcCosh[c*x])), x]","A",-1
291,0,0,31,1.0051986,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])), x]","A",-1
292,1,69,98,0.1182953,"\int \frac{x^4}{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx","Integrate[x^4/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]","\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(4 \text{Chi}\left(2 \cosh ^{-1}(a x)\right)+\text{Chi}\left(4 \cosh ^{-1}(a x)\right)+3 \log \left(\cosh ^{-1}(a x)\right)\right)}{8 a^5 \sqrt{-((a x-1) (a x+1))}}","\frac{\sqrt{a x-1} \text{Chi}\left(2 \cosh ^{-1}(a x)\right)}{2 a^5 \sqrt{1-a x}}+\frac{\sqrt{a x-1} \text{Chi}\left(4 \cosh ^{-1}(a x)\right)}{8 a^5 \sqrt{1-a x}}+\frac{3 \sqrt{a x-1} \log \left(\cosh ^{-1}(a x)\right)}{8 a^5 \sqrt{1-a x}}",1,"(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(4*CoshIntegral[2*ArcCosh[a*x]] + CoshIntegral[4*ArcCosh[a*x]] + 3*Log[ArcCosh[a*x]]))/(8*a^5*Sqrt[-((-1 + a*x)*(1 + a*x))])","A",1
293,1,60,65,0.0889453,"\int \frac{x^3}{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx","Integrate[x^3/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]","\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \left(3 \text{Chi}\left(\cosh ^{-1}(a x)\right)+\text{Chi}\left(3 \cosh ^{-1}(a x)\right)\right)}{4 a^4 \sqrt{-((a x-1) (a x+1))}}","\frac{3 \sqrt{a x-1} \text{Chi}\left(\cosh ^{-1}(a x)\right)}{4 a^4 \sqrt{1-a x}}+\frac{\sqrt{a x-1} \text{Chi}\left(3 \cosh ^{-1}(a x)\right)}{4 a^4 \sqrt{1-a x}}",1,"(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(3*CoshIntegral[ArcCosh[a*x]] + CoshIntegral[3*ArcCosh[a*x]]))/(4*a^4*Sqrt[-((-1 + a*x)*(1 + a*x))])","A",0
294,1,60,65,0.0995087,"\int \frac{x^2}{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx","Integrate[x^2/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]","-\frac{\sqrt{-((a x-1) (a x+1))} \left(\text{Chi}\left(2 \cosh ^{-1}(a x)\right)+\log \left(\cosh ^{-1}(a x)\right)\right)}{2 a^3 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","\frac{\sqrt{a x-1} \text{Chi}\left(2 \cosh ^{-1}(a x)\right)}{2 a^3 \sqrt{1-a x}}+\frac{\sqrt{a x-1} \log \left(\cosh ^{-1}(a x)\right)}{2 a^3 \sqrt{1-a x}}",1,"-1/2*(Sqrt[-((-1 + a*x)*(1 + a*x))]*(CoshIntegral[2*ArcCosh[a*x]] + Log[ArcCosh[a*x]]))/(a^3*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",1
295,1,50,28,0.0830384,"\int \frac{x}{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx","Integrate[x/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]","-\frac{\sqrt{-((a x-1) (a x+1))} \text{Chi}\left(\cosh ^{-1}(a x)\right)}{a^2 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","\frac{\sqrt{a x-1} \text{Chi}\left(\cosh ^{-1}(a x)\right)}{a^2 \sqrt{1-a x}}",1,"-((Sqrt[-((-1 + a*x)*(1 + a*x))]*CoshIntegral[ArcCosh[a*x]])/(a^2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)))","A",1
296,1,47,28,0.061526,"\int \frac{1}{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx","Integrate[1/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]","\frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \log \left(\cosh ^{-1}(a x)\right)}{a \sqrt{-((a x-1) (a x+1))}}","\frac{\sqrt{a x-1} \log \left(\cosh ^{-1}(a x)\right)}{a \sqrt{1-a x}}",1,"(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Log[ArcCosh[a*x]])/(a*Sqrt[-((-1 + a*x)*(1 + a*x))])","A",1
297,0,0,27,0.5868296,"\int \frac{1}{x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx","Integrate[1/(x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]","\int \frac{1}{x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)},x\right)",0,"Integrate[1/(x*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x]","A",-1
298,0,0,27,0.7208336,"\int \frac{1}{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx","Integrate[1/(x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]","\int \frac{1}{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)},x\right)",0,"Integrate[1/(x^2*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]), x]","A",-1
299,1,130,197,0.2925014,"\int \frac{x^3}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[x^3/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]","\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)}{4 b c^4 \sqrt{-((c x-1) (c x+1))}}","\frac{3 \sqrt{c x-1} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b c^4 \sqrt{1-c x}}+\frac{\sqrt{c x-1} \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b c^4 \sqrt{1-c x}}-\frac{3 \sqrt{c x-1} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b c^4 \sqrt{1-c x}}-\frac{\sqrt{c x-1} \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b c^4 \sqrt{1-c x}}",1,"(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(3*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] - 3*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])]))/(4*b*c^4*Sqrt[-((-1 + c*x)*(1 + c*x))])","A",1
300,1,99,139,0.2823458,"\int \frac{x^2}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]","-\frac{\sqrt{1-c^2 x^2} \left(\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\log \left(a+b \cosh ^{-1}(c x)\right)\right)}{2 c^3 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","\frac{\sqrt{c x-1} \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b c^3 \sqrt{1-c x}}-\frac{\sqrt{c x-1} \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b c^3 \sqrt{1-c x}}+\frac{\sqrt{c x-1} \log \left(a+b \cosh ^{-1}(c x)\right)}{2 b c^3 \sqrt{1-c x}}",1,"-1/2*(Sqrt[1 - c^2*x^2]*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] + Log[a + b*ArcCosh[c*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])]))/(c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",1
301,1,81,92,0.2170196,"\int \frac{x}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[x/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]","\frac{\sqrt{1-c^2 x^2} \left(\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)}{c^2 \sqrt{\frac{c x-1}{c x+1}} (b c x+b)}","\frac{\sqrt{c x-1} \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b c^2 \sqrt{1-c x}}-\frac{\sqrt{c x-1} \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b c^2 \sqrt{1-c x}}",1,"(Sqrt[1 - c^2*x^2]*(-(Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]]) + Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]]))/(c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))","A",1
302,1,54,35,0.1073521,"\int \frac{1}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]","\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left(a+b \cosh ^{-1}(c x)\right)}{b c \sqrt{-((c x-1) (c x+1))}}","\frac{\sqrt{c x-1} \log \left(a+b \cosh ^{-1}(c x)\right)}{b c \sqrt{1-c x}}",1,"(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[a + b*ArcCosh[c*x]])/(b*c*Sqrt[-((-1 + c*x)*(1 + c*x))])","A",1
303,0,0,31,0.748561,"\int \frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]","A",-1
304,0,0,31,1.3105188,"\int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]","A",-1
305,0,0,31,4.4294749,"\int \frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])),x]","\int \frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]","A",-1
306,0,0,29,7.155629,"\int \frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])),x]","\int \frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]","A",-1
307,0,0,28,0.1425745,"\int \frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]","A",-1
308,0,0,31,3.5646793,"\int \frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]","A",-1
309,0,0,31,2.223562,"\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]","A",-1
310,0,0,31,1.0678713,"\int \frac{x^m \left(1-c^2 x^2\right)^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]),x]","\int \frac{x^m \left(1-c^2 x^2\right)^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2} x^m}{a+b \cosh ^{-1}(c x)},x\right)",0,"Integrate[(x^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x]), x]","A",-1
311,0,0,31,0.1794434,"\int \frac{x^m \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]),x]","\int \frac{x^m \sqrt{1-c^2 x^2}}{a+b \cosh ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2} x^m}{a+b \cosh ^{-1}(c x)},x\right)",0,"Integrate[(x^m*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x]), x]","A",-1
312,0,0,31,0.6469839,"\int \frac{x^m}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[x^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])),x]","\int \frac{x^m}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[x^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])), x]","A",-1
313,0,0,31,1.1868607,"\int \frac{x^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])),x]","\int \frac{x^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[x^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]","A",-1
314,0,0,31,1.7195206,"\int \frac{x^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])),x]","\int \frac{x^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[x^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])), x]","A",-1
315,1,257,98,0.4111166,"\int \frac{\left(c-a^2 c x^2\right)^3}{\cosh ^{-1}(a x)^2} \, dx","Integrate[(c - a^2*c*x^2)^3/ArcCosh[a*x]^2,x]","\frac{c^3 \left(64 a^7 x^7 \sqrt{\frac{a x-1}{a x+1}}+64 a^6 x^6 \sqrt{\frac{a x-1}{a x+1}}-192 a^5 x^5 \sqrt{\frac{a x-1}{a x+1}}-192 a^4 x^4 \sqrt{\frac{a x-1}{a x+1}}+192 a^3 x^3 \sqrt{\frac{a x-1}{a x+1}}+192 a^2 x^2 \sqrt{\frac{a x-1}{a x+1}}+35 \cosh ^{-1}(a x) \text{Chi}\left(\cosh ^{-1}(a x)\right)-63 \cosh ^{-1}(a x) \text{Chi}\left(3 \cosh ^{-1}(a x)\right)+35 \cosh ^{-1}(a x) \text{Chi}\left(5 \cosh ^{-1}(a x)\right)-7 \cosh ^{-1}(a x) \text{Chi}\left(7 \cosh ^{-1}(a x)\right)-64 a x \sqrt{\frac{a x-1}{a x+1}}-64 \sqrt{\frac{a x-1}{a x+1}}\right)}{64 a \cosh ^{-1}(a x)}","\frac{35 c^3 \text{Chi}\left(\cosh ^{-1}(a x)\right)}{64 a}-\frac{63 c^3 \text{Chi}\left(3 \cosh ^{-1}(a x)\right)}{64 a}+\frac{35 c^3 \text{Chi}\left(5 \cosh ^{-1}(a x)\right)}{64 a}-\frac{7 c^3 \text{Chi}\left(7 \cosh ^{-1}(a x)\right)}{64 a}+\frac{c^3 (a x-1)^{7/2} (a x+1)^{7/2}}{a \cosh ^{-1}(a x)}",1,"(c^3*(-64*Sqrt[(-1 + a*x)/(1 + a*x)] - 64*a*x*Sqrt[(-1 + a*x)/(1 + a*x)] + 192*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)] + 192*a^3*x^3*Sqrt[(-1 + a*x)/(1 + a*x)] - 192*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] - 192*a^5*x^5*Sqrt[(-1 + a*x)/(1 + a*x)] + 64*a^6*x^6*Sqrt[(-1 + a*x)/(1 + a*x)] + 64*a^7*x^7*Sqrt[(-1 + a*x)/(1 + a*x)] + 35*ArcCosh[a*x]*CoshIntegral[ArcCosh[a*x]] - 63*ArcCosh[a*x]*CoshIntegral[3*ArcCosh[a*x]] + 35*ArcCosh[a*x]*CoshIntegral[5*ArcCosh[a*x]] - 7*ArcCosh[a*x]*CoshIntegral[7*ArcCosh[a*x]]))/(64*a*ArcCosh[a*x])","B",0
316,1,194,82,0.3179227,"\int \frac{\left(c-a^2 c x^2\right)^2}{\cosh ^{-1}(a x)^2} \, dx","Integrate[(c - a^2*c*x^2)^2/ArcCosh[a*x]^2,x]","-\frac{c^2 \left(16 a^5 x^5 \sqrt{\frac{a x-1}{a x+1}}+16 a^4 x^4 \sqrt{\frac{a x-1}{a x+1}}-32 a^3 x^3 \sqrt{\frac{a x-1}{a x+1}}-32 a^2 x^2 \sqrt{\frac{a x-1}{a x+1}}-10 \cosh ^{-1}(a x) \text{Chi}\left(\cosh ^{-1}(a x)\right)+15 \cosh ^{-1}(a x) \text{Chi}\left(3 \cosh ^{-1}(a x)\right)-5 \cosh ^{-1}(a x) \text{Chi}\left(5 \cosh ^{-1}(a x)\right)+16 a x \sqrt{\frac{a x-1}{a x+1}}+16 \sqrt{\frac{a x-1}{a x+1}}\right)}{16 a \cosh ^{-1}(a x)}","\frac{5 c^2 \text{Chi}\left(\cosh ^{-1}(a x)\right)}{8 a}-\frac{15 c^2 \text{Chi}\left(3 \cosh ^{-1}(a x)\right)}{16 a}+\frac{5 c^2 \text{Chi}\left(5 \cosh ^{-1}(a x)\right)}{16 a}-\frac{c^2 (a x-1)^{5/2} (a x+1)^{5/2}}{a \cosh ^{-1}(a x)}",1,"-1/16*(c^2*(16*Sqrt[(-1 + a*x)/(1 + a*x)] + 16*a*x*Sqrt[(-1 + a*x)/(1 + a*x)] - 32*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)] - 32*a^3*x^3*Sqrt[(-1 + a*x)/(1 + a*x)] + 16*a^4*x^4*Sqrt[(-1 + a*x)/(1 + a*x)] + 16*a^5*x^5*Sqrt[(-1 + a*x)/(1 + a*x)] - 10*ArcCosh[a*x]*CoshIntegral[ArcCosh[a*x]] + 15*ArcCosh[a*x]*CoshIntegral[3*ArcCosh[a*x]] - 5*ArcCosh[a*x]*CoshIntegral[5*ArcCosh[a*x]]))/(a*ArcCosh[a*x])","B",0
317,1,217,58,1.7763069,"\int \frac{c-a^2 c x^2}{\cosh ^{-1}(a x)^2} \, dx","Integrate[(c - a^2*c*x^2)/ArcCosh[a*x]^2,x]","\frac{c \sqrt{a x-1} \left(-3 \left(a^2 x^2-1\right) \cosh ^{-1}(a x) \text{Chi}\left(3 \cosh ^{-1}(a x)\right)-(a x-1) \cosh ^{-1}(a x) \text{Chi}\left(\cosh ^{-1}(a x)\right) \left(a x-4 \sqrt{a x-1} \sqrt{a x+1} \coth \left(\frac{1}{2} \cosh ^{-1}(a x)\right)+1\right)+4 \left(\frac{a x-1}{a x+1}\right)^{5/2} (a x+1)^5+4 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \log \left(\cosh ^{-1}(a x)\right) \left(\sqrt{\frac{a x-1}{a x+1}} (a x+1)+(1-a x) \coth \left(\frac{1}{2} \cosh ^{-1}(a x)\right)\right)\right)}{4 a \left(\frac{a x-1}{a x+1}\right)^{3/2} (a x+1)^{5/2} \cosh ^{-1}(a x)}","\frac{3 c \text{Chi}\left(\cosh ^{-1}(a x)\right)}{4 a}-\frac{3 c \text{Chi}\left(3 \cosh ^{-1}(a x)\right)}{4 a}+\frac{c (a x-1)^{3/2} (a x+1)^{3/2}}{a \cosh ^{-1}(a x)}",1,"(c*Sqrt[-1 + a*x]*(4*((-1 + a*x)/(1 + a*x))^(5/2)*(1 + a*x)^5 - 3*(-1 + a^2*x^2)*ArcCosh[a*x]*CoshIntegral[3*ArcCosh[a*x]] - (-1 + a*x)*ArcCosh[a*x]*CoshIntegral[ArcCosh[a*x]]*(1 + a*x - 4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Coth[ArcCosh[a*x]/2]) + 4*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*(Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x) + (1 - a*x)*Coth[ArcCosh[a*x]/2])*Log[ArcCosh[a*x]]))/(4*a*((-1 + a*x)/(1 + a*x))^(3/2)*(1 + a*x)^(5/2)*ArcCosh[a*x])","B",0
318,0,0,66,3.1154357,"\int \frac{1}{\left(c-a^2 c x^2\right) \cosh ^{-1}(a x)^2} \, dx","Integrate[1/((c - a^2*c*x^2)*ArcCosh[a*x]^2),x]","\int \frac{1}{\left(c-a^2 c x^2\right) \cosh ^{-1}(a x)^2} \, dx","\frac{a \text{Int}\left(\frac{x}{(a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)},x\right)}{c}+\frac{1}{a c \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)}",0,"Integrate[1/((c - a^2*c*x^2)*ArcCosh[a*x]^2), x]","A",-1
319,0,0,68,13.0524078,"\int \frac{1}{\left(c-a^2 c x^2\right)^2 \cosh ^{-1}(a x)^2} \, dx","Integrate[1/((c - a^2*c*x^2)^2*ArcCosh[a*x]^2),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^2 \cosh ^{-1}(a x)^2} \, dx","-\frac{3 a \text{Int}\left(\frac{x}{(a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)},x\right)}{c^2}-\frac{1}{a c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)}",0,"Integrate[1/((c - a^2*c*x^2)^2*ArcCosh[a*x]^2), x]","A",-1
320,1,322,350,0.7687558,"\int \frac{x^3 \sqrt{1-c^2 x^2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^3*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{1-c^2 x^2} \left(2 \sinh \left(\frac{a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-3 \sinh \left(\frac{3 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-5 a \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-5 b \sinh \left(\frac{5 a}{b}\right) \cosh ^{-1}(c x) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 a \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-2 b \cosh \left(\frac{a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 a \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 b \cosh \left(\frac{3 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+5 a \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+5 b \cosh \left(\frac{5 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-16 b c^5 x^5+16 b c^3 x^3\right)}{16 b^2 c^4 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}","\frac{\sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 c^4 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 c^4 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4 \sqrt{c x-1}}+\frac{5 \sqrt{1-c x} \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4 \sqrt{c x-1}}-\frac{x^3 \sqrt{c x-1} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[1 - c^2*x^2]*(16*b*c^3*x^3 - 16*b*c^5*x^5 + 2*(a + b*ArcCosh[c*x])*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - 3*(a + b*ArcCosh[c*x])*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - 5*a*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] - 5*b*ArcCosh[c*x]*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] - 2*a*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 2*b*ArcCosh[c*x]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 3*a*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 3*b*ArcCosh[c*x]*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 5*a*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + 5*b*ArcCosh[c*x]*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])]))/(16*b^2*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))","A",1
321,1,130,154,0.4894138,"\int \frac{x^2 \sqrt{1-c^2 x^2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^2*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{1-c^2 x^2} \left(-\sinh \left(\frac{4 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\cosh \left(\frac{4 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 b c^2 x^2 \left(c^2 x^2-1\right)\right)}{2 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}","-\frac{\sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3 \sqrt{c x-1}}-\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[1 - c^2*x^2]*(-2*b*c^2*x^2*(-1 + c^2*x^2) - (a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] + (a + b*ArcCosh[c*x])*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(2*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))","A",1
322,1,217,248,0.4304632,"\int \frac{x \sqrt{1-c^2 x^2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(x*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{1-c^2 x^2} \left(\sinh \left(\frac{a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-3 \sinh \left(\frac{3 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-a \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-b \cosh \left(\frac{a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 a \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 b \cosh \left(\frac{3 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-4 b c^3 x^3+4 b c x\right)}{4 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}","\frac{\sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b^2 c^2 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^2 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b^2 c^2 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^2 \sqrt{c x-1}}-\frac{x \sqrt{c x-1} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[1 - c^2*x^2]*(4*b*c*x - 4*b*c^3*x^3 + (a + b*ArcCosh[c*x])*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - 3*(a + b*ArcCosh[c*x])*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - a*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - b*ArcCosh[c*x]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 3*a*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 3*b*ArcCosh[c*x]*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])]))/(4*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))","A",1
323,1,121,146,0.2235189,"\int \frac{\sqrt{1-c^2 x^2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(a + b*ArcCosh[c*x])^2,x]","-\frac{\sqrt{1-c^2 x^2} \left(\sinh \left(\frac{2 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\cosh \left(\frac{2 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+b \left(c^2 x^2-1\right)\right)}{b^2 c \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}","-\frac{\sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b^2 c \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b^2 c \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"-((Sqrt[1 - c^2*x^2]*(b*(-1 + c^2*x^2) + (a + b*ArcCosh[c*x])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - (a + b*ArcCosh[c*x])*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])]))/(b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])))","A",1
324,0,0,182,33.1368135,"\int \frac{\sqrt{1-c^2 x^2}}{x \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\sqrt{1-c^2 x^2}}{x \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\frac{\sqrt{1-c x} \text{Int}\left(\frac{1}{x^2 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{b c x \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[Sqrt[1 - c^2*x^2]/(x*(a + b*ArcCosh[c*x])^2), x]","A",-1
325,0,0,98,10.0906505,"\int \frac{\sqrt{1-c^2 x^2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\frac{2 \sqrt{1-c x} \text{Int}\left(\frac{1}{x^3 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{b c x^2 \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^2*(a + b*ArcCosh[c*x])^2), x]","A",-1
326,0,0,31,163.2569596,"\int \frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[Sqrt[1 - c^2*x^2]/(x^3*(a + b*ArcCosh[c*x])^2), x]","A",-1
327,-1,0,31,180.0088424,"\int \frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcCosh[c*x])^2),x]","\text{\$Aborted}","\text{Int}\left(\frac{\sqrt{1-c^2 x^2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"$Aborted","F",-1
328,1,338,354,1.0705478,"\int \frac{x^2 \left(1-c^2 x^2\right)^{3/2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^2*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x])^2,x]","-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(-\sinh \left(\frac{2 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-4 \sinh \left(\frac{4 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 a \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 b \sinh \left(\frac{6 a}{b}\right) \cosh ^{-1}(c x) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+a \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+b \cosh \left(\frac{2 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+4 a \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+4 b \cosh \left(\frac{4 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-3 a \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-3 b \cosh \left(\frac{6 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+16 b c^6 x^6-32 b c^4 x^4+16 b c^2 x^2\right)}{16 b^2 c^3 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)}","-\frac{\sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}-\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"-1/16*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(16*b*c^2*x^2 - 32*b*c^4*x^4 + 16*b*c^6*x^6 - (a + b*ArcCosh[c*x])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - 4*(a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] + 3*a*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] + 3*b*ArcCosh[c*x]*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] + a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + b*ArcCosh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 4*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] + 4*b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] - 3*b*ArcCosh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])]))/(b^2*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))","A",0
329,1,327,348,0.9463479,"\int \frac{x \left(1-c^2 x^2\right)^{3/2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(x*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(-2 \sinh \left(\frac{a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+9 \sinh \left(\frac{3 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-5 a \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-5 b \sinh \left(\frac{5 a}{b}\right) \cosh ^{-1}(c x) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+2 a \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+2 b \cosh \left(\frac{a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-9 a \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-9 b \cosh \left(\frac{3 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+5 a \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+5 b \cosh \left(\frac{5 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-16 b c^5 x^5+32 b c^3 x^3-16 b c x\right)}{16 b^2 c^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)}","\frac{\sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 c^2 \sqrt{c x-1}}-\frac{9 \sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2 \sqrt{c x-1}}+\frac{5 \sqrt{1-c x} \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 c^2 \sqrt{c x-1}}+\frac{9 \sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2 \sqrt{c x-1}}-\frac{x \sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-16*b*c*x + 32*b*c^3*x^3 - 16*b*c^5*x^5 - 2*(a + b*ArcCosh[c*x])*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] + 9*(a + b*ArcCosh[c*x])*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - 5*a*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] - 5*b*ArcCosh[c*x]*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] + 2*a*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 2*b*ArcCosh[c*x]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 9*a*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - 9*b*ArcCosh[c*x]*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 5*a*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + 5*b*ArcCosh[c*x]*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])]))/(16*b^2*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))","A",0
330,1,232,246,0.6221123,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(2 \sinh \left(\frac{2 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\sinh \left(\frac{4 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 a \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 b \cosh \left(\frac{2 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+a \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+b \cosh \left(\frac{4 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 b c^4 x^4+4 b c^2 x^2-2 b\right)}{2 b^2 c \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)}","-\frac{\sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b^2 c \sqrt{c x-1}}+\frac{\sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b^2 c \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-2*b + 4*b*c^2*x^2 - 2*b*c^4*x^4 + 2*(a + b*ArcCosh[c*x])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - (a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] - 2*a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 2*b*ArcCosh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] + b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(2*b^2*c*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))","A",0
331,0,0,291,34.2969568,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","-\frac{\sqrt{1-c x} \text{Int}\left(\frac{c^2 x^2-1}{x^2 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{c x-1}}-\frac{9 \sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b^2 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 \sqrt{c x-1}}+\frac{9 \sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b^2 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2}}{b c x \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcCosh[c*x])^2), x]","A",-1
332,0,0,158,64.9410194,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","-\frac{2 c \sqrt{1-c x} \text{Int}\left(\frac{c^2 x^2-1}{x \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b \sqrt{c x-1}}-\frac{2 \sqrt{1-c x} \text{Int}\left(\frac{c^2 x^2-1}{x^3 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2}}{b c x^2 \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^2*(a + b*ArcCosh[c*x])^2), x]","A",-1
333,0,0,31,160.4912153,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(1 - c^2*x^2)^(3/2)/(x^3*(a + b*ArcCosh[c*x])^2), x]","A",-1
334,-1,0,107,180.0022878,"\int \frac{\left(1-c^2 x^2\right)^{3/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(3/2)/(x^4*(a + b*ArcCosh[c*x])^2),x]","\text{\$Aborted}","-\frac{4 \sqrt{1-c x} \text{Int}\left(\frac{c^2 x^2-1}{x^5 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{3/2}}{b c x^4 \left(a+b \cosh ^{-1}(c x)\right)}",0,"$Aborted","F",-1
335,1,446,454,1.736287,"\int \frac{x^2 \left(1-c^2 x^2\right)^{5/2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(\sinh \left(\frac{2 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+2 \sinh \left(\frac{4 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-3 a \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-3 b \sinh \left(\frac{6 a}{b}\right) \cosh ^{-1}(c x) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+a \sinh \left(\frac{8 a}{b}\right) \text{Chi}\left(8 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+b \sinh \left(\frac{8 a}{b}\right) \cosh ^{-1}(c x) \text{Chi}\left(8 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-a \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-b \cosh \left(\frac{2 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 a \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 b \cosh \left(\frac{4 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 a \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 b \cosh \left(\frac{6 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-a \cosh \left(\frac{8 a}{b}\right) \text{Shi}\left(8 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-b \cosh \left(\frac{8 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(8 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+16 b c^8 x^8-48 b c^6 x^6+48 b c^4 x^4-16 b c^2 x^2\right)}{16 b^2 c^3 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)}","-\frac{\sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 b^2 c^3 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left(\frac{8 a}{b}\right) \text{Chi}\left(\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 b^2 c^3 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left(\frac{8 a}{b}\right) \text{Shi}\left(\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3 \sqrt{c x-1}}-\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{5/2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-16*b*c^2*x^2 + 48*b*c^4*x^4 - 48*b*c^6*x^6 + 16*b*c^8*x^8 + (a + b*ArcCosh[c*x])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] + 2*(a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] - 3*a*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] - 3*b*ArcCosh[c*x]*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] + a*CoshIntegral[8*(a/b + ArcCosh[c*x])]*Sinh[(8*a)/b] + b*ArcCosh[c*x]*CoshIntegral[8*(a/b + ArcCosh[c*x])]*Sinh[(8*a)/b] - a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - b*ArcCosh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 2*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] - 2*b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] + 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] + 3*b*ArcCosh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] - a*Cosh[(8*a)/b]*SinhIntegral[8*(a/b + ArcCosh[c*x])] - b*ArcCosh[c*x]*Cosh[(8*a)/b]*SinhIntegral[8*(a/b + ArcCosh[c*x])]))/(16*b^2*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))","A",0
336,1,436,448,1.4335743,"\int \frac{x \left(1-c^2 x^2\right)^{5/2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(x*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(-5 \sinh \left(\frac{a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+27 \sinh \left(\frac{3 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-25 a \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-25 b \sinh \left(\frac{5 a}{b}\right) \cosh ^{-1}(c x) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+7 a \sinh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+7 b \sinh \left(\frac{7 a}{b}\right) \cosh ^{-1}(c x) \text{Chi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+5 a \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+5 b \cosh \left(\frac{a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-27 a \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-27 b \cosh \left(\frac{3 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+25 a \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+25 b \cosh \left(\frac{5 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-7 a \cosh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-7 b \cosh \left(\frac{7 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(7 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+64 b c^7 x^7-192 b c^5 x^5+192 b c^3 x^3-64 b c x\right)}{64 b^2 c^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)}","\frac{5 \sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{64 b^2 c^2 \sqrt{c x-1}}-\frac{27 \sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2 \sqrt{c x-1}}+\frac{25 \sqrt{1-c x} \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2 \sqrt{c x-1}}-\frac{7 \sqrt{1-c x} \sinh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{64 b^2 c^2 \sqrt{c x-1}}+\frac{27 \sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2 \sqrt{c x-1}}-\frac{25 \sqrt{1-c x} \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2 \sqrt{c x-1}}+\frac{7 \sqrt{1-c x} \cosh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2 \sqrt{c x-1}}-\frac{x \sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{5/2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-64*b*c*x + 192*b*c^3*x^3 - 192*b*c^5*x^5 + 64*b*c^7*x^7 - 5*(a + b*ArcCosh[c*x])*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] + 27*(a + b*ArcCosh[c*x])*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - 25*a*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] - 25*b*ArcCosh[c*x]*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] + 7*a*CoshIntegral[7*(a/b + ArcCosh[c*x])]*Sinh[(7*a)/b] + 7*b*ArcCosh[c*x]*CoshIntegral[7*(a/b + ArcCosh[c*x])]*Sinh[(7*a)/b] + 5*a*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 5*b*ArcCosh[c*x]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 27*a*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - 27*b*ArcCosh[c*x]*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 25*a*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + 25*b*ArcCosh[c*x]*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] - 7*a*Cosh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])] - 7*b*ArcCosh[c*x]*Cosh[(7*a)/b]*SinhIntegral[7*(a/b + ArcCosh[c*x])]))/(64*b^2*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))","A",0
337,1,343,351,1.040075,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(a + b*ArcCosh[c*x])^2,x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(15 \sinh \left(\frac{2 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-12 \sinh \left(\frac{4 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 a \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 b \sinh \left(\frac{6 a}{b}\right) \cosh ^{-1}(c x) \text{Chi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-15 a \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-15 b \cosh \left(\frac{2 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+12 a \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+12 b \cosh \left(\frac{4 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-3 a \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-3 b \cosh \left(\frac{6 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(6 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+16 b c^6 x^6-48 b c^4 x^4+48 b c^2 x^2-16 b\right)}{16 b^2 c \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)}","-\frac{15 \sqrt{1-c x} \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c \sqrt{c x-1}}+\frac{15 \sqrt{1-c x} \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{5/2}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-16*b + 48*b*c^2*x^2 - 48*b*c^4*x^4 + 16*b*c^6*x^6 + 15*(a + b*ArcCosh[c*x])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - 12*(a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] + 3*a*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] + 3*b*ArcCosh[c*x]*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] - 15*a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 15*b*ArcCosh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 12*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] + 12*b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] - 3*b*ArcCosh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])]))/(16*b^2*c*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))","A",0
338,0,0,386,9.0034983,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\frac{\sqrt{1-c x} \text{Int}\left(\frac{\left(c^2 x^2-1\right)^2}{x^2 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{c x-1}}-\frac{25 \sqrt{1-c x} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 \sqrt{c x-1}}+\frac{25 \sqrt{1-c x} \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 \sqrt{c x-1}}+\frac{25 \sqrt{1-c x} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 \sqrt{c x-1}}-\frac{25 \sqrt{1-c x} \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 \sqrt{c x-1}}+\frac{5 \sqrt{1-c x} \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{5/2}}{b c x \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcCosh[c*x])^2), x]","A",-1
339,0,0,162,17.8366939,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^2 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\frac{4 c \sqrt{1-c x} \text{Int}\left(\frac{\left(c^2 x^2-1\right)^2}{x \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b \sqrt{c x-1}}+\frac{2 \sqrt{1-c x} \text{Int}\left(\frac{\left(c^2 x^2-1\right)^2}{x^3 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{c x-1}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left(1-c^2 x^2\right)^{5/2}}{b c x^2 \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcCosh[c*x])^2), x]","A",-1
340,0,0,31,21.1816488,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2}}{x^3 \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^3*(a + b*ArcCosh[c*x])^2), x]","A",-1
341,0,0,31,179.6991914,"\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])^2),x]","\int \frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{5/2}}{x^4 \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(1 - c^2*x^2)^(5/2)/(x^4*(a + b*ArcCosh[c*x])^2), x]","A",-1
