1,1,87,124,0.1363251,"\int x^4 \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + c^2*d*x^2)*(a + b*ArcSinh[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) \sinh ^{-1}(c x)-\frac{b \sqrt{c^2 x^2+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 \sinh ^{-1}(c x)\right)+\frac{1}{5} d x^5 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d \left(c^2 x^2+1\right)^{7/2}}{49 c^5}+\frac{8 b d \left(c^2 x^2+1\right)^{5/2}}{175 c^5}-\frac{b d \left(c^2 x^2+1\right)^{3/2}}{105 c^5}-\frac{2 b d \sqrt{c^2 x^2+1}}{35 c^5}",1,"(d*(105*a*x^5*(7 + 5*c^2*x^2) - (b*Sqrt[1 + c^2*x^2]*(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)*ArcSinh[c*x]))/3675","A",1
2,1,88,120,0.0506046,"\int x^3 \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]","\frac{d \left(6 a c^4 x^4 \left(2 c^2 x^2+3\right)+3 b \left(4 c^6 x^6+6 c^4 x^4-1\right) \sinh ^{-1}(c x)+b c x \sqrt{c^2 x^2+1} \left(-2 c^4 x^4-2 c^2 x^2+3\right)\right)}{72 c^4}","\frac{1}{6} c^2 d x^6 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{4} d x^4 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d \sinh ^{-1}(c x)}{24 c^4}-\frac{1}{36} b c d x^5 \sqrt{c^2 x^2+1}-\frac{b d x^3 \sqrt{c^2 x^2+1}}{36 c}+\frac{b d x \sqrt{c^2 x^2+1}}{24 c^3}",1,"(d*(6*a*c^4*x^4*(3 + 2*c^2*x^2) + b*c*x*Sqrt[1 + c^2*x^2]*(3 - 2*c^2*x^2 - 2*c^4*x^4) + 3*b*(-1 + 6*c^4*x^4 + 4*c^6*x^6)*ArcSinh[c*x]))/(72*c^4)","A",1
3,1,78,102,0.0755464,"\int x^2 \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]","\frac{1}{225} d \left(15 a x^3 \left(3 c^2 x^2+5\right)+15 b x^3 \left(3 c^2 x^2+5\right) \sinh ^{-1}(c x)+\frac{b \sqrt{c^2 x^2+1} \left(-9 c^4 x^4-13 c^2 x^2+26\right)}{c^3}\right)","\frac{1}{5} c^2 d x^5 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{3} d x^3 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d \left(c^2 x^2+1\right)^{5/2}}{25 c^3}+\frac{b d \left(c^2 x^2+1\right)^{3/2}}{45 c^3}+\frac{2 b d \sqrt{c^2 x^2+1}}{15 c^3}",1,"(d*(15*a*x^3*(5 + 3*c^2*x^2) + (b*Sqrt[1 + c^2*x^2]*(26 - 13*c^2*x^2 - 9*c^4*x^4))/c^3 + 15*b*x^3*(5 + 3*c^2*x^2)*ArcSinh[c*x]))/225","A",1
4,1,77,87,0.0497002,"\int x \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]","\frac{d \left(c x \left(8 a c x \left(c^2 x^2+2\right)-b \sqrt{c^2 x^2+1} \left(2 c^2 x^2+5\right)\right)+b \left(8 c^4 x^4+16 c^2 x^2+5\right) \sinh ^{-1}(c x)\right)}{32 c^2}","\frac{d \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{4 c^2}-\frac{b d x \left(c^2 x^2+1\right)^{3/2}}{16 c}-\frac{3 b d x \sqrt{c^2 x^2+1}}{32 c}-\frac{3 b d \sinh ^{-1}(c x)}{32 c^2}",1,"(d*(c*x*(8*a*c*x*(2 + c^2*x^2) - b*Sqrt[1 + c^2*x^2]*(5 + 2*c^2*x^2)) + b*(5 + 16*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x]))/(32*c^2)","A",1
5,1,86,75,0.0420161,"\int \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]","\frac{1}{3} a c^2 d x^3+a d x+\frac{1}{3} b c^2 d x^3 \sinh ^{-1}(c x)-\frac{1}{9} b c d x^2 \sqrt{c^2 x^2+1}-\frac{7 b d \sqrt{c^2 x^2+1}}{9 c}+b d x \sinh ^{-1}(c x)","\frac{1}{3} c^2 d x^3 \left(a+b \sinh ^{-1}(c x)\right)+d x \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d \left(c^2 x^2+1\right)^{3/2}}{9 c}-\frac{2 b d \sqrt{c^2 x^2+1}}{3 c}",1,"a*d*x + (a*c^2*d*x^3)/3 - (7*b*d*Sqrt[1 + c^2*x^2])/(9*c) - (b*c*d*x^2*Sqrt[1 + c^2*x^2])/9 + b*d*x*ArcSinh[c*x] + (b*c^2*d*x^3*ArcSinh[c*x])/3","A",1
6,1,113,111,0.0574606,"\int \frac{\left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x,x]","\frac{1}{2} a c^2 d x^2+a d \log (x)-\frac{1}{4} b c d x \sqrt{c^2 x^2+1}+\frac{1}{2} b c^2 d x^2 \sinh ^{-1}(c x)+\frac{1}{2} b d \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-\frac{1}{2} b d \sinh ^{-1}(c x)^2+\frac{1}{4} b d \sinh ^{-1}(c x)+b d \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)","\frac{1}{2} d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{d \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b}+d \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{4} b c d x \sqrt{c^2 x^2+1}-\frac{1}{2} b d \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-\frac{1}{4} b d \sinh ^{-1}(c x)",1,"(a*c^2*d*x^2)/2 - (b*c*d*x*Sqrt[1 + c^2*x^2])/4 + (b*d*ArcSinh[c*x])/4 + (b*c^2*d*x^2*ArcSinh[c*x])/2 - (b*d*ArcSinh[c*x]^2)/2 + b*d*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + a*d*Log[x] + (b*d*PolyLog[2, E^(2*ArcSinh[c*x])])/2","A",0
7,1,74,66,0.0246886,"\int \frac{\left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x^2,x]","a c^2 d x-\frac{a d}{x}-b c d \sqrt{c^2 x^2+1}-b c d \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+b c^2 d x \sinh ^{-1}(c x)-\frac{b d \sinh ^{-1}(c x)}{x}","c^2 d x \left(a+b \sinh ^{-1}(c x)\right)-\frac{d \left(a+b \sinh ^{-1}(c x)\right)}{x}-b c d \sqrt{c^2 x^2+1}-b c d \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)",1,"-((a*d)/x) + a*c^2*d*x - b*c*d*Sqrt[1 + c^2*x^2] - (b*d*ArcSinh[c*x])/x + b*c^2*d*x*ArcSinh[c*x] - b*c*d*ArcTanh[Sqrt[1 + c^2*x^2]]","A",1
8,1,111,128,0.0540695,"\int \frac{\left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x^3,x]","a c^2 d \log (x)-\frac{a d}{2 x^2}+\frac{1}{2} b c^2 d \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-\frac{b c d \sqrt{c^2 x^2+1}}{2 x}-\frac{1}{2} b c^2 d \sinh ^{-1}(c x)^2+b c^2 d \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)-\frac{b d \sinh ^{-1}(c x)}{2 x^2}","-\frac{d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{2 x^2}+\frac{c^2 d \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b}+c^2 d \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{2} b c^2 d \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-\frac{b c d \sqrt{c^2 x^2+1}}{2 x}+\frac{1}{2} b c^2 d \sinh ^{-1}(c x)",1,"-1/2*(a*d)/x^2 - (b*c*d*Sqrt[1 + c^2*x^2])/(2*x) - (b*d*ArcSinh[c*x])/(2*x^2) - (b*c^2*d*ArcSinh[c*x]^2)/2 + b*c^2*d*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + a*c^2*d*Log[x] + (b*c^2*d*PolyLog[2, E^(2*ArcSinh[c*x])])/2","A",0
9,1,93,80,0.030873,"\int \frac{\left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x]))/x^4,x]","-\frac{a c^2 d}{x}-\frac{a d}{3 x^3}-\frac{b c d \sqrt{c^2 x^2+1}}{6 x^2}-\frac{b c^2 d \sinh ^{-1}(c x)}{x}-\frac{5}{6} b c^3 d \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-\frac{b d \sinh ^{-1}(c x)}{3 x^3}","-\frac{c^2 d \left(a+b \sinh ^{-1}(c x)\right)}{x}-\frac{d \left(a+b \sinh ^{-1}(c x)\right)}{3 x^3}-\frac{b c d \sqrt{c^2 x^2+1}}{6 x^2}-\frac{5}{6} b c^3 d \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)",1,"-1/3*(a*d)/x^3 - (a*c^2*d)/x - (b*c*d*Sqrt[1 + c^2*x^2])/(6*x^2) - (b*d*ArcSinh[c*x])/(3*x^3) - (b*c^2*d*ArcSinh[c*x])/x - (5*b*c^3*d*ArcTanh[Sqrt[1 + c^2*x^2]])/6","A",1
10,1,119,181,0.092904,"\int x^4 \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + c^2*d*x^2)^2*(a + b*ArcSinh[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) \sinh ^{-1}(c x)-b \sqrt{c^2 x^2+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 \sinh ^{-1}(c x)\right)+\frac{2}{7} c^2 d^2 x^7 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{5} d^2 x^5 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d^2 \left(c^2 x^2+1\right)^{9/2}}{81 c^5}+\frac{10 b d^2 \left(c^2 x^2+1\right)^{7/2}}{441 c^5}-\frac{b d^2 \left(c^2 x^2+1\right)^{5/2}}{525 c^5}-\frac{4 b d^2 \left(c^2 x^2+1\right)^{3/2}}{945 c^5}-\frac{8 b d^2 \sqrt{c^2 x^2+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^2*x^2]*(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)*ArcSinh[c*x]))/(99225*c^5)","A",1
11,1,115,180,0.0778162,"\int x^3 \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]),x]","\frac{d^2 \left(384 a c^4 x^4 \left(3 c^4 x^4+8 c^2 x^2+6\right)+3 b \left(384 c^8 x^8+1024 c^6 x^6+768 c^4 x^4-73\right) \sinh ^{-1}(c x)-b c x \sqrt{c^2 x^2+1} \left(144 c^6 x^6+344 c^4 x^4+146 c^2 x^2-219\right)\right)}{9216 c^4}","\frac{1}{8} c^4 d^2 x^8 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{3} c^2 d^2 x^6 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{4} d^2 x^4 \left(a+b \sinh ^{-1}(c x)\right)-\frac{73 b d^2 \sinh ^{-1}(c x)}{3072 c^4}-\frac{43 b c d^2 x^5 \sqrt{c^2 x^2+1}}{1152}-\frac{73 b d^2 x^3 \sqrt{c^2 x^2+1}}{4608 c}+\frac{73 b d^2 x \sqrt{c^2 x^2+1}}{3072 c^3}-\frac{1}{64} b c^3 d^2 x^7 \sqrt{c^2 x^2+1}",1,"(d^2*(384*a*c^4*x^4*(6 + 8*c^2*x^2 + 3*c^4*x^4) - b*c*x*Sqrt[1 + c^2*x^2]*(-219 + 146*c^2*x^2 + 344*c^4*x^4 + 144*c^6*x^6) + 3*b*(-73 + 768*c^4*x^4 + 1024*c^6*x^6 + 384*c^8*x^8)*ArcSinh[c*x]))/(9216*c^4)","A",1
12,1,111,157,0.0796193,"\int x^2 \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + c^2*d*x^2)^2*(a + b*ArcSinh[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^2 x^2+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) \sinh ^{-1}(c x)\right)}{11025 c^3}","\frac{1}{7} c^4 d^2 x^7 \left(a+b \sinh ^{-1}(c x)\right)+\frac{2}{5} c^2 d^2 x^5 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{3} d^2 x^3 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d^2 \left(c^2 x^2+1\right)^{7/2}}{49 c^3}+\frac{b d^2 \left(c^2 x^2+1\right)^{5/2}}{175 c^3}+\frac{4 b d^2 \left(c^2 x^2+1\right)^{3/2}}{315 c^3}+\frac{8 b d^2 \sqrt{c^2 x^2+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^2*x^2]*(-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)*ArcSinh[c*x]))/(11025*c^3)","A",1
13,1,104,120,0.1306023,"\int x \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x*(d + c^2*d*x^2)^2*(a + b*ArcSinh[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^2 x^2+1} \left(8 c^4 x^4+26 c^2 x^2+33\right)\right)+3 b \left(16 c^6 x^6+48 c^4 x^4+48 c^2 x^2+11\right) \sinh ^{-1}(c x)\right)}{288 c^2}","\frac{d^2 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)}{6 c^2}-\frac{b d^2 x \left(c^2 x^2+1\right)^{5/2}}{36 c}-\frac{5 b d^2 x \left(c^2 x^2+1\right)^{3/2}}{144 c}-\frac{5 b d^2 x \sqrt{c^2 x^2+1}}{96 c}-\frac{5 b d^2 \sinh ^{-1}(c x)}{96 c^2}",1,"(d^2*(c*x*(48*a*c*x*(3 + 3*c^2*x^2 + c^4*x^4) - b*Sqrt[1 + c^2*x^2]*(33 + 26*c^2*x^2 + 8*c^4*x^4)) + 3*b*(11 + 48*c^2*x^2 + 48*c^4*x^4 + 16*c^6*x^6)*ArcSinh[c*x]))/(288*c^2)","A",1
14,1,95,128,0.1133945,"\int \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + c^2*d*x^2)^2*(a + b*ArcSinh[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^2 x^2+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) \sinh ^{-1}(c x)\right)}{225 c}","\frac{1}{5} c^4 d^2 x^5 \left(a+b \sinh ^{-1}(c x)\right)+\frac{2}{3} c^2 d^2 x^3 \left(a+b \sinh ^{-1}(c x)\right)+d^2 x \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d^2 \left(c^2 x^2+1\right)^{5/2}}{25 c}-\frac{4 b d^2 \left(c^2 x^2+1\right)^{3/2}}{45 c}-\frac{8 b d^2 \sqrt{c^2 x^2+1}}{15 c}",1,"(d^2*(15*a*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) - b*Sqrt[1 + c^2*x^2]*(149 + 38*c^2*x^2 + 9*c^4*x^4) + 15*b*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]))/(225*c)","A",1
15,1,173,172,0.1828978,"\int \frac{\left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]))/x,x]","\frac{d^2 \left(-16 a^2+8 a b c^4 x^4+32 a b c^2 x^2+b \sinh ^{-1}(c x) \left(-32 a+b \left(8 c^4 x^4+32 c^2 x^2+13\right)+32 b \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)\right)+32 a b \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+24 a b-13 b^2 c x \sqrt{c^2 x^2+1}-2 b^2 c^3 x^3 \sqrt{c^2 x^2+1}+16 b^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-16 b^2 \sinh ^{-1}(c x)^2\right)}{32 b}","\frac{1}{4} d^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{2} d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{d^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b}+d^2 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{16} b c d^2 x \left(c^2 x^2+1\right)^{3/2}-\frac{11}{32} b c d^2 x \sqrt{c^2 x^2+1}-\frac{1}{2} b d^2 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-\frac{11}{32} b d^2 \sinh ^{-1}(c x)",1,"(d^2*(-16*a^2 + 24*a*b + 32*a*b*c^2*x^2 + 8*a*b*c^4*x^4 - 13*b^2*c*x*Sqrt[1 + c^2*x^2] - 2*b^2*c^3*x^3*Sqrt[1 + c^2*x^2] - 16*b^2*ArcSinh[c*x]^2 + 32*a*b*Log[1 - E^(2*ArcSinh[c*x])] + b*ArcSinh[c*x]*(-32*a + b*(13 + 32*c^2*x^2 + 8*c^4*x^4) + 32*b*Log[1 - E^(2*ArcSinh[c*x])]) + 16*b^2*PolyLog[2, E^(2*ArcSinh[c*x])]))/(32*b)","A",0
16,1,124,120,0.1162329,"\int \frac{\left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]))/x^2,x]","\frac{d^2 \left(3 a c^4 x^4+18 a c^2 x^2-9 a-16 b c x \sqrt{c^2 x^2+1}-9 b c x \log \left(\sqrt{c^2 x^2+1}+1\right)+3 b \left(c^4 x^4+6 c^2 x^2-3\right) \sinh ^{-1}(c x)-b c^3 x^3 \sqrt{c^2 x^2+1}+9 b c x \log (x)\right)}{9 x}","\frac{1}{3} c^4 d^2 x^3 \left(a+b \sinh ^{-1}(c x)\right)+2 c^2 d^2 x \left(a+b \sinh ^{-1}(c x)\right)-\frac{d^2 \left(a+b \sinh ^{-1}(c x)\right)}{x}-\frac{1}{9} b c d^2 \left(c^2 x^2+1\right)^{3/2}-\frac{5}{3} b c d^2 \sqrt{c^2 x^2+1}-b c d^2 \tanh ^{-1}\left(\sqrt{c^2 x^2+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^2*x^2] - b*c^3*x^3*Sqrt[1 + c^2*x^2] + 3*b*(-3 + 6*c^2*x^2 + c^4*x^4)*ArcSinh[c*x] + 9*b*c*x*Log[x] - 9*b*c*x*Log[1 + Sqrt[1 + c^2*x^2]]))/(9*x)","A",1
17,1,143,187,0.3236327,"\int \frac{\left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]))/x^3,x]","\frac{1}{4} d^2 \left(2 c^4 x^2 \left(a+b \sinh ^{-1}(c x)\right)+4 c^2 \left(2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)\right)-\frac{4 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{b}-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{x^2}-\frac{2 b c \sqrt{c^2 x^2+1}}{x}+b c^2 \left(\sinh ^{-1}(c x)-c x \sqrt{c^2 x^2+1}\right)\right)","c^2 d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{d^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 x^2}+\frac{c^2 d^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{b}+2 c^2 d^2 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-b c^2 d^2 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-\frac{b c d^2 \left(c^2 x^2+1\right)^{3/2}}{2 x}+\frac{1}{4} b c^2 d^2 \sinh ^{-1}(c x)+\frac{1}{4} b c^3 d^2 x \sqrt{c^2 x^2+1}",1,"(d^2*((-2*b*c*Sqrt[1 + c^2*x^2])/x + b*c^2*(-(c*x*Sqrt[1 + c^2*x^2]) + ArcSinh[c*x]) - (2*(a + b*ArcSinh[c*x]))/x^2 + 2*c^4*x^2*(a + b*ArcSinh[c*x]) - (4*c^2*(a + b*ArcSinh[c*x])^2)/b + 4*c^2*(2*(a + b*ArcSinh[c*x])*Log[1 - E^(2*ArcSinh[c*x])] + b*PolyLog[2, E^(2*ArcSinh[c*x])])))/4","A",0
18,1,133,126,0.1152128,"\int \frac{\left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]))/x^4,x]","\frac{d^2 \left(6 a c^4 x^4-12 a c^2 x^2-2 a+11 b c^3 x^3 \log (x)-b c x \sqrt{c^2 x^2+1}+2 b \left(3 c^4 x^4-6 c^2 x^2-1\right) \sinh ^{-1}(c x)-6 b c^3 x^3 \sqrt{c^2 x^2+1}-11 b c^3 x^3 \log \left(\sqrt{c^2 x^2+1}+1\right)\right)}{6 x^3}","c^4 d^2 x \left(a+b \sinh ^{-1}(c x)\right)-\frac{2 c^2 d^2 \left(a+b \sinh ^{-1}(c x)\right)}{x}-\frac{d^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 x^3}-\frac{b c d^2 \sqrt{c^2 x^2+1}}{6 x^2}-b c^3 d^2 \sqrt{c^2 x^2+1}-\frac{11}{6} b c^3 d^2 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)",1,"(d^2*(-2*a - 12*a*c^2*x^2 + 6*a*c^4*x^4 - b*c*x*Sqrt[1 + c^2*x^2] - 6*b*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*b*(-1 - 6*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x] + 11*b*c^3*x^3*Log[x] - 11*b*c^3*x^3*Log[1 + Sqrt[1 + c^2*x^2]]))/(6*x^3)","A",1
19,1,143,226,0.1140119,"\int x^4 \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[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) \sinh ^{-1}(c x)-b \sqrt{c^2 x^2+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 \sinh ^{-1}(c x)\right)+\frac{1}{3} c^4 d^3 x^9 \left(a+b \sinh ^{-1}(c x)\right)+\frac{3}{7} c^2 d^3 x^7 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{5} d^3 x^5 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d^3 \left(c^2 x^2+1\right)^{11/2}}{121 c^5}+\frac{4 b d^3 \left(c^2 x^2+1\right)^{9/2}}{297 c^5}-\frac{b d^3 \left(c^2 x^2+1\right)^{7/2}}{1617 c^5}-\frac{2 b d^3 \left(c^2 x^2+1\right)^{5/2}}{1925 c^5}-\frac{8 b d^3 \left(c^2 x^2+1\right)^{3/2}}{3465 c^5}-\frac{16 b d^3 \sqrt{c^2 x^2+1}}{1155 c^5}",1,"(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^2*x^2]*(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)*ArcSinh[c*x]))/(4002075*c^5)","A",1
20,1,139,199,0.1050089,"\int x^3 \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + c^2*d*x^2)^3*(a + b*ArcSinh[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)+15 b \left(512 c^{10} x^{10}+1920 c^8 x^8+2560 c^6 x^6+1280 c^4 x^4-79\right) \sinh ^{-1}(c x)-b c x \sqrt{c^2 x^2+1} \left(768 c^8 x^8+2736 c^6 x^6+3208 c^4 x^4+790 c^2 x^2-1185\right)\right)}{76800 c^4}","\frac{d^3 \left(c^2 x^2+1\right)^5 \left(a+b \sinh ^{-1}(c x)\right)}{10 c^4}-\frac{d^3 \left(c^2 x^2+1\right)^4 \left(a+b \sinh ^{-1}(c x)\right)}{8 c^4}+\frac{49 b d^3 \sinh ^{-1}(c x)}{5120 c^4}-\frac{b d^3 x \left(c^2 x^2+1\right)^{9/2}}{100 c^3}+\frac{7 b d^3 x \left(c^2 x^2+1\right)^{7/2}}{1600 c^3}+\frac{49 b d^3 x \left(c^2 x^2+1\right)^{5/2}}{9600 c^3}+\frac{49 b d^3 x \left(c^2 x^2+1\right)^{3/2}}{7680 c^3}+\frac{49 b d^3 x \sqrt{c^2 x^2+1}}{5120 c^3}",1,"(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^2*x^2]*(-1185 + 790*c^2*x^2 + 3208*c^4*x^4 + 2736*c^6*x^6 + 768*c^8*x^8) + 15*b*(-79 + 1280*c^4*x^4 + 2560*c^6*x^6 + 1920*c^8*x^8 + 512*c^10*x^10)*ArcSinh[c*x]))/(76800*c^4)","A",1