342,1,190,337,0.7032355,"\int \frac{x^5}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x^5/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]","\frac{\sqrt{1-c^2 x^2} \left(\frac{16 b c^5 x^5}{a+b \cosh ^{-1}(c x)}+5 \left(2 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-3 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)\right)}{16 b^2 c^6 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{5 \sqrt{c x-1} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 c^6 \sqrt{1-c x}}-\frac{15 \sqrt{c x-1} \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6 \sqrt{1-c x}}-\frac{5 \sqrt{c x-1} \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6 \sqrt{1-c x}}+\frac{5 \sqrt{c x-1} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 c^6 \sqrt{1-c x}}+\frac{15 \sqrt{c x-1} \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6 \sqrt{1-c x}}+\frac{5 \sqrt{c x-1} \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6 \sqrt{1-c x}}-\frac{x^5 \sqrt{c x-1}}{b c \sqrt{1-c x} \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[1 - c^2*x^2]*((16*b*c^5*x^5)/(a + b*ArcCosh[c*x]) + 5*(2*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] + 3*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] + CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] - 2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])])))/(16*b^2*c^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
343,1,149,236,0.5106866,"\int \frac{x^4}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x^4/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]","\frac{\sqrt{1-c^2 x^2} \left(\frac{2 b c^4 x^4}{a+b \cosh ^{-1}(c x)}+2 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)}{2 b^2 c^5 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{\sqrt{c x-1} \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b^2 c^5 \sqrt{1-c x}}-\frac{\sqrt{c x-1} \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^5 \sqrt{1-c x}}+\frac{\sqrt{c x-1} \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b^2 c^5 \sqrt{1-c x}}+\frac{\sqrt{c x-1} \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^5 \sqrt{1-c x}}-\frac{x^4 \sqrt{c x-1}}{b c \sqrt{1-c x} \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[1 - c^2*x^2]*((2*b*c^4*x^4)/(a + b*ArcCosh[c*x]) + 2*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] + CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] - 2*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]))/(2*b^2*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
344,1,144,237,0.4835243,"\int \frac{x^3}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x^3/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]","\frac{\sqrt{1-c^2 x^2} \left(\frac{4 b c^3 x^3}{a+b \cosh ^{-1}(c x)}+3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-3 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)\right)}{4 b^2 c^4 \sqrt{c x-1} \sqrt{c x+1}}","-\frac{3 \sqrt{c x-1} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b^2 c^4 \sqrt{1-c x}}-\frac{3 \sqrt{c x-1} \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^4 \sqrt{1-c x}}+\frac{3 \sqrt{c x-1} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b^2 c^4 \sqrt{1-c x}}+\frac{3 \sqrt{c x-1} \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^4 \sqrt{1-c x}}-\frac{x^3 \sqrt{c x-1}}{b c \sqrt{1-c x} \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[1 - c^2*x^2]*((4*b*c^3*x^3)/(a + b*ArcCosh[c*x]) + 3*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] + 3*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - 3*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] - 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])]))/(4*b^2*c^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
345,1,117,136,0.2939249,"\int \frac{x^2}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]","\frac{\sqrt{1-c^2 x^2} \left(\sinh \left(\frac{2 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-\cosh \left(\frac{2 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Shi}\left(2 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+b c^2 x^2\right)}{b^2 c^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}","-\frac{\sqrt{c x-1} \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b^2 c^3 \sqrt{1-c x}}+\frac{\sqrt{c x-1} \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{b^2 c^3 \sqrt{1-c x}}-\frac{x^2 \sqrt{c x-1}}{b c \sqrt{1-c x} \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[1 - c^2*x^2]*(b*c^2*x^2 + (a + b*ArcCosh[c*x])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - (a + b*ArcCosh[c*x])*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])]))/(b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))","A",1
346,1,107,130,0.2264912,"\int \frac{x}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]","\frac{\sqrt{1-c^2 x^2} \left(\sinh \left(\frac{a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-\cosh \left(\frac{a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+b c x\right)}{b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}","-\frac{\sqrt{c x-1} \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 c^2 \sqrt{1-c x}}+\frac{\sqrt{c x-1} \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 c^2 \sqrt{1-c x}}-\frac{x \sqrt{c x-1}}{b c \sqrt{1-c x} \left(a+b \cosh ^{-1}(c x)\right)}",1,"(Sqrt[1 - c^2*x^2]*(b*c*x + (a + b*ArcCosh[c*x])*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - (a + b*ArcCosh[c*x])*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]]))/(b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))","A",1
347,1,50,37,0.0329112,"\int \frac{1}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]","-\frac{\sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)}","-\frac{\sqrt{c x-1}}{b c \sqrt{1-c x} \left(a+b \cosh ^{-1}(c x)\right)}",1,"-((Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(b*c*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])))","A",1
348,0,0,85,4.8187846,"\int \frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{x \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","-\frac{\sqrt{c x-1} \text{Int}\left(\frac{1}{x^2 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{1-c x}}-\frac{\sqrt{c x-1}}{b c x \sqrt{1-c x} \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x]","A",-1
349,0,0,85,1.5941351,"\int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","-\frac{2 \sqrt{c x-1} \text{Int}\left(\frac{1}{x^3 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{1-c x}}-\frac{\sqrt{c x-1}}{b c x^2 \sqrt{1-c x} \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[1/(x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x]","A",-1
350,0,0,31,29.9896142,"\int \frac{x^3}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{x^3}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^3}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^3/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
351,0,0,107,5.9857586,"\int \frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{x^2}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\frac{2 \sqrt{c x-1} \text{Int}\left(\frac{x}{\left(c^2 x^2-1\right)^2 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{1-c x}}-\frac{x^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[x^2/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
352,0,0,29,22.7132991,"\int \frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
353,0,0,102,2.3200384,"\int \frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\frac{2 c \sqrt{c x-1} \text{Int}\left(\frac{x}{\left(c^2 x^2-1\right)^2 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b \sqrt{1-c x}}-\frac{\sqrt{c x-1} \sqrt{c x+1}}{b c \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
354,0,0,31,23.2463715,"\int \frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
355,0,0,31,21.043304,"\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x^2*(1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
356,0,0,109,5.1449921,"\int \frac{x^4}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x^4/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{x^4}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","-\frac{4 \sqrt{c x-1} \text{Int}\left(\frac{x^3}{\left(c^2 x^2-1\right)^3 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b c \sqrt{1-c x}}-\frac{x^4 \sqrt{c x-1} \sqrt{c x+1}}{b c \left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[x^4/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
357,0,0,31,50.7150926,"\int \frac{x^3}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{x^3}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^3}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
358,0,0,31,7.8274015,"\int \frac{x^2}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{x^2}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^2}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^2/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
359,0,0,29,45.9316515,"\int \frac{x}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[x/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{x}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
360,0,0,102,3.8384097,"\int \frac{1}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","-\frac{4 c \sqrt{c x-1} \text{Int}\left(\frac{x}{\left(c^2 x^2-1\right)^3 \left(a+b \cosh ^{-1}(c x)\right)},x\right)}{b \sqrt{1-c x}}-\frac{\sqrt{c x-1} \sqrt{c x+1}}{b c \left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[1/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
361,0,0,31,41.0189513,"\int \frac{1}{x \left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{x \left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
362,0,0,31,15.6759236,"\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{x^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x^2*(1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
363,0,0,33,1.1975948,"\int \frac{(f x)^m \left(1-c^2 x^2\right)^{3/2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[((f*x)^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x])^2,x]","\int \frac{(f x)^m \left(1-c^2 x^2\right)^{3/2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(1-c^2 x^2\right)^{3/2} (f x)^m}{\left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[((f*x)^m*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x])^2, x]","A",-1
364,0,0,33,0.2032331,"\int \frac{(f x)^m \sqrt{1-c^2 x^2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[((f*x)^m*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x])^2,x]","\int \frac{(f x)^m \sqrt{1-c^2 x^2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{1-c^2 x^2} (f x)^m}{\left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[((f*x)^m*Sqrt[1 - c^2*x^2])/(a + b*ArcCosh[c*x])^2, x]","A",-1
365,0,0,92,0.69073,"\int \frac{(f x)^m}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(f*x)^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2),x]","\int \frac{(f x)^m}{\sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\frac{f m \sqrt{c x-1} \text{Int}\left(\frac{(f x)^{m-1}}{a+b \cosh ^{-1}(c x)},x\right)}{b c \sqrt{1-c x}}-\frac{\sqrt{c x-1} (f x)^m}{b c \sqrt{1-c x} \left(a+b \cosh ^{-1}(c x)\right)}",0,"Integrate[(f*x)^m/(Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^2), x]","A",-1
366,0,0,33,1.2825624,"\int \frac{(f x)^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(f*x)^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{(f x)^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m}{\left(1-c^2 x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(f*x)^m/((1 - c^2*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
367,0,0,33,1.7984613,"\int \frac{(f x)^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(f*x)^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{(f x)^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m}{\left(1-c^2 x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(f*x)^m/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
368,1,45,32,0.025935,"\int \frac{1}{\sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^3} \, dx","Integrate[1/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3),x]","-\frac{\sqrt{a x-1} \sqrt{a x+1}}{2 a \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)^2}","-\frac{\sqrt{a x-1}}{2 a \sqrt{1-a x} \cosh ^{-1}(a x)^2}",1,"-1/2*(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^2)","A",1
369,1,300,259,2.6459766,"\int \frac{x^3 \left(d-c^2 d x^2\right)}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^3*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2),x]","\frac{d e^{-\frac{6 a}{b}} \left(e^{\frac{6 a}{b}} \left(-3 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+\sqrt{6} e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-64 c^4 x^4 \sqrt{\frac{c x-1}{c x+1}}-64 c^3 x^3 \sqrt{\frac{c x-1}{c x+1}}+10 \sinh \left(2 \cosh ^{-1}(c x)\right)+8 \sinh \left(4 \cosh ^{-1}(c x)\right)+2 \sinh \left(6 \cosh ^{-1}(c x)\right)\right)-\sqrt{6} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3 \sqrt{2} e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{32 b c^4 \sqrt{a+b \cosh ^{-1}(c x)}}","\frac{3 \sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^4}-\frac{\sqrt{\frac{3 \pi }{2}} d e^{\frac{6 a}{b}} \text{erf}\left(\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^4}+\frac{3 \sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^4}-\frac{\sqrt{\frac{3 \pi }{2}} d e^{-\frac{6 a}{b}} \text{erfi}\left(\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^4}+\frac{2 d x^3 (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"(d*(-(Sqrt[6]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcCosh[c*x]))/b]) + 3*Sqrt[2]*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcCosh[c*x]))/b] + E^((6*a)/b)*(-64*c^3*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] - 64*c^4*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - 3*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (2*(a + b*ArcCosh[c*x]))/b] + Sqrt[6]*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (6*(a + b*ArcCosh[c*x]))/b] + 10*Sinh[2*ArcCosh[c*x]] + 8*Sinh[4*ArcCosh[c*x]] + 2*Sinh[6*ArcCosh[c*x]])))/(32*b*c^4*E^((6*a)/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
370,1,384,340,1.5595681,"\int \frac{x^2 \left(d-c^2 d x^2\right)}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^2*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2),x]","\frac{d e^{-\frac{5 a}{b}} \left(-4 c x e^{\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}}-4 e^{\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}}-2 e^{\frac{5 a}{b}} \sinh \left(3 \cosh ^{-1}(c x)\right)+2 e^{\frac{5 a}{b}} \sinh \left(5 \cosh ^{-1}(c x)\right)-2 e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)-\sqrt{5} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+\sqrt{3} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)-\sqrt{3} e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+\sqrt{5} e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{16 b c^3 \sqrt{a+b \cosh ^{-1}(c x)}}","\frac{\sqrt{\pi } d e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}-\frac{\sqrt{5 \pi } d e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}+\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}-\frac{\sqrt{5 \pi } d e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}+\frac{2 d x^2 (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"(d*(-4*E^((5*a)/b)*Sqrt[(-1 + c*x)/(1 + c*x)] - 4*c*E^((5*a)/b)*x*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] - Sqrt[5]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcCosh[c*x]))/b] + Sqrt[3]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + 2*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] - Sqrt[3]*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + Sqrt[5]*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (5*(a + b*ArcCosh[c*x]))/b] - 2*E^((5*a)/b)*Sinh[3*ArcCosh[c*x]] + 2*E^((5*a)/b)*Sinh[5*ArcCosh[c*x]]))/(16*b*c^3*E^((5*a)/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
371,1,331,241,4.2730083,"\int \frac{x \left(d-c^2 d x^2\right)}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x*(d - c^2*d*x^2))/(a + b*ArcCosh[c*x])^(3/2),x]","\frac{d e^{-\frac{4 a}{b}} \left(2 \sqrt{2 \pi } e^{\frac{6 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)+2 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)+\frac{\sqrt{b} \left(-\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-\sqrt{2} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+e^{\frac{4 a}{b}} \left(\sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+8 c x \left(\frac{c x-1}{c x+1}\right)^{3/2} (c x+1)^3\right)\right)}{\sqrt{a+b \cosh ^{-1}(c x)}}\right)}{4 b^{3/2} c^2}","-\frac{\sqrt{\pi } d e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2} c^2}-\frac{\sqrt{\pi } d e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2} c^2}+\frac{2 d x (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"(d*(2*E^((6*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]] + 2*E^((2*a)/b)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]] + (Sqrt[b]*(-(Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-4*(a + b*ArcCosh[c*x]))/b]) - Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcCosh[c*x]))/b] + E^((4*a)/b)*(8*c*x*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3 + Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (2*(a + b*ArcCosh[c*x]))/b] + E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (4*(a + b*ArcCosh[c*x]))/b])))/Sqrt[a + b*ArcCosh[c*x]]))/(4*b^(3/2)*c^2*E^((4*a)/b))","A",0
372,1,246,233,1.6979965,"\int \frac{d-c^2 d x^2}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d - c^2*d*x^2)/(a + b*ArcCosh[c*x])^(3/2),x]","\frac{e^{-\frac{3 a}{b}} \left(-3 d e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)-\sqrt{3} d \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+d e^{\frac{2 a}{b}} \left(3 \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)+e^{a/b} \left(\sqrt{3} e^{\frac{3 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-6 \sqrt{\frac{c x-1}{c x+1}} (c x+1)+2 \sinh \left(3 \cosh ^{-1}(c x)\right)\right)\right)\right)}{4 b c \sqrt{a+b \cosh ^{-1}(c x)}}","\frac{3 \sqrt{\pi } d e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c}-\frac{\sqrt{3 \pi } d e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c}+\frac{3 \sqrt{\pi } d e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c}-\frac{\sqrt{3 \pi } d e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c}+\frac{2 d (c x-1)^{3/2} (c x+1)^{3/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"(-3*d*E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] - Sqrt[3]*d*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + d*E^((2*a)/b)*(3*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + E^(a/b)*(-6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + Sqrt[3]*E^((3*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + 2*Sinh[3*ArcCosh[c*x]])))/(4*b*c*E^((3*a)/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
373,0,0,187,4.4608328,"\int \frac{d-c^2 d x^2}{x \left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d - c^2*d*x^2)/(x*(a + b*ArcCosh[c*x])^(3/2)),x]","\int \frac{d-c^2 d x^2}{x \left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","\frac{2 d \text{Int}\left(\frac{1}{x^2 \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}},x\right)}{b c}-\frac{\sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2}}+\frac{2 d (c x-1)^{3/2} (c x+1)^{3/2}}{b c x \sqrt{a+b \cosh ^{-1}(c x)}}",0,"Integrate[(d - c^2*d*x^2)/(x*(a + b*ArcCosh[c*x])^(3/2)), x]","A",-1
374,1,527,479,3.7769475,"\int \frac{x^3 \left(d-c^2 d x^2\right)^2}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^3*(d - c^2*d*x^2)^2)/(a + b*ArcCosh[c*x])^(3/2),x]","-\frac{d^2 e^{-\frac{8 a}{b}} \left(128 c^4 x^4 e^{\frac{8 a}{b}} \sqrt{\frac{c x-1}{c x+1}}+128 c^3 x^3 e^{\frac{8 a}{b}} \sqrt{\frac{c x-1}{c x+1}}-26 e^{\frac{8 a}{b}} \sinh \left(2 \cosh ^{-1}(c x)\right)-18 e^{\frac{8 a}{b}} \sinh \left(4 \cosh ^{-1}(c x)\right)-2 e^{\frac{8 a}{b}} \sinh \left(6 \cosh ^{-1}(c x)\right)+e^{\frac{8 a}{b}} \sinh \left(8 \cosh ^{-1}(c x)\right)-\sqrt{2} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+\sqrt{6} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-3 \sqrt{2} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3 \sqrt{2} e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-2 e^{\frac{12 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-\sqrt{6} e^{\frac{14 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+\sqrt{2} e^{\frac{16 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{64 b c^4 \sqrt{a+b \cosh ^{-1}(c x)}}","-\frac{\sqrt{\pi } d^2 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 b^{3/2} c^4}+\frac{3 \sqrt{\frac{\pi }{2}} d^2 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 b^{3/2} c^4}+\frac{\sqrt{\frac{\pi }{2}} d^2 e^{\frac{8 a}{b}} \text{erf}\left(\frac{2 \sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 b^{3/2} c^4}-\frac{\sqrt{\frac{3 \pi }{2}} d^2 e^{\frac{6 a}{b}} \text{erf}\left(\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 b^{3/2} c^4}-\frac{\sqrt{\pi } d^2 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 b^{3/2} c^4}+\frac{3 \sqrt{\frac{\pi }{2}} d^2 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 b^{3/2} c^4}+\frac{\sqrt{\frac{\pi }{2}} d^2 e^{-\frac{8 a}{b}} \text{erfi}\left(\frac{2 \sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 b^{3/2} c^4}-\frac{\sqrt{\frac{3 \pi }{2}} d^2 e^{-\frac{6 a}{b}} \text{erfi}\left(\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 b^{3/2} c^4}-\frac{2 d^2 x^3 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"-1/64*(d^2*(128*c^3*E^((8*a)/b)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 128*c^4*E^((8*a)/b)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] - Sqrt[2]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-8*(a + b*ArcCosh[c*x]))/b] + Sqrt[6]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcCosh[c*x]))/b] + 2*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-4*(a + b*ArcCosh[c*x]))/b] - 3*Sqrt[2]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcCosh[c*x]))/b] + 3*Sqrt[2]*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (2*(a + b*ArcCosh[c*x]))/b] - 2*E^((12*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (4*(a + b*ArcCosh[c*x]))/b] - Sqrt[6]*E^((14*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (6*(a + b*ArcCosh[c*x]))/b] + Sqrt[2]*E^((16*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (8*(a + b*ArcCosh[c*x]))/b] - 26*E^((8*a)/b)*Sinh[2*ArcCosh[c*x]] - 18*E^((8*a)/b)*Sinh[4*ArcCosh[c*x]] - 2*E^((8*a)/b)*Sinh[6*ArcCosh[c*x]] + E^((8*a)/b)*Sinh[8*ArcCosh[c*x]]))/(b*c^4*E^((8*a)/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
375,1,498,462,2.9885115,"\int \frac{x^2 \left(d-c^2 d x^2\right)^2}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^2*(d - c^2*d*x^2)^2)/(a + b*ArcCosh[c*x])^(3/2),x]","-\frac{d^2 e^{-\frac{7 a}{b}} \left(10 c x e^{\frac{7 a}{b}} \sqrt{\frac{c x-1}{c x+1}}+10 e^{\frac{7 a}{b}} \sqrt{\frac{c x-1}{c x+1}}+2 e^{\frac{7 a}{b}} \sinh \left(3 \cosh ^{-1}(c x)\right)-6 e^{\frac{7 a}{b}} \sinh \left(5 \cosh ^{-1}(c x)\right)+2 e^{\frac{7 a}{b}} \sinh \left(7 \cosh ^{-1}(c x)\right)+5 e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)-\sqrt{7} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3 \sqrt{5} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-5 e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)+\sqrt{3} e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-3 \sqrt{5} e^{\frac{12 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+\sqrt{7} e^{\frac{14 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{64 b c^3 \sqrt{a+b \cosh ^{-1}(c x)}}","\frac{5 \sqrt{\pi } d^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 b^{3/2} c^3}-\frac{3 \sqrt{5 \pi } d^2 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 b^{3/2} c^3}+\frac{\sqrt{7 \pi } d^2 e^{\frac{7 a}{b}} \text{erf}\left(\frac{\sqrt{7} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 b^{3/2} c^3}+\frac{5 \sqrt{\pi } d^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 b^{3/2} c^3}-\frac{3 \sqrt{5 \pi } d^2 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 b^{3/2} c^3}+\frac{\sqrt{7 \pi } d^2 e^{-\frac{7 a}{b}} \text{erfi}\left(\frac{\sqrt{7} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 b^{3/2} c^3}-\frac{2 d^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"-1/64*(d^2*(10*E^((7*a)/b)*Sqrt[(-1 + c*x)/(1 + c*x)] + 10*c*E^((7*a)/b)*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 5*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] - Sqrt[7]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-7*(a + b*ArcCosh[c*x]))/b] + 3*Sqrt[5]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcCosh[c*x]))/b] - Sqrt[3]*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] - 5*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] - 3*Sqrt[5]*E^((12*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (5*(a + b*ArcCosh[c*x]))/b] + Sqrt[7]*E^((14*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (7*(a + b*ArcCosh[c*x]))/b] + 2*E^((7*a)/b)*Sinh[3*ArcCosh[c*x]] - 6*E^((7*a)/b)*Sinh[5*ArcCosh[c*x]] + 2*E^((7*a)/b)*Sinh[7*ArcCosh[c*x]]))/(b*c^3*E^((7*a)/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
376,1,508,363,7.2915325,"\int \frac{x \left(d-c^2 d x^2\right)^2}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x*(d - c^2*d*x^2)^2)/(a + b*ArcCosh[c*x])^(3/2),x]","\frac{d^2 e^{-\frac{6 a}{b}} \left(\frac{\sqrt{b} \left(128 c^4 x^4 e^{\frac{6 a}{b}} \sqrt{\frac{c x-1}{c x+1}}+128 c^3 x^3 e^{\frac{6 a}{b}} \sqrt{\frac{c x-1}{c x+1}}-42 e^{\frac{6 a}{b}} \sinh \left(2 \cosh ^{-1}(c x)\right)-8 e^{\frac{6 a}{b}} \sinh \left(4 \cosh ^{-1}(c x)\right)-2 e^{\frac{6 a}{b}} \sinh \left(6 \cosh ^{-1}(c x)\right)+\sqrt{6} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-8 e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-11 \sqrt{2} e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+11 \sqrt{2} e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+8 e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-\sqrt{6} e^{\frac{12 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{\sqrt{a+b \cosh ^{-1}(c x)}}+16 \sqrt{2 \pi } e^{\frac{8 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)+16 \sqrt{2 \pi } e^{\frac{4 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)\right)}{32 b^{3/2} c^2}","-\frac{\sqrt{\pi } d^2 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^2}+\frac{5 \sqrt{\frac{\pi }{2}} d^2 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^2}+\frac{\sqrt{\frac{3 \pi }{2}} d^2 e^{\frac{6 a}{b}} \text{erf}\left(\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^2}-\frac{\sqrt{\pi } d^2 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^2}+\frac{5 \sqrt{\frac{\pi }{2}} d^2 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^2}+\frac{\sqrt{\frac{3 \pi }{2}} d^2 e^{-\frac{6 a}{b}} \text{erfi}\left(\frac{\sqrt{6} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^2}-\frac{2 d^2 x (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"(d^2*(16*E^((8*a)/b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]] + 16*E^((4*a)/b)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]] + (Sqrt[b]*(128*c^3*E^((6*a)/b)*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 128*c^4*E^((6*a)/b)*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] + Sqrt[6]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcCosh[c*x]))/b] - 8*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-4*(a + b*ArcCosh[c*x]))/b] - 11*Sqrt[2]*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcCosh[c*x]))/b] + 11*Sqrt[2]*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (2*(a + b*ArcCosh[c*x]))/b] + 8*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (4*(a + b*ArcCosh[c*x]))/b] - Sqrt[6]*E^((12*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (6*(a + b*ArcCosh[c*x]))/b] - 42*E^((6*a)/b)*Sinh[2*ArcCosh[c*x]] - 8*E^((6*a)/b)*Sinh[4*ArcCosh[c*x]] - 2*E^((6*a)/b)*Sinh[6*ArcCosh[c*x]]))/Sqrt[a + b*ArcCosh[c*x]]))/(32*b^(3/2)*c^2*E^((6*a)/b))","A",0
377,1,387,351,1.8999592,"\int \frac{\left(d-c^2 d x^2\right)^2}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d - c^2*d*x^2)^2/(a + b*ArcCosh[c*x])^(3/2),x]","-\frac{d^2 e^{-\frac{5 a}{b}} \left(20 c x e^{\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}}+20 e^{\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}}-10 e^{\frac{5 a}{b}} \sinh \left(3 \cosh ^{-1}(c x)\right)+2 e^{\frac{5 a}{b}} \sinh \left(5 \cosh ^{-1}(c x)\right)+10 e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)-\sqrt{5} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+5 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-10 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)-5 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+\sqrt{5} e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{16 b c \sqrt{a+b \cosh ^{-1}(c x)}}","\frac{5 \sqrt{\pi } d^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c}-\frac{5 \sqrt{3 \pi } d^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c}+\frac{\sqrt{5 \pi } d^2 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c}+\frac{5 \sqrt{\pi } d^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 b^{3/2} c}-\frac{5 \sqrt{3 \pi } d^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c}+\frac{\sqrt{5 \pi } d^2 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c}-\frac{2 d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"-1/16*(d^2*(20*E^((5*a)/b)*Sqrt[(-1 + c*x)/(1 + c*x)] + 20*c*E^((5*a)/b)*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 10*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] - Sqrt[5]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcCosh[c*x]))/b] + 5*Sqrt[3]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] - 10*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] - 5*Sqrt[3]*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + Sqrt[5]*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (5*(a + b*ArcCosh[c*x]))/b] - 10*E^((5*a)/b)*Sinh[3*ArcCosh[c*x]] + 2*E^((5*a)/b)*Sinh[5*ArcCosh[c*x]]))/(b*c*E^((5*a)/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
378,0,0,289,3.4238123,"\int \frac{\left(d-c^2 d x^2\right)^2}{x \left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d - c^2*d*x^2)^2/(x*(a + b*ArcCosh[c*x])^(3/2)),x]","\int \frac{\left(d-c^2 d x^2\right)^2}{x \left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","\frac{2 d^2 \text{Int}\left(\frac{1}{x^2 \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}},x\right)}{b c}+\frac{\sqrt{\pi } d^2 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^2 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2}}+\frac{\sqrt{\pi } d^2 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} d^2 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2}}-\frac{2 d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c x \sqrt{a+b \cosh ^{-1}(c x)}}",0,"Integrate[(d - c^2*d*x^2)^2/(x*(a + b*ArcCosh[c*x])^(3/2)), x]","A",-1
379,1,154,351,0.2551468,"\int \left(c-a^2 c x^2\right)^{3/2} \sqrt{\cosh ^{-1}(a x)} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]],x]","-\frac{c \sqrt{c-a^2 c x^2} \left(-\sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{3}{2},-4 \cosh ^{-1}(a x)\right)+8 \sqrt{2} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{3}{2},-2 \cosh ^{-1}(a x)\right)+\sqrt{\cosh ^{-1}(a x)} \left(32 \cosh ^{-1}(a x)^{3/2}+8 \sqrt{2} \Gamma \left(\frac{3}{2},2 \cosh ^{-1}(a x)\right)-\Gamma \left(\frac{3}{2},4 \cosh ^{-1}(a x)\right)\right)\right)}{128 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{256 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{4} x \left(c-a^2 c x^2\right)^{3/2} \sqrt{\cosh ^{-1}(a x)}-\frac{c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{4 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}",1,"-1/128*(c*Sqrt[c - a^2*c*x^2]*(-(Sqrt[-ArcCosh[a*x]]*Gamma[3/2, -4*ArcCosh[a*x]]) + 8*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[3/2, -2*ArcCosh[a*x]] + Sqrt[ArcCosh[a*x]]*(32*ArcCosh[a*x]^(3/2) + 8*Sqrt[2]*Gamma[3/2, 2*ArcCosh[a*x]] - Gamma[3/2, 4*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
380,1,117,205,0.1408716,"\int \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)} \, dx","Integrate[Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]],x]","-\frac{\sqrt{-c (a x-1) (a x+1)} \left(16 \cosh ^{-1}(a x)^2+3 \sqrt{2} \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{3}{2},2 \cosh ^{-1}(a x)\right)+3 \sqrt{2} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{3}{2},-2 \cosh ^{-1}(a x)\right)\right)}{48 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}",1,"-1/48*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(16*ArcCosh[a*x]^2 + 3*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[3/2, -2*ArcCosh[a*x]] + 3*Sqrt[2]*Sqrt[ArcCosh[a*x]]*Gamma[3/2, 2*ArcCosh[a*x]]))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
381,1,48,48,0.0323544,"\int \frac{\sqrt{\cosh ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[Sqrt[ArcCosh[a*x]]/Sqrt[c - a^2*c*x^2],x]","\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}}","\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{3/2}}{3 a \sqrt{c-a^2 c x^2}}",1,"(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(3*a*Sqrt[c - a^2*c*x^2])","A",1
382,0,0,95,1.667232,"\int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[Sqrt[ArcCosh[a*x]]/(c - a^2*c*x^2)^(3/2),x]","\int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","\frac{a \sqrt{a x-1} \sqrt{a x+1} \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right) \sqrt{\cosh ^{-1}(a x)}},x\right)}{2 c \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{\cosh ^{-1}(a x)}}{c \sqrt{c-a^2 c x^2}}",0,"Integrate[Sqrt[ArcCosh[a*x]]/(c - a^2*c*x^2)^(3/2), x]","A",-1
383,0,0,194,2.2715552,"\int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","Integrate[Sqrt[ArcCosh[a*x]]/(c - a^2*c*x^2)^(5/2),x]","\int \frac{\sqrt{\cosh ^{-1}(a x)}}{\left(c-a^2 c x^2\right)^{5/2}} \, dx","\frac{a \sqrt{a x-1} \sqrt{a x+1} \text{Int}\left(\frac{x}{\left(1-a^2 x^2\right) \sqrt{\cosh ^{-1}(a x)}},x\right)}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{a x-1} \sqrt{a x+1} \text{Int}\left(\frac{x}{\left(a^2 x^2-1\right)^2 \sqrt{\cosh ^{-1}(a x)}},x\right)}{6 c^2 \sqrt{c-a^2 c x^2}}+\frac{2 x \sqrt{\cosh ^{-1}(a x)}}{3 c^2 \sqrt{c-a^2 c x^2}}+\frac{x \sqrt{\cosh ^{-1}(a x)}}{3 c \left(c-a^2 c x^2\right)^{3/2}}",0,"Integrate[Sqrt[ArcCosh[a*x]]/(c - a^2*c*x^2)^(5/2), x]","A",-1
384,1,198,511,0.479272,"\int \left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^{3/2} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2),x]","\frac{c \sqrt{c-a^2 c x^2} \left(60 \sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)+60 \sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)-384 \cosh ^{-1}(a x)^3-480 \cosh \left(2 \cosh ^{-1}(a x)\right) \cosh ^{-1}(a x)+640 \cosh ^{-1}(a x)^2 \sinh \left(2 \cosh ^{-1}(a x)\right)+5 \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{5}{2},4 \cosh ^{-1}(a x)\right)-5 \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{5}{2},-4 \cosh ^{-1}(a x)\right)\right)}{2560 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","-\frac{3 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{2048 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{2048 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{4} x \left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^{3/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{3 c \left(1-a^2 x^2\right)^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{a x-1} \sqrt{a x+1}}+\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{256 a \sqrt{a x-1} \sqrt{a x+1}}",1,"(c*Sqrt[c - a^2*c*x^2]*(-384*ArcCosh[a*x]^3 - 480*ArcCosh[a*x]*Cosh[2*ArcCosh[a*x]] + 60*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 60*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - 5*Sqrt[-ArcCosh[a*x]]*Gamma[5/2, -4*ArcCosh[a*x]] + 5*Sqrt[ArcCosh[a*x]]*Gamma[5/2, 4*ArcCosh[a*x]] + 640*ArcCosh[a*x]^2*Sinh[2*ArcCosh[a*x]]))/(2560*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
385,1,136,302,0.3532657,"\int \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2),x]","\frac{\sqrt{c-a^2 c x^2} \left(15 \sqrt{2 \pi } \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)+15 \sqrt{2 \pi } \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)-8 \sqrt{\cosh ^{-1}(a x)} \left(16 \cosh ^{-1}(a x)^2+15 \cosh \left(2 \cosh ^{-1}(a x)\right)-20 \cosh ^{-1}(a x) \sinh \left(2 \cosh ^{-1}(a x)\right)\right)\right)}{640 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 a x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{16 a \sqrt{a x-1} \sqrt{a x+1}}",1,"(Sqrt[c - a^2*c*x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 15*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - 8*Sqrt[ArcCosh[a*x]]*(16*ArcCosh[a*x]^2 + 15*Cosh[2*ArcCosh[a*x]] - 20*ArcCosh[a*x]*Sinh[2*ArcCosh[a*x]])))/(640*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",0
386,1,48,48,0.031587,"\int \frac{\cosh ^{-1}(a x)^{3/2}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[ArcCosh[a*x]^(3/2)/Sqrt[c - a^2*c*x^2],x]","\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{c-a^2 c x^2}}","\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{5/2}}{5 a \sqrt{c-a^2 c x^2}}",1,"(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(5/2))/(5*a*Sqrt[c - a^2*c*x^2])","A",1