21,1,135,202,0.0982232,"\int x^2 \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + c^2*d*x^2)^3*(a + b*ArcSinh[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^2 x^2+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) \sinh ^{-1}(c x)\right)}{99225 c^3}","\frac{1}{9} c^6 d^3 x^9 \left(a+b \sinh ^{-1}(c x)\right)+\frac{3}{7} c^4 d^3 x^7 \left(a+b \sinh ^{-1}(c x)\right)+\frac{3}{5} c^2 d^3 x^5 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{3} d^3 x^3 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d^3 \left(c^2 x^2+1\right)^{9/2}}{81 c^3}+\frac{b d^3 \left(c^2 x^2+1\right)^{7/2}}{441 c^3}+\frac{2 b d^3 \left(c^2 x^2+1\right)^{5/2}}{525 c^3}+\frac{8 b d^3 \left(c^2 x^2+1\right)^{3/2}}{945 c^3}+\frac{16 b d^3 \sqrt{c^2 x^2+1}}{315 c^3}",1,"(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^2*x^2]*(-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)*ArcSinh[c*x]))/(99225*c^3)","A",1
22,1,128,145,0.1617793,"\int x \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x*(d + c^2*d*x^2)^3*(a + b*ArcSinh[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^2 x^2+1} \left(48 c^6 x^6+200 c^4 x^4+326 c^2 x^2+279\right)\right)+3 b \left(128 c^8 x^8+512 c^6 x^6+768 c^4 x^4+512 c^2 x^2+93\right) \sinh ^{-1}(c x)\right)}{3072 c^2}","\frac{d^3 \left(c^2 x^2+1\right)^4 \left(a+b \sinh ^{-1}(c x)\right)}{8 c^2}-\frac{b d^3 x \left(c^2 x^2+1\right)^{7/2}}{64 c}-\frac{7 b d^3 x \left(c^2 x^2+1\right)^{5/2}}{384 c}-\frac{35 b d^3 x \left(c^2 x^2+1\right)^{3/2}}{1536 c}-\frac{35 b d^3 x \sqrt{c^2 x^2+1}}{1024 c}-\frac{35 b d^3 \sinh ^{-1}(c x)}{1024 c^2}",1,"(d^3*(c*x*(384*a*c*x*(4 + 6*c^2*x^2 + 4*c^4*x^4 + c^6*x^6) - b*Sqrt[1 + c^2*x^2]*(279 + 326*c^2*x^2 + 200*c^4*x^4 + 48*c^6*x^6)) + 3*b*(93 + 512*c^2*x^2 + 768*c^4*x^4 + 512*c^6*x^6 + 128*c^8*x^8)*ArcSinh[c*x]))/(3072*c^2)","A",1
23,1,119,170,0.1526884,"\int \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + c^2*d*x^2)^3*(a + b*ArcSinh[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^2 x^2+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) \sinh ^{-1}(c x)\right)}{3675 c}","\frac{1}{7} c^6 d^3 x^7 \left(a+b \sinh ^{-1}(c x)\right)+\frac{3}{5} c^4 d^3 x^5 \left(a+b \sinh ^{-1}(c x)\right)+c^2 d^3 x^3 \left(a+b \sinh ^{-1}(c x)\right)+d^3 x \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d^3 \left(c^2 x^2+1\right)^{7/2}}{49 c}-\frac{6 b d^3 \left(c^2 x^2+1\right)^{5/2}}{175 c}-\frac{8 b d^3 \left(c^2 x^2+1\right)^{3/2}}{105 c}-\frac{16 b d^3 \sqrt{c^2 x^2+1}}{35 c}",1,"(d^3*(105*a*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) - b*Sqrt[1 + c^2*x^2]*(2161 + 757*c^2*x^2 + 351*c^4*x^4 + 75*c^6*x^6) + 105*b*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6)*ArcSinh[c*x]))/(3675*c)","A",1
24,1,189,221,0.1245967,"\int \frac{\left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x,x]","\frac{1}{144} d^3 \left(3 \sinh ^{-1}(c x) \left(-48 a+b \left(8 c^6 x^6+36 c^4 x^4+72 c^2 x^2+25\right)+48 b \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)\right)+24 a c^6 x^6+108 a c^4 x^4+216 a c^2 x^2+144 a \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)-75 b c x \sqrt{c^2 x^2+1}-4 b c^5 x^5 \sqrt{c^2 x^2+1}-22 b c^3 x^3 \sqrt{c^2 x^2+1}+72 b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-72 b \sinh ^{-1}(c x)^2\right)","\frac{1}{6} d^3 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{4} d^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{2} d^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{d^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b}+d^3 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{36} b c d^3 x \left(c^2 x^2+1\right)^{5/2}-\frac{7}{72} b c d^3 x \left(c^2 x^2+1\right)^{3/2}-\frac{19}{48} b c d^3 x \sqrt{c^2 x^2+1}-\frac{1}{2} b d^3 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-\frac{19}{48} b d^3 \sinh ^{-1}(c x)",1,"(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^2*x^2] - 22*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 4*b*c^5*x^5*Sqrt[1 + c^2*x^2] - 72*b*ArcSinh[c*x]^2 + 144*a*Log[1 - E^(2*ArcSinh[c*x])] + 3*ArcSinh[c*x]*(-48*a + b*(25 + 72*c^2*x^2 + 36*c^4*x^4 + 8*c^6*x^6) + 48*b*Log[1 - E^(2*ArcSinh[c*x])]) + 72*b*PolyLog[2, E^(2*ArcSinh[c*x])]))/144","A",0
25,1,163,160,0.1493841,"\int \frac{\left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x^2,x]","\frac{d^3 \left(5 a c^6 x^6+25 a c^4 x^4+75 a c^2 x^2-25 a-61 b c x \sqrt{c^2 x^2+1}-25 b c x \log \left(\sqrt{c^2 x^2+1}+1\right)-b c^5 x^5 \sqrt{c^2 x^2+1}-7 b c^3 x^3 \sqrt{c^2 x^2+1}+5 b \left(c^6 x^6+5 c^4 x^4+15 c^2 x^2-5\right) \sinh ^{-1}(c x)+25 b c x \log (x)\right)}{25 x}","\frac{1}{5} c^6 d^3 x^5 \left(a+b \sinh ^{-1}(c x)\right)+c^4 d^3 x^3 \left(a+b \sinh ^{-1}(c x)\right)+3 c^2 d^3 x \left(a+b \sinh ^{-1}(c x)\right)-\frac{d^3 \left(a+b \sinh ^{-1}(c x)\right)}{x}-\frac{1}{25} b c d^3 \left(c^2 x^2+1\right)^{5/2}-\frac{1}{5} b c d^3 \left(c^2 x^2+1\right)^{3/2}-\frac{11}{5} b c d^3 \sqrt{c^2 x^2+1}-b c d^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)",1,"(d^3*(-25*a + 75*a*c^2*x^2 + 25*a*c^4*x^4 + 5*a*c^6*x^6 - 61*b*c*x*Sqrt[1 + c^2*x^2] - 7*b*c^3*x^3*Sqrt[1 + c^2*x^2] - b*c^5*x^5*Sqrt[1 + c^2*x^2] + 5*b*(-5 + 15*c^2*x^2 + 5*c^4*x^4 + c^6*x^6)*ArcSinh[c*x] + 25*b*c*x*Log[x] - 25*b*c*x*Log[1 + Sqrt[1 + c^2*x^2]]))/(25*x)","A",1
26,1,184,249,0.4113455,"\int \frac{\left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]))/x^3,x]","\frac{1}{32} d^3 \left(8 c^6 x^4 \left(a+b \sinh ^{-1}(c x)\right)+48 c^4 x^2 \left(a+b \sinh ^{-1}(c x)\right)-\frac{48 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{b}+96 c^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{16 \left(a+b \sinh ^{-1}(c x)\right)}{x^2}+48 b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-\frac{16 b c \sqrt{c^2 x^2+1}}{x}-21 b c^2 \left(c x \sqrt{c^2 x^2+1}-\sinh ^{-1}(c x)\right)-2 b c^5 x^3 \sqrt{c^2 x^2+1}\right)","-\frac{d^3 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)}{2 x^2}+\frac{3}{4} c^2 d^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)+\frac{3}{2} c^2 d^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{3 c^2 d^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b}+3 c^2 d^3 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{3}{2} b c^2 d^3 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-\frac{b c d^3 \left(c^2 x^2+1\right)^{5/2}}{2 x}-\frac{3}{32} b c^2 d^3 \sinh ^{-1}(c x)+\frac{7}{16} b c^3 d^3 x \left(c^2 x^2+1\right)^{3/2}-\frac{3}{32} b c^3 d^3 x \sqrt{c^2 x^2+1}",1,"(d^3*((-16*b*c*Sqrt[1 + c^2*x^2])/x - 2*b*c^5*x^3*Sqrt[1 + c^2*x^2] - 21*b*c^2*(c*x*Sqrt[1 + c^2*x^2] - ArcSinh[c*x]) - (16*(a + b*ArcSinh[c*x]))/x^2 + 48*c^4*x^2*(a + b*ArcSinh[c*x]) + 8*c^6*x^4*(a + b*ArcSinh[c*x]) - (48*c^2*(a + b*ArcSinh[c*x])^2)/b + 96*c^2*(a + b*ArcSinh[c*x])*Log[1 - E^(2*ArcSinh[c*x])] + 48*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])]))/32","A",0
27,1,171,174,0.1502815,"\int \frac{\left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[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 \log (x)-3 b c x \sqrt{c^2 x^2+1}-2 b c^5 x^5 \sqrt{c^2 x^2+1}-50 b c^3 x^3 \sqrt{c^2 x^2+1}-51 b c^3 x^3 \log \left(\sqrt{c^2 x^2+1}+1\right)+6 b \left(c^6 x^6+9 c^4 x^4-9 c^2 x^2-1\right) \sinh ^{-1}(c x)\right)}{18 x^3}","\frac{1}{3} c^6 d^3 x^3 \left(a+b \sinh ^{-1}(c x)\right)+3 c^4 d^3 x \left(a+b \sinh ^{-1}(c x)\right)-\frac{3 c^2 d^3 \left(a+b \sinh ^{-1}(c x)\right)}{x}-\frac{d^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 x^3}-\frac{b c d^3 \sqrt{c^2 x^2+1}}{6 x^2}-\frac{1}{9} b c^3 d^3 \left(c^2 x^2+1\right)^{3/2}-\frac{8}{3} b c^3 d^3 \sqrt{c^2 x^2+1}-\frac{17}{6} b c^3 d^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)",1,"(d^3*(-6*a - 54*a*c^2*x^2 + 54*a*c^4*x^4 + 6*a*c^6*x^6 - 3*b*c*x*Sqrt[1 + c^2*x^2] - 50*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 2*b*c^5*x^5*Sqrt[1 + c^2*x^2] + 6*b*(-1 - 9*c^2*x^2 + 9*c^4*x^4 + c^6*x^6)*ArcSinh[c*x] + 51*b*c^3*x^3*Log[x] - 51*b*c^3*x^3*Log[1 + Sqrt[1 + c^2*x^2]]))/(18*x^3)","A",1
28,1,170,156,0.2816284,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{d+c^2 d x^2} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2),x]","\frac{3 a c^3 x^3-9 a c x+9 a \tan ^{-1}(c x)+3 b c^3 x^3 \sinh ^{-1}(c x)-b c^2 x^2 \sqrt{c^2 x^2+1}+11 b \sqrt{c^2 x^2+1}-9 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)+9 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)-9 b c x \sinh ^{-1}(c x)+9 i b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-9 i b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{9 c^5 d}","\frac{2 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{c^4 d}+\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 c^2 d}-\frac{i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c^5 d}+\frac{i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c^5 d}-\frac{b \left(c^2 x^2+1\right)^{3/2}}{9 c^5 d}+\frac{4 b \sqrt{c^2 x^2+1}}{3 c^5 d}",1,"(-9*a*c*x + 3*a*c^3*x^3 + 11*b*Sqrt[1 + c^2*x^2] - b*c^2*x^2*Sqrt[1 + c^2*x^2] - 9*b*c*x*ArcSinh[c*x] + 3*b*c^3*x^3*ArcSinh[c*x] + 9*a*ArcTan[c*x] + (9*I)*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (9*I)*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (9*I)*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + (9*I)*b*PolyLog[2, I*E^ArcSinh[c*x]])/(9*c^5*d)","A",0
29,1,181,135,0.1881124,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{d+c^2 d x^2} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2),x]","-\frac{-2 a c^2 x^2+2 a \log \left(c^2 x^2+1\right)+4 b \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)+4 b \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)+b c x \sqrt{c^2 x^2+1}-2 b c^2 x^2 \sinh ^{-1}(c x)+4 b \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+4 b \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-2 b \sinh ^{-1}(c x)^2-b \sinh ^{-1}(c x)}{4 c^4 d}","\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c^4 d}-\frac{\log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^4 d}+\frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 c^2 d}-\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 c^4 d}+\frac{b \sinh ^{-1}(c x)}{4 c^4 d}-\frac{b x \sqrt{c^2 x^2+1}}{4 c^3 d}",1,"-1/4*(-2*a*c^2*x^2 + b*c*x*Sqrt[1 + c^2*x^2] - b*ArcSinh[c*x] - 2*b*c^2*x^2*ArcSinh[c*x] - 2*b*ArcSinh[c*x]^2 + 4*b*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 4*b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 2*a*Log[1 + c^2*x^2] + 4*b*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 4*b*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(c^4*d)","A",0
30,1,121,108,0.1508545,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{d+c^2 d x^2} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2),x]","\frac{a c x-a \tan ^{-1}(c x)-b \sqrt{c^2 x^2+1}+i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)-i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+b c x \sinh ^{-1}(c x)-i b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+i b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{c^3 d}","-\frac{2 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d}+\frac{i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c^3 d}-\frac{i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c^3 d}-\frac{b \sqrt{c^2 x^2+1}}{c^3 d}",1,"(a*c*x - b*Sqrt[1 + c^2*x^2] + b*c*x*ArcSinh[c*x] - a*ArcTan[c*x] - I*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + I*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + I*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] - I*b*PolyLog[2, I*E^ArcSinh[c*x]])/(c^3*d)","A",0
31,1,167,73,0.0629584,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)}{d+c^2 d x^2} \, dx","Integrate[(x*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2),x]","\frac{a \log \left(c^2 x^2+1\right)}{2 c^2 d}+\frac{b \text{Li}_2\left(-\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)}{c^2 d}+\frac{b \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)}{c^2 d}-\frac{b \sinh ^{-1}(c x)^2}{2 c^2 d}+\frac{b \sinh ^{-1}(c x) \log \left(1-\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)}{c^2 d}+\frac{b \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)}{c^2 d}","-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c^2 d}+\frac{\log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d}+\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 c^2 d}",1,"-1/2*(b*ArcSinh[c*x]^2)/(c^2*d) + (b*ArcSinh[c*x]*Log[1 - (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(c^2*d) + (b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(c^2*d) + (a*Log[1 + c^2*x^2])/(2*c^2*d) + (b*PolyLog[2, -((Sqrt[-c^2]*E^ArcSinh[c*x])/c)])/(c^2*d) + (b*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(c^2*d)","B",0
32,1,135,70,0.0844509,"\int \frac{a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2),x]","-\frac{c \left(a \sqrt{-c^2} \tan ^{-1}(c x)+b c \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)-b c \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)-b c \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+b c \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)\right)}{\left(-c^2\right)^{3/2} d}","\frac{2 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c d}-\frac{i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c d}+\frac{i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c d}",1,"-((c*(a*Sqrt[-c^2]*ArcTan[c*x] - b*c*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + b*c*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + b*c*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - b*c*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c]))/((-c^2)^(3/2)*d))","A",0
33,1,207,61,0.0807233,"\int \frac{a+b \sinh ^{-1}(c x)}{x \left(d+c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)),x]","-\frac{a \log \left(c^2 x^2+1\right)}{2 d}-\frac{a \sinh ^{-1}(c x)}{d}+\frac{a \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)}{d}-\frac{b \text{Li}_2\left(-\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)}{d}-\frac{b \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)}{d}-\frac{b \sinh ^{-1}(c x) \log \left(1-\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)}{d}-\frac{b \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)}{d}+\frac{b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d}+\frac{b \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)}{d}","-\frac{2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}-\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d}+\frac{b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d}",1,"-((a*ArcSinh[c*x])/d) - (b*ArcSinh[c*x]*Log[1 - (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/d - (b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/d + (a*Log[1 - E^(2*ArcSinh[c*x])])/d + (b*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])])/d - (a*Log[1 + c^2*x^2])/(2*d) - (b*PolyLog[2, -((Sqrt[-c^2]*E^ArcSinh[c*x])/c)])/d - (b*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/d + (b*PolyLog[2, E^(2*ArcSinh[c*x])])/(2*d)","B",0
34,1,182,101,0.1367284,"\int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left(d+c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)),x]","-\frac{a c x \tan ^{-1}(c x)+a-b \sqrt{-c^2} x \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)+b \sqrt{-c^2} x \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)+b c x \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+b \sqrt{-c^2} x \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)-b \sqrt{-c^2} x \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)+b \sinh ^{-1}(c x)}{d x}","-\frac{a+b \sinh ^{-1}(c x)}{d x}-\frac{2 c \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}-\frac{b c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{d}+\frac{i b c \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{d}-\frac{i b c \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{d}",1,"-((a + b*ArcSinh[c*x] + a*c*x*ArcTan[c*x] + b*c*x*ArcTanh[Sqrt[1 + c^2*x^2]] + b*Sqrt[-c^2]*x*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - b*Sqrt[-c^2]*x*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - b*Sqrt[-c^2]*x*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + b*Sqrt[-c^2]*x*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(d*x))","A",0