387,0,0,95,1.75204,"\int \frac{\cosh ^{-1}(a x)^{3/2}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcCosh[a*x]^(3/2)/(c - a^2*c*x^2)^(3/2),x]","\int \frac{\cosh ^{-1}(a x)^{3/2}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","\frac{3 a \sqrt{a x-1} \sqrt{a x+1} \text{Int}\left(\frac{x \sqrt{\cosh ^{-1}(a x)}}{1-a^2 x^2},x\right)}{2 c \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^{3/2}}{c \sqrt{c-a^2 c x^2}}",0,"Integrate[ArcCosh[a*x]^(3/2)/(c - a^2*c*x^2)^(3/2), x]","A",-1
388,1,213,580,0.5135762,"\int \left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^{5/2} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2),x]","\frac{c \sqrt{c-a^2 c x^2} \left(420 \sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)-420 \sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)-1536 \cosh ^{-1}(a x)^4-4480 \cosh \left(2 \cosh ^{-1}(a x)\right) \cosh ^{-1}(a x)^2+3584 \cosh ^{-1}(a x)^3 \sinh \left(2 \cosh ^{-1}(a x)\right)+3360 \cosh ^{-1}(a x) \sinh \left(2 \cosh ^{-1}(a x)\right)+7 \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{7}{2},4 \cosh ^{-1}(a x)\right)+7 \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{7}{2},-4 \cosh ^{-1}(a x)\right)\right)}{14336 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","-\frac{15 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{16384 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{256 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{16384 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{4} x \left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac{5 c \left(1-a^2 x^2\right)^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{a x-1} \sqrt{a x+1}}+\frac{45 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}",1,"(c*Sqrt[c - a^2*c*x^2]*(-1536*ArcCosh[a*x]^4 - 4480*ArcCosh[a*x]^2*Cosh[2*ArcCosh[a*x]] + 420*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - 420*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 7*Sqrt[-ArcCosh[a*x]]*Gamma[7/2, -4*ArcCosh[a*x]] + 7*Sqrt[ArcCosh[a*x]]*Gamma[7/2, 4*ArcCosh[a*x]] + 3360*ArcCosh[a*x]*Sinh[2*ArcCosh[a*x]] + 3584*ArcCosh[a*x]^3*Sinh[2*ArcCosh[a*x]]))/(14336*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
389,1,148,330,0.5176822,"\int \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2} \, dx","Integrate[Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2),x]","-\frac{\sqrt{-c (a x-1) (a x+1)} \left(-105 \sqrt{2 \pi } \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)+105 \sqrt{2 \pi } \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)+8 \sqrt{\cosh ^{-1}(a x)} \left(64 \cosh ^{-1}(a x)^3+140 \cosh \left(2 \cosh ^{-1}(a x)\right) \cosh ^{-1}(a x)-7 \left(16 \cosh ^{-1}(a x)^2+15\right) \sinh \left(2 \cosh ^{-1}(a x)\right)\right)\right)}{3584 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)}","\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{2} x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{5 a x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{8 \sqrt{a x-1} \sqrt{a x+1}}+\frac{5 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15}{32} x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}",1,"-1/3584*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(-105*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 105*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 8*Sqrt[ArcCosh[a*x]]*(64*ArcCosh[a*x]^3 + 140*ArcCosh[a*x]*Cosh[2*ArcCosh[a*x]] - 7*(15 + 16*ArcCosh[a*x]^2)*Sinh[2*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))","A",0
390,1,48,48,0.0322115,"\int \frac{\cosh ^{-1}(a x)^{5/2}}{\sqrt{c-a^2 c x^2}} \, dx","Integrate[ArcCosh[a*x]^(5/2)/Sqrt[c - a^2*c*x^2],x]","\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{c-a^2 c x^2}}","\frac{2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^{7/2}}{7 a \sqrt{c-a^2 c x^2}}",1,"(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(7/2))/(7*a*Sqrt[c - a^2*c*x^2])","A",1
391,0,0,95,1.6414752,"\int \frac{\cosh ^{-1}(a x)^{5/2}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcCosh[a*x]^(5/2)/(c - a^2*c*x^2)^(3/2),x]","\int \frac{\cosh ^{-1}(a x)^{5/2}}{\left(c-a^2 c x^2\right)^{3/2}} \, dx","\frac{5 a \sqrt{a x-1} \sqrt{a x+1} \text{Int}\left(\frac{x \cosh ^{-1}(a x)^{3/2}}{1-a^2 x^2},x\right)}{2 c \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^{5/2}}{c \sqrt{c-a^2 c x^2}}",0,"Integrate[ArcCosh[a*x]^(5/2)/(c - a^2*c*x^2)^(3/2), x]","A",-1
392,1,165,368,0.2870261,"\int \left(a^2-x^2\right)^{3/2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)} \, dx","Integrate[(a^2 - x^2)^(3/2)*Sqrt[ArcCosh[x/a]],x]","-\frac{a^4 \sqrt{a^2-x^2} \left(-\sqrt{-\cosh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},-4 \cosh ^{-1}\left(\frac{x}{a}\right)\right)+8 \sqrt{2} \sqrt{-\cosh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},-2 \cosh ^{-1}\left(\frac{x}{a}\right)\right)+\sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)} \left(32 \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}+8 \sqrt{2} \Gamma \left(\frac{3}{2},2 \cosh ^{-1}\left(\frac{x}{a}\right)\right)-\Gamma \left(\frac{3}{2},4 \cosh ^{-1}\left(\frac{x}{a}\right)\right)\right)\right)}{128 \sqrt{\frac{x-a}{a+x}} (a+x) \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}","\frac{3}{8} a^2 x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}+\frac{1}{4} x \left(a^2-x^2\right)^{3/2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}-\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{256 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{\sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{256 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{a^3 \sqrt{a^2-x^2} \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{4 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}",1,"-1/128*(a^4*Sqrt[a^2 - x^2]*(-(Sqrt[-ArcCosh[x/a]]*Gamma[3/2, -4*ArcCosh[x/a]]) + 8*Sqrt[2]*Sqrt[-ArcCosh[x/a]]*Gamma[3/2, -2*ArcCosh[x/a]] + Sqrt[ArcCosh[x/a]]*(32*ArcCosh[x/a]^(3/2) + 8*Sqrt[2]*Gamma[3/2, 2*ArcCosh[x/a]] - Gamma[3/2, 4*ArcCosh[x/a]])))/(Sqrt[(-a + x)/(a + x)]*(a + x)*Sqrt[ArcCosh[x/a]])","A",0
393,1,121,211,0.1511187,"\int \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)} \, dx","Integrate[Sqrt[a^2 - x^2]*Sqrt[ArcCosh[x/a]],x]","-\frac{a^2 \sqrt{a^2-x^2} \left(16 \cosh ^{-1}\left(\frac{x}{a}\right)^2+3 \sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},2 \cosh ^{-1}\left(\frac{x}{a}\right)\right)+3 \sqrt{2} \sqrt{-\cosh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},-2 \cosh ^{-1}\left(\frac{x}{a}\right)\right)\right)}{48 \sqrt{\frac{x-a}{a+x}} (a+x) \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}","\frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{3 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{1}{2} x \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}",1,"-1/48*(a^2*Sqrt[a^2 - x^2]*(16*ArcCosh[x/a]^2 + 3*Sqrt[2]*Sqrt[-ArcCosh[x/a]]*Gamma[3/2, -2*ArcCosh[x/a]] + 3*Sqrt[2]*Sqrt[ArcCosh[x/a]]*Gamma[3/2, 2*ArcCosh[x/a]]))/(Sqrt[(-a + x)/(a + x)]*(a + x)*Sqrt[ArcCosh[x/a]])","A",0
394,1,50,50,0.048861,"\int \frac{\sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{\sqrt{a^2-x^2}} \, dx","Integrate[Sqrt[ArcCosh[x/a]]/Sqrt[a^2 - x^2],x]","\frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{3 \sqrt{a^2-x^2}}","\frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{3 \sqrt{a^2-x^2}}",1,"(2*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*ArcCosh[x/a]^(3/2))/(3*Sqrt[a^2 - x^2])","A",1
395,0,0,98,0.9633084,"\int \frac{\sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2-x^2\right)^{3/2}} \, dx","Integrate[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(3/2),x]","\int \frac{\sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2-x^2\right)^{3/2}} \, dx","\frac{\sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \text{Int}\left(\frac{x}{\left(1-\frac{x^2}{a^2}\right) \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}},x\right)}{2 a^3 \sqrt{a^2-x^2}}+\frac{x \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{a^2 \sqrt{a^2-x^2}}",0,"Integrate[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(3/2), x]","A",-1
396,0,0,200,2.316155,"\int \frac{\sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2-x^2\right)^{5/2}} \, dx","Integrate[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(5/2),x]","\int \frac{\sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2-x^2\right)^{5/2}} \, dx","\frac{\sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \text{Int}\left(\frac{x}{\left(1-\frac{x^2}{a^2}\right) \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}},x\right)}{3 a^5 \sqrt{a^2-x^2}}+\frac{\sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \text{Int}\left(\frac{x}{\left(\frac{x^2}{a^2}-1\right)^2 \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}},x\right)}{6 a^5 \sqrt{a^2-x^2}}+\frac{x \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{3 a^2 \left(a^2-x^2\right)^{3/2}}+\frac{2 x \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{3 a^4 \sqrt{a^2-x^2}}",0,"Integrate[Sqrt[ArcCosh[x/a]]/(a^2 - x^2)^(5/2), x]","A",-1
397,1,219,525,0.477809,"\int \left(a^2-x^2\right)^{3/2} \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2} \, dx","Integrate[(a^2 - x^2)^(3/2)*ArcCosh[x/a]^(3/2),x]","\frac{a^4 \sqrt{a^2-x^2} \left(60 \sqrt{2 \pi } \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)+60 \sqrt{2 \pi } \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)-384 \cosh ^{-1}\left(\frac{x}{a}\right)^3-480 \cosh \left(2 \cosh ^{-1}\left(\frac{x}{a}\right)\right) \cosh ^{-1}\left(\frac{x}{a}\right)+640 \cosh ^{-1}\left(\frac{x}{a}\right)^2 \sinh \left(2 \cosh ^{-1}\left(\frac{x}{a}\right)\right)+5 \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{5}{2},4 \cosh ^{-1}\left(\frac{x}{a}\right)\right)-5 \sqrt{-\cosh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{5}{2},-4 \cosh ^{-1}\left(\frac{x}{a}\right)\right)\right)}{2560 \sqrt{\frac{x-a}{a+x}} (a+x) \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}","\frac{3}{8} a^2 x \sqrt{a^2-x^2} \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}-\frac{9 a x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{32 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{1}{4} x \left(a^2-x^2\right)^{3/2} \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}+\frac{3 \left(a^2-x^2\right)^{5/2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{32 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{3 \sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{2048 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{64 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{3 \sqrt{\pi } a^3 \sqrt{a^2-x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{2048 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2-x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{64 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{3 a^3 \sqrt{a^2-x^2} \cosh ^{-1}\left(\frac{x}{a}\right)^{5/2}}{20 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{27 a^3 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{256 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}",1,"(a^4*Sqrt[a^2 - x^2]*(-384*ArcCosh[x/a]^3 - 480*ArcCosh[x/a]*Cosh[2*ArcCosh[x/a]] + 60*Sqrt[2*Pi]*Sqrt[ArcCosh[x/a]]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]] + 60*Sqrt[2*Pi]*Sqrt[ArcCosh[x/a]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]] - 5*Sqrt[-ArcCosh[x/a]]*Gamma[5/2, -4*ArcCosh[x/a]] + 5*Sqrt[ArcCosh[x/a]]*Gamma[5/2, 4*ArcCosh[x/a]] + 640*ArcCosh[x/a]^2*Sinh[2*ArcCosh[x/a]]))/(2560*Sqrt[(-a + x)/(a + x)]*(a + x)*Sqrt[ArcCosh[x/a]])","A",0
398,1,144,316,0.3859192,"\int \sqrt{a^2-x^2} \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2} \, dx","Integrate[Sqrt[a^2 - x^2]*ArcCosh[x/a]^(3/2),x]","\frac{a^2 \sqrt{a^2-x^2} \left(15 \sqrt{2 \pi } \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)+15 \sqrt{2 \pi } \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)-8 \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)} \left(16 \cosh ^{-1}\left(\frac{x}{a}\right)^2+15 \cosh \left(2 \cosh ^{-1}\left(\frac{x}{a}\right)\right)-20 \cosh ^{-1}\left(\frac{x}{a}\right) \sinh \left(2 \cosh ^{-1}\left(\frac{x}{a}\right)\right)\right)\right)}{640 \sqrt{\frac{x-a}{a+x}} (a+x)}","\frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{64 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2-x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}\right)}{64 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}-\frac{a \sqrt{a^2-x^2} \cosh ^{-1}\left(\frac{x}{a}\right)^{5/2}}{5 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{1}{2} x \sqrt{a^2-x^2} \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}-\frac{3 x^2 \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{8 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}+\frac{3 a \sqrt{a^2-x^2} \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{16 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1}}",1,"(a^2*Sqrt[a^2 - x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcCosh[x/a]]] + 15*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcCosh[x/a]]] - 8*Sqrt[ArcCosh[x/a]]*(16*ArcCosh[x/a]^2 + 15*Cosh[2*ArcCosh[x/a]] - 20*ArcCosh[x/a]*Sinh[2*ArcCosh[x/a]])))/(640*Sqrt[(-a + x)/(a + x)]*(a + x))","A",0
399,1,50,50,0.0536003,"\int \frac{\cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{\sqrt{a^2-x^2}} \, dx","Integrate[ArcCosh[x/a]^(3/2)/Sqrt[a^2 - x^2],x]","\frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left(\frac{x}{a}\right)^{5/2}}{5 \sqrt{a^2-x^2}}","\frac{2 a \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \cosh ^{-1}\left(\frac{x}{a}\right)^{5/2}}{5 \sqrt{a^2-x^2}}",1,"(2*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*ArcCosh[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])","A",1
400,0,0,98,1.0420437,"\int \frac{\cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{\left(a^2-x^2\right)^{3/2}} \, dx","Integrate[ArcCosh[x/a]^(3/2)/(a^2 - x^2)^(3/2),x]","\int \frac{\cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{\left(a^2-x^2\right)^{3/2}} \, dx","\frac{3 \sqrt{\frac{x}{a}-1} \sqrt{\frac{x}{a}+1} \text{Int}\left(\frac{x \sqrt{\cosh ^{-1}\left(\frac{x}{a}\right)}}{1-\frac{x^2}{a^2}},x\right)}{2 a^3 \sqrt{a^2-x^2}}+\frac{x \cosh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{a^2 \sqrt{a^2-x^2}}",0,"Integrate[ArcCosh[x/a]^(3/2)/(a^2 - x^2)^(3/2), x]","A",-1
401,1,72,65,0.1059083,"\int \frac{x}{\sqrt{1-x^2} \sqrt{\cosh ^{-1}(x)}} \, dx","Integrate[x/(Sqrt[1 - x^2]*Sqrt[ArcCosh[x]]),x]","-\frac{\sqrt{-((x-1) (x+1))} \left(\sqrt{-\cosh ^{-1}(x)} \Gamma \left(\frac{1}{2},-\cosh ^{-1}(x)\right)-\sqrt{\cosh ^{-1}(x)} \Gamma \left(\frac{1}{2},\cosh ^{-1}(x)\right)\right)}{2 \sqrt{\frac{x-1}{x+1}} (x+1) \sqrt{\cosh ^{-1}(x)}}","\frac{\sqrt{\pi } \sqrt{x-1} \text{erf}\left(\sqrt{\cosh ^{-1}(x)}\right)}{2 \sqrt{1-x}}+\frac{\sqrt{\pi } \sqrt{x-1} \text{erfi}\left(\sqrt{\cosh ^{-1}(x)}\right)}{2 \sqrt{1-x}}",1,"-1/2*(Sqrt[-((-1 + x)*(1 + x))]*(Sqrt[-ArcCosh[x]]*Gamma[1/2, -ArcCosh[x]] - Sqrt[ArcCosh[x]]*Gamma[1/2, ArcCosh[x]]))/(Sqrt[(-1 + x)/(1 + x)]*(1 + x)*Sqrt[ArcCosh[x]])","A",0
402,1,209,438,0.4416074,"\int \frac{\left(c-a^2 c x^2\right)^{5/2}}{\sqrt{\cosh ^{-1}(a x)}} \, dx","Integrate[(c - a^2*c*x^2)^(5/2)/Sqrt[ArcCosh[a*x]],x]","-\frac{c^2 \sqrt{c-a^2 c x^2} \left(240 \cosh ^{-1}(a x)+45 \sqrt{2} \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 \cosh ^{-1}(a x)\right)-18 \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 \cosh ^{-1}(a x)\right)+\sqrt{6} \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},6 \cosh ^{-1}(a x)\right)-\sqrt{6} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-6 \cosh ^{-1}(a x)\right)+18 \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \cosh ^{-1}(a x)\right)-45 \sqrt{2} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \cosh ^{-1}(a x)\right)\right)}{384 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","-\frac{3 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{6}} c^2 \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{6}} c^2 \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right)}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{5 c^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{8 a \sqrt{a x-1} \sqrt{a x+1}}",1,"-1/384*(c^2*Sqrt[c - a^2*c*x^2]*(240*ArcCosh[a*x] - Sqrt[6]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -6*ArcCosh[a*x]] + 18*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] - 45*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x]] + 45*Sqrt[2]*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 2*ArcCosh[a*x]] - 18*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 4*ArcCosh[a*x]] + Sqrt[6]*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 6*ArcCosh[a*x]]))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
403,1,153,294,0.2648129,"\int \frac{\left(c-a^2 c x^2\right)^{3/2}}{\sqrt{\cosh ^{-1}(a x)}} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)/Sqrt[ArcCosh[a*x]],x]","-\frac{c \sqrt{c-a^2 c x^2} \left(\sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \cosh ^{-1}(a x)\right)-4 \sqrt{2} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \cosh ^{-1}(a x)\right)+\sqrt{\cosh ^{-1}(a x)} \left(24 \sqrt{\cosh ^{-1}(a x)}+4 \sqrt{2} \Gamma \left(\frac{1}{2},2 \cosh ^{-1}(a x)\right)-\Gamma \left(\frac{1}{2},4 \cosh ^{-1}(a x)\right)\right)\right)}{32 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{4 a \sqrt{a x-1} \sqrt{a x+1}}",1,"-1/32*(c*Sqrt[c - a^2*c*x^2]*(Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] - 4*Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x]] + Sqrt[ArcCosh[a*x]]*(24*Sqrt[ArcCosh[a*x]] + 4*Sqrt[2]*Gamma[1/2, 2*ArcCosh[a*x]] - Gamma[1/2, 4*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
404,1,114,175,0.1475238,"\int \frac{\sqrt{c-a^2 c x^2}}{\sqrt{\cosh ^{-1}(a x)}} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/Sqrt[ArcCosh[a*x]],x]","-\frac{\sqrt{-c (a x-1) (a x+1)} \left(8 \cosh ^{-1}(a x)+\sqrt{2} \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 \cosh ^{-1}(a x)\right)-\sqrt{2} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \cosh ^{-1}(a x)\right)\right)}{8 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{4 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{a \sqrt{a x-1} \sqrt{a x+1}}",1,"-1/8*(Sqrt[-(c*(-1 + a*x)*(1 + a*x))]*(8*ArcCosh[a*x] - Sqrt[2]*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x]] + Sqrt[2]*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 2*ArcCosh[a*x]]))/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
405,1,46,46,0.0374059,"\int \frac{1}{\sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}} \, dx","Integrate[1/(Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]]),x]","\frac{2 \sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{a \sqrt{c-a^2 c x^2}}","\frac{2 \sqrt{a x-1} \sqrt{a x+1} \sqrt{\cosh ^{-1}(a x)}}{a \sqrt{c-a^2 c x^2}}",1,"(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[ArcCosh[a*x]])/(a*Sqrt[c - a^2*c*x^2])","A",1
406,0,0,27,1.8070496,"\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sqrt{\cosh ^{-1}(a x)}} \, dx","Integrate[1/((c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]]),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sqrt{\cosh ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{\left(c-a^2 c x^2\right)^{3/2} \sqrt{\cosh ^{-1}(a x)}},x\right)",0,"Integrate[1/((c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]]), x]","A",-1
407,0,0,27,2.4805542,"\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sqrt{\cosh ^{-1}(a x)}} \, dx","Integrate[1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcCosh[a*x]]),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sqrt{\cosh ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{\left(c-a^2 c x^2\right)^{5/2} \sqrt{\cosh ^{-1}(a x)}},x\right)",0,"Integrate[1/((c - a^2*c*x^2)^(5/2)*Sqrt[ArcCosh[a*x]]), x]","A",-1
408,1,411,433,1.2745986,"\int \frac{\left(c-a^2 c x^2\right)^{5/2}}{\cosh ^{-1}(a x)^{3/2}} \, dx","Integrate[(c - a^2*c*x^2)^(5/2)/ArcCosh[a*x]^(3/2),x]","\frac{c^2 \sqrt{c-a^2 c x^2} e^{-6 \cosh ^{-1}(a x)} \left(-64 a^2 x^2 e^{6 \cosh ^{-1}(a x)}-16 \sqrt{2 \pi } e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)+16 \sqrt{2 \pi } e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)+6 e^{2 \cosh ^{-1}(a x)}+e^{4 \cosh ^{-1}(a x)}+52 e^{6 \cosh ^{-1}(a x)}+e^{8 \cosh ^{-1}(a x)}+6 e^{10 \cosh ^{-1}(a x)}-e^{12 \cosh ^{-1}(a x)}+\sqrt{6} e^{6 \cosh ^{-1}(a x)} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-6 \cosh ^{-1}(a x)\right)-12 e^{6 \cosh ^{-1}(a x)} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \cosh ^{-1}(a x)\right)-\sqrt{2} e^{6 \cosh ^{-1}(a x)} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \cosh ^{-1}(a x)\right)-\sqrt{2} e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 \cosh ^{-1}(a x)\right)-12 e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 \cosh ^{-1}(a x)\right)+\sqrt{6} e^{6 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},6 \cosh ^{-1}(a x)\right)-1\right)}{32 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","\frac{3 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{8 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{3 \pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right)}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 \sqrt{\pi } c^2 \sqrt{c-a^2 c x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{8 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{3 \pi }{2}} c^2 \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{6} \sqrt{\cosh ^{-1}(a x)}\right)}{16 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \left(c-a^2 c x^2\right)^{5/2}}{a \sqrt{\cosh ^{-1}(a x)}}",1,"(c^2*Sqrt[c - a^2*c*x^2]*(-1 + 6*E^(2*ArcCosh[a*x]) + E^(4*ArcCosh[a*x]) + 52*E^(6*ArcCosh[a*x]) + E^(8*ArcCosh[a*x]) + 6*E^(10*ArcCosh[a*x]) - E^(12*ArcCosh[a*x]) - 64*a^2*E^(6*ArcCosh[a*x])*x^2 - 16*E^(6*ArcCosh[a*x])*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 16*E^(6*ArcCosh[a*x])*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Sqrt[6]*E^(6*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -6*ArcCosh[a*x]] - 12*E^(6*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] - Sqrt[2]*E^(6*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -2*ArcCosh[a*x]] - Sqrt[2]*E^(6*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 2*ArcCosh[a*x]] - 12*E^(6*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 4*ArcCosh[a*x]] + Sqrt[6]*E^(6*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 6*ArcCosh[a*x]]))/(32*a*E^(6*ArcCosh[a*x])*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
409,1,239,286,0.482668,"\int \frac{\left(c-a^2 c x^2\right)^{3/2}}{\cosh ^{-1}(a x)^{3/2}} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)/ArcCosh[a*x]^(3/2),x]","-\frac{c \sqrt{c-a^2 c x^2} e^{-4 \cosh ^{-1}(a x)} \left(16 a^2 x^2 e^{4 \cosh ^{-1}(a x)}+4 \sqrt{2 \pi } e^{4 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)-4 \sqrt{2 \pi } e^{4 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)-14 e^{4 \cosh ^{-1}(a x)}-e^{8 \cosh ^{-1}(a x)}+2 e^{4 \cosh ^{-1}(a x)} \sqrt{-\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \cosh ^{-1}(a x)\right)+2 e^{4 \cosh ^{-1}(a x)} \sqrt{\cosh ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 \cosh ^{-1}(a x)\right)-1\right)}{8 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{4 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{a \sqrt{a x-1} \sqrt{a x+1}}-\frac{\sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{4 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \left(c-a^2 c x^2\right)^{3/2}}{a \sqrt{\cosh ^{-1}(a x)}}",1,"-1/8*(c*Sqrt[c - a^2*c*x^2]*(-1 - 14*E^(4*ArcCosh[a*x]) - E^(8*ArcCosh[a*x]) + 16*a^2*E^(4*ArcCosh[a*x])*x^2 + 4*E^(4*ArcCosh[a*x])*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - 4*E^(4*ArcCosh[a*x])*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 2*E^(4*ArcCosh[a*x])*Sqrt[-ArcCosh[a*x]]*Gamma[1/2, -4*ArcCosh[a*x]] + 2*E^(4*ArcCosh[a*x])*Sqrt[ArcCosh[a*x]]*Gamma[1/2, 4*ArcCosh[a*x]]))/(a*E^(4*ArcCosh[a*x])*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
410,1,127,170,0.2706633,"\int \frac{\sqrt{c-a^2 c x^2}}{\cosh ^{-1}(a x)^{3/2}} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/ArcCosh[a*x]^(3/2),x]","\frac{\sqrt{c-a^2 c x^2} \left(-4 a^2 x^2-\sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)+\sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)+4\right)}{2 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{a \sqrt{a x-1} \sqrt{a x+1}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \sqrt{c-a^2 c x^2}}{a \sqrt{\cosh ^{-1}(a x)}}",1,"(Sqrt[c - a^2*c*x^2]*(4 - 4*a^2*x^2 - Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]]))/(2*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])","A",0
411,1,46,46,0.0359288,"\int \frac{1}{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \, dx","Integrate[1/(Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2)),x]","-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}","-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{a \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}",1,"(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])","A",1
412,0,0,110,1.7923643,"\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2)),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^{3/2}} \, dx","\frac{4 a \sqrt{a x-1} \sqrt{a x+1} \text{Int}\left(\frac{x}{\left(a^2 x^2-1\right)^2 \sqrt{\cosh ^{-1}(a x)}},x\right)}{c \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{a \left(c-a^2 c x^2\right)^{3/2} \sqrt{\cosh ^{-1}(a x)}}",0,"Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2)), x]","A",-1
413,0,0,110,2.3434241,"\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \cosh ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^(3/2)),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \cosh ^{-1}(a x)^{3/2}} \, dx","-\frac{8 a \sqrt{a x-1} \sqrt{a x+1} \text{Int}\left(\frac{x}{\left(a^2 x^2-1\right)^3 \sqrt{\cosh ^{-1}(a x)}},x\right)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{a \left(c-a^2 c x^2\right)^{5/2} \sqrt{\cosh ^{-1}(a x)}}",0,"Integrate[1/((c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^(3/2)), x]","A",-1
414,1,317,329,0.5951828,"\int \frac{\left(c-a^2 c x^2\right)^{3/2}}{\cosh ^{-1}(a x)^{5/2}} \, dx","Integrate[(c - a^2*c*x^2)^(3/2)/ArcCosh[a*x]^(5/2),x]","-\frac{c \sqrt{c-a^2 c x^2} e^{-4 \cosh ^{-1}(a x)} \left(16 a^2 x^2 e^{4 \cosh ^{-1}(a x)}+64 a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+64 a x \sqrt{\frac{a x-1}{a x+1}} e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)-14 e^{4 \cosh ^{-1}(a x)}-e^{8 \cosh ^{-1}(a x)}-8 e^{8 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)+8 \cosh ^{-1}(a x)-16 e^{4 \cosh ^{-1}(a x)} \left(-\cosh ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-4 \cosh ^{-1}(a x)\right)+16 \sqrt{2} e^{4 \cosh ^{-1}(a x)} \left(-\cosh ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 \cosh ^{-1}(a x)\right)+16 \sqrt{2} e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{3/2} \Gamma \left(\frac{1}{2},2 \cosh ^{-1}(a x)\right)-16 e^{4 \cosh ^{-1}(a x)} \cosh ^{-1}(a x)^{3/2} \Gamma \left(\frac{1}{2},4 \cosh ^{-1}(a x)\right)-1\right)}{24 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^{3/2}}","-\frac{2 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erf}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{3 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{erfi}\left(2 \sqrt{\cosh ^{-1}(a x)}\right)}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{3 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \left(c-a^2 c x^2\right)^{3/2}}{3 a \cosh ^{-1}(a x)^{3/2}}-\frac{16 c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}",1,"-1/24*(c*Sqrt[c - a^2*c*x^2]*(-1 - 14*E^(4*ArcCosh[a*x]) - E^(8*ArcCosh[a*x]) + 16*a^2*E^(4*ArcCosh[a*x])*x^2 + 8*ArcCosh[a*x] - 8*E^(8*ArcCosh[a*x])*ArcCosh[a*x] + 64*a*E^(4*ArcCosh[a*x])*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] + 64*a^2*E^(4*ArcCosh[a*x])*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x] - 16*E^(4*ArcCosh[a*x])*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -4*ArcCosh[a*x]] + 16*Sqrt[2]*E^(4*ArcCosh[a*x])*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -2*ArcCosh[a*x]] + 16*Sqrt[2]*E^(4*ArcCosh[a*x])*ArcCosh[a*x]^(3/2)*Gamma[1/2, 2*ArcCosh[a*x]] - 16*E^(4*ArcCosh[a*x])*ArcCosh[a*x]^(3/2)*Gamma[1/2, 4*ArcCosh[a*x]]))/(a*E^(4*ArcCosh[a*x])*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^(3/2))","A",0
415,1,141,201,0.3209585,"\int \frac{\sqrt{c-a^2 c x^2}}{\cosh ^{-1}(a x)^{5/2}} \, dx","Integrate[Sqrt[c - a^2*c*x^2]/ArcCosh[a*x]^(5/2),x]","-\frac{2 \sqrt{c-a^2 c x^2} \left((a x+1) \left(a x+4 a x \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)-1\right)+\sqrt{2} \left(-\cosh ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 \cosh ^{-1}(a x)\right)+\sqrt{2} \cosh ^{-1}(a x)^{3/2} \Gamma \left(\frac{1}{2},2 \cosh ^{-1}(a x)\right)\right)}{3 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)^{3/2}}","\frac{2 \sqrt{2 \pi } \sqrt{c-a^2 c x^2} \text{erf}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{3 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2 \sqrt{2 \pi } \sqrt{c-a^2 c x^2} \text{erfi}\left(\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right)}{3 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{8 x \sqrt{c-a^2 c x^2}}{3 \sqrt{\cosh ^{-1}(a x)}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1} \sqrt{c-a^2 c x^2}}{3 a \cosh ^{-1}(a x)^{3/2}}",1,"(-2*Sqrt[c - a^2*c*x^2]*((1 + a*x)*(-1 + a*x + 4*a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]) + Sqrt[2]*(-ArcCosh[a*x])^(3/2)*Gamma[1/2, -2*ArcCosh[a*x]] + Sqrt[2]*ArcCosh[a*x]^(3/2)*Gamma[1/2, 2*ArcCosh[a*x]]))/(3*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*ArcCosh[a*x]^(3/2))","A",0
416,1,48,48,0.0341876,"\int \frac{1}{\sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}} \, dx","Integrate[1/(Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2)),x]","-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}","-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}",1,"(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))","A",1
417,0,0,114,1.8191757,"\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2)),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^{5/2}} \, dx","\frac{4 a \sqrt{a x-1} \sqrt{a x+1} \text{Int}\left(\frac{x}{\left(a^2 x^2-1\right)^2 \cosh ^{-1}(a x)^{3/2}},x\right)}{3 c \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \left(c-a^2 c x^2\right)^{3/2} \cosh ^{-1}(a x)^{3/2}}",0,"Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2)), x]","A",-1
418,0,0,114,2.3592596,"\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^(5/2)),x]","\int \frac{1}{\left(c-a^2 c x^2\right)^{5/2} \cosh ^{-1}(a x)^{5/2}} \, dx","-\frac{8 a \sqrt{a x-1} \sqrt{a x+1} \text{Int}\left(\frac{x}{\left(a^2 x^2-1\right)^3 \cosh ^{-1}(a x)^{3/2}},x\right)}{3 c^2 \sqrt{c-a^2 c x^2}}-\frac{2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \left(c-a^2 c x^2\right)^{5/2} \cosh ^{-1}(a x)^{3/2}}",0,"Integrate[1/((c - a^2*c*x^2)^(5/2)*ArcCosh[a*x]^(5/2)), x]","A",-1
419,1,181,253,1.0728696,"\int x^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n,x]","-\frac{d \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(4^{-n} e^{-\frac{4 a}{b}} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)-\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b n+b}\right)}{64 c^3 \sqrt{-d (c x-1) (c x+1)}}","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{8 b c^3 (n+1) \sqrt{c x-1} \sqrt{c x+1}}+\frac{2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/64*(d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*((-8*(a + b*ArcCosh[c*x]))/(b + b*n) + ((a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b] - E^((8*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(4^n*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^n)))/(c^3*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])","A",0
420,1,241,379,1.2825221,"\int x \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n,x]","-\frac{d e^{-\frac{3 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(\left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \left(-3^{-n} e^{\frac{6 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{2 n} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-3 e^{\frac{2 a}{b}} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)\right)+3 e^{\frac{4 a}{b}} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^{-n} \Gamma \left(n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)}{24 c^2 \sqrt{-d (c x-1) (c x+1)}}","\frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{e^{-\frac{a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{e^{a/b} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3^{-n-1} e^{\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 c^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/24*(d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*((3*E^((4*a)/b)*Gamma[1 + n, a/b + ArcCosh[c*x]])/(a/b + ArcCosh[c*x])^n + (Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b]/3^n - 3*E^((2*a)/b)*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)] - (E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^(2*n)*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(3^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n))/(-((a + b*ArcCosh[c*x])/b))^n))/(c^2*E^((3*a)/b)*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])","A",0
421,1,214,253,0.7022343,"\int \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n,x]","\frac{d 2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(2^{n+2} e^{\frac{2 a}{b}} \left(a+b \cosh ^{-1}(c x)\right) \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n-b (n+1) \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b (n+1) e^{\frac{4 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{b c (n+1) \sqrt{d-c^2 d x^2}}","-\frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{2 b c (n+1) \sqrt{c x-1} \sqrt{c x+1}}+\frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-n-3} e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}",1,"(2^(-3 - n)*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(2^(2 + n)*E^((2*a)/b)*(a + b*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])^2/b^2))^n - b*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] + b*E^((4*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b]))/(b*c*E^((2*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^n)","A",0
422,0,0,212,0.2499949,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n)/x,x]","\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x} \, dx","d \text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \sqrt{d-c^2 d x^2}},x\right)-\frac{d e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 \sqrt{d-c^2 d x^2}}+\frac{d e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 \sqrt{d-c^2 d x^2}}",0,"Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n)/x, x]","A",-1
423,0,0,92,0.2336487,"\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x^2} \, dx","Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n)/x^2,x]","\int \frac{\sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x^2} \, dx","d \text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \sqrt{d-c^2 d x^2}},x\right)-\frac{c d \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{b (n+1) \sqrt{d-c^2 d x^2}}",0,"Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n)/x^2, x]","A",-1
424,1,438,658,3.3668094,"\int x^2 \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[x^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]","\frac{d^2 2^{-2 n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(2^n e^{\frac{6 a}{b}} \left(2^{n+3} 3^{n+1} \left(a+b \cosh ^{-1}(c x)\right) \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n-b (n+1) e^{\frac{6 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)+b 2^n (n+1) \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b 3^{n+1} (n+1) e^{\frac{2 a}{b}} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b 2^n 3^{n+1} (n+1) e^{\frac{4 a}{b}} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b 2^n 3^{n+1} (n+1) e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b 3^{n+1} (n+1) e^{\frac{10 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{b c^3 (n+1) \sqrt{d-c^2 d x^2}}","-\frac{d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{16 b c^3 (n+1) \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 2^{-n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 2^{-2 n-7} e^{-\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 2^{-n-7} e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 2^{-n-7} e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 2^{-2 n-7} e^{\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 2^{-n-7} 3^{-n-1} e^{\frac{6 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(2^(-7 - 2*n)*3^(-1 - n)*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(2^n*b*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b] - 3^(1 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b] - 2^n*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] + 2^n*3^(1 + n)*b*E^((8*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*b*E^((10*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b] + 2^n*E^((6*a)/b)*(2^(3 + n)*3^(1 + n)*(a + b*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])^2/b^2))^n - b*E^((6*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b])))/(b*c^3*E^((6*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^n)","A",0