35,1,240,113,0.2366182,"\int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left(d+c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)),x]","\frac{-c^2 \left(2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)\right)+\frac{c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{b}-\frac{a+b \sinh ^{-1}(c x)}{x^2}+a c^2 \log \left(c^2 x^2+1\right)+2 b c^2 \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)+2 b c^2 \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)-\frac{b c \sqrt{c^2 x^2+1}}{x}-b c^2 \sinh ^{-1}(c x)^2+2 b c^2 \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+2 b c^2 \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)}{2 d}","\frac{2 c^2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2}+\frac{b c^2 \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d}-\frac{b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d}-\frac{b c \sqrt{c^2 x^2+1}}{2 d x}",1,"(-((b*c*Sqrt[1 + c^2*x^2])/x) - b*c^2*ArcSinh[c*x]^2 - (a + b*ArcSinh[c*x])/x^2 + (c^2*(a + b*ArcSinh[c*x])^2)/b + 2*b*c^2*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 2*b*c^2*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + a*c^2*Log[1 + c^2*x^2] + 2*b*c^2*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 2*b*c^2*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - c^2*(2*(a + b*ArcSinh[c*x])*Log[1 - E^(2*ArcSinh[c*x])] + b*PolyLog[2, E^(2*ArcSinh[c*x])]))/(2*d)","B",0
36,1,247,156,0.1912222,"\int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left(d+c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)),x]","\frac{6 a c^3 x^3 \tan ^{-1}(c x)+6 a c^2 x^2-2 a+6 b \left(-c^2\right)^{3/2} x^3 \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)-6 b \left(-c^2\right)^{3/2} x^3 \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)-6 b \left(-c^2\right)^{3/2} x^3 \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+6 b \left(-c^2\right)^{3/2} x^3 \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-b c x \sqrt{c^2 x^2+1}+6 b c^2 x^2 \sinh ^{-1}(c x)+7 b c^3 x^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-2 b \sinh ^{-1}(c x)}{6 d x^3}","\frac{2 c^3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}+\frac{c^2 \left(a+b \sinh ^{-1}(c x)\right)}{d x}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3}-\frac{i b c^3 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{d}+\frac{i b c^3 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{d}-\frac{b c \sqrt{c^2 x^2+1}}{6 d x^2}+\frac{7 b c^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{6 d}",1,"(-2*a + 6*a*c^2*x^2 - b*c*x*Sqrt[1 + c^2*x^2] - 2*b*ArcSinh[c*x] + 6*b*c^2*x^2*ArcSinh[c*x] + 6*a*c^3*x^3*ArcTan[c*x] + 7*b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] - 6*b*(-c^2)^(3/2)*x^3*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 6*b*(-c^2)^(3/2)*x^3*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 6*b*(-c^2)^(3/2)*x^3*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 6*b*(-c^2)^(3/2)*x^3*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(6*d*x^3)","A",0
37,1,268,171,0.3339405,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2,x]","\frac{2 a c^3 x^3-3 a c^2 x^2 \tan ^{-1}(c x)+3 a c x-3 a \tan ^{-1}(c x)+2 b c^3 x^3 \sinh ^{-1}(c x)+3 i b \left(c^2 x^2+1\right) \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)-3 i b \left(c^2 x^2+1\right) \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)-2 b c^2 x^2 \sqrt{c^2 x^2+1}-b \sqrt{c^2 x^2+1}-3 i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+3 i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+3 b c x \sinh ^{-1}(c x)-3 i b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+3 i b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{2 c^5 d^2 \left(c^2 x^2+1\right)}","-\frac{3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d^2}+\frac{3 x \left(a+b \sinh ^{-1}(c x)\right)}{2 c^4 d^2}-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{2 c^2 d^2 \left(c^2 x^2+1\right)}+\frac{3 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{2 c^5 d^2}-\frac{3 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{2 c^5 d^2}-\frac{b \sqrt{c^2 x^2+1}}{c^5 d^2}+\frac{b}{2 c^5 d^2 \sqrt{c^2 x^2+1}}",1,"(3*a*c*x + 2*a*c^3*x^3 - b*Sqrt[1 + c^2*x^2] - 2*b*c^2*x^2*Sqrt[1 + c^2*x^2] + 3*b*c*x*ArcSinh[c*x] + 2*b*c^3*x^3*ArcSinh[c*x] - 3*a*ArcTan[c*x] - 3*a*c^2*x^2*ArcTan[c*x] - (3*I)*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (3*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (3*I)*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + (3*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + (3*I)*b*(1 + c^2*x^2)*PolyLog[2, (-I)*E^ArcSinh[c*x]] - (3*I)*b*(1 + c^2*x^2)*PolyLog[2, I*E^ArcSinh[c*x]])/(2*c^5*d^2*(1 + c^2*x^2))","A",0
38,1,241,145,0.2147773,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2,x]","\frac{a c^2 x^2 \log \left(c^2 x^2+1\right)+a \log \left(c^2 x^2+1\right)+a+2 b \left(c^2 x^2+1\right) \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)+2 b \left(c^2 x^2+1\right) \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)-b c x \sqrt{c^2 x^2+1}-b c^2 x^2 \sinh ^{-1}(c x)^2+2 b c^2 x^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+2 b c^2 x^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)-b \sinh ^{-1}(c x)^2+b \sinh ^{-1}(c x)+2 b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+2 b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{2 c^4 d^2 \left(c^2 x^2+1\right)}","-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c^4 d^2}+\frac{\log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^4 d^2}-\frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 c^2 d^2 \left(c^2 x^2+1\right)}+\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 c^4 d^2}+\frac{b \sinh ^{-1}(c x)}{2 c^4 d^2}-\frac{b x}{2 c^3 d^2 \sqrt{c^2 x^2+1}}",1,"(a - b*c*x*Sqrt[1 + c^2*x^2] + b*ArcSinh[c*x] - b*ArcSinh[c*x]^2 - b*c^2*x^2*ArcSinh[c*x]^2 + 2*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 2*b*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 2*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + 2*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + a*Log[1 + c^2*x^2] + a*c^2*x^2*Log[1 + c^2*x^2] + 2*b*(1 + c^2*x^2)*PolyLog[2, (-I)*E^ArcSinh[c*x]] + 2*b*(1 + c^2*x^2)*PolyLog[2, I*E^ArcSinh[c*x]])/(2*c^4*d^2*(1 + c^2*x^2))","C",0
39,1,221,127,0.2583404,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2,x]","-\frac{-a c^2 x^2 \tan ^{-1}(c x)+a c x-a \tan ^{-1}(c x)+i b \left(c^2 x^2+1\right) \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)-i b \left(c^2 x^2+1\right) \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+b \sqrt{c^2 x^2+1}-i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+b c x \sinh ^{-1}(c x)-i b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+i b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{2 c^3 d^2 \left(c^2 x^2+1\right)}","\frac{\tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d^2}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{2 c^2 d^2 \left(c^2 x^2+1\right)}-\frac{i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{2 c^3 d^2}+\frac{i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{2 c^3 d^2}-\frac{b}{2 c^3 d^2 \sqrt{c^2 x^2+1}}",1,"-1/2*(a*c*x + b*Sqrt[1 + c^2*x^2] + b*c*x*ArcSinh[c*x] - a*ArcTan[c*x] - a*c^2*x^2*ArcTan[c*x] - I*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - I*b*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + I*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + I*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + I*b*(1 + c^2*x^2)*PolyLog[2, (-I)*E^ArcSinh[c*x]] - I*b*(1 + c^2*x^2)*PolyLog[2, I*E^ArcSinh[c*x]])/(c^3*d^2*(1 + c^2*x^2))","A",0
40,1,74,55,0.0610277,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2,x]","-\frac{a}{2 c^2 d^2 \left(c^2 x^2+1\right)}+\frac{b x}{2 c d^2 \sqrt{c^2 x^2+1}}-\frac{b \sinh ^{-1}(c x)}{2 c^2 d^2 \left(c^2 x^2+1\right)}","\frac{b x}{2 c d^2 \sqrt{c^2 x^2+1}}-\frac{a+b \sinh ^{-1}(c x)}{2 c^2 d^2 \left(c^2 x^2+1\right)}",1,"-1/2*a/(c^2*d^2*(1 + c^2*x^2)) + (b*x)/(2*c*d^2*Sqrt[1 + c^2*x^2]) - (b*ArcSinh[c*x])/(2*c^2*d^2*(1 + c^2*x^2))","A",1
41,1,216,124,0.1067295,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^2,x]","\frac{a c^2 x^2 \tan ^{-1}(c x)+a c x+a \tan ^{-1}(c x)-i b \left(c^2 x^2+1\right) \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)+i b \left(c^2 x^2+1\right) \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+b \sqrt{c^2 x^2+1}+i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+b c x \sinh ^{-1}(c x)+i b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-i b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{2 d^2 \left(c^3 x^2+c\right)}","\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{2 d^2 \left(c^2 x^2+1\right)}+\frac{\tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c d^2}+\frac{b}{2 c d^2 \sqrt{c^2 x^2+1}}-\frac{i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{2 c d^2}+\frac{i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{2 c d^2}",1,"(a*c*x + b*Sqrt[1 + c^2*x^2] + b*c*x*ArcSinh[c*x] + a*ArcTan[c*x] + a*c^2*x^2*ArcTan[c*x] + I*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + I*b*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - I*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - I*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - I*b*(1 + c^2*x^2)*PolyLog[2, (-I)*E^ArcSinh[c*x]] + I*b*(1 + c^2*x^2)*PolyLog[2, I*E^ArcSinh[c*x]])/(2*d^2*(c + c^3*x^2))","A",0
42,1,234,110,0.4150398,"\int \frac{a+b \sinh ^{-1}(c x)}{x \left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)^2),x]","-\frac{\frac{a^2}{b}-\frac{a}{c^2 x^2+1}+a \log \left(c^2 x^2+1\right)+2 a \sinh ^{-1}(c x)-2 a \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+2 b \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)+2 b \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)+\frac{b c x}{\sqrt{c^2 x^2+1}}-\frac{b \sinh ^{-1}(c x)}{c^2 x^2+1}+2 b \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+2 b \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-2 b \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)}{2 d^2}","\frac{a+b \sinh ^{-1}(c x)}{2 d^2 \left(c^2 x^2+1\right)}-\frac{2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}-\frac{b c x}{2 d^2 \sqrt{c^2 x^2+1}}-\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d^2}+\frac{b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d^2}",1,"-1/2*(a^2/b - a/(1 + c^2*x^2) + (b*c*x)/Sqrt[1 + c^2*x^2] + 2*a*ArcSinh[c*x] - (b*ArcSinh[c*x])/(1 + c^2*x^2) + 2*b*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 2*b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 2*a*Log[1 - E^(2*ArcSinh[c*x])] - 2*b*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + a*Log[1 + c^2*x^2] + 2*b*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 2*b*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - b*PolyLog[2, E^(2*ArcSinh[c*x])])/d^2","B",0
43,1,253,168,0.5492005,"\int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)^2),x]","-\frac{-\frac{a}{c^2 x^3+x}+3 a c \tan ^{-1}(c x)+\frac{3 a}{x}+\frac{b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};c^2 x^2+1\right)}{\sqrt{c^2 x^2+1}}-3 b \sqrt{-c^2} \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)+3 b \sqrt{-c^2} \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)-\frac{b \sinh ^{-1}(c x)}{c^2 x^3+x}+3 b c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+3 b \sqrt{-c^2} \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)-3 b \sqrt{-c^2} \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)+\frac{3 b \sinh ^{-1}(c x)}{x}}{2 d^2}","-\frac{3 c^2 x \left(a+b \sinh ^{-1}(c x)\right)}{2 d^2 \left(c^2 x^2+1\right)}-\frac{a+b \sinh ^{-1}(c x)}{d^2 x \left(c^2 x^2+1\right)}-\frac{3 c \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}-\frac{b c}{2 d^2 \sqrt{c^2 x^2+1}}-\frac{b c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{d^2}+\frac{3 i b c \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{2 d^2}-\frac{3 i b c \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{2 d^2}",1,"-1/2*((3*a)/x - a/(x + c^2*x^3) + (3*b*ArcSinh[c*x])/x - (b*ArcSinh[c*x])/(x + c^2*x^3) + 3*a*c*ArcTan[c*x] + 3*b*c*ArcTanh[Sqrt[1 + c^2*x^2]] + (b*c*Hypergeometric2F1[-1/2, 1, 1/2, 1 + c^2*x^2])/Sqrt[1 + c^2*x^2] + 3*b*Sqrt[-c^2]*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 3*b*Sqrt[-c^2]*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 3*b*Sqrt[-c^2]*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 3*b*Sqrt[-c^2]*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/d^2","C",0
44,1,326,146,0.4477732,"\int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^2),x]","\frac{\frac{2 a^2 c^2}{b}+2 a c^2 \log \left(c^2 x^2+1\right)+\frac{a}{c^2 x^4+x^2}+4 a c^2 \sinh ^{-1}(c x)-4 a c^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)-\frac{2 a}{x^2}+4 b c^2 \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)+4 b c^2 \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)-2 b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-\frac{2 b c \sqrt{c^2 x^2+1}}{x}+\frac{b c}{x \sqrt{c^2 x^2+1}}+\frac{b \sinh ^{-1}(c x)}{c^2 x^4+x^2}+4 b c^2 \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+4 b c^2 \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-4 b c^2 \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+\frac{2 b c^3 x}{\sqrt{c^2 x^2+1}}-\frac{2 b \sinh ^{-1}(c x)}{x^2}}{2 d^2}","-\frac{c^2 \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \left(c^2 x^2+1\right)}-\frac{a+b \sinh ^{-1}(c x)}{2 d^2 x^2 \left(c^2 x^2+1\right)}+\frac{4 c^2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}+\frac{b c^2 \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{d^2}-\frac{b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{d^2}-\frac{b c}{2 d^2 x \sqrt{c^2 x^2+1}}",1,"((2*a^2*c^2)/b - (2*a)/x^2 + (b*c)/(x*Sqrt[1 + c^2*x^2]) + (2*b*c^3*x)/Sqrt[1 + c^2*x^2] - (2*b*c*Sqrt[1 + c^2*x^2])/x + a/(x^2 + c^2*x^4) + 4*a*c^2*ArcSinh[c*x] - (2*b*ArcSinh[c*x])/x^2 + (b*ArcSinh[c*x])/(x^2 + c^2*x^4) + 4*b*c^2*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 4*b*c^2*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 4*a*c^2*Log[1 - E^(2*ArcSinh[c*x])] - 4*b*c^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 2*a*c^2*Log[1 + c^2*x^2] + 4*b*c^2*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 4*b*c^2*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 2*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])])/(2*d^2)","B",0
45,1,311,239,0.5766201,"\int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^2),x]","\frac{5 a c^3 \tan ^{-1}(c x)+\frac{a}{c^2 x^5+x^3}+\frac{5 a c^2}{x}-\frac{5 a}{3 x^3}+5 b \left(-c^2\right)^{3/2} \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)-5 b \left(-c^2\right)^{3/2} \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)-\frac{5 b c \sqrt{c^2 x^2+1}}{6 x^2}+\frac{b \sinh ^{-1}(c x)}{c^2 x^5+x^3}+\frac{5 b c^2 \sinh ^{-1}(c x)}{x}-5 b \left(-c^2\right)^{3/2} \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+5 b \left(-c^2\right)^{3/2} \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)+\frac{b c^3 \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};c^2 x^2+1\right)}{\sqrt{c^2 x^2+1}}+\frac{35}{6} b c^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-\frac{5 b \sinh ^{-1}(c x)}{3 x^3}}{2 d^2}","\frac{5 c^3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}+\frac{5 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 x \left(c^2 x^2+1\right)}-\frac{a+b \sinh ^{-1}(c x)}{3 d^2 x^3 \left(c^2 x^2+1\right)}+\frac{5 c^4 x \left(a+b \sinh ^{-1}(c x)\right)}{2 d^2 \left(c^2 x^2+1\right)}-\frac{5 i b c^3 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{2 d^2}+\frac{5 i b c^3 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{2 d^2}-\frac{b c}{6 d^2 x^2 \sqrt{c^2 x^2+1}}+\frac{b c^3}{3 d^2 \sqrt{c^2 x^2+1}}+\frac{13 b c^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{6 d^2}",1,"((-5*a)/(3*x^3) + (5*a*c^2)/x - (5*b*c*Sqrt[1 + c^2*x^2])/(6*x^2) + a/(x^3 + c^2*x^5) - (5*b*ArcSinh[c*x])/(3*x^3) + (5*b*c^2*ArcSinh[c*x])/x + (b*ArcSinh[c*x])/(x^3 + c^2*x^5) + 5*a*c^3*ArcTan[c*x] + (35*b*c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/6 + (b*c^3*Hypergeometric2F1[-1/2, 2, 1/2, 1 + c^2*x^2])/Sqrt[1 + c^2*x^2] - 5*b*(-c^2)^(3/2)*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 5*b*(-c^2)^(3/2)*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 5*b*(-c^2)^(3/2)*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 5*b*(-c^2)^(3/2)*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(2*d^2)","C",0
46,1,341,186,0.6801806,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3,x]","-\frac{-9 a c^4 x^4 \tan ^{-1}(c x)+15 a c^3 x^3-18 a c^2 x^2 \tan ^{-1}(c x)+9 a c x-9 a \tan ^{-1}(c x)-9 i b c^4 x^4 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+9 i b c^4 x^4 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+15 b c^3 x^3 \sinh ^{-1}(c x)+9 i b \left(c^2 x^2+1\right)^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)-9 i b \left(c^2 x^2+1\right)^2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+15 b c^2 x^2 \sqrt{c^2 x^2+1}+13 b \sqrt{c^2 x^2+1}-18 i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+18 i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+9 b c x \sinh ^{-1}(c x)-9 i b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+9 i b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{24 c^5 d^3 \left(c^2 x^2+1\right)^2}","\frac{3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 c^5 d^3}-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{4 c^2 d^3 \left(c^2 x^2+1\right)^2}-\frac{3 x \left(a+b \sinh ^{-1}(c x)\right)}{8 c^4 d^3 \left(c^2 x^2+1\right)}-\frac{3 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{8 c^5 d^3}+\frac{3 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{8 c^5 d^3}-\frac{5 b}{8 c^5 d^3 \sqrt{c^2 x^2+1}}+\frac{b}{12 c^5 d^3 \left(c^2 x^2+1\right)^{3/2}}",1,"-1/24*(9*a*c*x + 15*a*c^3*x^3 + 13*b*Sqrt[1 + c^2*x^2] + 15*b*c^2*x^2*Sqrt[1 + c^2*x^2] + 9*b*c*x*ArcSinh[c*x] + 15*b*c^3*x^3*ArcSinh[c*x] - 9*a*ArcTan[c*x] - 18*a*c^2*x^2*ArcTan[c*x] - 9*a*c^4*x^4*ArcTan[c*x] - (9*I)*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (18*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (9*I)*b*c^4*x^4*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (9*I)*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + (18*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + (9*I)*b*c^4*x^4*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + (9*I)*b*(1 + c^2*x^2)^2*PolyLog[2, (-I)*E^ArcSinh[c*x]] - (9*I)*b*(1 + c^2*x^2)^2*PolyLog[2, I*E^ArcSinh[c*x]])/(c^5*d^3*(1 + c^2*x^2)^2)","A",0
47,1,79,97,0.1421115,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3,x]","\frac{-3 a \left(2 c^2 x^2+1\right)+b c x \sqrt{c^2 x^2+1} \left(4 c^2 x^2+3\right)-3 \left(2 b c^2 x^2+b\right) \sinh ^{-1}(c x)}{12 c^4 d^3 \left(c^2 x^2+1\right)^2}","\frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{b \sinh ^{-1}(c x)}{4 c^4 d^3}+\frac{b x^3}{12 c d^3 \left(c^2 x^2+1\right)^{3/2}}+\frac{b x}{4 c^3 d^3 \sqrt{c^2 x^2+1}}",1,"(-3*a*(1 + 2*c^2*x^2) + b*c*x*Sqrt[1 + c^2*x^2]*(3 + 4*c^2*x^2) - 3*(b + 2*b*c^2*x^2)*ArcSinh[c*x])/(12*c^4*d^3*(1 + c^2*x^2)^2)","A",1
48,1,340,184,0.2470457,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3,x]","\frac{3 a c^4 x^4 \tan ^{-1}(c x)+3 a c^3 x^3+6 a c^2 x^2 \tan ^{-1}(c x)-3 a c x+3 a \tan ^{-1}(c x)+3 i b c^4 x^4 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-3 i b c^4 x^4 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+3 b c^3 x^3 \sinh ^{-1}(c x)-3 i b \left(c^2 x^2+1\right)^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)+3 i b \left(c^2 x^2+1\right)^2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+3 b c^2 x^2 \sqrt{c^2 x^2+1}+b \sqrt{c^2 x^2+1}+6 i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-6 i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)-3 b c x \sinh ^{-1}(c x)+3 i b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-3 i b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{24 c^3 d^3 \left(c^2 x^2+1\right)^2}","\frac{\tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 c^3 d^3}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{8 c^2 d^3 \left(c^2 x^2+1\right)}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{4 c^2 d^3 \left(c^2 x^2+1\right)^2}-\frac{i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{8 c^3 d^3}+\frac{i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{8 c^3 d^3}+\frac{b}{8 c^3 d^3 \sqrt{c^2 x^2+1}}-\frac{b}{12 c^3 d^3 \left(c^2 x^2+1\right)^{3/2}}",1,"(-3*a*c*x + 3*a*c^3*x^3 + b*Sqrt[1 + c^2*x^2] + 3*b*c^2*x^2*Sqrt[1 + c^2*x^2] - 3*b*c*x*ArcSinh[c*x] + 3*b*c^3*x^3*ArcSinh[c*x] + 3*a*ArcTan[c*x] + 6*a*c^2*x^2*ArcTan[c*x] + 3*a*c^4*x^4*ArcTan[c*x] + (3*I)*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (6*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (3*I)*b*c^4*x^4*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (3*I)*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (6*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (3*I)*b*c^4*x^4*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (3*I)*b*(1 + c^2*x^2)^2*PolyLog[2, (-I)*E^ArcSinh[c*x]] + (3*I)*b*(1 + c^2*x^2)^2*PolyLog[2, I*E^ArcSinh[c*x]])/(24*c^3*d^3*(1 + c^2*x^2)^2)","A",0
49,1,56,80,0.0717964,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3,x]","\frac{-3 a+b c x \sqrt{c^2 x^2+1} \left(2 c^2 x^2+3\right)-3 b \sinh ^{-1}(c x)}{12 d^3 \left(c^3 x^2+c\right)^2}","-\frac{a+b \sinh ^{-1}(c x)}{4 c^2 d^3 \left(c^2 x^2+1\right)^2}+\frac{b x}{6 c d^3 \sqrt{c^2 x^2+1}}+\frac{b x}{12 c d^3 \left(c^2 x^2+1\right)^{3/2}}",1,"(-3*a + b*c*x*Sqrt[1 + c^2*x^2]*(3 + 2*c^2*x^2) - 3*b*ArcSinh[c*x])/(12*d^3*(c + c^3*x^2)^2)","A",1
50,1,341,178,0.1516653,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^3,x]","\frac{9 a c^4 x^4 \tan ^{-1}(c x)+9 a c^3 x^3+18 a c^2 x^2 \tan ^{-1}(c x)+15 a c x+9 a \tan ^{-1}(c x)+9 i b c^4 x^4 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-9 i b c^4 x^4 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+9 b c^3 x^3 \sinh ^{-1}(c x)-9 i b \left(c^2 x^2+1\right)^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)+9 i b \left(c^2 x^2+1\right)^2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+9 b c^2 x^2 \sqrt{c^2 x^2+1}+11 b \sqrt{c^2 x^2+1}+18 i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-18 i b c^2 x^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+15 b c x \sinh ^{-1}(c x)+9 i b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-9 i b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{24 c d^3 \left(c^2 x^2+1\right)^2}","\frac{3 x \left(a+b \sinh ^{-1}(c x)\right)}{8 d^3 \left(c^2 x^2+1\right)}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3 \left(c^2 x^2+1\right)^2}+\frac{3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 c d^3}+\frac{3 b}{8 c d^3 \sqrt{c^2 x^2+1}}+\frac{b}{12 c d^3 \left(c^2 x^2+1\right)^{3/2}}-\frac{3 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{8 c d^3}+\frac{3 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{8 c d^3}",1,"(15*a*c*x + 9*a*c^3*x^3 + 11*b*Sqrt[1 + c^2*x^2] + 9*b*c^2*x^2*Sqrt[1 + c^2*x^2] + 15*b*c*x*ArcSinh[c*x] + 9*b*c^3*x^3*ArcSinh[c*x] + 9*a*ArcTan[c*x] + 18*a*c^2*x^2*ArcTan[c*x] + 9*a*c^4*x^4*ArcTan[c*x] + (9*I)*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (18*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (9*I)*b*c^4*x^4*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (9*I)*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (18*I)*b*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (9*I)*b*c^4*x^4*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (9*I)*b*(1 + c^2*x^2)^2*PolyLog[2, (-I)*E^ArcSinh[c*x]] + (9*I)*b*(1 + c^2*x^2)^2*PolyLog[2, I*E^ArcSinh[c*x]])/(24*c*d^3*(1 + c^2*x^2)^2)","A",0
51,1,289,159,0.5910305,"\int \frac{a+b \sinh ^{-1}(c x)}{x \left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)^3),x]","\frac{-\frac{2 a^2}{b}+\frac{2 a}{c^2 x^2+1}+\frac{a}{\left(c^2 x^2+1\right)^2}-2 a \log \left(c^2 x^2+1\right)-4 a \sinh ^{-1}(c x)+4 a \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)-4 b \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)-4 b \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)-\frac{8 b c x}{3 \sqrt{c^2 x^2+1}}-\frac{b c x}{3 \left(c^2 x^2+1\right)^{3/2}}+\frac{2 b \sinh ^{-1}(c x)}{c^2 x^2+1}+\frac{b \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^2}-4 b \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)-4 b \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)+2 b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+4 b \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)}{4 d^3}","\frac{a+b \sinh ^{-1}(c x)}{2 d^3 \left(c^2 x^2+1\right)}+\frac{a+b \sinh ^{-1}(c x)}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^3}-\frac{2 b c x}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{b c x}{12 d^3 \left(c^2 x^2+1\right)^{3/2}}-\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d^3}+\frac{b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d^3}",1,"((-2*a^2)/b + a/(1 + c^2*x^2)^2 - (b*c*x)/(3*(1 + c^2*x^2)^(3/2)) + (2*a)/(1 + c^2*x^2) - (8*b*c*x)/(3*Sqrt[1 + c^2*x^2]) - 4*a*ArcSinh[c*x] + (b*ArcSinh[c*x])/(1 + c^2*x^2)^2 + (2*b*ArcSinh[c*x])/(1 + c^2*x^2) - 4*b*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 4*b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 4*a*Log[1 - E^(2*ArcSinh[c*x])] + 4*b*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] - 2*a*Log[1 + c^2*x^2] - 4*b*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 4*b*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 2*b*PolyLog[2, E^(2*ArcSinh[c*x])])/(4*d^3)","A",0
52,1,298,222,1.0531207,"\int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)^3),x]","-\frac{-\frac{15 \left(a+b \sinh ^{-1}(c x)\right)}{c^2 x^3+x}-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{x \left(c^2 x^2+1\right)^2}+\frac{45 \left(a+b \sinh ^{-1}(c x)\right)}{x}+45 a c \tan ^{-1}(c x)+\frac{2 b c \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};c^2 x^2+1\right)}{\left(c^2 x^2+1\right)^{3/2}}+\frac{15 b c \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};c^2 x^2+1\right)}{\sqrt{c^2 x^2+1}}-45 b \sqrt{-c^2} \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)+45 b \sqrt{-c^2} \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)+45 b c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+45 b \sqrt{-c^2} \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)-45 b \sqrt{-c^2} \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)}{24 d^3}","-\frac{15 c^2 x \left(a+b \sinh ^{-1}(c x)\right)}{8 d^3 \left(c^2 x^2+1\right)}-\frac{5 c^2 x \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{a+b \sinh ^{-1}(c x)}{d^3 x \left(c^2 x^2+1\right)^2}-\frac{15 c \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3}-\frac{7 b c}{8 d^3 \sqrt{c^2 x^2+1}}-\frac{b c}{12 d^3 \left(c^2 x^2+1\right)^{3/2}}-\frac{b c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{d^3}+\frac{15 i b c \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{8 d^3}-\frac{15 i b c \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{8 d^3}",1,"-1/24*((45*(a + b*ArcSinh[c*x]))/x - (6*(a + b*ArcSinh[c*x]))/(x*(1 + c^2*x^2)^2) - (15*(a + b*ArcSinh[c*x]))/(x + c^2*x^3) + 45*a*c*ArcTan[c*x] + 45*b*c*ArcTanh[Sqrt[1 + c^2*x^2]] + (2*b*c*Hypergeometric2F1[-3/2, 1, -1/2, 1 + c^2*x^2])/(1 + c^2*x^2)^(3/2) + (15*b*c*Hypergeometric2F1[-1/2, 1, 1/2, 1 + c^2*x^2])/Sqrt[1 + c^2*x^2] + 45*b*Sqrt[-c^2]*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 45*b*Sqrt[-c^2]*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 45*b*Sqrt[-c^2]*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 45*b*Sqrt[-c^2]*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/d^3","C",0
53,1,353,232,0.8300173,"\int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^3),x]","\frac{-18 c^2 \left(2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)\right)+\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{\left(c^2 x^3+x\right)^2}+\frac{9 \left(a+b \sinh ^{-1}(c x)\right)}{c^2 x^4+x^2}+\frac{18 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{b}-\frac{18 \left(a+b \sinh ^{-1}(c x)\right)}{x^2}+18 a c^2 \log \left(c^2 x^2+1\right)+36 b c^2 \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)+36 b c^2 \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)+\frac{9 b c \left(2 c^2 x^2+1\right)}{x \sqrt{c^2 x^2+1}}-\frac{18 b c \sqrt{c^2 x^2+1}}{x}-18 b c^2 \sinh ^{-1}(c x)^2+36 b c^2 \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+36 b c^2 \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)+\frac{b c \left(8 c^4 x^4+12 c^2 x^2+3\right)}{x \left(c^2 x^2+1\right)^{3/2}}}{12 d^3}","-\frac{3 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 d^3 \left(c^2 x^2+1\right)}-\frac{3 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{a+b \sinh ^{-1}(c x)}{2 d^3 x^2 \left(c^2 x^2+1\right)^2}+\frac{6 c^2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^3}+\frac{3 b c^2 \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d^3}-\frac{3 b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d^3}-\frac{b c}{2 d^3 x \left(c^2 x^2+1\right)^{3/2}}+\frac{2 b c^3 x}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 x}{12 d^3 \left(c^2 x^2+1\right)^{3/2}}",1,"((-18*b*c*Sqrt[1 + c^2*x^2])/x + (9*b*c*(1 + 2*c^2*x^2))/(x*Sqrt[1 + c^2*x^2]) + (b*c*(3 + 12*c^2*x^2 + 8*c^4*x^4))/(x*(1 + c^2*x^2)^(3/2)) - 18*b*c^2*ArcSinh[c*x]^2 - (18*(a + b*ArcSinh[c*x]))/x^2 + (3*(a + b*ArcSinh[c*x]))/(x + c^2*x^3)^2 + (9*(a + b*ArcSinh[c*x]))/(x^2 + c^2*x^4) + (18*c^2*(a + b*ArcSinh[c*x])^2)/b + 36*b*c^2*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 36*b*c^2*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 18*a*c^2*Log[1 + c^2*x^2] + 36*b*c^2*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 36*b*c^2*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 18*c^2*(2*(a + b*ArcSinh[c*x])*Log[1 - E^(2*ArcSinh[c*x])] + b*PolyLog[2, E^(2*ArcSinh[c*x])]))/(12*d^3)","A",0
54,1,380,295,0.9288824,"\int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^3),x]","\frac{210 a c^3 \tan ^{-1}(c x)+\frac{42 a}{c^2 x^5+x^3}+\frac{12 a}{x^3 \left(c^2 x^2+1\right)^2}+\frac{210 a c^2}{x}-\frac{70 a}{x^3}+210 b \left(-c^2\right)^{3/2} \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)-210 b \left(-c^2\right)^{3/2} \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)-\frac{35 b c \sqrt{c^2 x^2+1}}{x^2}+\frac{42 b \sinh ^{-1}(c x)}{c^2 x^5+x^3}+\frac{12 b \sinh ^{-1}(c x)}{x^3 \left(c^2 x^2+1\right)^2}+\frac{210 b c^2 \sinh ^{-1}(c x)}{x}-210 b \left(-c^2\right)^{3/2} \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+210 b \left(-c^2\right)^{3/2} \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)+\frac{4 b c^3 \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};c^2 x^2+1\right)}{\left(c^2 x^2+1\right)^{3/2}}+\frac{42 b c^3 \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};c^2 x^2+1\right)}{\sqrt{c^2 x^2+1}}+245 b c^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-\frac{70 b \sinh ^{-1}(c x)}{x^3}}{48 d^3}","\frac{35 c^3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3}+\frac{7 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 d^3 x \left(c^2 x^2+1\right)^2}-\frac{a+b \sinh ^{-1}(c x)}{3 d^3 x^3 \left(c^2 x^2+1\right)^2}+\frac{35 c^4 x \left(a+b \sinh ^{-1}(c x)\right)}{8 d^3 \left(c^2 x^2+1\right)}+\frac{35 c^4 x \left(a+b \sinh ^{-1}(c x)\right)}{12 d^3 \left(c^2 x^2+1\right)^2}-\frac{35 i b c^3 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{8 d^3}+\frac{35 i b c^3 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{8 d^3}-\frac{b c}{6 d^3 x^2 \left(c^2 x^2+1\right)^{3/2}}+\frac{29 b c^3}{24 d^3 \sqrt{c^2 x^2+1}}-\frac{b c^3}{12 d^3 \left(c^2 x^2+1\right)^{3/2}}+\frac{19 b c^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{6 d^3}",1,"((-70*a)/x^3 + (210*a*c^2)/x + (12*a)/(x^3*(1 + c^2*x^2)^2) - (35*b*c*Sqrt[1 + c^2*x^2])/x^2 + (42*a)/(x^3 + c^2*x^5) - (70*b*ArcSinh[c*x])/x^3 + (210*b*c^2*ArcSinh[c*x])/x + (12*b*ArcSinh[c*x])/(x^3*(1 + c^2*x^2)^2) + (42*b*ArcSinh[c*x])/(x^3 + c^2*x^5) + 210*a*c^3*ArcTan[c*x] + 245*b*c^3*ArcTanh[Sqrt[1 + c^2*x^2]] + (4*b*c^3*Hypergeometric2F1[-3/2, 2, -1/2, 1 + c^2*x^2])/(1 + c^2*x^2)^(3/2) + (42*b*c^3*Hypergeometric2F1[-1/2, 2, 1/2, 1 + c^2*x^2])/Sqrt[1 + c^2*x^2] - 210*b*(-c^2)^(3/2)*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 210*b*(-c^2)^(3/2)*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 210*b*(-c^2)^(3/2)*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 210*b*(-c^2)^(3/2)*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(48*d^3)","C",0
55,1,106,109,0.1996315,"\int x^3 \sqrt{\pi +c^2 \pi  x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^3*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{\sqrt{\pi } \left(15 a \sqrt{c^2 x^2+1} \left(3 c^4 x^4+c^2 x^2-2\right)+b \left(-9 c^5 x^5-5 c^3 x^3+30 c x\right)+15 b \sqrt{c^2 x^2+1} \left(3 c^4 x^4+c^2 x^2-2\right) \sinh ^{-1}(c x)\right)}{225 c^4}","\frac{\left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 \pi ^2 c^4}-\frac{\left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  c^4}+\frac{2 \sqrt{\pi } b x}{15 c^3}-\frac{1}{25} \sqrt{\pi } b c x^5-\frac{\sqrt{\pi } b x^3}{45 c}",1,"(Sqrt[Pi]*(15*a*Sqrt[1 + c^2*x^2]*(-2 + c^2*x^2 + 3*c^4*x^4) + b*(30*c*x - 5*c^3*x^3 - 9*c^5*x^5) + 15*b*Sqrt[1 + c^2*x^2]*(-2 + c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]))/(225*c^4)","A",1