425,1,500,578,2.19165,"\int x \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]","-\frac{d^2 15^{-n-1} e^{-\frac{5 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-3 n} \left(\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \left(-3^{n+1} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3\ 5^{n+1} e^{\frac{2 a}{b}} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-2\ 15^{n+1} e^{\frac{4 a}{b}} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)-4\ 5^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{2 n} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+5^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{3 n} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3^{n+1} e^{\frac{10 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{3 n} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)+2\ 15^{n+1} e^{\frac{6 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)}{32 c^2 \sqrt{d-c^2 d x^2}}","-\frac{d 5^{-n-1} e^{-\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 3^{-n} e^{-\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d e^{-\frac{a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{16 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d e^{a/b} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{16 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 3^{-n} e^{\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 5^{-n-1} e^{\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/32*(15^(-1 - n)*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(2*15^(1 + n)*E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, a/b + ArcCosh[c*x]] + (a/b + ArcCosh[c*x])^n*(-(3^(1 + n)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-5*(a + b*ArcCosh[c*x]))/b]) + 3*5^(1 + n)*E^((2*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b] - 2*15^(1 + n)*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)] + 5^(1 + n)*E^((8*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b] - 4*5^(1 + n)*E^((8*a)/b)*(-((a + b*ArcCosh[c*x])/b))^(2*n)*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*E^((10*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (5*(a + b*ArcCosh[c*x]))/b])))/(c^2*E^((5*a)/b)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^(3*n))","A",0
426,1,384,450,2.2098039,"\int \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]","\frac{d^2 4^{-n-3} e^{-\frac{4 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-2 n} \left(3\ 2^{2 n+3} e^{\frac{4 a}{b}} \left(a+b \cosh ^{-1}(c x)\right) \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n}+b \left(-2^{n+3}\right) (n+1) e^{\frac{2 a}{b}} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b 2^{n+3} (n+1) e^{\frac{6 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b (n+1) e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{2 n} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b (n+1) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^{2 n} \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{b c (n+1) \sqrt{d-c^2 d x^2}}","-\frac{3 d \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{8 b c (n+1) \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d 2^{-n-3} e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}+\frac{d 2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}",1,"(4^(-3 - n)*d^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(3*2^(3 + 2*n)*E^((4*a)/b)*(a + b*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n) + b*(1 + n)*(a/b + ArcCosh[c*x])^(2*n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b] - 2^(3 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] + 2^(3 + n)*b*E^((6*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b] - b*E^((8*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(2*n)*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b]))/(b*c*E^((4*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n))","A",0
427,0,0,415,0.3021399,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n)/x,x]","\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x} \, dx","d^2 \text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \sqrt{d-c^2 d x^2}},x\right)+\frac{d^2 3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}-\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}+\frac{5 d^2 e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}-\frac{d^2 3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}",0,"Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n)/x, x]","A",-1
428,0,0,292,0.5924566,"\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n)/x^2,x]","\int \frac{\left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x^2} \, dx","d^2 \text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \sqrt{d-c^2 d x^2}},x\right)-\frac{3 c d^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{2 b (n+1) \sqrt{d-c^2 d x^2}}+\frac{c d^2 2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}-\frac{c d^2 2^{-n-3} e^{\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}",0,"Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n)/x^2, x]","A",-1
429,1,677,870,7.4052017,"\int x^2 \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[x^2*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n,x]","\frac{d^3 2^{-3 n-11} 3^{-n-1} e^{-\frac{8 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(e^{\frac{8 a}{b}} \left(5 a 2^{3 n+4} 3^{n+1} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n+5 b 2^{3 n+4} 3^{n+1} \cosh ^{-1}(c x) \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n+b 3^{n+1} 4^{n+2} (n+1) e^{\frac{2 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b 2^{n+3} 3^{n+1} (n+1) e^{\frac{4 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b 4^{n+2} e^{\frac{6 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b 4^{n+2} n e^{\frac{6 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b 3^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b 3^{n+1} n e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)+b \left(-3^{n+1}\right) (n+1) \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b 4^{n+2} (n+1) e^{\frac{2 a}{b}} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b 2^{n+3} 3^{n+1} (n+1) e^{\frac{4 a}{b}} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b 3^{n+1} 4^{n+2} (n+1) e^{\frac{6 a}{b}} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{b c^3 (n+1) \sqrt{d-c^2 d x^2}}","\frac{2^{-3 n-11} d^2 e^{-\frac{8 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n}}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-n-7} 3^{-n-1} d^2 e^{-\frac{6 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n}}{c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2^{-2 (n+4)} d^2 e^{-\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n}}{c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2^{-n-7} d^2 e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n}}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{128 b c^3 (n+1) \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-n-7} d^2 e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-2 (n+4)} d^2 e^{\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2^{-n-7} 3^{-n-1} d^2 e^{\frac{6 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2^{-3 n-11} d^2 e^{\frac{8 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{8 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"(2^(-11 - 3*n)*3^(-1 - n)*d^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(-(3^(1 + n)*b*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcCosh[c*x]))/b]) + 4^(2 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b] - 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b] - 3^(1 + n)*4^(2 + n)*b*E^((6*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] + E^((8*a)/b)*(5*2^(4 + 3*n)*3^(1 + n)*a*(-((a + b*ArcCosh[c*x])^2/b^2))^n + 5*2^(4 + 3*n)*3^(1 + n)*b*ArcCosh[c*x]*(-((a + b*ArcCosh[c*x])^2/b^2))^n + 3^(1 + n)*4^(2 + n)*b*E^((2*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b] + 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b] - 4^(2 + n)*b*E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b] - 4^(2 + n)*b*E^((6*a)/b)*n*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*n*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcCosh[c*x]))/b])))/(b*c^3*E^((8*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^n)","A",0
430,1,633,793,3.9762203,"\int x \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n,x]","\frac{d^3 5^{-n} 21^{-n-1} e^{-\frac{7 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-3 n} \left(\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \left(e^{\frac{2 a}{b}} \left(21^{n+1} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-9\ 5^n 7^{n+1} e^{\frac{2 a}{b}} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+105^{n+1} e^{\frac{4 a}{b}} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)+16\ 5^n 7^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{2 n} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-5^n 7^{n+2} e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{3 n} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-21^{n+1} e^{\frac{10 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{3 n} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3^{n+1} 5^n e^{\frac{12 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{3 n} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)-3^{n+1} 5^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)-105^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)}{128 c^2 \sqrt{d-c^2 d x^2}}","\frac{d^2 7^{-n-1} e^{-\frac{7 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 5^{-n} e^{-\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d^2 3^{1-n} e^{-\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 d^2 e^{a/b} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 3^{1-n} e^{\frac{3 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{d^2 5^{-n} e^{\frac{5 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 7^{-n-1} e^{\frac{7 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{7 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c x-1} \sqrt{c x+1}}",1,"(21^(-1 - n)*d^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(-(105^(1 + n)*E^((8*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, a/b + ArcCosh[c*x]]) + (a/b + ArcCosh[c*x])^n*(-(3^(1 + n)*5^n*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-7*(a + b*ArcCosh[c*x]))/b]) + E^((2*a)/b)*(21^(1 + n)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-5*(a + b*ArcCosh[c*x]))/b] - 9*5^n*7^(1 + n)*E^((2*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b] + 105^(1 + n)*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)] - 5^n*7^(2 + n)*E^((8*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b] + 16*5^n*7^(1 + n)*E^((8*a)/b)*(-((a + b*ArcCosh[c*x])/b))^(2*n)*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b] - 21^(1 + n)*E^((10*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (5*(a + b*ArcCosh[c*x]))/b] + 3^(1 + n)*5^n*E^((12*a)/b)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(3*n)*Gamma[1 + n, (7*(a + b*ArcCosh[c*x]))/b]))))/(128*5^n*c^2*E^((7*a)/b)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^(3*n))","A",0
431,1,538,674,5.4273264,"\int \left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n,x]","\frac{d^3 2^{-2 n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-2 n} \left(-5 b 2^n 3^{n+2} (n+1) e^{\frac{4 a}{b}} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+5 b 2^n 3^{n+2} (n+1) e^{\frac{8 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+2^n e^{\frac{6 a}{b}} \left(5\ 2^{n+3} 3^{n+1} \left(a+b \cosh ^{-1}(c x)\right) \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n}+b (n+1) e^{\frac{6 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{2 n} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)-b 3^{n+2} (n+1) e^{\frac{10 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{2 n} \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b 2^n (n+1) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^{2 n} \Gamma \left(n+1,-\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b 3^{n+2} (n+1) e^{\frac{2 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^{2 n} \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{b c (n+1) \sqrt{d-c^2 d x^2}}","-\frac{5 d^2 \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{16 b c (n+1) \sqrt{c x-1} \sqrt{c x+1}}+\frac{d^2 2^{-n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 d^2 2^{-2 n-7} e^{-\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}+\frac{15 d^2 2^{-n-7} e^{-\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{15 d^2 2^{-n-7} e^{\frac{2 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 d^2 2^{-2 n-7} e^{\frac{4 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}-\frac{d^2 2^{-n-7} 3^{-n-1} e^{\frac{6 a}{b}} \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{6 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c x-1} \sqrt{c x+1}}",1,"(2^(-7 - 2*n)*3^(-1 - n)*d^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(-(2^n*b*(1 + n)*(a/b + ArcCosh[c*x])^(2*n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (-6*(a + b*ArcCosh[c*x]))/b]) + 3^(2 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^(2*n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b] - 5*2^n*3^(2 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] + 5*2^n*3^(2 + n)*b*E^((8*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b] - 3^(2 + n)*b*E^((10*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(2*n)*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b] + 2^n*E^((6*a)/b)*(5*2^(3 + n)*3^(1 + n)*(a + b*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n) + b*E^((6*a)/b)*(1 + n)*(a/b + ArcCosh[c*x])^n*(-((a + b*ArcCosh[c*x])/b))^(2*n)*Gamma[1 + n, (6*(a + b*ArcCosh[c*x]))/b])))/(b*c*E^((6*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n))","A",0
432,0,0,805,0.3353089,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x,x]","\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x} \, dx","-\frac{5^{-n-1} d^3 e^{-\frac{5 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n}}{32 \sqrt{d-c^2 d x^2}}-\frac{5\ 3^{-n-1} d^3 e^{-\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n}}{32 \sqrt{d-c^2 d x^2}}+\frac{3^{-n} d^3 e^{-\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n}}{8 \sqrt{d-c^2 d x^2}}-\frac{11 d^3 e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right) \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n}}{16 \sqrt{d-c^2 d x^2}}+\frac{11 d^3 e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{16 \sqrt{d-c^2 d x^2}}+\frac{5\ 3^{-n-1} d^3 e^{\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 \sqrt{d-c^2 d x^2}}-\frac{3^{-n} d^3 e^{\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{d-c^2 d x^2}}+\frac{5^{-n-1} d^3 e^{\frac{5 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{32 \sqrt{d-c^2 d x^2}}+d^3 \text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \sqrt{d-c^2 d x^2}},x\right)",0,"Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x, x]","A",-1
433,0,0,486,0.6227165,"\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x^2} \, dx","Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x^2,x]","\int \frac{\left(d-c^2 d x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^n}{x^2} \, dx","d^3 \text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \sqrt{d-c^2 d x^2}},x\right)-\frac{15 c d^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{8 b (n+1) \sqrt{d-c^2 d x^2}}-\frac{c d^3 2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}+\frac{c d^3 2^{-n-2} e^{-\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}-\frac{c d^3 2^{-n-2} e^{\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}+\frac{c d^3 2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{\sqrt{d-c^2 d x^2}}",0,"Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x^2, x]","A",-1
434,1,292,323,1.3141917,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{1-c^2 x^2}} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x])^n)/Sqrt[1 - c^2*x^2],x]","\frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{1-c^2 x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-2 n} \left(3^{n+2} e^{\frac{4 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right)-\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \left(\left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3^{n+2} e^{\frac{2 a}{b}} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)-e^{\frac{6 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{2 n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)\right)}{8 c^4 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 c^4 \sqrt{1-c x}}+\frac{3 e^{-\frac{a}{b}} \sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 c^4 \sqrt{1-c x}}-\frac{3 e^{a/b} \sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 c^4 \sqrt{1-c x}}-\frac{3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 c^4 \sqrt{1-c x}}",1,"(3^(-1 - n)*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^n*(3^(2 + n)*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, a/b + ArcCosh[c*x]] - (a/b + ArcCosh[c*x])^n*((-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b] + 3^(2 + n)*E^((2*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)] - E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^(2*n)*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])))/(8*c^4*E^((3*a)/b)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n))","A",0
435,1,212,211,0.8142034,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{1-c^2 x^2}} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x])^n)/Sqrt[1 - c^2*x^2],x]","\frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(2^{n+2} e^{\frac{2 a}{b}} \left(a+b \cosh ^{-1}(c x)\right) \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n+b (n+1) \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b (n+1) e^{\frac{4 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{b c^3 (n+1) \sqrt{1-c^2 x^2}}","\frac{\sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{2 b c^3 (n+1) \sqrt{1-c x}}+\frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c x}}-\frac{2^{-n-3} e^{\frac{2 a}{b}} \sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{1-c x}}",1,"(2^(-3 - n)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(2^(2 + n)*E^((2*a)/b)*(a + b*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])^2/b^2))^n + b*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] - b*E^((4*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b]))/(b*c^3*E^((2*a)/b)*(1 + n)*Sqrt[1 - c^2*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^n)","A",0
436,1,154,154,0.2525747,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{1-c^2 x^2}} \, dx","Integrate[(x*(a + b*ArcCosh[c*x])^n)/Sqrt[1 - c^2*x^2],x]","-\frac{e^{-\frac{a}{b}} \sqrt{-((c x-1) (c x+1))} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)-e^{\frac{2 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)}{2 c^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1)}","\frac{e^{-\frac{a}{b}} \sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 c^2 \sqrt{1-c x}}-\frac{e^{a/b} \sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 c^2 \sqrt{1-c x}}",1,"-1/2*(Sqrt[-((-1 + c*x)*(1 + c*x))]*(a + b*ArcCosh[c*x])^n*(-(E^((2*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, a/b + ArcCosh[c*x]]) + (a/b + ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)]))/(c^2*E^(a/b)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-((a + b*ArcCosh[c*x])^2/b^2))^n)","A",0
437,1,56,43,0.0410786,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{1-c^2 x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^n/Sqrt[1 - c^2*x^2],x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{b c (n+1) \sqrt{1-c^2 x^2}}","\frac{\sqrt{c x-1} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{b c (n+1) \sqrt{1-c x}}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(b*c*(1 + n)*Sqrt[1 - c^2*x^2])","A",1
438,0,0,31,2.6635599,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \sqrt{1-c^2 x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^n/(x*Sqrt[1 - c^2*x^2]),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \sqrt{1-c^2 x^2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \sqrt{1-c^2 x^2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^n/(x*Sqrt[1 - c^2*x^2]), x]","A",-1
439,0,0,31,1.3957841,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \sqrt{1-c^2 x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[1 - c^2*x^2]),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \sqrt{1-c^2 x^2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \sqrt{1-c^2 x^2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[1 - c^2*x^2]), x]","A",-1
440,1,291,379,1.102576,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x])^n)/Sqrt[d - c^2*d*x^2],x]","-\frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-2 n} \left(3^{n+2} e^{\frac{4 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right)-\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \left(\left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3^{n+2} e^{\frac{2 a}{b}} \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)-e^{\frac{6 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{2 n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)\right)}{8 c^4 \sqrt{d-c^2 d x^2}}","\frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 c^4 \sqrt{d-c^2 d x^2}}+\frac{3 e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{3 e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{8 c^4 \sqrt{d-c^2 d x^2}}",1,"-1/8*(3^(-1 - n)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(3^(2 + n)*E^((4*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, a/b + ArcCosh[c*x]] - (a/b + ArcCosh[c*x])^n*((-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b] + 3^(2 + n)*E^((2*a)/b)*(-((a + b*ArcCosh[c*x])^2/b^2))^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)] - E^((6*a)/b)*(-((a + b*ArcCosh[c*x])/b))^(2*n)*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])))/(c^4*E^((3*a)/b)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^(2*n))","A",0
441,1,213,253,0.7736715,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x])^n)/Sqrt[d - c^2*d*x^2],x]","\frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(2^{n+2} e^{\frac{2 a}{b}} \left(a+b \cosh ^{-1}(c x)\right) \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^n+b (n+1) \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)-b (n+1) e^{\frac{4 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{b c^3 (n+1) \sqrt{d-c^2 d x^2}}","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{2 b c^3 (n+1) \sqrt{d-c^2 d x^2}}+\frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{d-c^2 d x^2}}-\frac{2^{-n-3} e^{\frac{2 a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{d-c^2 d x^2}}",1,"(2^(-3 - n)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(2^(2 + n)*E^((2*a)/b)*(a + b*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])^2/b^2))^n + b*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b] - b*E^((4*a)/b)*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcCosh[c*x]))/b]))/(b*c^3*E^((2*a)/b)*(1 + n)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^n)","A",0
442,1,153,182,0.2311669,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(x*(a + b*ArcCosh[c*x])^n)/Sqrt[d - c^2*d*x^2],x]","\frac{e^{-\frac{a}{b}} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)-e^{\frac{2 a}{b}} \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)}{2 c^2 \sqrt{-d (c x-1) (c x+1)}}","\frac{e^{-\frac{a}{b}} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(-\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 c^2 \sqrt{d-c^2 d x^2}}-\frac{e^{a/b} \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^n \left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 c^2 \sqrt{d-c^2 d x^2}}",1,"(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(-(E^((2*a)/b)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, a/b + ArcCosh[c*x]]) + (a/b + ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)]))/(2*c^2*E^(a/b)*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-((a + b*ArcCosh[c*x])^2/b^2))^n)","A",0
443,1,57,57,0.04543,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^n/Sqrt[d - c^2*d*x^2],x]","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{b c (n+1) \sqrt{d-c^2 d x^2}}","\frac{\sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)^{n+1}}{b c (n+1) \sqrt{d-c^2 d x^2}}",1,"(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(b*c*(1 + n)*Sqrt[d - c^2*d*x^2])","A",1
444,0,0,32,0.3315602,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^n/(x*Sqrt[d - c^2*d*x^2]),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \sqrt{d-c^2 d x^2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \sqrt{d-c^2 d x^2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^n/(x*Sqrt[d - c^2*d*x^2]), x]","A",-1
445,0,0,32,0.35139,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \sqrt{d-c^2 d x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \sqrt{d-c^2 d x^2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \sqrt{d-c^2 d x^2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[d - c^2*d*x^2]), x]","A",-1
446,0,0,32,0.6563439,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2),x]","\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}},x\right)",0,"Integrate[(x^2*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2), x]","A",-1
447,0,0,30,0.6422315,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2),x]","\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}},x\right)",0,"Integrate[(x*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2), x]","A",-1
448,0,0,29,0.0817404,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^(3/2),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^n/(d - c^2*d*x^2)^(3/2), x]","A",-1
449,0,0,32,0.5954185,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^n/(x*(d - c^2*d*x^2)^(3/2)),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \left(d-c^2 d x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x \left(d-c^2 d x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^n/(x*(d - c^2*d*x^2)^(3/2)), x]","A",-1
450,0,0,32,0.5564934,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^n/(x^2*(d - c^2*d*x^2)^(3/2)),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \left(d-c^2 d x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^n}{x^2 \left(d-c^2 d x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^n/(x^2*(d - c^2*d*x^2)^(3/2)), x]","A",-1
451,0,0,33,0.4737507,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{1-c^2 x^2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/Sqrt[1 - c^2*x^2],x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{1-c^2 x^2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{1-c^2 x^2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/Sqrt[1 - c^2*x^2], x]","A",-1
452,0,0,32,1.224029,"\int (f x)^m \left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^n,x]","\int (f x)^m \left(d-c^2 d x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","\text{Int}\left(\left(d-c^2 d x^2\right)^2 (f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n,x\right)",0,"Integrate[(f*x)^m*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^n, x]","A",-1
453,0,0,30,0.6234417,"\int (f x)^m \left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x])^n,x]","\int (f x)^m \left(d-c^2 d x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","\text{Int}\left(\left(d-c^2 d x^2\right) (f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n,x\right)",0,"Integrate[(f*x)^m*(d - c^2*d*x^2)*(a + b*ArcCosh[c*x])^n, x]","A",-1
454,0,0,19,0.0207666,"\int (f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[(f*x)^m*(a + b*ArcCosh[c*x])^n,x]","\int (f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","\text{Int}\left((f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n,x\right)",0,"Integrate[(f*x)^m*(a + b*ArcCosh[c*x])^n, x]","A",-1
455,0,0,32,0.6482154,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{d-c^2 d x^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2),x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{d-c^2 d x^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{d-c^2 d x^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2), x]","A",-1
456,0,0,32,0.983539,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^2,x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^2, x]","A",-1
457,0,0,34,0.6733919,"\int (f x)^m \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n,x]","\int (f x)^m \left(d-c^2 d x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","\text{Int}\left(\left(d-c^2 d x^2\right)^{3/2} (f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n,x\right)",0,"Integrate[(f*x)^m*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^n, x]","A",-1
458,0,0,34,0.1263821,"\int (f x)^m \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","Integrate[(f*x)^m*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n,x]","\int (f x)^m \sqrt{d-c^2 d x^2} \left(a+b \cosh ^{-1}(c x)\right)^n \, dx","\text{Int}\left(\sqrt{d-c^2 d x^2} (f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n,x\right)",0,"Integrate[(f*x)^m*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n, x]","A",-1
459,0,0,34,0.4624862,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{d-c^2 d x^2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/Sqrt[d - c^2*d*x^2],x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{d-c^2 d x^2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\sqrt{d-c^2 d x^2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/Sqrt[d - c^2*d*x^2], x]","A",-1
460,0,0,34,0.7194218,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2),x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)^n}{\left(d-c^2 d x^2\right)^{3/2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x])^n)/(d - c^2*d*x^2)^(3/2), x]","A",-1
461,1,122,177,0.1120741,"\int x^4 \left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]","\frac{1}{35} a x^5 \left(7 d+5 e x^2\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(3 c^6 \left(49 d x^4+25 e x^6\right)+2 c^4 \left(98 d x^2+45 e x^4\right)+8 c^2 \left(49 d+15 e x^2\right)+240 e\right)}{3675 c^7}+\frac{1}{35} b x^5 \cosh ^{-1}(c x) \left(7 d+5 e x^2\right)","\frac{1}{5} d x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{7} e x^7 \left(a+b \cosh ^{-1}(c x)\right)-\frac{8 b \sqrt{c x-1} \sqrt{c x+1} \left(49 c^2 d+30 e\right)}{3675 c^7}-\frac{4 b x^2 \sqrt{c x-1} \sqrt{c x+1} \left(49 c^2 d+30 e\right)}{3675 c^5}-\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1} \left(49 c^2 d+30 e\right)}{1225 c^3}-\frac{b e x^6 \sqrt{c x-1} \sqrt{c x+1}}{49 c}",1,"(a*x^5*(7*d + 5*e*x^2))/35 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e + 8*c^2*(49*d + 15*e*x^2) + 2*c^4*(98*d*x^2 + 45*e*x^4) + 3*c^6*(49*d*x^4 + 25*e*x^6)))/(3675*c^7) + (b*x^5*(7*d + 5*e*x^2)*ArcCosh[c*x])/35","A",1
462,1,140,161,0.1961193,"\int x^3 \left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]","\frac{24 a c^6 x^4 \left(3 d+2 e x^2\right)+24 b c^6 x^4 \cosh ^{-1}(c x) \left(3 d+2 e x^2\right)-6 b \left(9 c^2 d+5 e\right) \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)-b c x \sqrt{c x-1} \sqrt{c x+1} \left(2 c^4 \left(9 d x^2+4 e x^4\right)+c^2 \left(27 d+10 e x^2\right)+15 e\right)}{288 c^6}","\frac{1}{4} d x^4 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b \left(9 c^2 d+5 e\right) \cosh ^{-1}(c x)}{96 c^6}-\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left(9 c^2 d+5 e\right)}{96 c^5}-\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1} \left(9 c^2 d+5 e\right)}{144 c^3}-\frac{b e x^5 \sqrt{c x-1} \sqrt{c x+1}}{36 c}",1,"(24*a*c^6*x^4*(3*d + 2*e*x^2) - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15*e + c^2*(27*d + 10*e*x^2) + 2*c^4*(9*d*x^2 + 4*e*x^4)) + 24*b*c^6*x^4*(3*d + 2*e*x^2)*ArcCosh[c*x] - 6*b*(9*c^2*d + 5*e)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(288*c^6)","A",0
463,1,101,138,0.1045024,"\int x^2 \left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]","\frac{1}{225} \left(15 a x^3 \left(5 d+3 e x^2\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(c^4 \left(25 d x^2+9 e x^4\right)+2 c^2 \left(25 d+6 e x^2\right)+24 e\right)}{c^5}+15 b x^3 \cosh ^{-1}(c x) \left(5 d+3 e x^2\right)\right)","\frac{1}{3} d x^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \cosh ^{-1}(c x)\right)-\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \left(25 c^2 d+12 e\right)}{225 c^5}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \left(25 c^2 d+12 e\right)}{225 c^3}-\frac{b e x^4 \sqrt{c x-1} \sqrt{c x+1}}{25 c}",1,"(15*a*x^3*(5*d + 3*e*x^2) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(24*e + 2*c^2*(25*d + 6*e*x^2) + c^4*(25*d*x^2 + 9*e*x^4)))/c^5 + 15*b*x^3*(5*d + 3*e*x^2)*ArcCosh[c*x])/225","A",1
464,1,120,122,0.1355782,"\int x \left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]","\frac{c x \left(8 a c^3 x \left(2 d+e x^2\right)-b \sqrt{c x-1} \sqrt{c x+1} \left(2 c^2 \left(4 d+e x^2\right)+3 e\right)\right)+8 b c^4 x^2 \cosh ^{-1}(c x) \left(2 d+e x^2\right)-2 b \left(8 c^2 d+3 e\right) \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)}{32 c^4}","\frac{1}{2} d x^2 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{4} e x^4 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b \left(8 c^2 d+3 e\right) \cosh ^{-1}(c x)}{32 c^4}-\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left(8 c^2 d+3 e\right)}{32 c^3}-\frac{b e x^3 \sqrt{c x-1} \sqrt{c x+1}}{16 c}",1,"(c*x*(8*a*c^3*x*(2*d + e*x^2) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3*e + 2*c^2*(4*d + e*x^2))) + 8*b*c^4*x^2*(2*d + e*x^2)*ArcCosh[c*x] - 2*b*(8*c^2*d + 3*e)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(32*c^4)","A",0
465,1,76,94,0.0932485,"\int \left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)*(a + b*ArcCosh[c*x]),x]","\frac{1}{9} \left(3 a x \left(3 d+e x^2\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(c^2 \left(9 d+e x^2\right)+2 e\right)}{c^3}+3 b x \cosh ^{-1}(c x) \left(3 d+e x^2\right)\right)","d x \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(9 c^2 d+2 e\right)}{9 c^3}-\frac{b e x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c}",1,"(3*a*x*(3*d + e*x^2) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2*(9*d + e*x^2)))/c^3 + 3*b*x*(3*d + e*x^2)*ArcCosh[c*x])/9","A",1
466,1,119,264,0.2664377,"\int \frac{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCosh[c*x]))/x,x]","\frac{1}{2} \left(2 a d \log (x)+a e x^2-\frac{b e \left(c x \sqrt{c x-1} \sqrt{c x+1}+2 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{2 c^2}-b d \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+b d \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)+b e x^2 \cosh ^{-1}(c x)\right)","d \log (x) \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{2} e x^2 \left(a+b \cosh ^{-1}(c x)\right)-\frac{i b d \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b e \cosh ^{-1}(c x)}{4 c^2}-\frac{b e x \sqrt{c x-1} \sqrt{c x+1}}{4 c}",1,"(a*e*x^2 + b*e*x^2*ArcCosh[c*x] - (b*e*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(2*c^2) + b*d*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) + 2*a*d*Log[x] - b*d*PolyLog[2, -E^(-2*ArcCosh[c*x])])/2","A",0
467,1,105,75,0.1338605,"\int \frac{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCosh[c*x]))/x^2,x]","-\frac{a d}{x}+a e x+\frac{b c d \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d \cosh ^{-1}(c x)}{x}-\frac{b e \sqrt{c x-1} \sqrt{c x+1}}{c}+b e x \cosh ^{-1}(c x)","-\frac{d \left(a+b \cosh ^{-1}(c x)\right)}{x}+e x \left(a+b \cosh ^{-1}(c x)\right)+b c d \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)-\frac{b e \sqrt{c x-1} \sqrt{c x+1}}{c}",1,"-((a*d)/x) + a*e*x - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c - (b*d*ArcCosh[c*x])/x + b*e*x*ArcCosh[c*x] + (b*c*d*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])","A",1
468,1,101,251,0.1421184,"\int \frac{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCosh[c*x]))/x^3,x]","\frac{-a d+2 a e x^2 \log (x)-b \cosh ^{-1}(c x) \left(d-2 e x^2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)+b c d x \sqrt{c x-1} \sqrt{c x+1}-b e x^2 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+b e x^2 \cosh ^{-1}(c x)^2}{2 x^2}","-\frac{d \left(a+b \cosh ^{-1}(c x)\right)}{2 x^2}+e \log (x) \left(a+b \cosh ^{-1}(c x)\right)-\frac{i b e \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{i b e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b e \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d \sqrt{c x-1} \sqrt{c x+1}}{2 x}",1,"(-(a*d) + b*c*d*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + b*e*x^2*ArcCosh[c*x]^2 - b*ArcCosh[c*x]*(d - 2*e*x^2*Log[1 + E^(-2*ArcCosh[c*x])]) + 2*a*e*x^2*Log[x] - b*e*x^2*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*x^2)","A",0
469,1,128,94,0.2662871,"\int \frac{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCosh[c*x]))/x^4,x]","\frac{\frac{-2 a \sqrt{c x-1} \sqrt{c x+1} \left(d+3 e x^2\right)+b c x^3 \sqrt{c^2 x^2-1} \left(c^2 d+6 e\right) \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)+b c d x \left(c^2 x^2-1\right)}{\sqrt{c x-1} \sqrt{c x+1}}-2 b \cosh ^{-1}(c x) \left(d+3 e x^2\right)}{6 x^3}","-\frac{d \left(a+b \cosh ^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \cosh ^{-1}(c x)\right)}{x}+\frac{1}{6} b c \left(c^2 d+6 e\right) \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)+\frac{b c d \sqrt{c x-1} \sqrt{c x+1}}{6 x^2}",1,"(-2*b*(d + 3*e*x^2)*ArcCosh[c*x] + (b*c*d*x*(-1 + c^2*x^2) - 2*a*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + 3*e*x^2) + b*c*(c^2*d + 6*e)*x^3*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(6*x^3)","A",1