56,1,79,119,0.1704415,"\int x^2 \sqrt{\pi +c^2 \pi  x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{\sqrt{\pi } \left(\sinh ^{-1}(c x) \left(4 b \sinh \left(4 \sinh ^{-1}(c x)\right)-16 a\right)+16 a c x \sqrt{c^2 x^2+1} \left(2 c^2 x^2+1\right)-8 b \sinh ^{-1}(c x)^2-b \cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{128 c^3}","-\frac{\sqrt{\pi } \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c^3}+\frac{\sqrt{\pi } x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{8 c^2}+\frac{1}{4} x^3 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{16} \sqrt{\pi } b c x^4-\frac{\sqrt{\pi } b x^2}{16 c}",1,"(Sqrt[Pi]*(16*a*c*x*Sqrt[1 + c^2*x^2]*(1 + 2*c^2*x^2) - 8*b*ArcSinh[c*x]^2 - b*Cosh[4*ArcSinh[c*x]] + ArcSinh[c*x]*(-16*a + 4*b*Sinh[4*ArcSinh[c*x]])))/(128*c^3)","A",1
57,1,63,61,0.1202954,"\int x \sqrt{\pi +c^2 \pi  x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{\sqrt{\pi } \left(3 a \left(c^2 x^2+1\right)^{3/2}-b c x \left(c^2 x^2+3\right)+3 b \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)\right)}{9 c^2}","\frac{\left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  c^2}-\frac{1}{9} \sqrt{\pi } b c x^3-\frac{\sqrt{\pi } b x}{3 c}",1,"(Sqrt[Pi]*(3*a*(1 + c^2*x^2)^(3/2) - b*c*x*(3 + c^2*x^2) + 3*b*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]))/(9*c^2)","A",1
58,1,69,67,0.1414295,"\int \sqrt{\pi +c^2 \pi  x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{\sqrt{\pi } \left(2 \sinh ^{-1}(c x) \left(2 a+b \sinh \left(2 \sinh ^{-1}(c x)\right)\right)+4 a c x \sqrt{c^2 x^2+1}+2 b \sinh ^{-1}(c x)^2-b \cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{8 c}","\frac{1}{2} x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)+\frac{\sqrt{\pi } \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c}-\frac{1}{4} \sqrt{\pi } b c x^2",1,"(Sqrt[Pi]*(4*a*c*x*Sqrt[1 + c^2*x^2] + 2*b*ArcSinh[c*x]^2 - b*Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(2*a + b*Sinh[2*ArcSinh[c*x]])))/(8*c)","A",1
59,1,131,89,0.1972125,"\int \frac{\sqrt{\pi +c^2 \pi  x^2} \left(a+b \sinh ^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x,x]","\sqrt{\pi } \left(a \sqrt{c^2 x^2+1}-a \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)+a \log (x)+b \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)\right)","\sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-2 \sqrt{\pi } \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\sqrt{\pi } b \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+\sqrt{\pi } b \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)+\sqrt{\pi } (-b) c x",1,"Sqrt[Pi]*(a*Sqrt[1 + c^2*x^2] + a*Log[x] - a*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + b*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))","A",0
60,1,75,61,0.1674004,"\int \frac{\sqrt{\pi +c^2 \pi  x^2} \left(a+b \sinh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^2,x]","\frac{\sqrt{\pi } \left(2 \sinh ^{-1}(c x) \left(a c x-b \sqrt{c^2 x^2+1}\right)-2 a \sqrt{c^2 x^2+1}+2 b c x \log (c x)+b c x \sinh ^{-1}(c x)^2\right)}{2 x}","-\frac{\sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{x}+\frac{\sqrt{\pi } c \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b}+\sqrt{\pi } b c \log (x)",1,"(Sqrt[Pi]*(-2*a*Sqrt[1 + c^2*x^2] + 2*(a*c*x - b*Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + b*c*x*ArcSinh[c*x]^2 + 2*b*c*x*Log[c*x]))/(2*x)","A",1
61,1,185,113,3.3214863,"\int \frac{\sqrt{\pi +c^2 \pi  x^2} \left(a+b \sinh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^3,x]","\frac{1}{8} \sqrt{\pi } \left(-\frac{4 a \sqrt{c^2 x^2+1}}{x^2}-4 a c^2 \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)+4 a c^2 \log (x)+b c^2 \left(4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)","-\frac{\sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{2 x^2}-\sqrt{\pi } c^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{2} \sqrt{\pi } b c^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+\frac{1}{2} \sqrt{\pi } b c^2 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)-\frac{\sqrt{\pi } b c}{2 x}",1,"(Sqrt[Pi]*((-4*a*Sqrt[1 + c^2*x^2])/x^2 + 4*a*c^2*Log[x] - 4*a*c^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + b*c^2*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2])))/8","A",0
62,1,78,62,0.1430936,"\int \frac{\sqrt{\pi +c^2 \pi  x^2} \left(a+b \sinh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[(Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/x^4,x]","\frac{1}{3} \sqrt{\pi } b c^3 \log (x)-\frac{\sqrt{\pi } \left(2 a \left(c^2 x^2+1\right)^{3/2}+3 b c^3 x^3+2 b \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)+b c x\right)}{6 x^3}","-\frac{\left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  x^3}+\frac{1}{3} \sqrt{\pi } b c^3 \log (x)-\frac{\sqrt{\pi } b c}{6 x^2}",1,"-1/6*(Sqrt[Pi]*(b*c*x + 3*b*c^3*x^3 + 2*a*(1 + c^2*x^2)^(3/2) + 2*b*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]))/x^3 + (b*c^3*Sqrt[Pi]*Log[x])/3","A",1
63,1,100,125,0.1685259,"\int x^3 \left(\pi +c^2 \pi  x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^3*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{\pi ^{3/2} \left(105 a \left(5 c^2 x^2-2\right) \left(c^2 x^2+1\right)^{5/2}+105 b \left(5 c^2 x^2-2\right) \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)-b c x \left(75 c^6 x^6+168 c^4 x^4+35 c^2 x^2-210\right)\right)}{3675 c^4}","\frac{\left(\pi  c^2 x^2+\pi \right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{7 \pi ^2 c^4}-\frac{\left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 \pi  c^4}-\frac{1}{49} \pi ^{3/2} b c^3 x^7+\frac{2 \pi ^{3/2} b x}{35 c^3}-\frac{8}{175} \pi ^{3/2} b c x^5-\frac{\pi ^{3/2} b x^3}{105 c}",1,"(Pi^(3/2)*(105*a*(1 + c^2*x^2)^(5/2)*(-2 + 5*c^2*x^2) - b*c*x*(-210 + 35*c^2*x^2 + 168*c^4*x^4 + 75*c^6*x^6) + 105*b*(1 + c^2*x^2)^(5/2)*(-2 + 5*c^2*x^2)*ArcSinh[c*x]))/(3675*c^4)","A",1
64,1,154,165,0.3428653,"\int x^2 \left(\pi +c^2 \pi  x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^2*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{\pi ^{3/2} \left(-12 \sinh ^{-1}(c x) \left(12 a+3 b \sinh \left(2 \sinh ^{-1}(c x)\right)-3 b \sinh \left(4 \sinh ^{-1}(c x)\right)-b \sinh \left(6 \sinh ^{-1}(c x)\right)\right)+144 a c x \sqrt{c^2 x^2+1}+384 a c^5 x^5 \sqrt{c^2 x^2+1}+672 a c^3 x^3 \sqrt{c^2 x^2+1}-72 b \sinh ^{-1}(c x)^2+18 b \cosh \left(2 \sinh ^{-1}(c x)\right)-9 b \cosh \left(4 \sinh ^{-1}(c x)\right)-2 b \cosh \left(6 \sinh ^{-1}(c x)\right)\right)}{2304 c^3}","-\frac{\pi ^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{32 b c^3}+\frac{\pi ^{3/2} x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{16 c^2}+\frac{1}{6} x^3 \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{8} \pi  x^3 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{36} \pi ^{3/2} b c^3 x^6-\frac{7}{96} \pi ^{3/2} b c x^4-\frac{\pi ^{3/2} b x^2}{32 c}",1,"(Pi^(3/2)*(144*a*c*x*Sqrt[1 + c^2*x^2] + 672*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 384*a*c^5*x^5*Sqrt[1 + c^2*x^2] - 72*b*ArcSinh[c*x]^2 + 18*b*Cosh[2*ArcSinh[c*x]] - 9*b*Cosh[4*ArcSinh[c*x]] - 2*b*Cosh[6*ArcSinh[c*x]] - 12*ArcSinh[c*x]*(12*a + 3*b*Sinh[2*ArcSinh[c*x]] - 3*b*Sinh[4*ArcSinh[c*x]] - b*Sinh[6*ArcSinh[c*x]])))/(2304*c^3)","A",1
65,1,72,77,0.1247019,"\int x \left(\pi +c^2 \pi  x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{\pi ^{3/2} \left(15 a \left(c^2 x^2+1\right)^{5/2}+15 b \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)-b c x \left(3 c^4 x^4+10 c^2 x^2+15\right)\right)}{75 c^2}","\frac{\left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 \pi  c^2}-\frac{1}{25} \pi ^{3/2} b c^3 x^5-\frac{2}{15} \pi ^{3/2} b c x^3-\frac{\pi ^{3/2} b x}{5 c}",1,"(Pi^(3/2)*(15*a*(1 + c^2*x^2)^(5/2) - b*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) + 15*b*(1 + c^2*x^2)^(5/2)*ArcSinh[c*x]))/(75*c^2)","A",1
66,1,111,111,0.2368946,"\int \left(\pi +c^2 \pi  x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{\pi ^{3/2} \left(4 \sinh ^{-1}(c x) \left(12 a+8 b \sinh \left(2 \sinh ^{-1}(c x)\right)+b \sinh \left(4 \sinh ^{-1}(c x)\right)\right)+80 a c x \sqrt{c^2 x^2+1}+32 a c^3 x^3 \sqrt{c^2 x^2+1}+24 b \sinh ^{-1}(c x)^2-16 b \cosh \left(2 \sinh ^{-1}(c x)\right)-b \cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{128 c}","\frac{1}{4} x \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{3}{8} \pi  x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)+\frac{3 \pi ^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c}-\frac{1}{16} \pi ^{3/2} b c^3 x^4-\frac{5}{16} \pi ^{3/2} b c x^2",1,"(Pi^(3/2)*(80*a*c*x*Sqrt[1 + c^2*x^2] + 32*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 24*b*ArcSinh[c*x]^2 - 16*b*Cosh[2*ArcSinh[c*x]] - b*Cosh[4*ArcSinh[c*x]] + 4*ArcSinh[c*x]*(12*a + 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]])))/(128*c)","A",1
67,1,180,134,0.3705967,"\int \frac{\left(\pi +c^2 \pi  x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x} \, dx","Integrate[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x,x]","\frac{1}{9} \pi ^{3/2} \left(3 a \sqrt{c^2 x^2+1} \left(c^2 x^2+4\right)-9 a \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)+9 a \log (x)+9 b \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)-b \left(c^3 x^3-3 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)+3 c x\right)\right)","\frac{1}{3} \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\pi  \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-2 \pi ^{3/2} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{9} \pi ^{3/2} b c^3 x^3-\pi ^{3/2} b \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+\pi ^{3/2} b \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)-\frac{4}{3} \pi ^{3/2} b c x",1,"(Pi^(3/2)*(3*a*Sqrt[1 + c^2*x^2]*(4 + c^2*x^2) - b*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]) + 9*a*Log[x] - 9*a*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + 9*b*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])])))/9","A",0
68,1,122,108,0.2963193,"\int \frac{\left(\pi +c^2 \pi  x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^2,x]","\frac{\pi ^{3/2} \left(2 \sinh ^{-1}(c x) \left(6 a c x-4 b \sqrt{c^2 x^2+1}+b c x \sinh \left(2 \sinh ^{-1}(c x)\right)\right)+4 a c^2 x^2 \sqrt{c^2 x^2+1}-8 a \sqrt{c^2 x^2+1}+8 b c x \log (c x)+6 b c x \sinh ^{-1}(c x)^2-b c x \cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{8 x}","\frac{3}{2} \pi  c^2 x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-\frac{\left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x}+\frac{3 \pi ^{3/2} c \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b}-\frac{1}{4} \pi ^{3/2} b c^3 x^2+\pi ^{3/2} b c \log (x)",1,"(Pi^(3/2)*(-8*a*Sqrt[1 + c^2*x^2] + 4*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 6*b*c*x*ArcSinh[c*x]^2 - b*c*x*Cosh[2*ArcSinh[c*x]] + 8*b*c*x*Log[c*x] + 2*ArcSinh[c*x]*(6*a*c*x - 4*b*Sqrt[1 + c^2*x^2] + b*c*x*Sinh[2*ArcSinh[c*x]])))/(8*x)","A",1
69,1,292,155,1.6590104,"\int \frac{\left(\pi +c^2 \pi  x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^3,x]","\frac{\pi ^{3/2} \left(8 a c^2 x^2 \sqrt{c^2 x^2+1}-4 a \sqrt{c^2 x^2+1}+12 a c^2 x^2 \log (x)-12 a c^2 x^2 \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)-8 b c^3 x^3-b c^3 x^3 \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+12 b c^2 x^2 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-12 b c^2 x^2 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+8 b c^2 x^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+12 b c^2 x^2 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-12 b c^2 x^2 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-b c^2 x^2 \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+4 b c x \sinh ^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 b \sinh ^{-1}(c x) \sinh ^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{8 x^2}","\frac{3}{2} \pi  c^2 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-\frac{\left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{2 x^2}-3 \pi ^{3/2} c^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+\pi ^{3/2} (-b) c^3 x-\frac{3}{2} \pi ^{3/2} b c^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+\frac{3}{2} \pi ^{3/2} b c^2 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)-\frac{\pi ^{3/2} b c}{2 x}",1,"(Pi^(3/2)*(-8*b*c^3*x^3 - 4*a*Sqrt[1 + c^2*x^2] + 8*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 8*b*c^2*x^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - b*c^3*x^3*Csch[ArcSinh[c*x]/2]^2 - b*c^2*x^2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 12*b*c^2*x^2*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 12*b*c^2*x^2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 12*a*c^2*x^2*Log[x] - 12*a*c^2*x^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + 12*b*c^2*x^2*PolyLog[2, -E^(-ArcSinh[c*x])] - 12*b*c^2*x^2*PolyLog[2, E^(-ArcSinh[c*x])] + 4*b*c*x*Sinh[ArcSinh[c*x]/2]^2 - 4*b*ArcSinh[c*x]*Sinh[ArcSinh[c*x]/2]^2))/(8*x^2)","A",0
70,1,125,115,0.2325652,"\int \frac{\left(\pi +c^2 \pi  x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^4,x]","\frac{\pi ^{3/2} \left(\sinh ^{-1}(c x) \left(6 a c^3 x^3-2 b \sqrt{c^2 x^2+1} \left(4 c^2 x^2+1\right)\right)-8 a c^2 x^2 \sqrt{c^2 x^2+1}-2 a \sqrt{c^2 x^2+1}+8 b c^3 x^3 \log (c x)+3 b c^3 x^3 \sinh ^{-1}(c x)^2-b c x\right)}{6 x^3}","\frac{\pi ^{3/2} c^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b}-\frac{\pi  c^2 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{x}-\frac{\left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^3}+\frac{4}{3} \pi ^{3/2} b c^3 \log (x)-\frac{\pi ^{3/2} b c}{6 x^2}",1,"(Pi^(3/2)*(-(b*c*x) - 2*a*Sqrt[1 + c^2*x^2] - 8*a*c^2*x^2*Sqrt[1 + c^2*x^2] + (6*a*c^3*x^3 - 2*b*Sqrt[1 + c^2*x^2]*(1 + 4*c^2*x^2))*ArcSinh[c*x] + 3*b*c^3*x^3*ArcSinh[c*x]^2 + 8*b*c^3*x^3*Log[c*x]))/(6*x^3)","A",1
71,1,108,141,0.2033979,"\int x^3 \left(\pi +c^2 \pi  x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^3*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{\pi ^{5/2} \left(63 a \left(7 c^2 x^2-2\right) \left(c^2 x^2+1\right)^{7/2}+63 b \left(7 c^2 x^2-2\right) \left(c^2 x^2+1\right)^{7/2} \sinh ^{-1}(c x)-b c x \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 c^4}","\frac{\left(\pi  c^2 x^2+\pi \right)^{9/2} \left(a+b \sinh ^{-1}(c x)\right)}{9 \pi ^2 c^4}-\frac{\left(\pi  c^2 x^2+\pi \right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{7 \pi  c^4}-\frac{1}{81} \pi ^{5/2} b c^5 x^9-\frac{19}{441} \pi ^{5/2} b c^3 x^7+\frac{2 \pi ^{5/2} b x}{63 c^3}-\frac{1}{21} \pi ^{5/2} b c x^5-\frac{\pi ^{5/2} b x^3}{189 c}",1,"(Pi^(5/2)*(63*a*(1 + c^2*x^2)^(7/2)*(-2 + 7*c^2*x^2) - b*c*x*(-126 + 21*c^2*x^2 + 189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^8) + 63*b*(1 + c^2*x^2)^(7/2)*(-2 + 7*c^2*x^2)*ArcSinh[c*x]))/(3969*c^4)","A",1
72,1,196,213,0.605786,"\int x^2 \left(\pi +c^2 \pi  x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^2*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{\pi ^{5/2} \left(-24 \sinh ^{-1}(c x) \left(120 a+48 b \sinh \left(2 \sinh ^{-1}(c x)\right)-24 b \sinh \left(4 \sinh ^{-1}(c x)\right)-16 b \sinh \left(6 \sinh ^{-1}(c x)\right)-3 b \sinh \left(8 \sinh ^{-1}(c x)\right)\right)+2880 a c x \sqrt{c^2 x^2+1}+9216 a c^7 x^7 \sqrt{c^2 x^2+1}+26112 a c^5 x^5 \sqrt{c^2 x^2+1}+22656 a c^3 x^3 \sqrt{c^2 x^2+1}-1440 b \sinh ^{-1}(c x)^2+576 b \cosh \left(2 \sinh ^{-1}(c x)\right)-144 b \cosh \left(4 \sinh ^{-1}(c x)\right)-64 b \cosh \left(6 \sinh ^{-1}(c x)\right)-9 b \cosh \left(8 \sinh ^{-1}(c x)\right)\right)}{73728 c^3}","-\frac{5 \pi ^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{256 b c^3}+\frac{5 \pi ^{5/2} x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{128 c^2}+\frac{1}{8} x^3 \left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5}{48} \pi  x^3 \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5}{64} \pi ^2 x^3 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{64} \pi ^{5/2} b c^5 x^8-\frac{17}{288} \pi ^{5/2} b c^3 x^6-\frac{59}{768} \pi ^{5/2} b c x^4-\frac{5 \pi ^{5/2} b x^2}{256 c}",1,"(Pi^(5/2)*(2880*a*c*x*Sqrt[1 + c^2*x^2] + 22656*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 26112*a*c^5*x^5*Sqrt[1 + c^2*x^2] + 9216*a*c^7*x^7*Sqrt[1 + c^2*x^2] - 1440*b*ArcSinh[c*x]^2 + 576*b*Cosh[2*ArcSinh[c*x]] - 144*b*Cosh[4*ArcSinh[c*x]] - 64*b*Cosh[6*ArcSinh[c*x]] - 9*b*Cosh[8*ArcSinh[c*x]] - 24*ArcSinh[c*x]*(120*a + 48*b*Sinh[2*ArcSinh[c*x]] - 24*b*Sinh[4*ArcSinh[c*x]] - 16*b*Sinh[6*ArcSinh[c*x]] - 3*b*Sinh[8*ArcSinh[c*x]])))/(73728*c^3)","A",1
73,1,80,93,0.1516704,"\int x \left(\pi +c^2 \pi  x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x*(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{\pi ^{5/2} \left(35 a \left(c^2 x^2+1\right)^{7/2}+35 b \left(c^2 x^2+1\right)^{7/2} \sinh ^{-1}(c x)-b c x \left(5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right)\right)}{245 c^2}","\frac{\left(\pi  c^2 x^2+\pi \right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{7 \pi  c^2}-\frac{1}{49} \pi ^{5/2} b c^5 x^7-\frac{3}{35} \pi ^{5/2} b c^3 x^5-\frac{1}{7} \pi ^{5/2} b c x^3-\frac{\pi ^{5/2} b x}{7 c}",1,"(Pi^(5/2)*(35*a*(1 + c^2*x^2)^(7/2) - b*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)^(7/2)*ArcSinh[c*x]))/(245*c^2)","A",1
74,1,153,165,0.3836804,"\int \left(\pi +c^2 \pi  x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{\pi ^{5/2} \left(12 \sinh ^{-1}(c x) \left(60 a+45 b \sinh \left(2 \sinh ^{-1}(c x)\right)+9 b \sinh \left(4 \sinh ^{-1}(c x)\right)+b \sinh \left(6 \sinh ^{-1}(c x)\right)\right)+1584 a c x \sqrt{c^2 x^2+1}+384 a c^5 x^5 \sqrt{c^2 x^2+1}+1248 a c^3 x^3 \sqrt{c^2 x^2+1}+360 b \sinh ^{-1}(c x)^2-270 b \cosh \left(2 \sinh ^{-1}(c x)\right)-27 b \cosh \left(4 \sinh ^{-1}(c x)\right)-2 b \cosh \left(6 \sinh ^{-1}(c x)\right)\right)}{2304 c}","\frac{1}{6} x \left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5}{24} \pi  x \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5}{16} \pi ^2 x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)+\frac{5 \pi ^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{32 b c}-\frac{5}{96} \pi ^{5/2} b c^3 x^4-\frac{\pi ^{5/2} b \left(c^2 x^2+1\right)^3}{36 c}-\frac{25}{96} \pi ^{5/2} b c x^2",1,"(Pi^(5/2)*(1584*a*c*x*Sqrt[1 + c^2*x^2] + 1248*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 384*a*c^5*x^5*Sqrt[1 + c^2*x^2] + 360*b*ArcSinh[c*x]^2 - 270*b*Cosh[2*ArcSinh[c*x]] - 27*b*Cosh[4*ArcSinh[c*x]] - 2*b*Cosh[6*ArcSinh[c*x]] + 12*ArcSinh[c*x]*(60*a + 45*b*Sinh[2*ArcSinh[c*x]] + 9*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]])))/(2304*c)","A",1
75,1,257,179,0.3633078,"\int \frac{\left(\pi +c^2 \pi  x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x} \, dx","Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x,x]","\frac{1}{225} \pi ^{5/2} \left(165 a c^2 x^2 \sqrt{c^2 x^2+1}+345 a \sqrt{c^2 x^2+1}-225 a \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)+45 a c^4 x^4 \sqrt{c^2 x^2+1}+225 a \log (x)-9 b c^5 x^5-55 b c^3 x^3+165 b c^2 x^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+345 b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+45 b c^4 x^4 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+225 b \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-225 b \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-345 b c x+225 b \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-225 b \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)","\frac{1}{5} \left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{3} \pi  \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\pi ^2 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-2 \pi ^{5/2} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{25} \pi ^{5/2} b c^5 x^5-\frac{11}{45} \pi ^{5/2} b c^3 x^3-\pi ^{5/2} b \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+\pi ^{5/2} b \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)-\frac{23}{15} \pi ^{5/2} b c x",1,"(Pi^(5/2)*(-345*b*c*x - 55*b*c^3*x^3 - 9*b*c^5*x^5 + 345*a*Sqrt[1 + c^2*x^2] + 165*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 45*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 345*b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 165*b*c^2*x^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 45*b*c^4*x^4*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 225*b*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 225*b*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 225*a*Log[x] - 225*a*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + 225*b*PolyLog[2, -E^(-ArcSinh[c*x])] - 225*b*PolyLog[2, E^(-ArcSinh[c*x])]))/225","A",0
76,1,168,157,0.4212279,"\int \frac{\left(\pi +c^2 \pi  x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^2,x]","\frac{\pi ^{5/2} \left(4 \sinh ^{-1}(c x) \left(60 a c x-32 b \sqrt{c^2 x^2+1}+16 b c x \sinh \left(2 \sinh ^{-1}(c x)\right)+b c x \sinh \left(4 \sinh ^{-1}(c x)\right)\right)+144 a c^2 x^2 \sqrt{c^2 x^2+1}-128 a \sqrt{c^2 x^2+1}+32 a c^4 x^4 \sqrt{c^2 x^2+1}+128 b c x \log (c x)+120 b c x \sinh ^{-1}(c x)^2-32 b c x \cosh \left(2 \sinh ^{-1}(c x)\right)-b c x \cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{128 x}","\frac{5}{4} \pi  c^2 x \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{15}{8} \pi ^2 c^2 x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-\frac{\left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x}+\frac{15 \pi ^{5/2} c \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b}-\frac{1}{16} \pi ^{5/2} b c^5 x^4-\frac{9}{16} \pi ^{5/2} b c^3 x^2+\pi ^{5/2} b c \log (x)",1,"(Pi^(5/2)*(-128*a*Sqrt[1 + c^2*x^2] + 144*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 32*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 120*b*c*x*ArcSinh[c*x]^2 - 32*b*c*x*Cosh[2*ArcSinh[c*x]] - b*c*x*Cosh[4*ArcSinh[c*x]] + 128*b*c*x*Log[c*x] + 4*ArcSinh[c*x]*(60*a*c*x - 32*b*Sqrt[1 + c^2*x^2] + 16*b*c*x*Sinh[2*ArcSinh[c*x]] + b*c*x*Sinh[4*ArcSinh[c*x]])))/(128*x)","A",1
77,1,349,205,1.9535354,"\int \frac{\left(\pi +c^2 \pi  x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3,x]","\frac{\pi ^{5/2} \left(168 a c^2 x^2 \sqrt{c^2 x^2+1}-36 a \sqrt{c^2 x^2+1}+180 a c^2 x^2 \log (x)-180 a c^2 x^2 \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)+24 a c^4 x^4 \sqrt{c^2 x^2+1}-8 b c^5 x^5-168 b c^3 x^3-9 b c^3 x^3 \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+180 b c^2 x^2 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-180 b c^2 x^2 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+168 b c^2 x^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+180 b c^2 x^2 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-180 b c^2 x^2 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-9 b c^2 x^2 \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+24 b c^4 x^4 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+36 b c x \sinh ^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-36 b \sinh ^{-1}(c x) \sinh ^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{72 x^2}","\frac{5}{6} \pi  c^2 \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5}{2} \pi ^2 c^2 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-\frac{\left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{2 x^2}-5 \pi ^{5/2} c^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{9} \pi ^{5/2} b c^5 x^3-\frac{7}{3} \pi ^{5/2} b c^3 x-\frac{5}{2} \pi ^{5/2} b c^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+\frac{5}{2} \pi ^{5/2} b c^2 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)-\frac{\pi ^{5/2} b c}{2 x}",1,"(Pi^(5/2)*(-168*b*c^3*x^3 - 8*b*c^5*x^5 - 36*a*Sqrt[1 + c^2*x^2] + 168*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 24*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 168*b*c^2*x^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 24*b*c^4*x^4*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 9*b*c^3*x^3*Csch[ArcSinh[c*x]/2]^2 - 9*b*c^2*x^2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 180*b*c^2*x^2*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 180*b*c^2*x^2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 180*a*c^2*x^2*Log[x] - 180*a*c^2*x^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + 180*b*c^2*x^2*PolyLog[2, -E^(-ArcSinh[c*x])] - 180*b*c^2*x^2*PolyLog[2, E^(-ArcSinh[c*x])] + 36*b*c*x*Sinh[ArcSinh[c*x]/2]^2 - 36*b*ArcSinh[c*x]*Sinh[ArcSinh[c*x]/2]^2))/(72*x^2)","A",0
78,1,179,166,0.3765803,"\int \frac{\left(\pi +c^2 \pi  x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^4,x]","\frac{\pi ^{5/2} \left(\sinh ^{-1}(c x) \left(60 a c^3 x^3+6 b c^3 x^3 \sinh \left(2 \sinh ^{-1}(c x)\right)-8 b \sqrt{c^2 x^2+1} \left(7 c^2 x^2+1\right)\right)-56 a c^2 x^2 \sqrt{c^2 x^2+1}-8 a \sqrt{c^2 x^2+1}+12 a c^4 x^4 \sqrt{c^2 x^2+1}+56 b c^3 x^3 \log (c x)+30 b c^3 x^3 \sinh ^{-1}(c x)^2-3 b c^3 x^3 \cosh \left(2 \sinh ^{-1}(c x)\right)-4 b c x\right)}{24 x^3}","\frac{5 \pi ^{5/2} c^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b}-\frac{5 \pi  c^2 \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 x}-\frac{\left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^3}+\frac{5}{2} \pi ^2 c^4 x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{4} \pi ^{5/2} b c^5 x^2+\frac{7}{3} \pi ^{5/2} b c^3 \log (x)-\frac{\pi ^{5/2} b c}{6 x^2}",1,"(Pi^(5/2)*(-4*b*c*x - 8*a*Sqrt[1 + c^2*x^2] - 56*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 12*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 30*b*c^3*x^3*ArcSinh[c*x]^2 - 3*b*c^3*x^3*Cosh[2*ArcSinh[c*x]] + 56*b*c^3*x^3*Log[c*x] + ArcSinh[c*x]*(60*a*c^3*x^3 - 8*b*Sqrt[1 + c^2*x^2]*(1 + 7*c^2*x^2) + 6*b*c^3*x^3*Sinh[2*ArcSinh[c*x]])))/(24*x^3)","A",1
79,1,28,32,0.0131136,"\int \sqrt{1+x^2} \sinh ^{-1}(x) \, dx","Integrate[Sqrt[1 + x^2]*ArcSinh[x],x]","\frac{1}{4} \left(-x^2+2 \sqrt{x^2+1} x \sinh ^{-1}(x)+\sinh ^{-1}(x)^2\right)","-\frac{x^2}{4}+\frac{1}{2} \sqrt{x^2+1} x \sinh ^{-1}(x)+\frac{1}{4} \sinh ^{-1}(x)^2",1,"(-x^2 + 2*x*Sqrt[1 + x^2]*ArcSinh[x] + ArcSinh[x]^2)/4","A",1
80,1,108,149,0.1807962,"\int \frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(x^5*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2],x]","\frac{15 a \sqrt{c^2 x^2+1} \left(3 c^4 x^4-4 c^2 x^2+8\right)+b \left(-9 c^5 x^5+20 c^3 x^3-120 c x\right)+15 b \sqrt{c^2 x^2+1} \left(3 c^4 x^4-4 c^2 x^2+8\right) \sinh ^{-1}(c x)}{225 \sqrt{\pi } c^6}","\frac{x^4 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{5 \pi  c^2}+\frac{8 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{15 \pi  c^6}-\frac{4 x^2 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{15 \pi  c^4}-\frac{8 b x}{15 \sqrt{\pi } c^5}+\frac{4 b x^3}{45 \sqrt{\pi } c^3}-\frac{b x^5}{25 \sqrt{\pi } c}",1,"(15*a*Sqrt[1 + c^2*x^2]*(8 - 4*c^2*x^2 + 3*c^4*x^4) + b*(-120*c*x + 20*c^3*x^3 - 9*c^5*x^5) + 15*b*Sqrt[1 + c^2*x^2]*(8 - 4*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x])/(225*c^6*Sqrt[Pi])","A",1
81,1,111,126,0.2328056,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2],x]","\frac{4 \sinh ^{-1}(c x) \left(12 a-8 b \sinh \left(2 \sinh ^{-1}(c x)\right)+b \sinh \left(4 \sinh ^{-1}(c x)\right)\right)-48 a c x \sqrt{c^2 x^2+1}+32 a c^3 x^3 \sqrt{c^2 x^2+1}+24 b \sinh ^{-1}(c x)^2+16 b \cosh \left(2 \sinh ^{-1}(c x)\right)-b \cosh \left(4 \sinh ^{-1}(c x)\right)}{128 \sqrt{\pi } c^5}","\frac{3 \left(a+b \sinh ^{-1}(c x)\right)^2}{16 \sqrt{\pi } b c^5}+\frac{x^3 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{4 \pi  c^2}-\frac{3 x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{8 \pi  c^4}+\frac{3 b x^2}{16 \sqrt{\pi } c^3}-\frac{b x^4}{16 \sqrt{\pi } c}",1,"(-48*a*c*x*Sqrt[1 + c^2*x^2] + 32*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 24*b*ArcSinh[c*x]^2 + 16*b*Cosh[2*ArcSinh[c*x]] - b*Cosh[4*ArcSinh[c*x]] + 4*ArcSinh[c*x]*(12*a - 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]))/(128*c^5*Sqrt[Pi])","A",1
82,1,82,98,0.1395011,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2],x]","\frac{3 a \sqrt{c^2 x^2+1} \left(c^2 x^2-2\right)+b \left(6 c x-c^3 x^3\right)+3 b \sqrt{c^2 x^2+1} \left(c^2 x^2-2\right) \sinh ^{-1}(c x)}{9 \sqrt{\pi } c^4}","\frac{x^2 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  c^2}-\frac{2 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  c^4}+\frac{2 b x}{3 \sqrt{\pi } c^3}-\frac{b x^3}{9 \sqrt{\pi } c}",1,"(3*a*(-2 + c^2*x^2)*Sqrt[1 + c^2*x^2] + b*(6*c*x - c^3*x^3) + 3*b*(-2 + c^2*x^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(9*c^4*Sqrt[Pi])","A",1
83,1,69,75,0.1792108,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2],x]","\frac{\sinh ^{-1}(c x) \left(2 b \sinh \left(2 \sinh ^{-1}(c x)\right)-4 a\right)+4 a c x \sqrt{c^2 x^2+1}-2 b \sinh ^{-1}(c x)^2-b \cosh \left(2 \sinh ^{-1}(c x)\right)}{8 \sqrt{\pi } c^3}","-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{4 \sqrt{\pi } b c^3}+\frac{x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{2 \pi  c^2}-\frac{b x^2}{4 \sqrt{\pi } c}",1,"(4*a*c*x*Sqrt[1 + c^2*x^2] - 2*b*ArcSinh[c*x]^2 - b*Cosh[2*ArcSinh[c*x]] + ArcSinh[c*x]*(-4*a + 2*b*Sinh[2*ArcSinh[c*x]]))/(8*c^3*Sqrt[Pi])","A",1
84,1,49,42,0.0800551,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(x*(a + b*ArcSinh[c*x]))/Sqrt[Pi + c^2*Pi*x^2],x]","\frac{a \sqrt{c^2 x^2+1}+b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-b c x}{\sqrt{\pi } c^2}","\frac{\sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{\pi  c^2}-\frac{b x}{\sqrt{\pi } c}",1,"(-(b*c*x) + a*Sqrt[1 + c^2*x^2] + b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(c^2*Sqrt[Pi])","A",1
85,1,25,25,0.0163475,"\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/Sqrt[Pi + c^2*Pi*x^2],x]","\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 \sqrt{\pi } b c}","\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 \sqrt{\pi } b c}",1,"(a + b*ArcSinh[c*x])^2/(2*b*c*Sqrt[Pi])","A",1
86,1,96,56,0.1554932,"\int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x*Sqrt[Pi + c^2*Pi*x^2]),x]","\frac{-a \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)+a \log (x)+b \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-b \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+b \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-b \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)}{\sqrt{\pi }}","-\frac{2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{\pi }}-\frac{b \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{\pi }}+\frac{b \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{\pi }}",1,"(b*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - b*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + a*Log[x] - a*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + b*PolyLog[2, -E^(-ArcSinh[c*x])] - b*PolyLog[2, E^(-ArcSinh[c*x])])/Sqrt[Pi]","A",0
87,1,42,41,0.1112608,"\int \frac{a+b \sinh ^{-1}(c x)}{x^2 \sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^2*Sqrt[Pi + c^2*Pi*x^2]),x]","\frac{b c \log (x)}{\sqrt{\pi }}-\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{\pi } x}","\frac{b c \log (x)}{\sqrt{\pi }}-\frac{\sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{\pi  x}",1,"-((Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(Sqrt[Pi]*x)) + (b*c*Log[x])/Sqrt[Pi]","A",1
88,1,185,115,2.5946492,"\int \frac{a+b \sinh ^{-1}(c x)}{x^3 \sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^3*Sqrt[Pi + c^2*Pi*x^2]),x]","\frac{-\frac{4 a \sqrt{c^2 x^2+1}}{x^2}+4 a c^2 \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)-4 a c^2 \log (x)+b c^2 \left(-4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{8 \sqrt{\pi }}","-\frac{\sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{2 \pi  x^2}+\frac{c^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{\pi }}+\frac{b c^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{\pi }}-\frac{b c^2 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{\pi }}-\frac{b c}{2 \sqrt{\pi } x}",1,"((-4*a*Sqrt[1 + c^2*x^2])/x^2 - 4*a*c^2*Log[x] + 4*a*c^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + b*c^2*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 4*PolyLog[2, -E^(-ArcSinh[c*x])] + 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(8*Sqrt[Pi])","A",0
89,1,99,97,0.1584074,"\int \frac{a+b \sinh ^{-1}(c x)}{x^4 \sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^4*Sqrt[Pi + c^2*Pi*x^2]),x]","\frac{2 a \sqrt{c^2 x^2+1} \left(2 c^2 x^2-1\right)+b c x \left(6 c^2 x^2-1\right)+2 b \sqrt{c^2 x^2+1} \left(2 c^2 x^2-1\right) \sinh ^{-1}(c x)}{6 \sqrt{\pi } x^3}-\frac{2 b c^3 \log (x)}{3 \sqrt{\pi }}","\frac{2 c^2 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  x}-\frac{\sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  x^3}-\frac{2 b c^3 \log (x)}{3 \sqrt{\pi }}-\frac{b c}{6 \sqrt{\pi } x^2}",1,"(2*a*Sqrt[1 + c^2*x^2]*(-1 + 2*c^2*x^2) + b*c*x*(-1 + 6*c^2*x^2) + 2*b*Sqrt[1 + c^2*x^2]*(-1 + 2*c^2*x^2)*ArcSinh[c*x])/(6*Sqrt[Pi]*x^3) - (2*b*c^3*Log[x])/(3*Sqrt[Pi])","A",1