470,1,192,319,0.2780971,"\int x^4 \left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{315 a x^5 \left(63 d^2+90 d e x^2+35 e^2 x^4\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(c^8 \left(3969 d^2 x^4+4050 d e x^6+1225 e^2 x^8\right)+4 c^6 \left(1323 d^2 x^2+1215 d e x^4+350 e^2 x^6\right)+24 c^4 \left(441 d^2+270 d e x^2+70 e^2 x^4\right)+160 c^2 e \left(81 d+14 e x^2\right)+4480 e^2\right)}{c^9}+315 b x^5 \cosh ^{-1}(c x) \left(63 d^2+90 d e x^2+35 e^2 x^4\right)}{99225}","\frac{1}{5} d^2 x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{2}{7} d e x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{9} e^2 x^9 \left(a+b \cosh ^{-1}(c x)\right)-\frac{2 b e \left(1-c^2 x^2\right)^4 \left(9 c^2 d+14 e\right)}{441 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left(1-c^2 x^2\right)^5}{81 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left(1-c^2 x^2\right)^3 \left(21 c^4 d^2+90 c^2 d e+70 e^2\right)}{525 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b \left(1-c^2 x^2\right)^2 \left(63 c^4 d^2+135 c^2 d e+70 e^2\right)}{945 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left(1-c^2 x^2\right) \left(63 c^4 d^2+90 c^2 d e+35 e^2\right)}{315 c^9 \sqrt{c x-1} \sqrt{c x+1}}",1,"(315*a*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4480*e^2 + 160*c^2*e*(81*d + 14*e*x^2) + 24*c^4*(441*d^2 + 270*d*e*x^2 + 70*e^2*x^4) + 4*c^6*(1323*d^2*x^2 + 1215*d*e*x^4 + 350*e^2*x^6) + c^8*(3969*d^2*x^4 + 4050*d*e*x^6 + 1225*e^2*x^8)))/c^9 + 315*b*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4)*ArcCosh[c*x])/99225","A",1
471,1,214,341,0.3542778,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{384 a c^8 x^4 \left(6 d^2+8 d e x^2+3 e^2 x^4\right)+384 b c^8 x^4 \cosh ^{-1}(c x) \left(6 d^2+8 d e x^2+3 e^2 x^4\right)-6 b \left(288 c^4 d^2+320 c^2 d e+105 e^2\right) \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)-b c x \sqrt{c x-1} \sqrt{c x+1} \left(16 c^6 \left(36 d^2 x^2+32 d e x^4+9 e^2 x^6\right)+8 c^4 \left(108 d^2+80 d e x^2+21 e^2 x^4\right)+30 c^2 e \left(32 d+7 e x^2\right)+315 e^2\right)}{9216 c^8}","\frac{1}{4} d^2 x^4 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} d e x^6 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{8} e^2 x^8 \left(a+b \cosh ^{-1}(c x)\right)+\frac{b e^2 x^7 \left(1-c^2 x^2\right)}{64 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e x^5 \left(1-c^2 x^2\right) \left(64 c^2 d+21 e\right)}{1152 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left(288 c^4 d^2+320 c^2 d e+105 e^2\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{3072 c^8 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left(1-c^2 x^2\right) \left(288 c^4 d^2+320 c^2 d e+105 e^2\right)}{3072 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x^3 \left(1-c^2 x^2\right) \left(288 c^4 d^2+320 c^2 d e+105 e^2\right)}{4608 c^5 \sqrt{c x-1} \sqrt{c x+1}}",1,"(384*a*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) - b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(315*e^2 + 30*c^2*e*(32*d + 7*e*x^2) + 8*c^4*(108*d^2 + 80*d*e*x^2 + 21*e^2*x^4) + 16*c^6*(36*d^2*x^2 + 32*d*e*x^4 + 9*e^2*x^6)) + 384*b*c^8*x^4*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCosh[c*x] - 6*b*(288*c^4*d^2 + 320*c^2*d*e + 105*e^2)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(9216*c^8)","A",0
472,1,163,260,0.2567899,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{105 a x^3 \left(35 d^2+42 d e x^2+15 e^2 x^4\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(c^6 \left(1225 d^2 x^2+882 d e x^4+225 e^2 x^6\right)+2 c^4 \left(1225 d^2+588 d e x^2+135 e^2 x^4\right)+24 c^2 e \left(98 d+15 e x^2\right)+720 e^2\right)}{c^7}+105 b x^3 \cosh ^{-1}(c x) \left(35 d^2+42 d e x^2+15 e^2 x^4\right)}{11025}","\frac{1}{3} d^2 x^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{2}{5} d e x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{7} e^2 x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{b e \left(1-c^2 x^2\right)^3 \left(14 c^2 d+15 e\right)}{175 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \left(1-c^2 x^2\right)^4}{49 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \left(1-c^2 x^2\right)^2 \left(35 c^4 d^2+84 c^2 d e+45 e^2\right)}{315 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left(1-c^2 x^2\right) \left(35 c^4 d^2+42 c^2 d e+15 e^2\right)}{105 c^7 \sqrt{c x-1} \sqrt{c x+1}}",1,"(105*a*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(720*e^2 + 24*c^2*e*(98*d + 15*e*x^2) + 2*c^4*(1225*d^2 + 588*d*e*x^2 + 135*e^2*x^4) + c^6*(1225*d^2*x^2 + 882*d*e*x^4 + 225*e^2*x^6)))/c^7 + 105*b*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcCosh[c*x])/11025","A",1
473,1,183,269,0.3442925,"\int x \left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{c x \left(48 a c^5 x \left(3 d^2+3 d e x^2+e^2 x^4\right)-b \sqrt{c x-1} \sqrt{c x+1} \left(4 c^4 \left(18 d^2+9 d e x^2+2 e^2 x^4\right)+2 c^2 e \left(27 d+5 e x^2\right)+15 e^2\right)\right)+48 b c^6 x^2 \cosh ^{-1}(c x) \left(3 d^2+3 d e x^2+e^2 x^4\right)-6 b \left(24 c^4 d^2+18 c^2 d e+5 e^2\right) \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)}{288 c^6}","\frac{\left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{6 e}+\frac{b x \left(1-c^2 x^2\right) \left(d+e x^2\right)^2}{36 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b x \left(1-c^2 x^2\right) \left(2 c^2 d+e\right) \left(d+e x^2\right)}{144 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left(2 c^2 d+e\right) \left(8 c^4 d^2+8 c^2 d e+5 e^2\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{96 c^6 e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left(1-c^2 x^2\right) \left(44 c^4 d^2+44 c^2 d e+15 e^2\right)}{288 c^5 \sqrt{c x-1} \sqrt{c x+1}}",1,"(c*x*(48*a*c^5*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15*e^2 + 2*c^2*e*(27*d + 5*e*x^2) + 4*c^4*(18*d^2 + 9*d*e*x^2 + 2*e^2*x^4))) + 48*b*c^6*x^2*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcCosh[c*x] - 6*b*(24*c^4*d^2 + 18*c^2*d*e + 5*e^2)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(288*c^6)","A",0
474,1,130,196,0.2092511,"\int \left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]","\frac{1}{225} \left(15 a x \left(15 d^2+10 d e x^2+3 e^2 x^4\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(c^4 \left(225 d^2+50 d e x^2+9 e^2 x^4\right)+4 c^2 e \left(25 d+3 e x^2\right)+24 e^2\right)}{c^5}+15 b x \cosh ^{-1}(c x) \left(15 d^2+10 d e x^2+3 e^2 x^4\right)\right)","d^2 x \left(a+b \cosh ^{-1}(c x)\right)+\frac{2}{3} d e x^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 \left(a+b \cosh ^{-1}(c x)\right)-\frac{2 b e \left(1-c^2 x^2\right)^2 \left(5 c^2 d+3 e\right)}{45 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left(1-c^2 x^2\right)^3}{25 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left(1-c^2 x^2\right) \left(15 c^4 d^2+10 c^2 d e+3 e^2\right)}{15 c^5 \sqrt{c x-1} \sqrt{c x+1}}",1,"(15*a*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))/c^5 + 15*b*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcCosh[c*x])/225","A",1
475,1,217,342,0.4439311,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCosh[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 \left(c x \sqrt{c x-1} \sqrt{c x+1}+2 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{2 c^2}-\frac{b e^2 \left(c x \sqrt{c x-1} \sqrt{c x+1} \left(2 c^2 x^2+3\right)+6 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{32 c^4}-\frac{1}{2} b d^2 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+\frac{1}{2} b d^2 \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)+b d e x^2 \cosh ^{-1}(c x)+\frac{1}{4} b e^2 x^4 \cosh ^{-1}(c x)","d^2 \log (x) \left(a+b \cosh ^{-1}(c x)\right)+d e x^2 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{4} e^2 x^4 \left(a+b \cosh ^{-1}(c x)\right)-\frac{i b d^2 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d^2 \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b e \left(16 c^2 d+3 e\right) \cosh ^{-1}(c x)}{32 c^4}-\frac{b e x \sqrt{c x-1} \sqrt{c x+1} \left(16 c^2 d+3 e\right)}{32 c^3}-\frac{b e^2 x^3 \sqrt{c x-1} \sqrt{c x+1}}{16 c}",1,"a*d*e*x^2 + (a*e^2*x^4)/4 + b*d*e*x^2*ArcCosh[c*x] + (b*e^2*x^4*ArcCosh[c*x])/4 - (b*d*e*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(2*c^2) - (b*e^2*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 + 2*c^2*x^2) + 6*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(32*c^4) + (b*d^2*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]))/2 + a*d^2*Log[x] - (b*d^2*PolyLog[2, -E^(-2*ArcCosh[c*x])])/2","A",0
476,1,128,160,0.2634581,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCosh[c*x]))/x^2,x]","\frac{1}{3} \left(-\frac{3 a d^2}{x}+6 a d e x+a e^2 x^3-\frac{b e \sqrt{c x-1} \sqrt{c x+1} \left(c^2 \left(18 d+e x^2\right)+2 e\right)}{3 c^3}+\frac{b \cosh ^{-1}(c x) \left(-3 d^2+6 d e x^2+e^2 x^4\right)}{x}-3 b c d^2 \tan ^{-1}\left(\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right)\right)","-\frac{d^2 \left(a+b \cosh ^{-1}(c x)\right)}{x}+2 d e x \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} e^2 x^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{b e \left(1-c^2 x^2\right) \left(6 c^2 d+e\right)}{3 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \left(1-c^2 x^2\right)^2}{9 c^3 \sqrt{c x-1} \sqrt{c x+1}}+b c d^2 \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)",1,"((-3*a*d^2)/x + 6*a*d*e*x + a*e^2*x^3 - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2*(18*d + e*x^2)))/(3*c^3) + (b*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*ArcCosh[c*x])/x - 3*b*c*d^2*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])])/3","A",1
477,1,173,321,0.4545491,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCosh[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 e^2 \left(2 c^2 x^2 \cosh ^{-1}(c x)-c x \sqrt{c x-1} \sqrt{c x+1}-2 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{c^2}+\frac{2 b d^2 \left(c x \sqrt{c x-1} \sqrt{c x+1}-\cosh ^{-1}(c x)\right)}{x^2}+4 b d e \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)-\text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)\right)\right)","-\frac{d^2 \left(a+b \cosh ^{-1}(c x)\right)}{2 x^2}+2 d e \log (x) \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{2} e^2 x^2 \left(a+b \cosh ^{-1}(c x)\right)-\frac{i b d e \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b d e \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \cosh ^{-1}(c x)}{4 c^2}+\frac{b c d^2 \sqrt{c x-1} \sqrt{c x+1}}{2 x}-\frac{b e^2 x \sqrt{c x-1} \sqrt{c x+1}}{4 c}",1,"((-2*a*d^2)/x^2 + 2*a*e^2*x^2 + (2*b*d^2*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - ArcCosh[c*x]))/x^2 + (b*e^2*(-(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + 2*c^2*x^2*ArcCosh[c*x] - 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/c^2 + 8*a*d*e*Log[x] + 4*b*d*e*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]))/4","A",0
478,1,133,184,0.26521,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCosh[c*x]))/x^4,x]","-\frac{a d^2}{3 x^3}-\frac{2 a d e}{x}+a e^2 x-\frac{1}{6} b c d \left(c^2 d+12 e\right) \tan ^{-1}\left(\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right)+b \sqrt{c x-1} \sqrt{c x+1} \left(\frac{c d^2}{6 x^2}-\frac{e^2}{c}\right)-\frac{b \cosh ^{-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 \cosh ^{-1}(c x)\right)}{3 x^3}-\frac{2 d e \left(a+b \cosh ^{-1}(c x)\right)}{x}+e^2 x \left(a+b \cosh ^{-1}(c x)\right)-\frac{b c d^2 \left(1-c^2 x^2\right)}{6 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d \sqrt{c^2 x^2-1} \left(c^2 d+12 e\right) \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{6 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left(1-c^2 x^2\right)}{c \sqrt{c x-1} \sqrt{c x+1}}",1,"-1/3*(a*d^2)/x^3 - (2*a*d*e)/x + a*e^2*x + b*(-(e^2/c) + (c*d^2)/(6*x^2))*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - (b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcCosh[c*x])/(3*x^3) - (b*c*d*(c^2*d + 12*e)*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])])/6","A",1
479,1,276,435,0.4116628,"\int x^4 \left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]","\frac{3465 a x^5 \left(231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(c^{10} x^4 \left(160083 d^3+245025 d^2 e x^2+148225 d e^2 x^4+33075 e^3 x^6\right)+2 c^8 \left(106722 d^3 x^2+147015 d^2 e x^4+84700 d e^2 x^6+18375 e^3 x^8\right)+24 c^6 \left(17787 d^3+16335 d^2 e x^2+8470 d e^2 x^4+1750 e^3 x^6\right)+80 c^4 e \left(9801 d^2+3388 d e x^2+630 e^2 x^4\right)+4480 c^2 e^2 \left(121 d+15 e x^2\right)+134400 e^3\right)}{c^{11}}+3465 b x^5 \cosh ^{-1}(c x) \left(231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right)}{4002075}","\frac{1}{5} d^3 x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{7} d^2 e x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} d e^2 x^9 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{11} e^3 x^{11} \left(a+b \cosh ^{-1}(c x)\right)+\frac{b e^2 \left(1-c^2 x^2\right)^5 \left(11 c^2 d+15 e\right)}{297 c^{11} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^3 \left(1-c^2 x^2\right)^6}{121 c^{11} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e \left(1-c^2 x^2\right)^4 \left(99 c^4 d^2+308 c^2 d e+210 e^2\right)}{1617 c^{11} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left(1-c^2 x^2\right)^3 \left(77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right)}{1925 c^{11} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \left(1-c^2 x^2\right)^2 \left(462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right)}{3465 c^{11} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left(1-c^2 x^2\right) \left(231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right)}{1155 c^{11} \sqrt{c x-1} \sqrt{c x+1}}",1,"(3465*a*x^5*(231*d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(134400*e^3 + 4480*c^2*e^2*(121*d + 15*e*x^2) + 80*c^4*e*(9801*d^2 + 3388*d*e*x^2 + 630*e^2*x^4) + 24*c^6*(17787*d^3 + 16335*d^2*e*x^2 + 8470*d*e^2*x^4 + 1750*e^3*x^6) + c^10*x^4*(160083*d^3 + 245025*d^2*e*x^2 + 148225*d*e^2*x^4 + 33075*e^3*x^6) + 2*c^8*(106722*d^3*x^2 + 147015*d^2*e*x^4 + 84700*d*e^2*x^6 + 18375*e^3*x^8)))/c^11 + 3465*b*x^5*(231*d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6)*ArcCosh[c*x])/4002075","A",1
480,1,294,494,0.5501543,"\int x^3 \left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]","\frac{1920 a x^4 \left(10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right)-\frac{30 b \left(480 c^6 d^3+800 c^4 d^2 e+525 c^2 d e^2+126 e^3\right) \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)}{c^{10}}-\frac{b x \sqrt{c x-1} \sqrt{c x+1} \left(16 c^8 \left(300 d^3 x^2+400 d^2 e x^4+225 d e^2 x^6+48 e^3 x^8\right)+8 c^6 \left(900 d^3+1000 d^2 e x^2+525 d e^2 x^4+108 e^3 x^6\right)+6 c^4 e \left(2000 d^2+875 d e x^2+168 e^2 x^4\right)+315 c^2 e^2 \left(25 d+4 e x^2\right)+1890 e^3\right)}{c^9}+1920 b x^4 \cosh ^{-1}(c x) \left(10 d^3+20 d^2 e x^2+15 d e^2 x^4+4 e^3 x^6\right)}{76800}","\frac{\left(d+e x^2\right)^5 \left(a+b \cosh ^{-1}(c x)\right)}{10 e^2}-\frac{d \left(d+e x^2\right)^4 \left(a+b \cosh ^{-1}(c x)\right)}{8 e^2}+\frac{b x \left(1-c^2 x^2\right) \left(d+e x^2\right)^4}{100 c e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left(1-c^2 x^2\right) \left(11 c^2 d+18 e\right) \left(d+e x^2\right)^3}{1600 c^3 e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left(1-c^2 x^2\right) \left(26 c^4 d^2+201 c^2 d e+126 e^2\right) \left(d+e x^2\right)^2}{9600 c^5 e \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \sqrt{c^2 x^2-1} \left(128 c^{10} d^5-480 c^6 d^3 e^2-800 c^4 d^2 e^3-525 c^2 d e^4-126 e^5\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{5120 c^{10} e^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x \left(1-c^2 x^2\right) \left(136 c^6 d^3-1096 c^4 d^2 e-1617 c^2 d e^2-630 e^3\right) \left(d+e x^2\right)}{38400 c^7 e \sqrt{c x-1} \sqrt{c x+1}}-\frac{b x \left(1-c^2 x^2\right) \left(1232 c^8 d^4-2536 c^6 d^3 e-7758 c^4 d^2 e^2-6615 c^2 d e^3-1890 e^4\right)}{76800 c^9 e \sqrt{c x-1} \sqrt{c x+1}}",1,"(1920*a*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6) - (b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1890*e^3 + 315*c^2*e^2*(25*d + 4*e*x^2) + 6*c^4*e*(2000*d^2 + 875*d*e*x^2 + 168*e^2*x^4) + 8*c^6*(900*d^3 + 1000*d^2*e*x^2 + 525*d*e^2*x^4 + 108*e^3*x^6) + 16*c^8*(300*d^3*x^2 + 400*d^2*e*x^4 + 225*d*e^2*x^6 + 48*e^3*x^8)))/c^9 + 1920*b*x^4*(10*d^3 + 20*d^2*e*x^2 + 15*d*e^2*x^4 + 4*e^3*x^6)*ArcCosh[c*x] - (30*b*(480*c^6*d^3 + 800*c^4*d^2*e + 525*c^2*d*e^2 + 126*e^3)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/c^10)/76800","A",0
481,1,236,365,0.3326286,"\int x^2 \left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]","\frac{315 a x^3 \left(105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(c^8 \left(11025 d^3 x^2+11907 d^2 e x^4+6075 d e^2 x^6+1225 e^3 x^8\right)+2 c^6 \left(11025 d^3+7938 d^2 e x^2+3645 d e^2 x^4+700 e^3 x^6\right)+24 c^4 e \left(1323 d^2+405 d e x^2+70 e^2 x^4\right)+80 c^2 e^2 \left(243 d+28 e x^2\right)+4480 e^3\right)}{c^9}+315 b x^3 \cosh ^{-1}(c x) \left(105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right)}{99225}","\frac{1}{3} d^3 x^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{5} d^2 e x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{7} d e^2 x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{9} e^3 x^9 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b e^2 \left(1-c^2 x^2\right)^4 \left(27 c^2 d+28 e\right)}{441 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^3 \left(1-c^2 x^2\right)^5}{81 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e \left(1-c^2 x^2\right)^3 \left(63 c^4 d^2+135 c^2 d e+70 e^2\right)}{525 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \left(1-c^2 x^2\right)^2 \left(105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right)}{945 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left(1-c^2 x^2\right) \left(105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right)}{315 c^9 \sqrt{c x-1} \sqrt{c x+1}}",1,"(315*a*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4480*e^3 + 80*c^2*e^2*(243*d + 28*e*x^2) + 24*c^4*e*(1323*d^2 + 405*d*e*x^2 + 70*e^2*x^4) + 2*c^6*(11025*d^3 + 7938*d^2*e*x^2 + 3645*d*e^2*x^4 + 700*e^3*x^6) + c^8*(11025*d^3*x^2 + 11907*d^2*e*x^4 + 6075*d*e^2*x^6 + 1225*e^3*x^8)))/c^9 + 315*b*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6)*ArcCosh[c*x])/99225","A",1
482,1,256,358,0.4087276,"\int x \left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]","\frac{c x \left(384 a c^7 x \left(4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right)-b \sqrt{c x-1} \sqrt{c x+1} \left(16 c^6 \left(48 d^3+36 d^2 e x^2+16 d e^2 x^4+3 e^3 x^6\right)+8 c^4 e \left(108 d^2+40 d e x^2+7 e^2 x^4\right)+10 c^2 e^2 \left(48 d+7 e x^2\right)+105 e^3\right)\right)+384 b c^8 x^2 \cosh ^{-1}(c x) \left(4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right)-6 b \left(256 c^6 d^3+288 c^4 d^2 e+160 c^2 d e^2+35 e^3\right) \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)}{3072 c^8}","\frac{\left(d+e x^2\right)^4 \left(a+b \cosh ^{-1}(c x)\right)}{8 e}+\frac{b x \left(1-c^2 x^2\right) \left(d+e x^2\right)^3}{64 c \sqrt{c x-1} \sqrt{c x+1}}+\frac{7 b x \left(1-c^2 x^2\right) \left(2 c^2 d+e\right) \left(d+e x^2\right)^2}{384 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 b x \left(1-c^2 x^2\right) \left(2 c^2 d+e\right) \left(40 c^4 d^2+40 c^2 d e+21 e^2\right)}{3072 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b x \left(1-c^2 x^2\right) \left(104 c^4 d^2+104 c^2 d e+35 e^2\right) \left(d+e x^2\right)}{1536 c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{c^2 x^2-1} \left(128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{1024 c^8 e \sqrt{c x-1} \sqrt{c x+1}}",1,"(c*x*(384*a*c^7*x*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(105*e^3 + 10*c^2*e^2*(48*d + 7*e*x^2) + 8*c^4*e*(108*d^2 + 40*d*e*x^2 + 7*e^2*x^4) + 16*c^6*(48*d^3 + 36*d^2*e*x^2 + 16*d*e^2*x^4 + 3*e^3*x^6))) + 384*b*c^8*x^2*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6)*ArcCosh[c*x] - 6*b*(256*c^6*d^3 + 288*c^4*d^2*e + 160*c^2*d*e^2 + 35*e^3)*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/(3072*c^8)","A",0
483,1,193,287,0.2891972,"\int \left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]","a \left(d^3 x+d^2 e x^3+\frac{3}{5} d e^2 x^5+\frac{e^3 x^7}{7}\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(c^6 \left(3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right)+2 c^4 e \left(1225 d^2+294 d e x^2+45 e^2 x^4\right)+24 c^2 e^2 \left(49 d+5 e x^2\right)+240 e^3\right)}{3675 c^7}+\frac{1}{35} b x \cosh ^{-1}(c x) \left(35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right)","d^3 x \left(a+b \cosh ^{-1}(c x)\right)+d^2 e x^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{5} d e^2 x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{7} e^3 x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{3 b e^2 \left(1-c^2 x^2\right)^3 \left(7 c^2 d+5 e\right)}{175 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^3 \left(1-c^2 x^2\right)^4}{49 c^7 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e \left(1-c^2 x^2\right)^2 \left(35 c^4 d^2+42 c^2 d e+15 e^2\right)}{105 c^7 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left(1-c^2 x^2\right) \left(35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right)}{35 c^7 \sqrt{c x-1} \sqrt{c x+1}}",1,"a*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e^3*x^6)))/(3675*c^7) + (b*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcCosh[c*x])/35","A",1
484,1,314,509,0.7580057,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x,x]","a d^3 \log (x)+\frac{3}{2} a d^2 e x^2+\frac{3}{4} a d e^2 x^4+\frac{1}{6} a e^3 x^6-\frac{3 b d^2 e \left(-2 c^2 x^2 \cosh ^{-1}(c x)+c x \sqrt{c x-1} \sqrt{c x+1}+2 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{4 c^2}-\frac{3 b d e^2 \left(-8 c^4 x^4 \cosh ^{-1}(c x)+c x \sqrt{c x-1} \sqrt{c x+1} \left(2 c^2 x^2+3\right)+6 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{32 c^4}-\frac{b e^3 \left(-48 c^6 x^6 \cosh ^{-1}(c x)+c x \sqrt{c x-1} \sqrt{c x+1} \left(8 c^4 x^4+10 c^2 x^2+15\right)+30 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{288 c^6}+\frac{1}{2} b d^3 \left(\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)-\text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)\right)","d^3 \log (x) \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{2} d^2 e x^2 \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{4} d e^2 x^4 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{6} e^3 x^6 \left(a+b \cosh ^{-1}(c x)\right)-\frac{5 b e^3 \cosh ^{-1}(c x)}{96 c^6}-\frac{5 b e^3 x \sqrt{c x-1} \sqrt{c x+1}}{96 c^5}-\frac{9 b d e^2 \cosh ^{-1}(c x)}{32 c^4}-\frac{9 b d e^2 x \sqrt{c x-1} \sqrt{c x+1}}{32 c^3}-\frac{5 b e^3 x^3 \sqrt{c x-1} \sqrt{c x+1}}{144 c^3}-\frac{i b d^3 \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{i b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b d^3 \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b d^2 e \cosh ^{-1}(c x)}{4 c^2}-\frac{3 b d^2 e x \sqrt{c x-1} \sqrt{c x+1}}{4 c}-\frac{3 b d e^2 x^3 \sqrt{c x-1} \sqrt{c x+1}}{16 c}-\frac{b e^3 x^5 \sqrt{c x-1} \sqrt{c x+1}}{36 c}",1,"(3*a*d^2*e*x^2)/2 + (3*a*d*e^2*x^4)/4 + (a*e^3*x^6)/6 - (3*b*d^2*e*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*c^2*x^2*ArcCosh[c*x] + 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(4*c^2) - (3*b*d*e^2*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 + 2*c^2*x^2) - 8*c^4*x^4*ArcCosh[c*x] + 6*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(32*c^4) - (b*e^3*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(15 + 10*c^2*x^2 + 8*c^4*x^4) - 48*c^6*x^6*ArcCosh[c*x] + 30*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(288*c^6) + a*d^3*Log[x] + (b*d^3*(ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) - PolyLog[2, -E^(-2*ArcCosh[c*x])]))/2","A",0
485,1,182,265,0.3681639,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x^2,x]","-\frac{a d^3}{x}+3 a d^2 e x+a d e^2 x^3+\frac{1}{5} a e^3 x^5-\frac{b e \sqrt{c x-1} \sqrt{c x+1} \left(c^4 \left(225 d^2+25 d e x^2+3 e^2 x^4\right)+2 c^2 e \left(25 d+2 e x^2\right)+8 e^2\right)}{75 c^5}-b c d^3 \tan ^{-1}\left(\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right)+\frac{b \cosh ^{-1}(c x) \left(-5 d^3+15 d^2 e x^2+5 d e^2 x^4+e^3 x^6\right)}{5 x}","-\frac{d^3 \left(a+b \cosh ^{-1}(c x)\right)}{x}+3 d^2 e x \left(a+b \cosh ^{-1}(c x)\right)+d e^2 x^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{5} e^3 x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{b c d^3 \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^2 \left(1-c^2 x^2\right)^2 \left(5 c^2 d+2 e\right)}{15 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^3 \left(1-c^2 x^2\right)^3}{25 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e \left(1-c^2 x^2\right) \left(15 c^4 d^2+5 c^2 d e+e^2\right)}{5 c^5 \sqrt{c x-1} \sqrt{c x+1}}",1,"-((a*d^3)/x) + 3*a*d^2*e*x + a*d*e^2*x^3 + (a*e^3*x^5)/5 - (b*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(8*e^2 + 2*c^2*e*(25*d + 2*e*x^2) + c^4*(225*d^2 + 25*d*e*x^2 + 3*e^2*x^4)))/(75*c^5) + (b*(-5*d^3 + 15*d^2*e*x^2 + 5*d*e^2*x^4 + e^3*x^6)*ArcCosh[c*x])/(5*x) - b*c*d^3*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])]","A",1
486,1,267,476,0.7064285,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x^3,x]","\frac{1}{4} \left(-\frac{2 a d^3}{x^2}+12 a d^2 e \log (x)+6 a d e^2 x^2+a e^3 x^4-\frac{3 b d e^2 \left(c x \sqrt{c x-1} \sqrt{c x+1}+2 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{c^2}-\frac{b e^3 \left(c x \sqrt{c x-1} \sqrt{c x+1} \left(2 c^2 x^2+3\right)+6 \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)\right)}{8 c^4}+\frac{2 b d^3 \left(c x \sqrt{c x-1} \sqrt{c x+1}-\cosh ^{-1}(c x)\right)}{x^2}-6 b d^2 e \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+6 b d^2 e \cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)+2 \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)+6 b d e^2 x^2 \cosh ^{-1}(c x)+b e^3 x^4 \cosh ^{-1}(c x)\right)","-\frac{d^3 \left(a+b \cosh ^{-1}(c x)\right)}{2 x^2}+3 d^2 e \log (x) \left(a+b \cosh ^{-1}(c x)\right)+\frac{3}{2} d e^2 x^2 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{4} e^3 x^4 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b c d^3 \left(1-c^2 x^2\right)}{2 x \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \text{Li}_2\left(e^{2 i \sin ^{-1}(c x)}\right)}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 i b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x)^2}{2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b d^2 e \sqrt{1-c^2 x^2} \sin ^{-1}(c x) \log \left(1-e^{2 i \sin ^{-1}(c x)}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b d^2 e \sqrt{1-c^2 x^2} \log (x) \sin ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^3 x^3 \left(1-c^2 x^2\right)}{16 c \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b e^2 \sqrt{c^2 x^2-1} \left(8 c^2 d+e\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{32 c^4 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b e^2 x \left(1-c^2 x^2\right) \left(8 c^2 d+e\right)}{32 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"((-2*a*d^3)/x^2 + 6*a*d*e^2*x^2 + a*e^3*x^4 + (2*b*d^3*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - ArcCosh[c*x]))/x^2 + 6*b*d*e^2*x^2*ArcCosh[c*x] + b*e^3*x^4*ArcCosh[c*x] - (3*b*d*e^2*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + 2*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/c^2 - (b*e^3*(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(3 + 2*c^2*x^2) + 6*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]]))/(8*c^4) + 6*b*d^2*e*ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 + E^(-2*ArcCosh[c*x])]) + 12*a*d^2*e*Log[x] - 6*b*d^2*e*PolyLog[2, -E^(-2*ArcCosh[c*x])])/4","A",0
487,1,184,260,0.3936185,"\int \frac{\left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^3*(a + b*ArcCosh[c*x]))/x^4,x]","\frac{1}{6} \left(-\frac{2 a d^3}{x^3}-\frac{18 a d^2 e}{x}+18 a d e^2 x+2 a e^3 x^3-b c d^2 \left(c^2 d+18 e\right) \tan ^{-1}\left(\frac{1}{\sqrt{c x-1} \sqrt{c x+1}}\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(-3 c^4 d^3+2 c^2 e^2 x^2 \left(27 d+e x^2\right)+4 e^3 x^2\right)}{3 c^3 x^2}+\frac{2 b \cosh ^{-1}(c x) \left(-d^3-9 d^2 e x^2+9 d e^2 x^4+e^3 x^6\right)}{x^3}\right)","-\frac{d^3 \left(a+b \cosh ^{-1}(c x)\right)}{3 x^3}-\frac{3 d^2 e \left(a+b \cosh ^{-1}(c x)\right)}{x}+3 d e^2 x \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{3} e^3 x^3 \left(a+b \cosh ^{-1}(c x)\right)-\frac{b c d^3 \left(1-c^2 x^2\right)}{6 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d^2 \sqrt{c^2 x^2-1} \left(c^2 d+18 e\right) \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{6 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left(1-c^2 x^2\right) \left(9 c^2 d+e\right)}{3 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e^3 \left(1-c^2 x^2\right)^2}{9 c^3 \sqrt{c x-1} \sqrt{c x+1}}",1,"((-2*a*d^3)/x^3 - (18*a*d^2*e)/x + 18*a*d*e^2*x + 2*a*e^3*x^3 - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-3*c^4*d^3 + 4*e^3*x^2 + 2*c^2*e^2*x^2*(27*d + e*x^2)))/(3*c^3*x^2) + (2*b*(-d^3 - 9*d^2*e*x^2 + 9*d*e^2*x^4 + e^3*x^6)*ArcCosh[c*x])/x^3 - b*c*d^2*(c^2*d + 18*e)*ArcTan[1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])])/6","A",1
488,1,265,395,0.4025744,"\int \left(d+e x^2\right)^4 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^4*(a + b*ArcCosh[c*x]),x]","\frac{315 a x \left(315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right)-\frac{b \sqrt{c x-1} \sqrt{c x+1} \left(c^8 \left(99225 d^4+44100 d^3 e x^2+23814 d^2 e^2 x^4+8100 d e^3 x^6+1225 e^4 x^8\right)+8 c^6 e \left(11025 d^3+3969 d^2 e x^2+1215 d e^2 x^4+175 e^3 x^6\right)+48 c^4 e^2 \left(1323 d^2+270 d e x^2+35 e^2 x^4\right)+320 c^2 e^3 \left(81 d+7 e x^2\right)+4480 e^4\right)}{c^9}+315 b x \cosh ^{-1}(c x) \left(315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right)}{99225}","d^4 x \left(a+b \cosh ^{-1}(c x)\right)+\frac{4}{3} d^3 e x^3 \left(a+b \cosh ^{-1}(c x)\right)+\frac{6}{5} d^2 e^2 x^5 \left(a+b \cosh ^{-1}(c x)\right)+\frac{4}{7} d e^3 x^7 \left(a+b \cosh ^{-1}(c x)\right)+\frac{1}{9} e^4 x^9 \left(a+b \cosh ^{-1}(c x)\right)-\frac{4 b e^3 \left(1-c^2 x^2\right)^4 \left(9 c^2 d+7 e\right)}{441 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^4 \left(1-c^2 x^2\right)^5}{81 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b e^2 \left(1-c^2 x^2\right)^3 \left(63 c^4 d^2+90 c^2 d e+35 e^2\right)}{525 c^9 \sqrt{c x-1} \sqrt{c x+1}}-\frac{4 b e \left(1-c^2 x^2\right)^2 \left(105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right)}{945 c^9 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b \left(1-c^2 x^2\right) \left(315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right)}{315 c^9 \sqrt{c x-1} \sqrt{c x+1}}",1,"(315*a*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x^8) - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4480*e^4 + 320*c^2*e^3*(81*d + 7*e*x^2) + 48*c^4*e^2*(1323*d^2 + 270*d*e*x^2 + 35*e^2*x^4) + 8*c^6*e*(11025*d^3 + 3969*d^2*e*x^2 + 1215*d*e^2*x^4 + 175*e^3*x^6) + c^8*(99225*d^4 + 44100*d^3*e*x^2 + 23814*d^2*e^2*x^4 + 8100*d*e^3*x^6 + 1225*e^4*x^8)))/c^9 + 315*b*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x^8)*ArcCosh[c*x])/99225","A",1
489,1,524,627,1.5114698,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2),x]","\frac{a d^{3/2} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{e^{5/2}}-\frac{a d x}{e^2}+\frac{a x^3}{3 e}+\frac{b \left(-i d^{3/2} \left(2 \text{Li}_2\left(\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\cosh ^{-1}(c x) \left(-\cosh ^{-1}(c x)+2 \left(\log \left(1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)+i d^{3/2} \left(2 \text{Li}_2\left(\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)+2 \text{Li}_2\left(\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\cosh ^{-1}(c x) \left(-\cosh ^{-1}(c x)+2 \left(\log \left(1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)-\frac{4 e^{3/2} \left(\sqrt{c x-1} \sqrt{c x+1} \left(c^2 x^2+2\right)-3 c^3 x^3 \cosh ^{-1}(c x)\right)}{9 c^3}+\frac{4 d \sqrt{e} \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)-c x \cosh ^{-1}(c x)\right)}{c}\right)}{4 e^{5/2}}","\frac{(-d)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 e^{5/2}}-\frac{(-d)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 e^{5/2}}+\frac{(-d)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 e^{5/2}}-\frac{(-d)^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 e^{5/2}}+\frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{3 e}-\frac{a d x}{e^2}-\frac{2 b \sqrt{c x-1} \sqrt{c x+1}}{9 c^3 e}-\frac{b (-d)^{3/2} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^{5/2}}+\frac{b (-d)^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^{5/2}}-\frac{b (-d)^{3/2} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^{5/2}}+\frac{b (-d)^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^{5/2}}+\frac{b d \sqrt{c x-1} \sqrt{c x+1}}{c e^2}-\frac{b d x \cosh ^{-1}(c x)}{e^2}-\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{9 c e}",1,"-((a*d*x)/e^2) + (a*x^3)/(3*e) + (a*d^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/e^(5/2) + (b*((4*d*Sqrt[e]*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - c*x*ArcCosh[c*x]))/c - (4*e^(3/2)*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c^2*x^2) - 3*c^3*x^3*ArcCosh[c*x]))/(9*c^3) - I*d^(3/2)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, ((-I)*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])]) + I*d^(3/2)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*e^(5/2))","C",0
490,1,512,521,0.5630174,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x]))/(d + e*x^2),x]","-\frac{2 a c^2 d \log \left(d+e x^2\right)-2 a c^2 e x^2+2 b c^2 d \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)+2 b c^2 d \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-c \sqrt{-d}}\right)+2 b c^2 d \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)+2 b c^2 d \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)+2 b c^2 d \cosh ^{-1}(c x) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)+2 b c^2 d \cosh ^{-1}(c x) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}+1\right)+2 b c^2 d \cosh ^{-1}(c x) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)+2 b c^2 d \cosh ^{-1}(c x) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)-2 b c^2 d \cosh ^{-1}(c x)^2-2 b c^2 e x^2 \cosh ^{-1}(c x)+b c e x \sqrt{c x-1} \sqrt{c x+1}+2 b e \tanh ^{-1}\left(\sqrt{\frac{c x-1}{c x+1}}\right)}{4 c^2 e^2}","-\frac{d \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 e^2}-\frac{d \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 e^2}-\frac{d \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 e^2}-\frac{d \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 e^2}+\frac{d \left(a+b \cosh ^{-1}(c x)\right)^2}{2 b e^2}+\frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{2 e}-\frac{b d \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^2}-\frac{b d \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^2}-\frac{b d \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^2}-\frac{b d \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^2}-\frac{b \cosh ^{-1}(c x)}{4 c^2 e}-\frac{b x \sqrt{c x-1} \sqrt{c x+1}}{4 c e}",1,"-1/4*(-2*a*c^2*e*x^2 + b*c*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 2*b*c^2*e*x^2*ArcCosh[c*x] - 2*b*c^2*d*ArcCosh[c*x]^2 + 2*b*e*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]] + 2*b*c^2*d*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])] + 2*b*c^2*d*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + 2*b*c^2*d*ArcCosh[c*x]*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] + 2*b*c^2*d*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] + 2*a*c^2*d*Log[d + e*x^2] + 2*b*c^2*d*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])] + 2*b*c^2*d*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + 2*b*c^2*d*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))] + 2*b*c^2*d*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(c^2*e^2)","A",0
491,1,457,544,0.7475636,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d + e*x^2),x]","\frac{-4 a c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+4 a c \sqrt{e} x+i b \left(-c \sqrt{d} \left(-2 \text{Li}_2\left(\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{d c^2+e}-c \sqrt{d}}\right)-2 \text{Li}_2\left(-\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-2 \left(\log \left(1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{d}-\sqrt{c^2 d+e}}\right)+\log \left(1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)+c \sqrt{d} \left(-2 \text{Li}_2\left(\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{d}-\sqrt{d c^2+e}}\right)-2 \text{Li}_2\left(\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{d} c+\sqrt{d c^2+e}}\right)+\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-2 \left(\log \left(1+\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}-c \sqrt{d}}\right)+\log \left(1-\frac{i \sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 d+e}+c \sqrt{d}}\right)\right)\right)\right)+4 i \sqrt{e} \left(\sqrt{\frac{c x-1}{c x+1}} (c x+1)-c x \cosh ^{-1}(c x)\right)\right)}{4 c e^{3/2}}","\frac{\sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 e^{3/2}}+\frac{\sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 e^{3/2}}+\frac{a x}{e}-\frac{b \sqrt{-d} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^{3/2}}-\frac{b \sqrt{-d} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^{3/2}}+\frac{b \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^{3/2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1}}{c e}+\frac{b x \cosh ^{-1}(c x)}{e}",1,"(4*a*c*Sqrt[e]*x - 4*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + I*b*((4*I)*Sqrt[e]*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - c*x*ArcCosh[c*x]) - c*Sqrt[d]*(ArcCosh[c*x]*(ArcCosh[c*x] - 2*(Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])) - 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] - 2*PolyLog[2, ((-I)*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])]) + c*Sqrt[d]*(ArcCosh[c*x]*(ArcCosh[c*x] - 2*(Log[1 + (I*Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])) - 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] - Sqrt[c^2*d + e])] - 2*PolyLog[2, (I*Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[d] + Sqrt[c^2*d + e])])))/(4*c*e^(3/2))","C",0
492,1,447,449,0.1380721,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x*(a + b*ArcCosh[c*x]))/(d + e*x^2),x]","\frac{a \log \left(d+e x^2\right)}{2 e}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e}+\frac{b \cosh ^{-1}(c x) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 e}+\frac{b \cosh ^{-1}(c x) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 e}+\frac{b \cosh ^{-1}(c x) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 e}+\frac{b \cosh ^{-1}(c x) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 e}-\frac{b \cosh ^{-1}(c x)^2}{2 e}","\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 e}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 e}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 e}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 e}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{2 b e}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e}",1,"-1/2*(b*ArcCosh[c*x]^2)/e + (b*ArcCosh[c*x]*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e) + (b*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e) + (b*ArcCosh[c*x]*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e) + (b*ArcCosh[c*x]*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e) + (a*Log[d + e*x^2])/(2*e) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*e) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*e) + (b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(2*e) + (b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*e)","A",0
493,1,397,501,0.3451772,"\int \frac{a+b \cosh ^{-1}(c x)}{d+e x^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d + e*x^2),x]","\frac{-\left(\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)\right)+\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}+1\right)+\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)-\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)+b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)-b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-c \sqrt{-d}}\right)-b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)+b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 \sqrt{-d} \sqrt{e}}","\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 \sqrt{-d} \sqrt{e}}",1,"(-((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])]) + (a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + (a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] - (a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] + b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])] - b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] - b*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))] + b*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*Sqrt[-d]*Sqrt[e])","A",1
494,1,418,489,0.8162238,"\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcCosh[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(2 d c^2+e-2 \sqrt{c^2 d \left(d c^2+e\right)}\right) e^{-2 \cosh ^{-1}(c x)}}{e}\right)+\text{Li}_2\left(-\frac{\left(2 d c^2+e+2 \sqrt{c^2 d \left(d c^2+e\right)}\right) e^{-2 \cosh ^{-1}(c x)}}{e}\right)-4 i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{e}+1}\right) \tanh ^{-1}\left(\frac{c d \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{x \sqrt{c^2 d \left(c^2 d+e\right)}}\right)-2 \cosh ^{-1}(c x) \log \left(\frac{\left(-2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right)-2 \cosh ^{-1}(c x) \log \left(\frac{\left(2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right)+2 i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{e}+1}\right) \log \left(\frac{\left(-2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right)-2 i \sin ^{-1}\left(\sqrt{\frac{c^2 d}{e}+1}\right) \log \left(\frac{\left(2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right)-2 \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)+4 \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)}{4 d}","-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 d}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 d}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 d}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 d}+\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b d}+\frac{\log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{d}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 d}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 d}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 d}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 d}-\frac{b \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{2 d}",1,"(4*a*Log[x] - 2*a*Log[d + e*x^2] + b*((-4*I)*ArcSin[Sqrt[1 + (c^2*d)/e]]*ArcTanh[(c*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[c^2*d*(c^2*d + e)]*x)] + 4*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] - 2*ArcCosh[c*x]*Log[1 + (2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] + (2*I)*ArcSin[Sqrt[1 + (c^2*d)/e]]*Log[1 + (2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] - 2*ArcCosh[c*x]*Log[1 + (2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] - (2*I)*ArcSin[Sqrt[1 + (c^2*d)/e]]*Log[1 + (2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] - 2*PolyLog[2, -E^(-2*ArcCosh[c*x])] + PolyLog[2, -((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x])))] + PolyLog[2, -((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x])))]))/(4*d)","C",0
495,1,549,543,1.6900905,"\int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^2*(d + e*x^2)),x]","\frac{1}{2} \left(\frac{d \sqrt{e} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{(-d)^{5/2}}+\frac{\sqrt{e} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}+1\right)}{(-d)^{3/2}}+\frac{\sqrt{e} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{(-d)^{3/2}}+\frac{d \sqrt{e} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{(-d)^{5/2}}-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{d x}+\frac{b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{(-d)^{3/2}}+\frac{b d \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-c \sqrt{-d}}\right)}{(-d)^{5/2}}+\frac{b d \sqrt{e} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{(-d)^{5/2}}+\frac{b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{(-d)^{3/2}}+\frac{2 b c \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{d \sqrt{c x-1} \sqrt{c x+1}}\right)","\frac{\sqrt{e} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 (-d)^{3/2}}-\frac{a+b \cosh ^{-1}(c x)}{d x}-\frac{b \sqrt{e} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 (-d)^{3/2}}-\frac{b \sqrt{e} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 (-d)^{3/2}}+\frac{b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 (-d)^{3/2}}+\frac{b c \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{d}",1,"((-2*(a + b*ArcCosh[c*x]))/(d*x) + (2*b*c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(d*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(-d)^(5/2) + (Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])])/(-d)^(3/2) + (Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(3/2) + (d*Sqrt[e]*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) + (b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(-d)^(3/2) + (b*d*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) + (b*d*Sqrt[e]*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(-d)^(5/2) + (b*Sqrt[e]*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(3/2))/2","A",0
496,1,479,550,1.4381851,"\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)),x]","\frac{2 a e \log \left(d+e x^2\right)-\frac{2 a d}{x^2}-4 a e \log (x)+b \left(-e \text{Li}_2\left(-\frac{\left(2 d c^2+e-2 \sqrt{c^2 d \left(d c^2+e\right)}\right) e^{-2 \cosh ^{-1}(c x)}}{e}\right)-e \text{Li}_2\left(-\frac{\left(2 d c^2+e+2 \sqrt{c^2 d \left(d c^2+e\right)}\right) e^{-2 \cosh ^{-1}(c x)}}{e}\right)+4 i e \sin ^{-1}\left(\sqrt{\frac{c^2 d}{e}+1}\right) \tanh ^{-1}\left(\frac{c d \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{x \sqrt{c^2 d \left(c^2 d+e\right)}}\right)+2 e \cosh ^{-1}(c x) \log \left(\frac{\left(-2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right)+2 e \cosh ^{-1}(c x) \log \left(\frac{\left(2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right)-2 i e \sin ^{-1}\left(\sqrt{\frac{c^2 d}{e}+1}\right) \log \left(\frac{\left(-2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right)+2 i e \sin ^{-1}\left(\sqrt{\frac{c^2 d}{e}+1}\right) \log \left(\frac{\left(2 \sqrt{c^2 d \left(c^2 d+e\right)}+2 c^2 d+e\right) e^{-2 \cosh ^{-1}(c x)}}{e}+1\right)-\frac{2 d \cosh ^{-1}(c x)}{x^2}+\frac{2 c d \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{x}+2 e \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)-4 e \cosh ^{-1}(c x) \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right)\right)}{4 d^2}","\frac{e \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 d^2}+\frac{e \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 d^2}+\frac{e \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 d^2}+\frac{e \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 d^2}-\frac{e \left(a+b \cosh ^{-1}(c x)\right)^2}{b d^2}-\frac{e \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2}-\frac{a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac{b e \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 d^2}+\frac{b e \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 d^2}+\frac{b e \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 d^2}+\frac{b e \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 d^2}+\frac{b e \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{2 d^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d x}",1,"((-2*a*d)/x^2 - 4*a*e*Log[x] + 2*a*e*Log[d + e*x^2] + b*((2*c*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/x - (2*d*ArcCosh[c*x])/x^2 + (4*I)*e*ArcSin[Sqrt[1 + (c^2*d)/e]]*ArcTanh[(c*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[c^2*d*(c^2*d + e)]*x)] - 4*e*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] + 2*e*ArcCosh[c*x]*Log[1 + (2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] - (2*I)*e*ArcSin[Sqrt[1 + (c^2*d)/e]]*Log[1 + (2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] + 2*e*ArcCosh[c*x]*Log[1 + (2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] + (2*I)*e*ArcSin[Sqrt[1 + (c^2*d)/e]]*Log[1 + (2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] + 2*e*PolyLog[2, -E^(-2*ArcCosh[c*x])] - e*PolyLog[2, -((2*c^2*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x])))] - e*PolyLog[2, -((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x])))]))/(4*d^2)","C",0
497,1,641,624,1.6107737,"\int \frac{a+b \cosh ^{-1}(c x)}{x^4 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^4*(d + e*x^2)),x]","\frac{1}{6} \left(-\frac{3 e^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{(-d)^{5/2}}+\frac{3 e^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}+1\right)}{(-d)^{5/2}}+\frac{3 e^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{(-d)^{5/2}}-\frac{3 e^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{(-d)^{5/2}}+\frac{6 e \left(a+b \cosh ^{-1}(c x)\right)}{d^2 x}-\frac{2 \left(a+b \cosh ^{-1}(c x)\right)}{d x^3}-\frac{6 b c e \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{d^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b e^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{(-d)^{5/2}}-\frac{3 b e^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-c \sqrt{-d}}\right)}{(-d)^{5/2}}-\frac{3 b e^{3/2} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{(-d)^{5/2}}+\frac{3 b e^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{(-d)^{5/2}}+\frac{b c \left(c^2 x^2+c^2 x^2 \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)-1\right)}{d x^2 \sqrt{c x-1} \sqrt{c x+1}}\right)","\frac{e^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 (-d)^{5/2}}-\frac{e^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 (-d)^{5/2}}+\frac{e^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 (-d)^{5/2}}-\frac{e^{3/2} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 (-d)^{5/2}}+\frac{e \left(a+b \cosh ^{-1}(c x)\right)}{d^2 x}-\frac{a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac{b c^3 \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{6 d}-\frac{b e^{3/2} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 (-d)^{5/2}}+\frac{b e^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 (-d)^{5/2}}-\frac{b e^{3/2} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 (-d)^{5/2}}+\frac{b e^{3/2} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 (-d)^{5/2}}-\frac{b c e \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{d^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{6 d x^2}",1,"((-2*(a + b*ArcCosh[c*x]))/(d*x^3) + (6*e*(a + b*ArcCosh[c*x]))/(d^2*x) - (6*b*c*e*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*(-1 + c^2*x^2 + c^2*x^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]]))/(d*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(-d)^(5/2) + (3*e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) + (3*e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) - (3*e^(3/2)*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) + (3*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(-d)^(5/2) - (3*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])])/(-d)^(5/2) - (3*b*e^(3/2)*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))])/(-d)^(5/2) + (3*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(-d)^(5/2))/6","A",0
498,1,693,562,2.146057,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2,x]","\frac{\frac{2 a d}{d+e x^2}+2 a \log \left(d+e x^2\right)+b \left(2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)+2 \cosh ^{-1}(c x) \left(\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-i c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)\right)+2 \cosh ^{-1}(c x) \left(\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{-\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)+\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)\right)-i \sqrt{d} \left(\frac{c \log \left(\frac{2 e \left(-i \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}+c^2 \sqrt{d} x+i \sqrt{e}\right)}{c \sqrt{c^2 (-d)-e} \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{\sqrt{c^2 (-d)-e}}+\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}\right)-i \sqrt{d} \left(-\frac{c \log \left(\frac{2 e \left(\sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}-i c^2 \sqrt{d} x-\sqrt{e}\right)}{c \sqrt{c^2 (-d)-e} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{c^2 (-d)-e}}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)-2 \cosh ^{-1}(c x)^2\right)}{4 e^2}","\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 e^2}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 e^2}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 e^2}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 e^2}+\frac{d \left(a+b \cosh ^{-1}(c x)\right)}{2 e^2 \left(d+e x^2\right)}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{2 b e^2}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^2}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^2}-\frac{b c \sqrt{d} \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{2 e^2 \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}",1,"((2*a*d)/(d + e*x^2) + 2*a*Log[d + e*x^2] + b*(-2*ArcCosh[c*x]^2 + 2*ArcCosh[c*x]*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]) + 2*ArcCosh[c*x]*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]) - I*Sqrt[d]*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) - I*Sqrt[d]*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/(4*e^2)","C",0
499,1,123,113,0.3282842,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2,x]","-\frac{\frac{a}{d+e x^2}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left(\frac{x \sqrt{c^2 (-d)-e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{\sqrt{d} \sqrt{c^2 x^2-1} \sqrt{c^2 (-d)-e}}+\frac{b \cosh ^{-1}(c x)}{d+e x^2}}{2 e}","\frac{b c \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{2 \sqrt{d} e \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}-\frac{a+b \cosh ^{-1}(c x)}{2 e \left(d+e x^2\right)}",1,"-1/2*(a/(d + e*x^2) + (b*ArcCosh[c*x])/(d + e*x^2) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[(Sqrt[-(c^2*d) - e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(Sqrt[d]*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c^2*x^2]))/e","A",1
500,0,0,598,5.0744207,"\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x*(d + e*x^2)^2),x]","\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 d^2}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 d^2}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 d^2}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 d^2}+\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b d^2}+\frac{\log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^2}+\frac{a+b \cosh ^{-1}(c x)}{2 d \left(d+e x^2\right)}-\frac{b c \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{2 d^{3/2} \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 d^2}-\frac{b \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{2 d^2}",1,"Integrate[(a + b*ArcCosh[c*x])/(x*(d + e*x^2)^2), x]","F",-1
501,0,0,634,6.027413,"\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)^2),x]","\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d+e x^2\right)^2} \, dx","\frac{e \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{d^3}+\frac{e \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{d^3}+\frac{e \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{d^3}+\frac{e \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{d^3}-\frac{2 e \left(a+b \cosh ^{-1}(c x)\right)^2}{b d^3}-\frac{2 e \log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^3}-\frac{e \left(a+b \cosh ^{-1}(c x)\right)}{2 d^2 \left(d+e x^2\right)}-\frac{a+b \cosh ^{-1}(c x)}{2 d^2 x^2}+\frac{b c e \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{2 d^{5/2} \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}+\frac{b e \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{d^3}+\frac{b e \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{d^3}+\frac{b e \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{d^3}+\frac{b e \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{d^3}+\frac{b e \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{d^3}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d^2 x}",1,"Integrate[(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)^2), x]","F",-1
502,1,777,839,2.4105197,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2,x]","\frac{\frac{4 a d \sqrt{e} x}{d+e x^2}-12 a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+8 a \sqrt{e} x+b \left(-3 i \sqrt{d} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)+\cosh ^{-1}(c x) \left(-\cosh ^{-1}(c x)+2 \left(\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{-\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)+\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)\right)\right)\right)+3 i \sqrt{d} \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i c \sqrt{d}-\sqrt{-d c^2-e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)+\cosh ^{-1}(c x) \left(-\cosh ^{-1}(c x)+2 \left(\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-i c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)\right)\right)\right)+2 d \left(\frac{c \log \left(\frac{2 e \left(-i \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}+c^2 \sqrt{d} x+i \sqrt{e}\right)}{c \sqrt{c^2 (-d)-e} \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{\sqrt{c^2 (-d)-e}}+\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}\right)+2 d \left(\frac{c \log \left(\frac{2 e \left(\sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}-i c^2 \sqrt{d} x-\sqrt{e}\right)}{c \sqrt{c^2 (-d)-e} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{c^2 (-d)-e}}+\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)+\frac{8 \sqrt{e} \left(c x \cosh ^{-1}(c x)-\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right)}{c}\right)}{8 e^{5/2}}","\frac{x \cosh ^{-1}(c x) b}{e^2}+\frac{c d \tanh ^{-1}\left(\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right) b}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}-\frac{c d \tanh ^{-1}\left(\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right) b}{2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}-\frac{3 \sqrt{-d} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right) b}{4 e^{5/2}}+\frac{3 \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right) b}{4 e^{5/2}}-\frac{3 \sqrt{-d} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right) b}{4 e^{5/2}}+\frac{3 \sqrt{-d} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right) b}{4 e^{5/2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} b}{c e^2}+\frac{a x}{e^2}-\frac{d \left(a+b \cosh ^{-1}(c x)\right)}{4 e^{5/2} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{d \left(a+b \cosh ^{-1}(c x)\right)}{4 e^{5/2} \left(\sqrt{e} x+\sqrt{-d}\right)}+\frac{3 \sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right)}{4 e^{5/2}}+\frac{3 \sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right)}{4 e^{5/2}}",1,"(8*a*Sqrt[e]*x + (4*a*d*Sqrt[e]*x)/(d + e*x^2) - 12*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*((8*Sqrt[e]*(-(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + c*x*ArcCosh[c*x]))/c + 2*d*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) + 2*d*(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) - (3*I)*Sqrt[d]*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]) + (3*I)*Sqrt[d]*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])))/(8*e^(5/2))","C",0
503,1,720,792,1.9161706,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2,x]","\frac{-\frac{4 a \sqrt{e} x}{d+e x^2}+\frac{4 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}+b \left(\frac{i \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)+\cosh ^{-1}(c x) \left(-\cosh ^{-1}(c x)+2 \left(\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{-\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)+\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)\right)\right)\right)}{\sqrt{d}}+\frac{i \left(-2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i c \sqrt{d}-\sqrt{-d c^2-e}}\right)-2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)+\cosh ^{-1}(c x) \left(\cosh ^{-1}(c x)-2 \left(\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-i c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)\right)\right)\right)}{\sqrt{d}}-\frac{2 c \log \left(\frac{2 e \left(\sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}-i c^2 \sqrt{d} x-\sqrt{e}\right)}{c \sqrt{c^2 (-d)-e} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{c^2 (-d)-e}}-2 \left(\frac{c \log \left(\frac{2 e \left(-i \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}+c^2 \sqrt{d} x+i \sqrt{e}\right)}{c \sqrt{c^2 (-d)-e} \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{\sqrt{c^2 (-d)-e}}+\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}\right)-\frac{2 \cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)}{8 e^{3/2}}","\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{4 \sqrt{-d} e^{3/2}}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{4 \sqrt{-d} e^{3/2}}+\frac{a+b \cosh ^{-1}(c x)}{4 e^{3/2} \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{a+b \cosh ^{-1}(c x)}{4 e^{3/2} \left(\sqrt{-d}+\sqrt{e} x\right)}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{c x+1} \sqrt{c \sqrt{-d}-\sqrt{e}}}{\sqrt{c x-1} \sqrt{c \sqrt{-d}+\sqrt{e}}}\right)}{2 e^{3/2} \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}}}+\frac{b c \tanh ^{-1}\left(\frac{\sqrt{c x+1} \sqrt{c \sqrt{-d}+\sqrt{e}}}{\sqrt{c x-1} \sqrt{c \sqrt{-d}-\sqrt{e}}}\right)}{2 e^{3/2} \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c \sqrt{-d}+\sqrt{e}}}",1,"((-4*a*Sqrt[e]*x)/(d + e*x^2) + (4*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + b*((-2*ArcCosh[c*x])/(I*Sqrt[d] + Sqrt[e]*x) - 2*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) - (2*c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + (I*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]))/Sqrt[d] + (I*(ArcCosh[c*x]*(ArcCosh[c*x] - 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) - 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] - 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/Sqrt[d]))/(8*e^(3/2))","C",0
504,1,734,804,2.0874356,"\int \frac{a+b \cosh ^{-1}(c x)}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d + e*x^2)^2,x]","\frac{1}{2} \left(\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \sqrt{e}}+\frac{a x}{d^2+d e x^2}+\frac{b \left(i \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)+\cosh ^{-1}(c x) \left(-\cosh ^{-1}(c x)+2 \left(\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{-\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)+\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)\right)\right)\right)-i \left(2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i c \sqrt{d}-\sqrt{-d c^2-e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)+\cosh ^{-1}(c x) \left(-\cosh ^{-1}(c x)+2 \left(\log \left(1+\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-i c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+i c \sqrt{d}}\right)\right)\right)\right)+2 \sqrt{d} \left(\frac{c \log \left(\frac{2 e \left(-i \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}+c^2 \sqrt{d} x+i \sqrt{e}\right)}{c \sqrt{c^2 (-d)-e} \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{\sqrt{c^2 (-d)-e}}+\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}\right)-2 \sqrt{d} \left(-\frac{c \log \left(\frac{2 e \left(\sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 (-d)-e}-i c^2 \sqrt{d} x-\sqrt{e}\right)}{c \sqrt{c^2 (-d)-e} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{c^2 (-d)-e}}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)\right)}{4 d^{3/2} \sqrt{e}}\right)","-\frac{\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \cosh ^{-1}(c x)}{4 d \sqrt{e} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{a+b \cosh ^{-1}(c x)}{4 d \sqrt{e} \left(\sqrt{e} x+\sqrt{-d}\right)}+\frac{b c \tanh ^{-1}\left(\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right)}{2 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{e}}-\frac{b c \tanh ^{-1}\left(\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right)}{2 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{e}}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\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]*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) - 2*Sqrt[d]*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) + I*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]) - I*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])))/(4*d^(3/2)*Sqrt[e]))/2","C",0
505,1,821,846,2.8518011,"\int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[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} \left(\frac{c \sqrt{c^2 x^2-1} \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{\cosh ^{-1}(c x)}{x}\right)-2 \sqrt{d} \sqrt{e} \left(\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{c \log \left(\frac{2 e \left(\sqrt{d} x c^2+i \sqrt{e}-i \sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)+2 \sqrt{d} \sqrt{e} \left(-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \log \left(\frac{2 e \left(-i \sqrt{d} x c^2-\sqrt{e}+\sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)-3 i \sqrt{e} \left(\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i c \sqrt{d}-\sqrt{-d c^2-e}}+1\right)+\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i \sqrt{d} c+\sqrt{-d c^2-e}}+1\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)+3 i \sqrt{e} \left(\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d c^2-e}-i c \sqrt{d}}+1\right)+\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i c \sqrt{d}-\sqrt{-d c^2-e}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)\right)}{8 d^{5/2}}","-\frac{3 \sqrt{e} \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{4 (-d)^{5/2}}-\frac{a+b \cosh ^{-1}(c x)}{d^2 x}+\frac{\sqrt{e} \left(a+b \cosh ^{-1}(c x)\right)}{4 d^2 \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{\sqrt{e} \left(a+b \cosh ^{-1}(c x)\right)}{4 d^2 \left(\sqrt{e} x+\sqrt{-d}\right)}+\frac{b c \tan ^{-1}\left(\sqrt{c x-1} \sqrt{c x+1}\right)}{d^2}-\frac{b c \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right)}{2 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}}}+\frac{b c \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right)}{2 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}}}+\frac{3 b \sqrt{e} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{4 (-d)^{5/2}}+\frac{3 b \sqrt{e} \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{4 (-d)^{5/2}}-\frac{3 b \sqrt{e} \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{4 (-d)^{5/2}}",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*Sqrt[d]*(-(ArcCosh[c*x]/x) + (c*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - 2*Sqrt[d]*Sqrt[e]*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) + 2*Sqrt[d]*Sqrt[e]*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]) - (3*I)*Sqrt[e]*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]) + (3*I)*Sqrt[e]*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])))/(8*d^(5/2))","C",0
506,1,1155,737,7.25973,"\int \frac{x^5 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^5*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]","-\frac{a d^2}{4 e^3 \left(e x^2+d\right)^2}+\frac{a d}{e^3 \left(e x^2+d\right)}+\frac{a \log \left(e x^2+d\right)}{2 e^3}+b \left(-\frac{7 i \sqrt{d} \left(\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{c \log \left(\frac{2 e \left(\sqrt{d} x c^2+i \sqrt{e}-i \sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{16 e^3}-\frac{7 i \sqrt{d} \left(-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \log \left(\frac{2 e \left(-i \sqrt{d} x c^2-\sqrt{e}+\sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{16 e^3}-\frac{d \left(\frac{\sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(-\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c^3 \left(d+i \sqrt{e} x \sqrt{d}\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 e^{5/2}}-\frac{d \left(-\frac{\sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c^3 \left(d-i \sqrt{d} \sqrt{e} x\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}\right)}{16 e^{5/2}}+\frac{\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i c \sqrt{d}-\sqrt{-d c^2-e}}+1\right)+\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i \sqrt{d} c+\sqrt{-d c^2-e}}+1\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)}{4 e^3}+\frac{\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d c^2-e}-i c \sqrt{d}}+1\right)+\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)}{4 e^3}\right)","\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 e^3}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 e^3}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 e^3}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 e^3}-\frac{d^2 \left(a+b \cosh ^{-1}(c x)\right)}{4 e^3 \left(d+e x^2\right)^2}+\frac{d \left(a+b \cosh ^{-1}(c x)\right)}{e^3 \left(d+e x^2\right)}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{2 b e^3}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^3}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 e^3}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^3}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 e^3}+\frac{b c \sqrt{d} \sqrt{c^2 x^2-1} \left(2 c^2 d+e\right) \tanh ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{8 e^3 \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right)^{3/2}}-\frac{b c \sqrt{d} \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{e^3 \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}+\frac{b c d x \left(1-c^2 x^2\right)}{8 e^2 \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right) \left(d+e x^2\right)}",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*((((-7*I)/16)*Sqrt[d]*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/e^3 - (((7*I)/16)*Sqrt[d]*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/e^3 - (d*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] - c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(16*e^(5/2)) - (d*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(16*e^(5/2)) + (ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))])/(4*e^3) + (ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])/(4*e^3))","C",0
507,1,192,231,0.8434904,"\int \frac{x^3 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]","\frac{\frac{\frac{b c e x \sqrt{c x-1} \sqrt{c x+1} \left(d+e x^2\right)}{c^2 d+e}-2 a \left(d+2 e x^2\right)}{\left(d+e x^2\right)^2}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left(2 c^2 d+3 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 (-d)-e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{\sqrt{d} \sqrt{c^2 x^2-1} \left(c^2 (-d)-e\right)^{3/2}}-\frac{2 b \cosh ^{-1}(c x) \left(d+2 e x^2\right)}{\left(d+e x^2\right)^2}}{8 e^2}","\frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{4 d \left(d+e x^2\right)^2}-\frac{b \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 d e^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \sqrt{1-c^2 x^2} \left(2 c^2 d+3 e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right)}{8 \sqrt{d} e^2 \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right)^{3/2}}-\frac{b c x \left(1-c^2 x^2\right)}{8 e \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"(((b*c*e*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))/(c^2*d + e) - 2*a*(d + 2*e*x^2))/(d + e*x^2)^2 - (2*b*(d + 2*e*x^2)*ArcCosh[c*x])/(d + e*x^2)^2 - (b*c*(2*c^2*d + 3*e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[(Sqrt[-(c^2*d) - e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(Sqrt[d]*(-(c^2*d) - e)^(3/2)*Sqrt[-1 + c^2*x^2]))/(8*e^2)","A",1
508,1,183,177,0.9353166,"\int \frac{x \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]","\frac{1}{8} \left(-\frac{\frac{2 a}{e}+\frac{b c x \sqrt{c x-1} \sqrt{c x+1} \left(d+e x^2\right)}{d \left(c^2 d+e\right)}}{\left(d+e x^2\right)^2}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left(2 c^2 d+e\right) \tan ^{-1}\left(\frac{x \sqrt{c^2 (-d)-e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{d^{3/2} e \sqrt{c^2 x^2-1} \left(c^2 (-d)-e\right)^{3/2}}-\frac{2 b \cosh ^{-1}(c x)}{e \left(d+e x^2\right)^2}\right)","-\frac{a+b \cosh ^{-1}(c x)}{4 e \left(d+e x^2\right)^2}+\frac{b c \sqrt{c^2 x^2-1} \left(2 c^2 d+e\right) \tanh ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{8 d^{3/2} e \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right)^{3/2}}+\frac{b c x \left(1-c^2 x^2\right)}{8 d \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"(-(((2*a)/e + (b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))/(d*(c^2*d + e)))/(d + e*x^2)^2) - (2*b*ArcCosh[c*x])/(e*(d + e*x^2)^2) - (b*c*(2*c^2*d + e)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[(Sqrt[-(c^2*d) - e]*x)/(Sqrt[d]*Sqrt[-1 + c^2*x^2])])/(d^(3/2)*(-(c^2*d) - e)^(3/2)*e*Sqrt[-1 + c^2*x^2]))/8","A",1
509,0,0,772,8.336636,"\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x*(d + e*x^2)^3),x]","\int \frac{a+b \cosh ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 d^3}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 d^3}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 d^3}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 d^3}+\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b d^3}+\frac{\log \left(e^{-2 \cosh ^{-1}(c x)}+1\right) \left(a+b \cosh ^{-1}(c x)\right)}{d^3}+\frac{a+b \cosh ^{-1}(c x)}{2 d^2 \left(d+e x^2\right)}+\frac{a+b \cosh ^{-1}(c x)}{4 d \left(d+e x^2\right)^2}-\frac{b c \sqrt{c^2 x^2-1} \left(2 c^2 d+e\right) \tanh ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{8 d^{5/2} \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right)^{3/2}}-\frac{b c \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{x \sqrt{c^2 d+e}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{2 d^{5/2} \sqrt{c x-1} \sqrt{c x+1} \sqrt{c^2 d+e}}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 d^3}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{2 d^3}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 d^3}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 d^3}-\frac{b c e x \left(1-c^2 x^2\right)}{8 d^2 \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right) \left(d+e x^2\right)}-\frac{b \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right)}{2 d^3}",1,"Integrate[(a + b*ArcCosh[c*x])/(x*(d + e*x^2)^3), x]","F",-1