90,1,131,137,0.2157316,"\int \frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]","\frac{3 a c^4 x^4-12 a c^2 x^2-24 a+15 b c x \sqrt{c^2 x^2+1}+9 b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)+3 b \left(c^4 x^4-4 c^2 x^2-8\right) \sinh ^{-1}(c x)-b c^3 x^3 \sqrt{c^2 x^2+1}}{9 \pi ^{3/2} c^6 \sqrt{c^2 x^2+1}}","\frac{\left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi ^3 c^6}-\frac{2 \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^2 c^6}-\frac{a+b \sinh ^{-1}(c x)}{\pi  c^6 \sqrt{\pi  c^2 x^2+\pi }}+\frac{b \tan ^{-1}(c x)}{\pi ^{3/2} c^6}+\frac{5 b x}{3 \pi ^{3/2} c^5}-\frac{b x^3}{9 \pi ^{3/2} c^3}",1,"(-24*a - 12*a*c^2*x^2 + 3*a*c^4*x^4 + 15*b*c*x*Sqrt[1 + c^2*x^2] - b*c^3*x^3*Sqrt[1 + c^2*x^2] + 3*b*(-8 - 4*c^2*x^2 + c^4*x^4)*ArcSinh[c*x] + 9*b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(9*c^6*Pi^(3/2)*Sqrt[1 + c^2*x^2])","A",1
91,1,147,131,0.3561232,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]","\frac{\sinh ^{-1}(c x) \left(-12 a \sqrt{c^2 x^2+1}+9 b c x+b \sinh \left(3 \sinh ^{-1}(c x)\right)\right)+4 a c^3 x^3+12 a c x-4 b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)-6 b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2-b \sqrt{c^2 x^2+1} \cosh \left(2 \sinh ^{-1}(c x)\right)}{8 \pi ^{3/2} c^5 \sqrt{c^2 x^2+1}}","-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)^2}{4 \pi ^{3/2} b c^5}-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{\pi  c^2 \sqrt{\pi  c^2 x^2+\pi }}+\frac{3 x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{2 \pi ^2 c^4}-\frac{b x^2}{4 \pi ^{3/2} c^3}-\frac{b \log \left(c^2 x^2+1\right)}{2 \pi ^{3/2} c^5}",1,"(12*a*c*x + 4*a*c^3*x^3 - 6*b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 - b*Sqrt[1 + c^2*x^2]*Cosh[2*ArcSinh[c*x]] - 4*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + ArcSinh[c*x]*(9*b*c*x - 12*a*Sqrt[1 + c^2*x^2] + b*Sinh[3*ArcSinh[c*x]]))/(8*c^5*Pi^(3/2)*Sqrt[1 + c^2*x^2])","A",1
92,1,87,86,0.1577145,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]","\frac{a c^2 x^2+2 a-b c x \sqrt{c^2 x^2+1}-b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)+b \left(c^2 x^2+2\right) \sinh ^{-1}(c x)}{\pi ^{3/2} c^4 \sqrt{c^2 x^2+1}}","\frac{\sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^2 c^4}+\frac{a+b \sinh ^{-1}(c x)}{\pi  c^4 \sqrt{\pi  c^2 x^2+\pi }}-\frac{b \tan ^{-1}(c x)}{\pi ^{3/2} c^4}-\frac{b x}{\pi ^{3/2} c^3}",1,"(2*a + a*c^2*x^2 - b*c*x*Sqrt[1 + c^2*x^2] + b*(2 + c^2*x^2)*ArcSinh[c*x] - b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(c^4*Pi^(3/2)*Sqrt[1 + c^2*x^2])","A",1
93,1,78,80,0.2907468,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]","\frac{\sinh ^{-1}(c x) \left(2 a-\frac{2 b c x}{\sqrt{c^2 x^2+1}}\right)-\frac{2 a c x}{\sqrt{c^2 x^2+1}}+b \log \left(c^2 x^2+1\right)+b \sinh ^{-1}(c x)^2}{2 \pi ^{3/2} c^3}","\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 \pi ^{3/2} b c^3}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\pi  c^2 \sqrt{\pi  c^2 x^2+\pi }}+\frac{b \log \left(c^2 x^2+1\right)}{2 \pi ^{3/2} c^3}",1,"((-2*a*c*x)/Sqrt[1 + c^2*x^2] + (2*a - (2*b*c*x)/Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + b*ArcSinh[c*x]^2 + b*Log[1 + c^2*x^2])/(2*c^3*Pi^(3/2))","A",1
94,1,52,45,0.1116761,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]","\frac{-a+b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)-b \sinh ^{-1}(c x)}{\pi ^{3/2} c^2 \sqrt{c^2 x^2+1}}","\frac{b \tan ^{-1}(c x)}{\pi ^{3/2} c^2}-\frac{a+b \sinh ^{-1}(c x)}{\pi  c^2 \sqrt{\pi  c^2 x^2+\pi }}",1,"(-a - b*ArcSinh[c*x] + b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(c^2*Pi^(3/2)*Sqrt[1 + c^2*x^2])","A",1
95,1,66,51,0.0987421,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(Pi + c^2*Pi*x^2)^(3/2),x]","\frac{2 a c x-b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)+2 b c x \sinh ^{-1}(c x)}{2 \pi ^{3/2} c \sqrt{c^2 x^2+1}}","\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\pi  \sqrt{\pi  c^2 x^2+\pi }}-\frac{b \log \left(c^2 x^2+1\right)}{2 \pi ^{3/2} c}",1,"(2*a*c*x + 2*b*c*x*ArcSinh[c*x] - b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2])/(2*c*Pi^(3/2)*Sqrt[1 + c^2*x^2])","A",1
96,1,143,94,0.3177063,"\int \frac{a+b \sinh ^{-1}(c x)}{x \left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x*(Pi + c^2*Pi*x^2)^(3/2)),x]","\frac{\frac{a}{\sqrt{c^2 x^2+1}}-a \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)+a \log (x)+\frac{b \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+b \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-b \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+b \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-b \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-2 b \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{\pi ^{3/2}}","\frac{a+b \sinh ^{-1}(c x)}{\pi  \sqrt{\pi  c^2 x^2+\pi }}-\frac{2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^{3/2}}-\frac{b \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{\pi ^{3/2}}+\frac{b \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{\pi ^{3/2}}-\frac{b \tan ^{-1}(c x)}{\pi ^{3/2}}",1,"(a/Sqrt[1 + c^2*x^2] + (b*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - 2*b*ArcTan[Tanh[ArcSinh[c*x]/2]] + b*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - b*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + a*Log[x] - a*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + b*PolyLog[2, -E^(-ArcSinh[c*x])] - b*PolyLog[2, E^(-ArcSinh[c*x])])/Pi^(3/2)","A",0
97,1,69,93,0.1522271,"\int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^2*(Pi + c^2*Pi*x^2)^(3/2)),x]","\frac{b \left(\frac{1}{2} c \log \left(c^2 x^2+1\right)+c \log (x)\right)}{\pi ^{3/2}}-\frac{\left(2 c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^{3/2} x \sqrt{c^2 x^2+1}}","-\frac{2 c^2 x \left(a+b \sinh ^{-1}(c x)\right)}{\pi  \sqrt{\pi  c^2 x^2+\pi }}-\frac{a+b \sinh ^{-1}(c x)}{\pi  x \sqrt{\pi  c^2 x^2+\pi }}+\frac{b c \log \left(c^2 x^2+1\right)}{2 \pi ^{3/2}}+\frac{b c \log (x)}{\pi ^{3/2}}",1,"-(((1 + 2*c^2*x^2)*(a + b*ArcSinh[c*x]))/(Pi^(3/2)*x*Sqrt[1 + c^2*x^2])) + (b*(c*Log[x] + (c*Log[1 + c^2*x^2])/2))/Pi^(3/2)","A",1
98,1,269,162,3.8303762,"\int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^3*(Pi + c^2*Pi*x^2)^(3/2)),x]","\frac{-\frac{8 a c^2}{\sqrt{c^2 x^2+1}}-\frac{4 a \sqrt{c^2 x^2+1}}{x^2}+12 a c^2 \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)-12 a c^2 \log (x)-12 b c^2 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+12 b c^2 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-\frac{8 b c^2 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-12 b c^2 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+12 b c^2 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 b c^2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 b c^2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-b c^2 \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-b c^2 \sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+16 b c^2 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{8 \pi ^{3/2}}","-\frac{3 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 \pi  \sqrt{\pi  c^2 x^2+\pi }}-\frac{a+b \sinh ^{-1}(c x)}{2 \pi  x^2 \sqrt{\pi  c^2 x^2+\pi }}+\frac{3 c^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^{3/2}}+\frac{3 b c^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{2 \pi ^{3/2}}-\frac{3 b c^2 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{2 \pi ^{3/2}}+\frac{b c^2 \tan ^{-1}(c x)}{\pi ^{3/2}}-\frac{b c}{2 \pi ^{3/2} x}",1,"((-8*a*c^2)/Sqrt[1 + c^2*x^2] - (4*a*Sqrt[1 + c^2*x^2])/x^2 - (8*b*c^2*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + 16*b*c^2*ArcTan[Tanh[ArcSinh[c*x]/2]] - 2*b*c^2*Coth[ArcSinh[c*x]/2] - b*c^2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 12*b*c^2*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 12*b*c^2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 12*a*c^2*Log[x] + 12*a*c^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] - 12*b*c^2*PolyLog[2, -E^(-ArcSinh[c*x])] + 12*b*c^2*PolyLog[2, E^(-ArcSinh[c*x])] - b*c^2*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*b*c^2*Tanh[ArcSinh[c*x]/2])/(8*Pi^(3/2))","A",0
99,1,127,153,0.2004334,"\int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^4*(Pi + c^2*Pi*x^2)^(3/2)),x]","\frac{2 a \left(8 c^4 x^4+4 c^2 x^2-1\right)-b c x \sqrt{c^2 x^2+1}+2 b \left(8 c^4 x^4+4 c^2 x^2-1\right) \sinh ^{-1}(c x)}{6 \pi ^{3/2} x^3 \sqrt{c^2 x^2+1}}+\frac{-5 b c^3 \log (x)-\frac{3}{2} b c^3 \log \left(c^2 x^2+1\right)}{3 \pi ^{3/2}}","\frac{4 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  x \sqrt{\pi  c^2 x^2+\pi }}-\frac{a+b \sinh ^{-1}(c x)}{3 \pi  x^3 \sqrt{\pi  c^2 x^2+\pi }}+\frac{8 c^4 x \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  \sqrt{\pi  c^2 x^2+\pi }}-\frac{5 b c^3 \log (x)}{3 \pi ^{3/2}}-\frac{b c^3 \log \left(c^2 x^2+1\right)}{2 \pi ^{3/2}}-\frac{b c}{6 \pi ^{3/2} x^2}",1,"(-(b*c*x*Sqrt[1 + c^2*x^2]) + 2*a*(-1 + 4*c^2*x^2 + 8*c^4*x^4) + 2*b*(-1 + 4*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x])/(6*Pi^(3/2)*x^3*Sqrt[1 + c^2*x^2]) + (-5*b*c^3*Log[x] - (3*b*c^3*Log[1 + c^2*x^2])/2)/(3*Pi^(3/2))","A",1
100,1,202,192,0.4740561,"\int \frac{x^6 \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(x^6*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]","\frac{4 \sinh ^{-1}(c x) \left(b c x \left(3 c^4 x^4+20 c^2 x^2+15\right)-15 a \left(c^2 x^2+1\right)^{3/2}\right)+12 a c^5 x^5+80 a c^3 x^3+60 a c x-9 b c^2 x^2 \sqrt{c^2 x^2+1}-7 b \sqrt{c^2 x^2+1}-28 b \left(c^2 x^2+1\right)^{3/2} \log \left(c^2 x^2+1\right)-30 b \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)^2-6 b c^4 x^4 \sqrt{c^2 x^2+1}}{24 \pi ^{5/2} c^7 \left(c^2 x^2+1\right)^{3/2}}","-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)^2}{4 \pi ^{5/2} b c^7}-\frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  c^2 \left(\pi  c^2 x^2+\pi \right)^{3/2}}+\frac{5 x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{2 \pi ^3 c^6}-\frac{5 x^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi ^2 c^4 \sqrt{\pi  c^2 x^2+\pi }}-\frac{b x^2}{4 \pi ^{5/2} c^5}-\frac{b}{6 \pi ^{5/2} c^7 \left(c^2 x^2+1\right)}-\frac{7 b \log \left(c^2 x^2+1\right)}{6 \pi ^{5/2} c^7}",1,"(60*a*c*x + 80*a*c^3*x^3 + 12*a*c^5*x^5 - 7*b*Sqrt[1 + c^2*x^2] - 9*b*c^2*x^2*Sqrt[1 + c^2*x^2] - 6*b*c^4*x^4*Sqrt[1 + c^2*x^2] + 4*(-15*a*(1 + c^2*x^2)^(3/2) + b*c*x*(15 + 20*c^2*x^2 + 3*c^4*x^4))*ArcSinh[c*x] - 30*b*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2 - 28*b*(1 + c^2*x^2)^(3/2)*Log[1 + c^2*x^2])/(24*c^7*Pi^(5/2)*(1 + c^2*x^2)^(3/2))","A",1
101,1,132,146,0.2146522,"\int \frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]","\frac{6 a c^4 x^4+24 a c^2 x^2+16 a-5 b c x \sqrt{c^2 x^2+1}-11 b \left(c^2 x^2+1\right)^{3/2} \tan ^{-1}(c x)+2 b \left(3 c^4 x^4+12 c^2 x^2+8\right) \sinh ^{-1}(c x)-6 b c^3 x^3 \sqrt{c^2 x^2+1}}{6 \pi ^{5/2} c^6 \left(c^2 x^2+1\right)^{3/2}}","\frac{\sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^3 c^6}+\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^2 c^6 \sqrt{\pi  c^2 x^2+\pi }}-\frac{a+b \sinh ^{-1}(c x)}{3 \pi  c^6 \left(\pi  c^2 x^2+\pi \right)^{3/2}}-\frac{11 b \tan ^{-1}(c x)}{6 \pi ^{5/2} c^6}-\frac{b x}{\pi ^{5/2} c^5}+\frac{b x}{6 \pi ^{5/2} c^5 \left(c^2 x^2+1\right)}",1,"(16*a + 24*a*c^2*x^2 + 6*a*c^4*x^4 - 5*b*c*x*Sqrt[1 + c^2*x^2] - 6*b*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*b*(8 + 12*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x] - 11*b*(1 + c^2*x^2)^(3/2)*ArcTan[c*x])/(6*c^6*Pi^(5/2)*(1 + c^2*x^2)^(3/2))","A",1
102,1,166,139,0.3628158,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]","\frac{2 \sinh ^{-1}(c x) \left(3 a \left(c^2 x^2+1\right)^2-b c x \sqrt{c^2 x^2+1} \left(4 c^2 x^2+3\right)\right)-6 a c x \sqrt{c^2 x^2+1}-8 a c^3 x^3 \sqrt{c^2 x^2+1}+b c^2 x^2+4 b \left(c^2 x^2+1\right)^2 \log \left(c^2 x^2+1\right)+3 b \left(c^2 x^2+1\right)^2 \sinh ^{-1}(c x)^2+b}{6 \pi ^{5/2} c^5 \left(c^2 x^2+1\right)^2}","\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 \pi ^{5/2} b c^5}-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  c^2 \left(\pi  c^2 x^2+\pi \right)^{3/2}}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^2 c^4 \sqrt{\pi  c^2 x^2+\pi }}+\frac{b}{6 \pi ^{5/2} c^5 \left(c^2 x^2+1\right)}+\frac{2 b \log \left(c^2 x^2+1\right)}{3 \pi ^{5/2} c^5}",1,"(b + b*c^2*x^2 - 6*a*c*x*Sqrt[1 + c^2*x^2] - 8*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*(3*a*(1 + c^2*x^2)^2 - b*c*x*Sqrt[1 + c^2*x^2]*(3 + 4*c^2*x^2))*ArcSinh[c*x] + 3*b*(1 + c^2*x^2)^2*ArcSinh[c*x]^2 + 4*b*(1 + c^2*x^2)^2*Log[1 + c^2*x^2])/(6*c^5*Pi^(5/2)*(1 + c^2*x^2)^2)","A",1
103,1,93,105,0.1828474,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]","\frac{-6 a c^2 x^2-4 a-b c x \sqrt{c^2 x^2+1}+5 b \left(c^2 x^2+1\right)^{3/2} \tan ^{-1}(c x)-2 b \left(3 c^2 x^2+2\right) \sinh ^{-1}(c x)}{6 \pi ^{5/2} c^4 \left(c^2 x^2+1\right)^{3/2}}","-\frac{a+b \sinh ^{-1}(c x)}{\pi ^2 c^4 \sqrt{\pi  c^2 x^2+\pi }}+\frac{a+b \sinh ^{-1}(c x)}{3 \pi  c^4 \left(\pi  c^2 x^2+\pi \right)^{3/2}}+\frac{5 b \tan ^{-1}(c x)}{6 \pi ^{5/2} c^4}-\frac{b x}{6 \pi ^{5/2} c^3 \left(c^2 x^2+1\right)}",1,"(-4*a - 6*a*c^2*x^2 - b*c*x*Sqrt[1 + c^2*x^2] - 2*b*(2 + 3*c^2*x^2)*ArcSinh[c*x] + 5*b*(1 + c^2*x^2)^(3/2)*ArcTan[c*x])/(6*c^4*Pi^(5/2)*(1 + c^2*x^2)^(3/2))","A",1
104,1,88,80,0.1700945,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]","-\frac{-2 a c^3 x^3-2 b c^3 x^3 \sinh ^{-1}(c x)+b \sqrt{c^2 x^2+1}+b \left(c^2 x^2+1\right)^{3/2} \log \left(c^2 x^2+1\right)}{6 \pi ^{5/2} c^3 \left(c^2 x^2+1\right)^{3/2}}","\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  \left(\pi  c^2 x^2+\pi \right)^{3/2}}-\frac{b}{6 \pi ^{5/2} c^3 \left(c^2 x^2+1\right)}-\frac{b \log \left(c^2 x^2+1\right)}{6 \pi ^{5/2} c^3}",1,"-1/6*(-2*a*c^3*x^3 + b*Sqrt[1 + c^2*x^2] - 2*b*c^3*x^3*ArcSinh[c*x] + b*(1 + c^2*x^2)^(3/2)*Log[1 + c^2*x^2])/(c^3*Pi^(5/2)*(1 + c^2*x^2)^(3/2))","A",1
105,1,72,75,0.1364538,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]","\frac{-2 a+b c x \sqrt{c^2 x^2+1}+b \left(c^2 x^2+1\right)^{3/2} \tan ^{-1}(c x)-2 b \sinh ^{-1}(c x)}{6 \pi ^{5/2} c^2 \left(c^2 x^2+1\right)^{3/2}}","-\frac{a+b \sinh ^{-1}(c x)}{3 \pi  c^2 \left(\pi  c^2 x^2+\pi \right)^{3/2}}+\frac{b x}{6 \pi ^{5/2} c \left(c^2 x^2+1\right)}+\frac{b \tan ^{-1}(c x)}{6 \pi ^{5/2} c^2}",1,"(-2*a + b*c*x*Sqrt[1 + c^2*x^2] - 2*b*ArcSinh[c*x] + b*(1 + c^2*x^2)^(3/2)*ArcTan[c*x])/(6*c^2*Pi^(5/2)*(1 + c^2*x^2)^(3/2))","A",1
106,1,100,108,0.143187,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(Pi + c^2*Pi*x^2)^(5/2),x]","\frac{4 a c^3 x^3+6 a c x+b \sqrt{c^2 x^2+1}-2 b \left(c^2 x^2+1\right)^{3/2} \log \left(c^2 x^2+1\right)+2 b c x \left(2 c^2 x^2+3\right) \sinh ^{-1}(c x)}{6 \pi ^{5/2} c \left(c^2 x^2+1\right)^{3/2}}","\frac{2 x \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi ^2 \sqrt{\pi  c^2 x^2+\pi }}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  \left(\pi  c^2 x^2+\pi \right)^{3/2}}+\frac{b}{6 \pi ^{5/2} c \left(c^2 x^2+1\right)}-\frac{b \log \left(c^2 x^2+1\right)}{3 \pi ^{5/2} c}",1,"(6*a*c*x + 4*a*c^3*x^3 + b*Sqrt[1 + c^2*x^2] + 2*b*c*x*(3 + 2*c^2*x^2)*ArcSinh[c*x] - 2*b*(1 + c^2*x^2)^(3/2)*Log[1 + c^2*x^2])/(6*c*Pi^(5/2)*(1 + c^2*x^2)^(3/2))","A",1
107,1,209,148,0.7664675,"\int \frac{a+b \sinh ^{-1}(c x)}{x \left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x*(Pi + c^2*Pi*x^2)^(5/2)),x]","\frac{\frac{6 a}{\sqrt{c^2 x^2+1}}+\frac{2 a}{\left(c^2 x^2+1\right)^{3/2}}-6 a \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)+6 a \log (x)-\frac{b c x}{c^2 x^2+1}+\frac{6 b c^2 x^2 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}+\frac{8 b \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}+6 b \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-6 b \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+6 b \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-6 b \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-14 b \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{6 \pi ^{5/2}}","\frac{a+b \sinh ^{-1}(c x)}{\pi ^2 \sqrt{\pi  c^2 x^2+\pi }}+\frac{a+b \sinh ^{-1}(c x)}{3 \pi  \left(\pi  c^2 x^2+\pi \right)^{3/2}}-\frac{2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^{5/2}}-\frac{b c x}{6 \pi ^{5/2} \left(c^2 x^2+1\right)}-\frac{b \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{\pi ^{5/2}}+\frac{b \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{\pi ^{5/2}}-\frac{7 b \tan ^{-1}(c x)}{6 \pi ^{5/2}}",1,"((2*a)/(1 + c^2*x^2)^(3/2) - (b*c*x)/(1 + c^2*x^2) + (6*a)/Sqrt[1 + c^2*x^2] + (8*b*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + (6*b*c^2*x^2*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) - 14*b*ArcTan[Tanh[ArcSinh[c*x]/2]] + 6*b*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 6*b*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 6*a*Log[x] - 6*a*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + 6*b*PolyLog[2, -E^(-ArcSinh[c*x])] - 6*b*PolyLog[2, E^(-ArcSinh[c*x])])/(6*Pi^(5/2))","A",0