510,0,0,834,12.2128542,"\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)^3),x]","\int \frac{a+b \cosh ^{-1}(c x)}{x^3 \left(d+e x^2\right)^3} \, dx","\frac{b c x \left(1-c^2 x^2\right) e^2}{8 d^3 \left(d c^2+e\right) \sqrt{c x-1} \sqrt{c x+1} \left(e x^2+d\right)}-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)^2 e}{b d^4}-\frac{\left(a+b \cosh ^{-1}(c x)\right) e}{d^3 \left(e x^2+d\right)}-\frac{\left(a+b \cosh ^{-1}(c x)\right) e}{4 d^2 \left(e x^2+d\right)^2}+\frac{b c \left(2 d c^2+e\right) \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{\sqrt{d c^2+e} x}{\sqrt{d} \sqrt{c^2 x^2-1}}\right) e}{8 d^{7/2} \left(d c^2+e\right)^{3/2} \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{\sqrt{d c^2+e} x}{\sqrt{d} \sqrt{c^2 x^2-1}}\right) e}{d^{7/2} \sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(1+e^{-2 \cosh ^{-1}(c x)}\right) e}{d^4}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right) e}{2 d^4}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right) e}{2 d^4}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right) e}{2 d^4}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right) e}{2 d^4}+\frac{3 b \text{Li}_2\left(-e^{-2 \cosh ^{-1}(c x)}\right) e}{2 d^4}+\frac{3 b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right) e}{2 d^4}+\frac{3 b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right) e}{2 d^4}+\frac{3 b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right) e}{2 d^4}+\frac{3 b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right) e}{2 d^4}-\frac{a+b \cosh ^{-1}(c x)}{2 d^3 x^2}+\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{2 d^3 x}",1,"Integrate[(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)^3), x]","F",-1
511,1,1185,1224,6.9028623,"\int \frac{x^4 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]","-\frac{5 a x}{8 e^2 \left(e x^2+d\right)}+\frac{a d x}{4 e^2 \left(e x^2+d\right)^2}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 \sqrt{d} e^{5/2}}+b \left(-\frac{5 \left(\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{c \log \left(\frac{2 e \left(\sqrt{d} x c^2+i \sqrt{e}-i \sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{16 e^{5/2}}+\frac{5 \left(-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \log \left(\frac{2 e \left(-i \sqrt{d} x c^2-\sqrt{e}+\sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{16 e^{5/2}}+\frac{i \sqrt{d} \left(\frac{\sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(-\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c^3 \left(d+i \sqrt{e} x \sqrt{d}\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 e^2}-\frac{i \sqrt{d} \left(-\frac{\sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c^3 \left(d-i \sqrt{d} \sqrt{e} x\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}\right)}{16 e^2}+\frac{3 i \left(\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i c \sqrt{d}-\sqrt{-d c^2-e}}+1\right)+\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i \sqrt{d} c+\sqrt{-d c^2-e}}+1\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)}{32 \sqrt{d} e^{5/2}}-\frac{3 i \left(\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d c^2-e}-i c \sqrt{d}}+1\right)+\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)}{32 \sqrt{d} e^{5/2}}\right)","-\frac{b d \tanh ^{-1}\left(\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right) c^3}{8 \left(c \sqrt{-d}-\sqrt{e}\right)^{3/2} \left(\sqrt{-d} c+\sqrt{e}\right)^{3/2} e^{5/2}}+\frac{b d \tanh ^{-1}\left(\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right) c^3}{8 \left(c \sqrt{-d}-\sqrt{e}\right)^{3/2} \left(\sqrt{-d} c+\sqrt{e}\right)^{3/2} e^{5/2}}-\frac{5 b \tanh ^{-1}\left(\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right) c}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}+\frac{5 b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right) c}{8 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{5/2}}-\frac{b \sqrt{-d} \sqrt{c x-1} \sqrt{c x+1} c}{16 e^2 \left(d c^2+e\right) \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{b \sqrt{-d} \sqrt{c x-1} \sqrt{c x+1} c}{16 e^2 \left(d c^2+e\right) \left(\sqrt{e} x+\sqrt{-d}\right)}+\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{16 e^{5/2} \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{16 e^{5/2} \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{\sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right)}{16 e^{5/2} \left(\sqrt{-d}-\sqrt{e} x\right)^2}+\frac{\sqrt{-d} \left(a+b \cosh ^{-1}(c x)\right)}{16 e^{5/2} \left(\sqrt{e} x+\sqrt{-d}\right)^2}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\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*((-5*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/(16*e^(5/2)) + (5*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/(16*e^(5/2)) + ((I/16)*Sqrt[d]*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] - c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/e^2 - ((I/16)*Sqrt[d]*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/e^2 + (((3*I)/32)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]))/(Sqrt[d]*e^(5/2)) - (((3*I)/32)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/(Sqrt[d]*e^(5/2)))","C",0
512,1,1193,1234,6.7240923,"\int \frac{x^2 \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]","\frac{a x}{8 d e \left(e x^2+d\right)}-\frac{a x}{4 e \left(e x^2+d\right)^2}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{3/2} e^{3/2}}+b \left(\frac{\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{c \log \left(\frac{2 e \left(\sqrt{d} x c^2+i \sqrt{e}-i \sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}}{16 d e^{3/2}}-\frac{-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \log \left(\frac{2 e \left(-i \sqrt{d} x c^2-\sqrt{e}+\sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}}{16 d e^{3/2}}-\frac{i \left(\frac{\sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(-\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c^3 \left(d+i \sqrt{e} x \sqrt{d}\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 \sqrt{d} e}+\frac{i \left(-\frac{\sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c^3 \left(d-i \sqrt{d} \sqrt{e} x\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}\right)}{16 \sqrt{d} e}+\frac{i \left(\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i c \sqrt{d}-\sqrt{-d c^2-e}}+1\right)+\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i \sqrt{d} c+\sqrt{-d c^2-e}}+1\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)}{32 d^{3/2} e^{3/2}}-\frac{i \left(\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d c^2-e}-i c \sqrt{d}}+1\right)+\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)}{32 d^{3/2} e^{3/2}}\right)","\frac{b \tanh ^{-1}\left(\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right) c^3}{8 \left(c \sqrt{-d}-\sqrt{e}\right)^{3/2} \left(\sqrt{-d} c+\sqrt{e}\right)^{3/2} e^{3/2}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right) c^3}{8 \left(c \sqrt{-d}-\sqrt{e}\right)^{3/2} \left(\sqrt{-d} c+\sqrt{e}\right)^{3/2} e^{3/2}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right) c}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{3/2}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right) c}{8 d \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} e^{3/2}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} c}{16 \sqrt{-d} e \left(d c^2+e\right) \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{b \sqrt{c x-1} \sqrt{c x+1} c}{16 \sqrt{-d} e \left(d c^2+e\right) \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{a+b \cosh ^{-1}(c x)}{16 d e^{3/2} \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{a+b \cosh ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left(\sqrt{-d}-\sqrt{e} x\right)^2}+\frac{a+b \cosh ^{-1}(c x)}{16 \sqrt{-d} e^{3/2} \left(\sqrt{e} x+\sqrt{-d}\right)^2}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\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*((ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e])/(16*d*e^(3/2)) - (-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e])/(16*d*e^(3/2)) - ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] - c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + ((I/32)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]))/(d^(3/2)*e^(3/2)) - ((I/32)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/(d^(3/2)*e^(3/2)))","C",0
513,1,1184,1234,6.3945038,"\int \frac{a+b \cosh ^{-1}(c x)}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d + e*x^2)^3,x]","\frac{3 a x}{8 d^2 \left(e x^2+d\right)}+\frac{a x}{4 d \left(e x^2+d\right)^2}+\frac{3 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{8 d^{5/2} \sqrt{e}}+b \left(\frac{3 \left(\frac{\cosh ^{-1}(c x)}{\sqrt{e} x-i \sqrt{d}}+\frac{c \log \left(\frac{2 e \left(\sqrt{d} x c^2+i \sqrt{e}-i \sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{16 d^2 \sqrt{e}}-\frac{3 \left(-\frac{\cosh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}-\frac{c \log \left(\frac{2 e \left(-i \sqrt{d} x c^2-\sqrt{e}+\sqrt{-d c^2-e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c \sqrt{-d c^2-e} \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{16 d^2 \sqrt{e}}+\frac{i \left(\frac{\sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(-\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c^3 \left(d+i \sqrt{e} x \sqrt{d}\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left(d c^2+e\right) \left(\sqrt{e} x-i \sqrt{d}\right)}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x-i \sqrt{d}\right)^2}\right)}{16 d^{3/2}}-\frac{i \left(-\frac{\sqrt{d} \left(\log \left(\frac{e \sqrt{d c^2+e} \left(\sqrt{d} x c^2-i \sqrt{e}+\sqrt{d c^2+e} \sqrt{c x-1} \sqrt{c x+1}\right)}{c^3 \left(d-i \sqrt{d} \sqrt{e} x\right)}\right)+\log (4)\right) c^3}{\sqrt{e} \left(d c^2+e\right)^{3/2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} c}{\left(d c^2+e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}-\frac{\cosh ^{-1}(c x)}{\sqrt{e} \left(\sqrt{e} x+i \sqrt{d}\right)^2}\right)}{16 d^{3/2}}+\frac{3 i \left(\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i c \sqrt{d}-\sqrt{-d c^2-e}}+1\right)+\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{i \sqrt{d} c+\sqrt{-d c^2-e}}+1\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)}{32 d^{5/2} \sqrt{e}}-\frac{3 i \left(\cosh ^{-1}(c x) \left(2 \left(\log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d c^2-e}-i c \sqrt{d}}+1\right)+\log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)-\cosh ^{-1}(c x)\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{-d c^2-e}}\right)\right)}{32 d^{5/2} \sqrt{e}}\right)","-\frac{b \tanh ^{-1}\left(\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right) c^3}{8 d \left(c \sqrt{-d}-\sqrt{e}\right)^{3/2} \left(\sqrt{-d} c+\sqrt{e}\right)^{3/2} \sqrt{e}}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right) c^3}{8 d \left(c \sqrt{-d}-\sqrt{e}\right)^{3/2} \left(\sqrt{-d} c+\sqrt{e}\right)^{3/2} \sqrt{e}}+\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x+1}}{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x-1}}\right) c}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{e}}-\frac{3 b \tanh ^{-1}\left(\frac{\sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{c x+1}}{\sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{c x-1}}\right) c}{8 d^2 \sqrt{c \sqrt{-d}-\sqrt{e}} \sqrt{\sqrt{-d} c+\sqrt{e}} \sqrt{e}}-\frac{b \sqrt{c x-1} \sqrt{c x+1} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\sqrt{-d}-\sqrt{e} x\right)}-\frac{b \sqrt{c x-1} \sqrt{c x+1} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{16 d^2 \sqrt{e} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{16 d^2 \sqrt{e} \left(\sqrt{e} x+\sqrt{-d}\right)}-\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left(\sqrt{-d}-\sqrt{e} x\right)^2}+\frac{a+b \cosh ^{-1}(c x)}{16 (-d)^{3/2} \sqrt{e} \left(\sqrt{e} x+\sqrt{-d}\right)^2}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{-d c^2-e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \cosh ^{-1}(c x)\right) \log \left(\frac{e^{\cosh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{-d c^2-e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 b \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 b \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\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*((3*(ArcCosh[c*x]/((-I)*Sqrt[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/(16*d^2*Sqrt[e]) - (3*(-(ArcCosh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x))])/Sqrt[-(c^2*d) - e]))/(16*d^2*Sqrt[e]) + ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] - c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/d^(3/2) - ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/d^(3/2) + (((3*I)/32)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))]))/(d^(5/2)*Sqrt[e]) - (((3*I)/32)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e])] + Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e])]))/(d^(5/2)*Sqrt[e]))","C",0
514,0,0,23,5.7799187,"\int \sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x]),x]","\int \sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right),x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x]), x]","A",-1
515,0,0,23,3.8087719,"\int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/Sqrt[d + e*x^2],x]","\int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \cosh ^{-1}(c x)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])/Sqrt[d + e*x^2], x]","A",-1
516,1,556,101,3.3579446,"\int \frac{a+b \cosh ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d + e*x^2)^(3/2),x]","\frac{a x+\frac{2 b (c x-1)^{3/2} \sqrt{\frac{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x-1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(c \sqrt{d} \left(-c \sqrt{d}+i \sqrt{e}\right) \sqrt{\frac{\left(c^2 d+e\right) \left(d+e x^2\right)}{d e (c x-1)^2}} \sqrt{-\frac{c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}} \Pi \left(\frac{2 c \sqrt{d}}{\sqrt{d} c+i \sqrt{e}};\sin ^{-1}\left(\sqrt{-\frac{\frac{i \sqrt{e} x}{\sqrt{d}}+c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)-1}{2-2 c x}}\right)|\frac{4 i c \sqrt{d} \sqrt{e}}{\left(\sqrt{d} c+i \sqrt{e}\right)^2}\right)+\frac{c \left(\sqrt{e}-i c \sqrt{d}\right) \left(\sqrt{e} x+i \sqrt{d}\right) \sqrt{\frac{\frac{i c \sqrt{d}}{\sqrt{e}}+c (-x)+\frac{i \sqrt{e} x}{\sqrt{d}}+1}{1-c x}} F\left(\sin ^{-1}\left(\sqrt{-\frac{\frac{i \sqrt{e} x}{\sqrt{d}}+c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)-1}{2-2 c x}}\right)|\frac{4 i c \sqrt{d} \sqrt{e}}{\left(\sqrt{d} c+i \sqrt{e}\right)^2}\right)}{c x-1}\right)}{c \sqrt{c x+1} \left(c^2 d+e\right) \sqrt{-\frac{c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}}}+b x \cosh ^{-1}(c x)}{d \sqrt{d+e x^2}}","\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{d \sqrt{d+e x^2}}-\frac{b \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{d \sqrt{e} \sqrt{c x-1} \sqrt{c x+1}}",1,"(a*x + b*x*ArcCosh[c*x] + (2*b*(-1 + c*x)^(3/2)*Sqrt[((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(-1 + c*x))]*((c*((-I)*c*Sqrt[d] + Sqrt[e])*(I*Sqrt[d] + Sqrt[e]*x)*Sqrt[(1 + (I*c*Sqrt[d])/Sqrt[e] - c*x + (I*Sqrt[e]*x)/Sqrt[d])/(1 - c*x)]*EllipticF[ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2])/(-1 + c*x) + c*Sqrt[d]*(-(c*Sqrt[d]) + I*Sqrt[e])*Sqrt[((c^2*d + e)*(d + e*x^2))/(d*e*(-1 + c*x)^2)]*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]*EllipticPi[(2*c*Sqrt[d])/(c*Sqrt[d] + I*Sqrt[e]), ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2]))/(c*(c^2*d + e)*Sqrt[1 + c*x]*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]))/(d*Sqrt[d + e*x^2])","C",1
517,1,633,182,2.4074855,"\int \frac{a+b \cosh ^{-1}(c x)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d + e*x^2)^(5/2),x]","\frac{\frac{a x \left(3 d+2 e x^2\right)}{d^2}+\frac{4 b (c x-1)^{3/2} \left(d+e x^2\right) \sqrt{\frac{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x-1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(c \sqrt{d} \left(-c \sqrt{d}+i \sqrt{e}\right) \sqrt{\frac{\left(c^2 d+e\right) \left(d+e x^2\right)}{d e (c x-1)^2}} \sqrt{-\frac{c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}} \Pi \left(\frac{2 c \sqrt{d}}{\sqrt{d} c+i \sqrt{e}};\sin ^{-1}\left(\sqrt{-\frac{\frac{i \sqrt{e} x}{\sqrt{d}}+c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)-1}{2-2 c x}}\right)|\frac{4 i c \sqrt{d} \sqrt{e}}{\left(\sqrt{d} c+i \sqrt{e}\right)^2}\right)+\frac{c \left(\sqrt{e}-i c \sqrt{d}\right) \left(\sqrt{e} x+i \sqrt{d}\right) \sqrt{\frac{\frac{i c \sqrt{d}}{\sqrt{e}}+c (-x)+\frac{i \sqrt{e} x}{\sqrt{d}}+1}{1-c x}} F\left(\sin ^{-1}\left(\sqrt{-\frac{\frac{i \sqrt{e} x}{\sqrt{d}}+c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)-1}{2-2 c x}}\right)|\frac{4 i c \sqrt{d} \sqrt{e}}{\left(\sqrt{d} c+i \sqrt{e}\right)^2}\right)}{c x-1}\right)}{c d^2 \sqrt{c x+1} \left(c^2 d+e\right) \sqrt{-\frac{c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}}}-\frac{b c \sqrt{c x+1} \sqrt{c x-1} \left(d+e x^2\right)}{d \left(c^2 d+e\right)}+\frac{b x \cosh ^{-1}(c x) \left(3 d+2 e x^2\right)}{d^2}}{3 \left(d+e x^2\right)^{3/2}}","\frac{2 x \left(a+b \cosh ^{-1}(c x)\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{2 b \sqrt{1-c^2 x^2} \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{1-c^2 x^2}}{c \sqrt{d+e x^2}}\right)}{3 d^2 \sqrt{e} \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{c x-1} \sqrt{c x+1}}{3 d \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(-((b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2))/(d*(c^2*d + e))) + (a*x*(3*d + 2*e*x^2))/d^2 + (b*x*(3*d + 2*e*x^2)*ArcCosh[c*x])/d^2 + (4*b*(-1 + c*x)^(3/2)*Sqrt[((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(-1 + c*x))]*(d + e*x^2)*((c*((-I)*c*Sqrt[d] + Sqrt[e])*(I*Sqrt[d] + Sqrt[e]*x)*Sqrt[(1 + (I*c*Sqrt[d])/Sqrt[e] - c*x + (I*Sqrt[e]*x)/Sqrt[d])/(1 - c*x)]*EllipticF[ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2])/(-1 + c*x) + c*Sqrt[d]*(-(c*Sqrt[d]) + I*Sqrt[e])*Sqrt[((c^2*d + e)*(d + e*x^2))/(d*e*(-1 + c*x)^2)]*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]*EllipticPi[(2*c*Sqrt[d])/(c*Sqrt[d] + I*Sqrt[e]), ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2]))/(c*d^2*(c^2*d + e)*Sqrt[1 + c*x]*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]))/(3*(d + e*x^2)^(3/2))","C",1
518,1,685,284,4.2382543,"\int \frac{a+b \cosh ^{-1}(c x)}{\left(d+e x^2\right)^{7/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])/(d + e*x^2)^(7/2),x]","\frac{\frac{a x \left(15 d^2+20 d e x^2+8 e^2 x^4\right)}{d^3}+\frac{16 b (c x-1)^{3/2} \left(d+e x^2\right)^2 \sqrt{\frac{(c x+1) \left(c \sqrt{d}-i \sqrt{e}\right)}{(c x-1) \left(c \sqrt{d}+i \sqrt{e}\right)}} \left(c \sqrt{d} \left(-c \sqrt{d}+i \sqrt{e}\right) \sqrt{\frac{\left(c^2 d+e\right) \left(d+e x^2\right)}{d e (c x-1)^2}} \sqrt{-\frac{c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}} \Pi \left(\frac{2 c \sqrt{d}}{\sqrt{d} c+i \sqrt{e}};\sin ^{-1}\left(\sqrt{-\frac{\frac{i \sqrt{e} x}{\sqrt{d}}+c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)-1}{2-2 c x}}\right)|\frac{4 i c \sqrt{d} \sqrt{e}}{\left(\sqrt{d} c+i \sqrt{e}\right)^2}\right)+\frac{c \left(\sqrt{e}-i c \sqrt{d}\right) \left(\sqrt{e} x+i \sqrt{d}\right) \sqrt{\frac{\frac{i c \sqrt{d}}{\sqrt{e}}+c (-x)+\frac{i \sqrt{e} x}{\sqrt{d}}+1}{1-c x}} F\left(\sin ^{-1}\left(\sqrt{-\frac{\frac{i \sqrt{e} x}{\sqrt{d}}+c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)-1}{2-2 c x}}\right)|\frac{4 i c \sqrt{d} \sqrt{e}}{\left(\sqrt{d} c+i \sqrt{e}\right)^2}\right)}{c x-1}\right)}{c d^3 \sqrt{c x+1} \left(c^2 d+e\right) \sqrt{-\frac{c \left(x+\frac{i \sqrt{d}}{\sqrt{e}}\right)+\frac{i \sqrt{e} x}{\sqrt{d}}-1}{1-c x}}}-\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left(d+e x^2\right) \left(c^2 d \left(7 d+6 e x^2\right)+e \left(5 d+4 e x^2\right)\right)}{d^2 \left(c^2 d+e\right)^2}+\frac{b x \cosh ^{-1}(c x) \left(15 d^2+20 d e x^2+8 e^2 x^4\right)}{d^3}}{15 \left(d+e x^2\right)^{5/2}}","\frac{8 x \left(a+b \cosh ^{-1}(c x)\right)}{15 d^3 \sqrt{d+e x^2}}+\frac{4 x \left(a+b \cosh ^{-1}(c x)\right)}{15 d^2 \left(d+e x^2\right)^{3/2}}+\frac{x \left(a+b \cosh ^{-1}(c x)\right)}{5 d \left(d+e x^2\right)^{5/2}}-\frac{8 b \sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{15 d^3 \sqrt{e} \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b c \left(1-c^2 x^2\right) \left(3 c^2 d+2 e\right)}{15 d^2 \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right)^2 \sqrt{d+e x^2}}+\frac{b c \left(1-c^2 x^2\right)}{15 d \sqrt{c x-1} \sqrt{c x+1} \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}",1,"((a*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4))/d^3 - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x^2)*(e*(5*d + 4*e*x^2) + c^2*d*(7*d + 6*e*x^2)))/(d^2*(c^2*d + e)^2) + (b*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4)*ArcCosh[c*x])/d^3 + (16*b*(-1 + c*x)^(3/2)*Sqrt[((c*Sqrt[d] - I*Sqrt[e])*(1 + c*x))/((c*Sqrt[d] + I*Sqrt[e])*(-1 + c*x))]*(d + e*x^2)^2*((c*((-I)*c*Sqrt[d] + Sqrt[e])*(I*Sqrt[d] + Sqrt[e]*x)*Sqrt[(1 + (I*c*Sqrt[d])/Sqrt[e] - c*x + (I*Sqrt[e]*x)/Sqrt[d])/(1 - c*x)]*EllipticF[ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2])/(-1 + c*x) + c*Sqrt[d]*(-(c*Sqrt[d]) + I*Sqrt[e])*Sqrt[((c^2*d + e)*(d + e*x^2))/(d*e*(-1 + c*x)^2)]*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]*EllipticPi[(2*c*Sqrt[d])/(c*Sqrt[d] + I*Sqrt[e]), ArcSin[Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(2 - 2*c*x))]], ((4*I)*c*Sqrt[d]*Sqrt[e])/(c*Sqrt[d] + I*Sqrt[e])^2]))/(c*d^3*(c^2*d + e)*Sqrt[1 + c*x]*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]))/(15*(d + e*x^2)^(5/2))","C",1
519,1,397,558,1.4171248,"\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcCosh[c*x]),x]","x (f x)^m \left(\frac{d^3 \left(a+b \cosh ^{-1}(c x)\right)}{m+1}+\frac{3 d^2 e x^2 \left(a+b \cosh ^{-1}(c x)\right)}{m+3}+\frac{3 d e^2 x^4 \left(a+b \cosh ^{-1}(c x)\right)}{m+5}+\frac{e^3 x^6 \left(a+b \cosh ^{-1}(c x)\right)}{m+7}-\frac{b c d^3 x \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{\left(m^2+3 m+2\right) \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c d^2 e x^3 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};c^2 x^2\right)}{\left(m^2+7 m+12\right) \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 b c d e^2 x^5 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+6}{2};\frac{m+8}{2};c^2 x^2\right)}{(m+5) (m+6) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c e^3 x^7 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m}{2}+4;\frac{m}{2}+5;c^2 x^2\right)}{(m+7) (m+8) \sqrt{c x-1} \sqrt{c x+1}}\right)","\frac{d^3 (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+1)}+\frac{3 d^2 e (f x)^{m+3} \left(a+b \cosh ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{3 d e^2 (f x)^{m+5} \left(a+b \cosh ^{-1}(c x)\right)}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left(a+b \cosh ^{-1}(c x)\right)}{f^7 (m+7)}+\frac{b e^3 \left(1-c^2 x^2\right) (f x)^{m+6}}{c f^6 (m+7)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e^2 \left(1-c^2 x^2\right) (f x)^{m+4} \left(3 c^2 d (m+7)^2+e \left(m^2+11 m+30\right)\right)}{c^3 f^4 (m+5)^2 (m+7)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e \left(1-c^2 x^2\right) (f x)^{m+2} \left(3 c^4 d^2 \left(m^2+12 m+35\right)^2+3 c^2 d e (m+7)^2 \left(m^2+7 m+12\right)+e^2 \left(m^4+18 m^3+119 m^2+342 m+360\right)\right)}{c^5 f^2 (m+3)^2 (m+5)^2 (m+7)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{1-c^2 x^2} (f x)^{m+2} \left(\frac{c^6 d^3 (m+3) (m+5) (m+7)}{m+1}+\frac{e (m+2) \left(3 c^4 d^2 \left(m^2+12 m+35\right)^2+3 c^2 d e (m+7)^2 \left(m^2+7 m+12\right)+e^2 \left(m^4+18 m^3+119 m^2+342 m+360\right)\right)}{(m+3) (m+5) (m+7)}\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{c^5 f^2 (m+2) (m+3) (m+5) (m+7) \sqrt{c x-1} \sqrt{c x+1}}",1,"x*(f*x)^m*((d^3*(a + b*ArcCosh[c*x]))/(1 + m) + (3*d^2*e*x^2*(a + b*ArcCosh[c*x]))/(3 + m) + (3*d*e^2*x^4*(a + b*ArcCosh[c*x]))/(5 + m) + (e^3*x^6*(a + b*ArcCosh[c*x]))/(7 + m) - (b*c*e^3*x^7*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, 4 + m/2, 5 + m/2, c^2*x^2])/((7 + m)*(8 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^3*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c*d^2*e*x^3*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((12 + 7*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c*d*e^2*x^5*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (6 + m)/2, (8 + m)/2, c^2*x^2])/((5 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))","A",1
520,1,293,353,0.4851929,"\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcCosh[c*x]),x]","x (f x)^m \left(\frac{d^2 \left(a+b \cosh ^{-1}(c x)\right)}{m+1}+\frac{2 d e x^2 \left(a+b \cosh ^{-1}(c x)\right)}{m+3}+\frac{e^2 x^4 \left(a+b \cosh ^{-1}(c x)\right)}{m+5}-\frac{b c d^2 x \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{\left(m^2+3 m+2\right) \sqrt{c x-1} \sqrt{c x+1}}-\frac{2 b c d e x^3 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};c^2 x^2\right)}{\left(m^2+7 m+12\right) \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c e^2 x^5 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+6}{2};\frac{m+8}{2};c^2 x^2\right)}{(m+5) (m+6) \sqrt{c x-1} \sqrt{c x+1}}\right)","\frac{d^2 (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left(a+b \cosh ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left(a+b \cosh ^{-1}(c x)\right)}{f^5 (m+5)}+\frac{b e^2 \left(1-c^2 x^2\right) (f x)^{m+4}}{c f^4 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b e \left(1-c^2 x^2\right) (f x)^{m+2} \left(2 c^2 d (m+5)^2+e \left(m^2+7 m+12\right)\right)}{c^3 f^2 (m+3)^2 (m+5)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b \sqrt{1-c^2 x^2} (f x)^{m+2} \left(\frac{c^4 d^2 (m+3) (m+5)}{m+1}+\frac{e (m+2) \left(2 c^2 d (m+5)^2+e \left(m^2+7 m+12\right)\right)}{(m+3) (m+5)}\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{c^3 f^2 (m+2) (m+3) (m+5) \sqrt{c x-1} \sqrt{c x+1}}",1,"x*(f*x)^m*((d^2*(a + b*ArcCosh[c*x]))/(1 + m) + (2*d*e*x^2*(a + b*ArcCosh[c*x]))/(3 + m) + (e^2*x^4*(a + b*ArcCosh[c*x]))/(5 + m) - (b*c*d^2*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c*d*e*x^3*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((12 + 7*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*e^2*x^5*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (6 + m)/2, (8 + m)/2, c^2*x^2])/((5 + m)*(6 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))","A",1
521,1,186,198,0.6065878,"\int (f x)^m \left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcCosh[c*x]),x]","x (f x)^m \left(\frac{\frac{\left(d (m+3)+e (m+1) x^2\right) \left(a+b \cosh ^{-1}(c x)\right)}{m+1}-\frac{b c e x^3 \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};c^2 x^2\right)}{(m+4) \sqrt{c x-1} \sqrt{c x+1}}}{m+3}-\frac{b c d x \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{\left(m^2+3 m+2\right) \sqrt{c x-1} \sqrt{c x+1}}\right)","\frac{d (f x)^{m+1} \left(a+b \cosh ^{-1}(c x)\right)}{f (m+1)}+\frac{e (f x)^{m+3} \left(a+b \cosh ^{-1}(c x)\right)}{f^3 (m+3)}-\frac{b \sqrt{1-c^2 x^2} (f x)^{m+2} \left(c^2 d (m+3)^2+e (m+1) (m+2)\right) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};c^2 x^2\right)}{c f^2 (m+1) (m+2) (m+3)^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b e \sqrt{c x-1} \sqrt{c x+1} (f x)^{m+2}}{c f^2 (m+3)^2}",1,"x*(f*x)^m*(-((b*c*d*x*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, c^2*x^2])/((2 + 3*m + m^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (((d*(3 + m) + e*(1 + m)*x^2)*(a + b*ArcCosh[c*x]))/(1 + m) - (b*c*e*x^3*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, c^2*x^2])/((4 + m)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(3 + m))","A",1
522,0,0,26,9.9938923,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d + e*x^2),x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{d+e x^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{d+e x^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d + e*x^2), x]","A",-1
523,0,0,26,9.3756988,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2,x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d + e*x^2)^2, x]","A",-1
524,0,0,26,20.3983193,"\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3,x]","\int \frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \cosh ^{-1}(c x)\right)}{\left(d+e x^2\right)^3},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCosh[c*x]))/(d + e*x^2)^3, x]","A",-1
525,1,453,609,0.846927,"\int \left(d+e x^2\right)^3 \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)^3*(a + b*ArcCosh[c*x])^2,x]","\frac{11025 a^2 c^7 x \left(35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right)-210 a b \sqrt{c x-1} \sqrt{c x+1} \left(c^6 \left(3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right)+2 c^4 e \left(1225 d^2+294 d e x^2+45 e^2 x^4\right)+24 c^2 e^2 \left(49 d+5 e x^2\right)+240 e^3\right)-210 b \cosh ^{-1}(c x) \left(b \sqrt{c x-1} \sqrt{c x+1} \left(c^6 \left(3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right)+2 c^4 e \left(1225 d^2+294 d e x^2+45 e^2 x^4\right)+24 c^2 e^2 \left(49 d+5 e x^2\right)+240 e^3\right)-105 a c^7 x \left(35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right)\right)+11025 b^2 c^7 x \cosh ^{-1}(c x)^2 \left(35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right)+2 b^2 c x \left(c^6 \left(385875 d^3+42875 d^2 e x^2+9261 d e^2 x^4+1125 e^3 x^6\right)+210 c^4 e \left(1225 d^2+98 d e x^2+9 e^2 x^4\right)+840 c^2 e^2 \left(147 d+5 e x^2\right)+25200 e^3\right)}{385875 c^7}","-\frac{32 b e^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{245 c^7}-\frac{16 b d e^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{25 c^5}-\frac{16 b e^3 x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{245 c^5}-\frac{4 b d^2 e \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 c^3}-\frac{8 b d e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{25 c^3}-\frac{12 b e^3 x^4 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{245 c^3}+d^3 x \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 b d^3 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{c}+d^2 e x^3 \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 b d^2 e x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{3 c}+\frac{3}{5} d e^2 x^5 \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{6 b d e^2 x^4 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{25 c}+\frac{1}{7} e^3 x^7 \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 b e^3 x^6 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{49 c}+\frac{32 b^2 e^3 x}{245 c^6}+\frac{16 b^2 d e^2 x}{25 c^4}+\frac{16 b^2 e^3 x^3}{735 c^4}+\frac{4 b^2 d^2 e x}{3 c^2}+\frac{8 b^2 d e^2 x^3}{75 c^2}+\frac{12 b^2 e^3 x^5}{1225 c^2}+2 b^2 d^3 x+\frac{2}{9} b^2 d^2 e x^3+\frac{6}{125} b^2 d e^2 x^5+\frac{2}{343} b^2 e^3 x^7",1,"(11025*a^2*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) - 210*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e^3*x^6)) + 2*b^2*c*x*(25200*e^3 + 840*c^2*e^2*(147*d + 5*e*x^2) + 210*c^4*e*(1225*d^2 + 98*d*e*x^2 + 9*e^2*x^4) + c^6*(385875*d^3 + 42875*d^2*e*x^2 + 9261*d*e^2*x^4 + 1125*e^3*x^6)) - 210*b*(-105*a*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(240*e^3 + 24*c^2*e^2*(49*d + 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 + 75*e^3*x^6)))*ArcCosh[c*x] + 11025*b^2*c^7*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcCosh[c*x]^2)/(385875*c^7)","A",1
526,1,299,359,0.5411459,"\int \left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcCosh[c*x])^2,x]","\frac{225 a^2 c^5 x \left(15 d^2+10 d e x^2+3 e^2 x^4\right)-30 a b \sqrt{c x-1} \sqrt{c x+1} \left(c^4 \left(225 d^2+50 d e x^2+9 e^2 x^4\right)+4 c^2 e \left(25 d+3 e x^2\right)+24 e^2\right)-30 b \cosh ^{-1}(c x) \left(b \sqrt{c x-1} \sqrt{c x+1} \left(c^4 \left(225 d^2+50 d e x^2+9 e^2 x^4\right)+4 c^2 e \left(25 d+3 e x^2\right)+24 e^2\right)-15 a c^5 x \left(15 d^2+10 d e x^2+3 e^2 x^4\right)\right)+225 b^2 c^5 x \cosh ^{-1}(c x)^2 \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+2 b^2 c x \left(c^4 \left(3375 d^2+250 d e x^2+27 e^2 x^4\right)+60 c^2 e \left(25 d+e x^2\right)+360 e^2\right)}{3375 c^5}","-\frac{16 b e^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{75 c^5}-\frac{8 b d e \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{9 c^3}-\frac{8 b e^2 x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{75 c^3}+d^2 x \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 b d^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{c}+\frac{2}{3} d e x^3 \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{4 b d e x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{9 c}+\frac{1}{5} e^2 x^5 \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 b e^2 x^4 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{25 c}+\frac{16 b^2 e^2 x}{75 c^4}+\frac{8 b^2 d e x}{9 c^2}+\frac{8 b^2 e^2 x^3}{225 c^2}+2 b^2 d^2 x+\frac{4}{27} b^2 d e x^3+\frac{2}{125} b^2 e^2 x^5",1,"(225*a^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - 30*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)) + 2*b^2*c*x*(360*e^2 + 60*c^2*e*(25*d + e*x^2) + c^4*(3375*d^2 + 250*d*e*x^2 + 27*e^2*x^4)) - 30*b*(-15*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(24*e^2 + 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))*ArcCosh[c*x] + 225*b^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcCosh[c*x]^2)/(3375*c^5)","A",1
527,1,174,168,0.2807159,"\int \left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)*(a + b*ArcCosh[c*x])^2,x]","\frac{9 a^2 c^3 x \left(3 d+e x^2\right)-6 a b \sqrt{c x-1} \sqrt{c x+1} \left(c^2 \left(9 d+e x^2\right)+2 e\right)-6 b \cosh ^{-1}(c x) \left(b \sqrt{c x-1} \sqrt{c x+1} \left(c^2 \left(9 d+e x^2\right)+2 e\right)-3 a c^3 x \left(3 d+e x^2\right)\right)+9 b^2 c^3 x \cosh ^{-1}(c x)^2 \left(3 d+e x^2\right)+2 b^2 c x \left(c^2 \left(27 d+e x^2\right)+6 e\right)}{27 c^3}","-\frac{4 b e \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{9 c^3}+d x \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 b d \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{c}+\frac{1}{3} e x^3 \left(a+b \cosh ^{-1}(c x)\right)^2-\frac{2 b e x^2 \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{9 c}+\frac{4 b^2 e x}{9 c^2}+2 b^2 d x+\frac{2}{27} b^2 e x^3",1,"(9*a^2*c^3*x*(3*d + e*x^2) - 6*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2*(9*d + e*x^2)) + 2*b^2*c*x*(6*e + c^2*(27*d + e*x^2)) - 6*b*(-3*a*c^3*x*(3*d + e*x^2) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2*e + c^2*(9*d + e*x^2)))*ArcCosh[c*x] + 9*b^2*c^3*x*(3*d + e*x^2)*ArcCosh[c*x]^2)/(27*c^3)","A",1
528,1,84,51,0.0838662,"\int \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcCosh[c*x])^2,x]","x \left(a^2+2 b^2\right)-\frac{2 a b \sqrt{c x-1} \sqrt{c x+1}}{c}+\frac{2 b \cosh ^{-1}(c x) \left(a c x-b \sqrt{c x-1} \sqrt{c x+1}\right)}{c}+b^2 x \cosh ^{-1}(c x)^2","-\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \left(a+b \cosh ^{-1}(c x)\right)}{c}+x \left(a+b \cosh ^{-1}(c x)\right)^2+2 b^2 x",1,"(a^2 + 2*b^2)*x - (2*a*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c + (2*b*(a*c*x - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])*ArcCosh[c*x])/c + b^2*x*ArcCosh[c*x]^2","A",1
529,1,623,763,0.5746724,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2),x]","\frac{2 b \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)-2 b \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-c \sqrt{-d}}\right)-2 b \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)+2 b \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)-\left(a+b \cosh ^{-1}(c x)\right)^2 \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)+\left(a+b \cosh ^{-1}(c x)\right)^2 \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}-c \sqrt{-d}}+1\right)+\left(a+b \cosh ^{-1}(c x)\right)^2 \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)-\left(a+b \cosh ^{-1}(c x)\right)^2 \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)-2 b^2 \text{Li}_3\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)+2 b^2 \text{Li}_3\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d c^2-e}-c \sqrt{-d}}\right)+2 b^2 \text{Li}_3\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)-2 b^2 \text{Li}_3\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{2 \sqrt{-d} \sqrt{e}}","-\frac{b \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{\sqrt{-d} \sqrt{e}}+\frac{b \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{\sqrt{-d} \sqrt{e}}-\frac{b \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{\sqrt{-d} \sqrt{e}}+\frac{b \left(a+b \cosh ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{\sqrt{-d} \sqrt{e}}+\frac{\left(a+b \cosh ^{-1}(c x)\right)^2 \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2 \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{c^2 (-d)-e}}+1\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \cosh ^{-1}(c x)\right)^2 \log \left(1-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2 \log \left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{c^2 (-d)-e}+c \sqrt{-d}}+1\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{-d c^2-e}}\right)}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left(-\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left(\frac{\sqrt{e} e^{\cosh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{-d c^2-e}}\right)}{\sqrt{-d} \sqrt{e}}",1,"(-((a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])]) + (a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + (a + b*ArcCosh[c*x])^2*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] - (a + b*ArcCosh[c*x])^2*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] + 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])] - 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] - 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))] + 2*b*(a + b*ArcCosh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])] - 2*b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])] + 2*b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) - e])] + 2*b^2*PolyLog[3, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e]))] - 2*b^2*PolyLog[3, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*Sqrt[-d]*Sqrt[e])","A",1
530,0,0,25,15.8575432,"\int \sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^2,x]","\int \sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right)^2 \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right)^2,x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^2, x]","A",-1
531,0,0,25,11.4170542,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/Sqrt[d + e*x^2],x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^2/Sqrt[d + e*x^2], x]","A",-1
532,0,0,25,3.7660739,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(3/2),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(3/2), x]","A",-1
533,0,0,25,7.5291889,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^2/(d + e*x^2)^(5/2), x]","A",-1
534,1,254,388,0.534999,"\int \frac{\left(d+e x^2\right)^2}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)^2/(a + b*ArcCosh[c*x]),x]","\frac{16 c^4 d^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-e \sinh \left(\frac{3 a}{b}\right) \left(8 c^2 d+3 e\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+8 c^2 d e \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+8 c^2 d e \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 \sinh \left(\frac{a}{b}\right) \left(8 c^4 d^2+4 c^2 d e+e^2\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-e^2 \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+2 e^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 e^2 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+e^2 \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)}{16 b c^5}","-\frac{e^2 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b c^5}-\frac{3 e^2 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}-\frac{e^2 \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{e^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b c^5}+\frac{3 e^2 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{e^2 \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}-\frac{d e \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 b c^3}-\frac{d e \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b c^3}+\frac{d e \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 b c^3}+\frac{d e \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b c^3}-\frac{d^2 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b c}+\frac{d^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b c}",1,"(-2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - e*(8*c^2*d + 3*e)*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - e^2*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] + 16*c^4*d^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 8*c^2*d*e*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 2*e^2*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 8*c^2*d*e*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 3*e^2*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + e^2*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])])/(16*b*c^5)","A",1
535,1,125,139,0.2261165,"\int \frac{d+e x^2}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)/(a + b*ArcCosh[c*x]),x]","\frac{-\sinh \left(\frac{a}{b}\right) \left(4 c^2 d+e\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+4 c^2 d \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-e \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+e \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+e \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)}{4 b c^3}","-\frac{e \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b c^3}+\frac{e \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b c^3}-\frac{\sinh \left(\frac{a}{b}\right) \left(4 c^2 d+e\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b c^3}+\frac{\cosh \left(\frac{a}{b}\right) \left(4 c^2 d+e\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b c^3}",1,"(-((4*c^2*d + e)*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b]) - e*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] + 4*c^2*d*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + e*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + e*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])])/(4*b*c^3)","A",1
536,1,46,54,0.0643836,"\int \frac{1}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(a + b*ArcCosh[c*x])^(-1),x]","-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)}{b c}","\frac{\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b c}-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b c}",1,"-((CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(b*c))","A",1
537,0,0,23,0.6034342,"\int \frac{1}{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)*(a + b*ArcCosh[c*x])), x]","A",-1
538,0,0,23,3.3023517,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])), x]","A",-1
539,0,0,25,1.178695,"\int \frac{\sqrt{d+e x^2}}{a+b \cosh ^{-1}(c x)} \, dx","Integrate[Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x]),x]","\int \frac{\sqrt{d+e x^2}}{a+b \cosh ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2}}{a+b \cosh ^{-1}(c x)},x\right)",0,"Integrate[Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x]), x]","A",-1
540,0,0,25,1.1057442,"\int \frac{1}{\sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{\sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])), x]","A",-1
541,0,0,25,1.5624799,"\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])), x]","A",-1
542,0,0,25,3.9215304,"\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcCosh[c*x])),x]","\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcCosh[c*x])), x]","A",-1
543,1,663,510,2.0582853,"\int \frac{\left(d+e x^2\right)^2}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x^2)^2/(a + b*ArcCosh[c*x])^2,x]","-\frac{16 a c^4 d^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+16 b c^4 d^2 \sinh \left(\frac{a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-3 e \cosh \left(\frac{3 a}{b}\right) \left(8 c^2 d+3 e\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+8 a c^2 d e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+8 b c^2 d e \sinh \left(\frac{a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+24 a c^2 d e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+24 b c^2 d e \sinh \left(\frac{3 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-2 \cosh \left(\frac{a}{b}\right) \left(8 c^4 d^2+4 c^2 d e+e^2\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-5 a e^2 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)-5 b e^2 \cosh \left(\frac{5 a}{b}\right) \cosh ^{-1}(c x) \text{Chi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+2 a e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+2 b e^2 \sinh \left(\frac{a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+9 a e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+9 b e^2 \sinh \left(\frac{3 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+5 a e^2 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+5 b e^2 \sinh \left(\frac{5 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(5 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+16 b c^5 d^2 x \sqrt{\frac{c x-1}{c x+1}}+32 b c^5 d e x^3 \sqrt{\frac{c x-1}{c x+1}}+16 b c^5 e^2 x^5 \sqrt{\frac{c x-1}{c x+1}}+16 b c^4 d^2 \sqrt{\frac{c x-1}{c x+1}}+32 b c^4 d e x^2 \sqrt{\frac{c x-1}{c x+1}}+16 b c^4 e^2 x^4 \sqrt{\frac{c x-1}{c x+1}}}{16 b^2 c^5 \left(a+b \cosh ^{-1}(c x)\right)}","\frac{e^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 c^5}+\frac{9 e^2 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}+\frac{5 e^2 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}-\frac{e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{8 b^2 c^5}-\frac{9 e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}-\frac{5 e^2 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}+\frac{d e \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 b^2 c^3}+\frac{3 d e \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}-\frac{d e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{2 b^2 c^3}-\frac{3 d e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}+\frac{d^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \left(a+b \cosh ^{-1}(c x)\right)}-\frac{2 d e x^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \left(a+b \cosh ^{-1}(c x)\right)}-\frac{e^2 x^4 \sqrt{c x-1} \sqrt{c x+1}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"-1/16*(16*b*c^4*d^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 16*b*c^5*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 32*b*c^4*d*e*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 32*b*c^5*d*e*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] + 16*b*c^4*e^2*x^4*Sqrt[(-1 + c*x)/(1 + c*x)] + 16*b*c^5*e^2*x^5*Sqrt[(-1 + c*x)/(1 + c*x)] - 2*(8*c^4*d^2 + 4*c^2*d*e + e^2)*(a + b*ArcCosh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] - 3*e*(8*c^2*d + 3*e)*(a + b*ArcCosh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] - 5*a*e^2*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] - 5*b*e^2*ArcCosh[c*x]*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcCosh[c*x])] + 16*a*c^4*d^2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 8*a*c^2*d*e*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 2*a*e^2*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 16*b*c^4*d^2*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 8*b*c^2*d*e*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 2*b*e^2*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 24*a*c^2*d*e*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 9*a*e^2*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 24*b*c^2*d*e*ArcCosh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 9*b*e^2*ArcCosh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 5*a*e^2*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + 5*b*e^2*ArcCosh[c*x]*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])])/(b^2*c^5*(a + b*ArcCosh[c*x]))","A",0
544,1,338,257,0.9662391,"\int \frac{d+e x^2}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x^2)/(a + b*ArcCosh[c*x])^2,x]","-\frac{-\cosh \left(\frac{a}{b}\right) \left(4 c^2 d+e\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+4 a c^2 d \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+4 b c^2 d \sinh \left(\frac{a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-3 e \cosh \left(\frac{3 a}{b}\right) \left(a+b \cosh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+a e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+b e \sinh \left(\frac{a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 a e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+3 b e \sinh \left(\frac{3 a}{b}\right) \cosh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+4 b c^3 d x \sqrt{\frac{c x-1}{c x+1}}+4 b c^3 e x^3 \sqrt{\frac{c x-1}{c x+1}}+4 b c^2 d \sqrt{\frac{c x-1}{c x+1}}+4 b c^2 e x^2 \sqrt{\frac{c x-1}{c x+1}}}{4 b^2 c^3 \left(a+b \cosh ^{-1}(c x)\right)}","\frac{e \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b^2 c^3}+\frac{3 e \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}-\frac{e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{4 b^2 c^3}-\frac{3 e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}+\frac{d \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d \sqrt{c x-1} \sqrt{c x+1}}{b c \left(a+b \cosh ^{-1}(c x)\right)}-\frac{e x^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"-1/4*(4*b*c^2*d*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*b*c^3*d*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*b*c^2*e*x^2*Sqrt[(-1 + c*x)/(1 + c*x)] + 4*b*c^3*e*x^3*Sqrt[(-1 + c*x)/(1 + c*x)] - (4*c^2*d + e)*(a + b*ArcCosh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] - 3*e*(a + b*ArcCosh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcCosh[c*x])] + 4*a*c^2*d*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + a*e*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 4*b*c^2*d*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + b*e*ArcCosh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 3*a*e*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 3*b*e*ArcCosh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])])/(b^2*c^3*(a + b*ArcCosh[c*x]))","A",0
545,1,80,90,0.3531259,"\int \frac{1}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])^(-2),x]","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\cosh ^{-1}(c x)\right)-\frac{b \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{a+b \cosh ^{-1}(c x)}}{b^2 c}","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{\sqrt{c x-1} \sqrt{c x+1}}{b c \left(a+b \cosh ^{-1}(c x)\right)}",1,"(-((b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(a + b*ArcCosh[c*x])) + Cosh[a/b]*CoshIntegral[a/b + ArcCosh[c*x]] - Sinh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(b^2*c)","A",0
546,0,0,23,178.6220542,"\int \frac{1}{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x^2)*(a + b*ArcCosh[c*x])^2), x]","A",-1
547,-1,0,23,180.0017338,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^2),x]","\text{\$Aborted}","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"$Aborted","F",-1
548,0,0,25,28.2265817,"\int \frac{\sqrt{d+e x^2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x])^2,x]","\int \frac{\sqrt{d+e x^2}}{\left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2}}{\left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[Sqrt[d + e*x^2]/(a + b*ArcCosh[c*x])^2, x]","A",-1
549,0,0,25,24.8429444,"\int \frac{1}{\sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^2),x]","\int \frac{1}{\sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\sqrt{d+e x^2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcCosh[c*x])^2), x]","A",-1
550,-1,0,25,180.0009133,"\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2),x]","\text{\$Aborted}","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"$Aborted","F",-1
551,-1,0,25,180.0044556,"\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]","\text{\$Aborted}","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \cosh ^{-1}(c x)\right)^2},x\right)",0,"$Aborted","F",-1
552,1,536,672,6.6927187,"\int \left(d+e x^2\right)^2 \sqrt{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)^2*Sqrt[a + b*ArcCosh[c*x]],x]","\frac{b e^{-\frac{5 a}{b}} \left(450 e^{\frac{6 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right) \left(-b e \left(4 c^2 d+e\right) \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \sqrt{-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}}+8 a c^4 d^2 \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)}+8 b c^4 d^2 \cosh ^{-1}(c x) \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)}\right)-e^{\frac{2 a}{b}} \left(25 \sqrt{3} b e \left(8 c^2 d+3 e\right) \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \sqrt{-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}} \Gamma \left(\frac{3}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+b e e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \sqrt{-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}} \left(25 \sqrt{3} \left(8 c^2 d+3 e\right) \Gamma \left(\frac{3}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+9 \sqrt{5} e e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)+450 e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right) \left(b e \left(4 c^2 d+e\right) \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \sqrt{-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}}+8 a c^4 d^2 \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}}+8 b c^4 d^2 \cosh ^{-1}(c x) \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}}\right)\right)-9 \sqrt{5} b e^2 \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \sqrt{-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}} \Gamma \left(\frac{3}{2},-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{7200 c^5 \left(a+b \cosh ^{-1}(c x)\right)^{3/2}}","-\frac{\sqrt{\pi } \sqrt{b} e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 c^5}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 c^5}-\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^2 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{320 c^5}-\frac{\sqrt{\pi } \sqrt{b} e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 c^5}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 c^5}-\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^2 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{320 c^5}-\frac{\sqrt{\pi } \sqrt{b} d e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} d e e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{24 c^3}-\frac{\sqrt{\pi } \sqrt{b} d e e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} d e e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{24 c^3}-\frac{\sqrt{\pi } \sqrt{b} d^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}-\frac{\sqrt{\pi } \sqrt{b} d^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}+d^2 x \sqrt{a+b \cosh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \cosh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \cosh ^{-1}(c x)}",1,"(b*(450*E^((6*a)/b)*(8*a*c^4*d^2*Sqrt[a/b + ArcCosh[c*x]] + 8*b*c^4*d^2*ArcCosh[c*x]*Sqrt[a/b + ArcCosh[c*x]] - b*e*(4*c^2*d + e)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)])*Gamma[3/2, a/b + ArcCosh[c*x]] - 9*Sqrt[5]*b*e^2*Sqrt[a/b + ArcCosh[c*x]]*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]*Gamma[3/2, (-5*(a + b*ArcCosh[c*x]))/b] - E^((2*a)/b)*(25*Sqrt[3]*b*e*(8*c^2*d + 3*e)*Sqrt[a/b + ArcCosh[c*x]]*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]*Gamma[3/2, (-3*(a + b*ArcCosh[c*x]))/b] + 450*E^((2*a)/b)*(8*a*c^4*d^2*Sqrt[-((a + b*ArcCosh[c*x])/b)] + 8*b*c^4*d^2*ArcCosh[c*x]*Sqrt[-((a + b*ArcCosh[c*x])/b)] + b*e*(4*c^2*d + e)*Sqrt[a/b + ArcCosh[c*x]]*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)])*Gamma[3/2, -((a + b*ArcCosh[c*x])/b)] + b*e*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]*(25*Sqrt[3]*(8*c^2*d + 3*e)*Gamma[3/2, (3*(a + b*ArcCosh[c*x]))/b] + 9*Sqrt[5]*e*E^((2*a)/b)*Gamma[3/2, (5*(a + b*ArcCosh[c*x]))/b]))))/(7200*c^5*E^((5*a)/b)*(a + b*ArcCosh[c*x])^(3/2))","A",0
553,1,317,322,2.8182266,"\int \left(d+e x^2\right) \sqrt{a+b \cosh ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)*Sqrt[a + b*ArcCosh[c*x]],x]","\frac{e e^{-\frac{3 a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left(9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)+\sqrt{3} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{3}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)+\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{3}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{72 c^3 \sqrt{-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}}}+\frac{d e^{-\frac{a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left(\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)}{\sqrt{\frac{a}{b}+\cosh ^{-1}(c x)}}+\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}}}\right)}{2 c}","-\frac{\sqrt{\pi } \sqrt{b} e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{48 c^3}-\frac{\sqrt{\pi } \sqrt{b} e e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 c^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{48 c^3}-\frac{\sqrt{\pi } \sqrt{b} d e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}-\frac{\sqrt{\pi } \sqrt{b} d e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}+d x \sqrt{a+b \cosh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \cosh ^{-1}(c x)}",1,"(d*Sqrt[a + b*ArcCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]])/Sqrt[a/b + ArcCosh[c*x]] + Gamma[3/2, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-((a + b*ArcCosh[c*x])/b)]))/(2*c*E^(a/b)) + (e*Sqrt[a + b*ArcCosh[c*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, a/b + ArcCosh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c*x]))/b] + 9*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcCosh[c*x]))/b]))/(72*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)])","A",0
554,1,100,102,0.1961084,"\int \sqrt{a+b \cosh ^{-1}(c x)} \, dx","Integrate[Sqrt[a + b*ArcCosh[c*x]],x]","\frac{e^{-\frac{a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left(\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)}{\sqrt{\frac{a}{b}+\cosh ^{-1}(c x)}}+\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}}}\right)}{2 c}","-\frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}-\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}+x \sqrt{a+b \cosh ^{-1}(c x)}",1,"(Sqrt[a + b*ArcCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]])/Sqrt[a/b + ArcCosh[c*x]] + Gamma[3/2, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-((a + b*ArcCosh[c*x])/b)]))/(2*c*E^(a/b))","A",0
555,0,0,25,4.894156,"\int \frac{\sqrt{a+b \cosh ^{-1}(c x)}}{d+e x^2} \, dx","Integrate[Sqrt[a + b*ArcCosh[c*x]]/(d + e*x^2),x]","\int \frac{\sqrt{a+b \cosh ^{-1}(c x)}}{d+e x^2} \, dx","\text{Int}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{d+e x^2},x\right)",0,"Integrate[Sqrt[a + b*ArcCosh[c*x]]/(d + e*x^2), x]","A",-1
556,0,0,25,21.7276397,"\int \frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\left(d+e x^2\right)^2} \, dx","Integrate[Sqrt[a + b*ArcCosh[c*x]]/(d + e*x^2)^2,x]","\int \frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[Sqrt[a + b*ArcCosh[c*x]]/(d + e*x^2)^2, x]","A",-1
557,1,812,442,3.4787618,"\int \left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^{3/2} \, dx","Integrate[(d + e*x^2)*(a + b*ArcCosh[c*x])^(3/2),x]","\frac{a d e^{-\frac{a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left(\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)}{\sqrt{\frac{a}{b}+\cosh ^{-1}(c x)}}+\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}}}\right)}{2 c}+\frac{a e e^{-\frac{3 a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left(9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)+\sqrt{3} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{3}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)+\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{3}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{72 c^3 \sqrt{-\frac{\left(a+b \cosh ^{-1}(c x)\right)^2}{b^2}}}+\frac{b d \left(-12 \sqrt{\frac{c x-1}{c x+1}} \sqrt{a+b \cosh ^{-1}(c x)} (c x+1)+\frac{(2 a+3 b) \sqrt{\pi } \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right) \left(\cosh \left(\frac{a}{b}\right)-\sinh \left(\frac{a}{b}\right)\right)}{\sqrt{b}}+\frac{(2 a-3 b) \sqrt{\pi } \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right) \left(\cosh \left(\frac{a}{b}\right)+\sinh \left(\frac{a}{b}\right)\right)}{\sqrt{b}}+8 c x \cosh ^{-1}(c x) \sqrt{a+b \cosh ^{-1}(c x)}\right)}{8 c}+\frac{\sqrt{b} e \left(9 \left(-12 \sqrt{b} \sqrt{\frac{c x-1}{c x+1}} \sqrt{a+b \cosh ^{-1}(c x)} (c x+1)+(2 a+3 b) \sqrt{\pi } \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right) \left(\cosh \left(\frac{a}{b}\right)-\sinh \left(\frac{a}{b}\right)\right)+(2 a-3 b) \sqrt{\pi } \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right) \left(\cosh \left(\frac{a}{b}\right)+\sinh \left(\frac{a}{b}\right)\right)+8 \sqrt{b} c x \cosh ^{-1}(c x) \sqrt{a+b \cosh ^{-1}(c x)}\right)+(2 a+b) \sqrt{3 \pi } \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right) \left(\cosh \left(\frac{3 a}{b}\right)-\sinh \left(\frac{3 a}{b}\right)\right)+(2 a-b) \sqrt{3 \pi } \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right) \left(\cosh \left(\frac{3 a}{b}\right)+\sinh \left(\frac{3 a}{b}\right)\right)+12 \sqrt{b} \sqrt{a+b \cosh ^{-1}(c x)} \left(2 \cosh ^{-1}(c x) \cosh \left(3 \cosh ^{-1}(c x)\right)-\sinh \left(3 \cosh ^{-1}(c x)\right)\right)\right)}{288 c^3}","-\frac{3 \sqrt{\pi } b^{3/2} e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 c^3}-\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{96 c^3}+\frac{3 \sqrt{\pi } b^{3/2} e e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 c^3}+\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{96 c^3}-\frac{3 \sqrt{\pi } b^{3/2} d e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c}+\frac{3 \sqrt{\pi } b^{3/2} d e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c}-\frac{b e \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{3 c^3}+d x \left(a+b \cosh ^{-1}(c x)\right)^{3/2}-\frac{3 b d \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{2 c}+\frac{1}{3} e x^3 \left(a+b \cosh ^{-1}(c x)\right)^{3/2}-\frac{b e x^2 \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{6 c}",1,"(a*d*Sqrt[a + b*ArcCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]])/Sqrt[a/b + ArcCosh[c*x]] + Gamma[3/2, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-((a + b*ArcCosh[c*x])/b)]))/(2*c*E^(a/b)) + (a*e*Sqrt[a + b*ArcCosh[c*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, a/b + ArcCosh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, (-3*(a + b*ArcCosh[c*x]))/b] + 9*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcCosh[c*x]))/b]))/(72*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]) + (b*d*(-12*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*ArcCosh[c*x]] + 8*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + ((2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqrt[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(8*c) + (Sqrt[b]*e*(9*(-12*Sqrt[b]*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*ArcCosh[c*x]] + 8*Sqrt[b]*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] - Sinh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b*ArcCosh[c*x]]*(2*ArcCosh[c*x]*Cosh[3*ArcCosh[c*x]] - Sinh[3*ArcCosh[c*x]])))/(288*c^3)","A",0
558,1,269,140,0.7138546,"\int \left(a+b \cosh ^{-1}(c x)\right)^{3/2} \, dx","Integrate[(a + b*ArcCosh[c*x])^(3/2),x]","\frac{b \left(\frac{\sqrt{\pi } (2 a-3 b) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b}}+\frac{\sqrt{\pi } (2 a+3 b) \left(\cosh \left(\frac{a}{b}\right)-\sinh \left(\frac{a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{\sqrt{b}}-12 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \sqrt{a+b \cosh ^{-1}(c x)}+8 c x \cosh ^{-1}(c x) \sqrt{a+b \cosh ^{-1}(c x)}\right)}{8 c}+\frac{a e^{-\frac{a}{b}} \sqrt{a+b \cosh ^{-1}(c x)} \left(\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)}{\sqrt{\frac{a}{b}+\cosh ^{-1}(c x)}}+\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)}{\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}}}\right)}{2 c}","-\frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c}-\frac{3 b \sqrt{c x-1} \sqrt{c x+1} \sqrt{a+b \cosh ^{-1}(c x)}}{2 c}+x \left(a+b \cosh ^{-1}(c x)\right)^{3/2}",1,"(a*Sqrt[a + b*ArcCosh[c*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c*x]])/Sqrt[a/b + ArcCosh[c*x]] + Gamma[3/2, -((a + b*ArcCosh[c*x])/b)]/Sqrt[-((a + b*ArcCosh[c*x])/b)]))/(2*c*E^(a/b)) + (b*(-12*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*ArcCosh[c*x]] + 8*c*x*ArcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + ((2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqrt[b] + ((2*a - 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sqrt[b]))/(8*c)","A",0
559,0,0,25,2.3361658,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}}{d+e x^2} \, dx","Integrate[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2),x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}}{d+e x^2} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}}{d+e x^2},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2), x]","A",-1
560,0,0,25,14.2576729,"\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2)^2,x]","\int \frac{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[(a + b*ArcCosh[c*x])^(3/2)/(d + e*x^2)^2, x]","A",-1
561,1,530,608,1.1417708,"\int \frac{\left(d+e x^2\right)^2}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx","Integrate[(d + e*x^2)^2/Sqrt[a + b*ArcCosh[c*x]],x]","\frac{e^{-\frac{5 a}{b}} \left(240 c^4 d^2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)+40 \sqrt{3} c^2 d e e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+120 c^2 d e e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)+40 \sqrt{3} c^2 d e e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+30 e^{\frac{6 a}{b}} \left(8 c^4 d^2+4 c^2 d e+e^2\right) \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 \sqrt{5} e^2 \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+15 \sqrt{3} e^2 e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+30 e^2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)+15 \sqrt{3} e^2 e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+3 \sqrt{5} e^2 e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{480 c^5 \sqrt{a+b \cosh ^{-1}(c x)}}","-\frac{\sqrt{\pi } e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 \sqrt{b} c^5}-\frac{\sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 \sqrt{b} c^5}-\frac{\sqrt{\frac{\pi }{5}} e^2 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 \sqrt{b} c^5}+\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 \sqrt{b} c^5}+\frac{\sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 \sqrt{b} c^5}+\frac{\sqrt{\frac{\pi }{5}} e^2 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 \sqrt{b} c^5}-\frac{\sqrt{\pi } d e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{3}} d e e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 \sqrt{b} c^3}+\frac{\sqrt{\pi } d e e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{3}} d e e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 \sqrt{b} c^3}-\frac{\sqrt{\pi } d^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c}+\frac{\sqrt{\pi } d^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c}",1,"(30*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] + 3*Sqrt[5]*e^2*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcCosh[c*x]))/b] + 40*Sqrt[3]*c^2*d*e*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + 15*Sqrt[3]*e^2*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + 240*c^4*d^2*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + 120*c^2*d*e*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + 30*e^2*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + 40*Sqrt[3]*c^2*d*e*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + 15*Sqrt[3]*e^2*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + 3*Sqrt[5]*e^2*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (5*(a + b*ArcCosh[c*x]))/b])/(480*c^5*E^((5*a)/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
562,1,213,287,0.6424633,"\int \frac{d+e x^2}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx","Integrate[(d + e*x^2)/Sqrt[a + b*ArcCosh[c*x]],x]","\frac{e^{-\frac{3 a}{b}} \left(3 e^{\frac{4 a}{b}} \left(4 c^2 d+e\right) \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)+3 e^{\frac{2 a}{b}} \left(4 c^2 d+e\right) \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)+\sqrt{3} e \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+\sqrt{3} e e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{24 c^3 \sqrt{a+b \cosh ^{-1}(c x)}}","-\frac{\sqrt{\pi } e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{3}} e e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 \sqrt{b} c^3}+\frac{\sqrt{\pi } e e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{3}} e e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 \sqrt{b} c^3}-\frac{\sqrt{\pi } d e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c}+\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c}",1,"(3*(4*c^2*d + e)*E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] + Sqrt[3]*e*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + 3*(4*c^2*d + e)*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*e*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b])/(24*c^3*E^((3*a)/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
563,1,100,88,0.1175546,"\int \frac{1}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx","Integrate[1/Sqrt[a + b*ArcCosh[c*x]],x]","\frac{e^{-\frac{a}{b}} \left(e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)+\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)\right)}{2 c \sqrt{a+b \cosh ^{-1}(c x)}}","\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c}-\frac{\sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c}",1,"(E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] + Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)])/(2*c*E^(a/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
564,0,0,25,0.1584019,"\int \frac{1}{\left(d+e x^2\right) \sqrt{a+b \cosh ^{-1}(c x)}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[a + b*ArcCosh[c*x]]),x]","\int \frac{1}{\left(d+e x^2\right) \sqrt{a+b \cosh ^{-1}(c x)}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \sqrt{a+b \cosh ^{-1}(c x)}},x\right)",0,"Integrate[1/((d + e*x^2)*Sqrt[a + b*ArcCosh[c*x]]), x]","A",-1
565,0,0,25,0.2949122,"\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b \cosh ^{-1}(c x)}} \, dx","Integrate[1/((d + e*x^2)^2*Sqrt[a + b*ArcCosh[c*x]]),x]","\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b \cosh ^{-1}(c x)}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b \cosh ^{-1}(c x)}},x\right)",0,"Integrate[1/((d + e*x^2)^2*Sqrt[a + b*ArcCosh[c*x]]), x]","A",-1
566,1,268,358,1.9692638,"\int \frac{d+e x^2}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d + e*x^2)/(a + b*ArcCosh[c*x])^(3/2),x]","\frac{e^{-\frac{3 a}{b}} \left(-\left(e^{\frac{4 a}{b}} \left(4 c^2 d+e\right) \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)\right)+e^{\frac{2 a}{b}} \left(4 c^2 d+e\right) \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)-e^{\frac{3 a}{b}} \left(\sqrt{3} e e^{\frac{3 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)+2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left(4 c^2 d+e\right)+2 e \sinh \left(3 \cosh ^{-1}(c x)\right)\right)+\sqrt{3} e \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \cosh ^{-1}(c x)\right)}{b}\right)\right)}{4 b c^3 \sqrt{a+b \cosh ^{-1}(c x)}}","\frac{\sqrt{\pi } e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^3}+\frac{\sqrt{3 \pi } e e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^3}+\frac{\sqrt{\pi } e e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^3}+\frac{\sqrt{3 \pi } e e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c^3}+\frac{\sqrt{\pi } d e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}+\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 d \sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"(-((4*c^2*d + e)*E^((4*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]]) + Sqrt[3]*e*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + (4*c^2*d + e)*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] - E^((3*a)/b)*(2*(4*c^2*d + e)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + Sqrt[3]*e*E^((3*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + 2*e*Sinh[3*ArcCosh[c*x]]))/(4*b*c^3*E^((3*a)/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
567,1,132,120,0.1381951,"\int \frac{1}{\left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(a + b*ArcCosh[c*x])^(-3/2),x]","\frac{e^{-\frac{a}{b}} \left(-2 e^{a/b} \sqrt{\frac{c x-1}{c x+1}} (c x+1)-e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right)+\sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right)\right)}{b c \sqrt{a+b \cosh ^{-1}(c x)}}","\frac{\sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 \sqrt{c x-1} \sqrt{c x+1}}{b c \sqrt{a+b \cosh ^{-1}(c x)}}",1,"(-2*E^(a/b)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) - E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] + Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)])/(b*c*E^(a/b)*Sqrt[a + b*ArcCosh[c*x]])","A",0
568,0,0,25,0.1771865,"\int \frac{1}{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[1/((d + e*x^2)*(a + b*ArcCosh[c*x])^(3/2)),x]","\int \frac{1}{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \left(a+b \cosh ^{-1}(c x)\right)^{3/2}},x\right)",0,"Integrate[1/((d + e*x^2)*(a + b*ArcCosh[c*x])^(3/2)), x]","A",-1
569,0,0,25,0.2980985,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^(3/2)),x]","\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)^{3/2}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \left(a+b \cosh ^{-1}(c x)\right)^{3/2}},x\right)",0,"Integrate[1/((d + e*x^2)^2*(a + b*ArcCosh[c*x])^(3/2)), x]","A",-1