108,1,123,150,0.2017252,"\int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^2*(Pi + c^2*Pi*x^2)^(5/2)),x]","\frac{-2 a \left(8 c^4 x^4+12 c^2 x^2+3\right)-b c x \sqrt{c^2 x^2+1}-2 b \left(8 c^4 x^4+12 c^2 x^2+3\right) \sinh ^{-1}(c x)}{6 \pi ^{5/2} x \left(c^2 x^2+1\right)^{3/2}}+\frac{\frac{5}{2} b c \log \left(c^2 x^2+1\right)+3 b c \log (x)}{3 \pi ^{5/2}}","-\frac{8 c^2 x \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi ^2 \sqrt{\pi  c^2 x^2+\pi }}-\frac{4 c^2 x \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  \left(\pi  c^2 x^2+\pi \right)^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{\pi  x \left(\pi  c^2 x^2+\pi \right)^{3/2}}-\frac{b c}{6 \pi ^{5/2} \left(c^2 x^2+1\right)}+\frac{5 b c \log \left(c^2 x^2+1\right)}{6 \pi ^{5/2}}+\frac{b c \log (x)}{\pi ^{5/2}}",1,"(-(b*c*x*Sqrt[1 + c^2*x^2]) - 2*a*(3 + 12*c^2*x^2 + 8*c^4*x^4) - 2*b*(3 + 12*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x])/(6*Pi^(5/2)*x*(1 + c^2*x^2)^(3/2)) + (3*b*c*Log[x] + (5*b*c*Log[1 + c^2*x^2])/2)/(3*Pi^(5/2))","A",1
109,1,331,247,6.3085433,"\int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^3*(Pi + c^2*Pi*x^2)^(5/2)),x]","\frac{-\frac{48 a c^2}{\sqrt{c^2 x^2+1}}-\frac{8 a c^2}{\left(c^2 x^2+1\right)^{3/2}}-\frac{12 a \sqrt{c^2 x^2+1}}{x^2}+60 a c^2 \log \left(\pi  \left(\sqrt{c^2 x^2+1}+1\right)\right)-60 a c^2 \log (x)-60 b c^2 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+60 b c^2 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-\frac{56 b c^2 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}-60 b c^2 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+60 b c^2 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+6 b c^2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-6 b c^2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-3 b c^2 \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+104 b c^2 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-\frac{48 b c^4 x^2 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}+\frac{4 b c^3 x}{c^2 x^2+1}-\frac{6 b c \sinh ^{-1}(c x) \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{x}}{24 \pi ^{5/2}}","-\frac{5 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 \pi ^2 \sqrt{\pi  c^2 x^2+\pi }}-\frac{5 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{6 \pi  \left(\pi  c^2 x^2+\pi \right)^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{2 \pi  x^2 \left(\pi  c^2 x^2+\pi \right)^{3/2}}+\frac{5 c^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^{5/2}}+\frac{5 b c^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{2 \pi ^{5/2}}-\frac{5 b c^2 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{2 \pi ^{5/2}}+\frac{b c}{4 \pi ^{5/2} x \left(c^2 x^2+1\right)}+\frac{13 b c^2 \tan ^{-1}(c x)}{6 \pi ^{5/2}}+\frac{5 b c^3 x}{12 \pi ^{5/2} \left(c^2 x^2+1\right)}-\frac{3 b c}{4 \pi ^{5/2} x}",1,"((-8*a*c^2)/(1 + c^2*x^2)^(3/2) + (4*b*c^3*x)/(1 + c^2*x^2) - (48*a*c^2)/Sqrt[1 + c^2*x^2] - (12*a*Sqrt[1 + c^2*x^2])/x^2 - (56*b*c^2*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) - (48*b*c^4*x^2*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + 104*b*c^2*ArcTan[Tanh[ArcSinh[c*x]/2]] - 6*b*c^2*Coth[ArcSinh[c*x]/2] - 3*b*c^2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 60*b*c^2*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 60*b*c^2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 60*a*c^2*Log[x] + 60*a*c^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] - 60*b*c^2*PolyLog[2, -E^(-ArcSinh[c*x])] + 60*b*c^2*PolyLog[2, E^(-ArcSinh[c*x])] + 6*b*c^2*Tanh[ArcSinh[c*x]/2] - (6*b*c*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2])/x)/(24*Pi^(5/2))","A",0
110,1,142,208,0.2282322,"\int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^4*(Pi + c^2*Pi*x^2)^(5/2)),x]","\frac{2 a \left(16 c^6 x^6+24 c^4 x^4+6 c^2 x^2-1\right)-b c x \sqrt{c^2 x^2+1}+2 b \left(16 c^6 x^6+24 c^4 x^4+6 c^2 x^2-1\right) \sinh ^{-1}(c x)}{6 \pi ^{5/2} x^3 \left(c^2 x^2+1\right)^{3/2}}-\frac{8 \left(b c^3 \log (x)+\frac{1}{2} b c^3 \log \left(c^2 x^2+1\right)\right)}{3 \pi ^{5/2}}","\frac{2 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{\pi  x \left(\pi  c^2 x^2+\pi \right)^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{3 \pi  x^3 \left(\pi  c^2 x^2+\pi \right)^{3/2}}+\frac{16 c^4 x \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi ^2 \sqrt{\pi  c^2 x^2+\pi }}+\frac{8 c^4 x \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi  \left(\pi  c^2 x^2+\pi \right)^{3/2}}-\frac{8 b c^3 \log (x)}{3 \pi ^{5/2}}+\frac{b c^3}{6 \pi ^{5/2} \left(c^2 x^2+1\right)}-\frac{4 b c^3 \log \left(c^2 x^2+1\right)}{3 \pi ^{5/2}}-\frac{b c}{6 \pi ^{5/2} x^2}",1,"(-(b*c*x*Sqrt[1 + c^2*x^2]) + 2*a*(-1 + 6*c^2*x^2 + 24*c^4*x^4 + 16*c^6*x^6) + 2*b*(-1 + 6*c^2*x^2 + 24*c^4*x^4 + 16*c^6*x^6)*ArcSinh[c*x])/(6*Pi^(5/2)*x^3*(1 + c^2*x^2)^(3/2)) - (8*(b*c^3*Log[x] + (b*c^3*Log[1 + c^2*x^2])/2))/(3*Pi^(5/2))","A",1
111,1,121,200,0.1797119,"\int \frac{\sinh ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcSinh[a*x]/(c + a^2*c*x^2)^(7/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(4 a x \sqrt{a^2 x^2+1} \left(8 a^4 x^4+20 a^2 x^2+15\right) \sinh ^{-1}(a x)-\left(a^2 x^2+1\right) \left(-8 a^2 x^2+16 \left(a^2 x^2+1\right)^2 \log \left(a^2 x^2+1\right)-11\right)\right)}{60 a c^4 \left(a^2 x^2+1\right)^{7/2}}","\frac{2}{15 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{1}{20 a c^3 \left(a^2 x^2+1\right)^{3/2} \sqrt{a^2 c x^2+c}}-\frac{4 \sqrt{a^2 x^2+1} \log \left(a^2 x^2+1\right)}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{8 x \sinh ^{-1}(a x)}{15 c^3 \sqrt{a^2 c x^2+c}}+\frac{4 x \sinh ^{-1}(a x)}{15 c^2 \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \sinh ^{-1}(a x)}{5 c \left(a^2 c x^2+c\right)^{5/2}}",1,"(Sqrt[c + a^2*c*x^2]*(4*a*x*Sqrt[1 + a^2*x^2]*(15 + 20*a^2*x^2 + 8*a^4*x^4)*ArcSinh[a*x] - (1 + a^2*x^2)*(-11 - 8*a^2*x^2 + 16*(1 + a^2*x^2)^2*Log[1 + a^2*x^2])))/(60*a*c^4*(1 + a^2*x^2)^(7/2))","A",1
112,1,63,86,0.0507001,"\int \frac{x^4 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^4*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{-a^4 x^4+3 a^2 x^2+2 a x \sqrt{a^2 x^2+1} \left(2 a^2 x^2-3\right) \sinh ^{-1}(a x)+3 \sinh ^{-1}(a x)^2}{16 a^5}","\frac{3 \sinh ^{-1}(a x)^2}{16 a^5}+\frac{3 x^2}{16 a^3}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{8 a^4}-\frac{x^4}{16 a}",1,"(3*a^2*x^2 - a^4*x^4 + 2*a*x*Sqrt[1 + a^2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a*x] + 3*ArcSinh[a*x]^2)/(16*a^5)","A",1
113,1,48,70,0.0412977,"\int \frac{x^3 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^3*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{-a^3 x^3+3 \left(a^2 x^2-2\right) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+6 a x}{9 a^4}","\frac{2 x}{3 a^3}+\frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^4}-\frac{x^3}{9 a}",1,"(6*a*x - a^3*x^3 + 3*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^4)","A",1
114,1,42,49,0.0401337,"\int \frac{x^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^2*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]","-\frac{a^2 x^2-2 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+\sinh ^{-1}(a x)^2}{4 a^3}","-\frac{\sinh ^{-1}(a x)^2}{4 a^3}+\frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 a^2}-\frac{x^2}{4 a}",1,"-1/4*(a^2*x^2 - 2*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + ArcSinh[a*x]^2)/a^3","A",1
115,1,28,28,0.0276508,"\int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{x}{a}","\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{x}{a}",1,"-(x/a) + (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a^2","A",1
116,1,13,13,0.0062975,"\int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]/Sqrt[1 + a^2*x^2],x]","\frac{\sinh ^{-1}(a x)^2}{2 a}","\frac{\sinh ^{-1}(a x)^2}{2 a}",1,"ArcSinh[a*x]^2/(2*a)","A",1
117,1,57,34,0.1018459,"\int \frac{\sinh ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]/(x*Sqrt[1 + a^2*x^2]),x]","\text{Li}_2\left(-e^{-\sinh ^{-1}(a x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(a x)}\right)+\sinh ^{-1}(a x) \left(\log \left(1-e^{-\sinh ^{-1}(a x)}\right)-\log \left(e^{-\sinh ^{-1}(a x)}+1\right)\right)","-\text{Li}_2\left(-e^{\sinh ^{-1}(a x)}\right)+\text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)-2 \sinh ^{-1}(a x) \tanh ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)",1,"ArcSinh[a*x]*(Log[1 - E^(-ArcSinh[a*x])] - Log[1 + E^(-ArcSinh[a*x])]) + PolyLog[2, -E^(-ArcSinh[a*x])] - PolyLog[2, E^(-ArcSinh[a*x])]","A",0
118,1,29,27,0.0364743,"\int \frac{\sinh ^{-1}(a x)}{x^2 \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]/(x^2*Sqrt[1 + a^2*x^2]),x]","a \log (a x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}","a \log (x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}",1,"-((Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/x) + a*Log[a*x]","A",1
119,1,126,80,0.6603418,"\int \frac{\sinh ^{-1}(a x)}{x^3 \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]/(x^3*Sqrt[1 + a^2*x^2]),x]","\frac{1}{8} a^2 \left(-4 \text{Li}_2\left(-e^{-\sinh ^{-1}(a x)}\right)+4 \text{Li}_2\left(e^{-\sinh ^{-1}(a x)}\right)-4 \sinh ^{-1}(a x) \log \left(1-e^{-\sinh ^{-1}(a x)}\right)+4 \sinh ^{-1}(a x) \log \left(e^{-\sinh ^{-1}(a x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(a x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(a x)\right)-\sinh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(a x)\right)-\sinh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(a x)\right)\right)","\frac{1}{2} a^2 \text{Li}_2\left(-e^{\sinh ^{-1}(a x)}\right)-\frac{1}{2} a^2 \text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 x^2}+a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)-\frac{a}{2 x}",1,"(a^2*(-2*Coth[ArcSinh[a*x]/2] - ArcSinh[a*x]*Csch[ArcSinh[a*x]/2]^2 - 4*ArcSinh[a*x]*Log[1 - E^(-ArcSinh[a*x])] + 4*ArcSinh[a*x]*Log[1 + E^(-ArcSinh[a*x])] - 4*PolyLog[2, -E^(-ArcSinh[a*x])] + 4*PolyLog[2, E^(-ArcSinh[a*x])] - ArcSinh[a*x]*Sech[ArcSinh[a*x]/2]^2 + 2*Tanh[ArcSinh[a*x]/2]))/8","A",0
120,1,120,175,0.1109149,"\int x^3 \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{\sqrt{c^2 d x^2+d} \left(15 a \left(3 c^2 x^2-2\right) \left(c^2 x^2+1\right)^2+15 b \left(3 c^2 x^2-2\right) \left(c^2 x^2+1\right)^2 \sinh ^{-1}(c x)+b c x \left(-9 c^4 x^4-5 c^2 x^2+30\right) \sqrt{c^2 x^2+1}\right)}{225 c^4 \left(c^2 x^2+1\right)}","\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 c^4 d^2}-\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^4 d}-\frac{b c x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}-\frac{b x^3 \sqrt{c^2 d x^2+d}}{45 c \sqrt{c^2 x^2+1}}+\frac{2 b x \sqrt{c^2 d x^2+d}}{15 c^3 \sqrt{c^2 x^2+1}}",1,"(Sqrt[d + c^2*d*x^2]*(15*a*(1 + c^2*x^2)^2*(-2 + 3*c^2*x^2) + b*c*x*Sqrt[1 + c^2*x^2]*(30 - 5*c^2*x^2 - 9*c^4*x^4) + 15*b*(1 + c^2*x^2)^2*(-2 + 3*c^2*x^2)*ArcSinh[c*x]))/(225*c^4*(1 + c^2*x^2))","A",1
121,1,129,181,0.809817,"\int x^2 \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]","-\frac{-16 a c x \left(2 c^2 x^2+1\right) \sqrt{c^2 d x^2+d}+16 a \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+\frac{b \sqrt{c^2 d x^2+d} \left(8 \sinh ^{-1}(c x)^2-4 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}}{128 c^3}","\frac{x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{8 c^2}+\frac{1}{4} x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c^3 \sqrt{c^2 x^2+1}}-\frac{b x^2 \sqrt{c^2 d x^2+d}}{16 c \sqrt{c^2 x^2+1}}-\frac{b c x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+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]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2])/c^3","A",1
122,1,92,105,0.0911372,"\int x \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{\sqrt{c^2 d x^2+d} \left(3 a \left(c^2 x^2+1\right)^2-b c x \left(c^2 x^2+3\right) \sqrt{c^2 x^2+1}+3 b \left(c^2 x^2+1\right)^2 \sinh ^{-1}(c x)\right)}{9 c^2 \left(c^2 x^2+1\right)}","\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^2 d}-\frac{b x \sqrt{c^2 d x^2+d}}{3 c \sqrt{c^2 x^2+1}}-\frac{b c x^3 \sqrt{c^2 d x^2+d}}{9 \sqrt{c^2 x^2+1}}",1,"(Sqrt[d + c^2*d*x^2]*(3*a*(1 + c^2*x^2)^2 - b*c*x*Sqrt[1 + c^2*x^2]*(3 + c^2*x^2) + 3*b*(1 + c^2*x^2)^2*ArcSinh[c*x]))/(9*c^2*(1 + c^2*x^2))","A",1
123,1,120,111,0.4508284,"\int \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{1}{8} \left(4 a x \sqrt{c^2 d x^2+d}+\frac{4 a \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)}{c}+\frac{b \sqrt{c^2 d x^2+d} \left(2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)-\cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{c \sqrt{c^2 x^2+1}}\right)","\frac{1}{2} x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c \sqrt{c^2 x^2+1}}-\frac{b c x^2 \sqrt{c^2 d x^2+d}}{4 \sqrt{c^2 x^2+1}}",1,"(4*a*x*Sqrt[d + c^2*d*x^2] + (4*a*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/c + (b*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(c*Sqrt[1 + c^2*x^2]))/8","A",1
124,1,168,177,0.4532843,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x,x]","a \sqrt{c^2 d x^2+d}-a \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+a \sqrt{d} \log (x)+\frac{b \sqrt{c^2 d x^2+d} \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}","\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+\frac{b \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{b c x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+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^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2]","A",0
125,1,129,105,0.3242406,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^2,x]","-\frac{a \sqrt{d \left(c^2 x^2+1\right)}}{x}+a c \sqrt{d} \log \left(\sqrt{d} \sqrt{d \left(c^2 x^2+1\right)}+c d x\right)+\frac{b c \sqrt{d \left(c^2 x^2+1\right)} \left(-\frac{2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{c x}+2 \log (c x)+\sinh ^{-1}(c x)^2\right)}{2 \sqrt{c^2 x^2+1}}","\frac{c \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{x}+\frac{b c \log (x) \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}",1,"-((a*Sqrt[d*(1 + c^2*x^2)])/x) + (b*c*Sqrt[d*(1 + c^2*x^2)]*((-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(c*x) + ArcSinh[c*x]^2 + 2*Log[c*x]))/(2*Sqrt[1 + c^2*x^2]) + a*c*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]]","A",1
126,1,223,201,2.9272374,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^3,x]","\frac{1}{8} \left(-\frac{4 a \sqrt{c^2 d x^2+d}}{x^2}-4 a c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+4 a c^2 \sqrt{d} \log (x)+\frac{b c^2 \sqrt{c^2 d x^2+d} \left(4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}\right)","-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{2 x^2}-\frac{c^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{b c^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{c^2 x^2+1}}+\frac{b c^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{c^2 x^2+1}}-\frac{b c \sqrt{c^2 d x^2+d}}{2 x \sqrt{c^2 x^2+1}}",1,"((-4*a*Sqrt[d + c^2*d*x^2])/x^2 + 4*a*c^2*Sqrt[d]*Log[x] - 4*a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b*c^2*Sqrt[d + c^2*d*x^2]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/Sqrt[1 + c^2*x^2])/8","A",1
127,1,131,106,0.2416098,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/x^4,x]","\frac{b c^3 \log (x) \sqrt{d \left(c^2 x^2+1\right)}}{3 \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left(2 a \left(c^2 x^2+1\right)^2+b c x \left(3 c^2 x^2+1\right) \sqrt{c^2 x^2+1}+2 b \left(c^2 x^2+1\right)^2 \sinh ^{-1}(c x)\right)}{6 x^3 \left(c^2 x^2+1\right)}","-\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 d x^3}-\frac{b c \sqrt{c^2 d x^2+d}}{6 x^2 \sqrt{c^2 x^2+1}}+\frac{b c^3 \log (x) \sqrt{c^2 d x^2+d}}{3 \sqrt{c^2 x^2+1}}",1,"-1/6*(Sqrt[d + c^2*d*x^2]*(2*a*(1 + c^2*x^2)^2 + b*c*x*Sqrt[1 + c^2*x^2]*(1 + 3*c^2*x^2) + 2*b*(1 + c^2*x^2)^2*ArcSinh[c*x]))/(x^3*(1 + c^2*x^2)) + (b*c^3*Sqrt[d*(1 + c^2*x^2)]*Log[x])/(3*Sqrt[1 + c^2*x^2])","A",1
128,1,130,217,0.1620317,"\int x^3 \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{d \sqrt{c^2 d x^2+d} \left(105 a \left(5 c^2 x^2-2\right) \left(c^2 x^2+1\right)^3+105 b \left(5 c^2 x^2-2\right) \left(c^2 x^2+1\right)^3 \sinh ^{-1}(c x)-b c x \left(75 c^6 x^6+168 c^4 x^4+35 c^2 x^2-210\right) \sqrt{c^2 x^2+1}\right)}{3675 c^4 \left(c^2 x^2+1\right)}","\frac{\left(c^2 d x^2+d\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{7 c^4 d^2}-\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 c^4 d}-\frac{8 b c d x^5 \sqrt{c^2 d x^2+d}}{175 \sqrt{c^2 x^2+1}}-\frac{b d x^3 \sqrt{c^2 d x^2+d}}{105 c \sqrt{c^2 x^2+1}}+\frac{2 b d x \sqrt{c^2 d x^2+d}}{35 c^3 \sqrt{c^2 x^2+1}}-\frac{b c^3 d x^7 \sqrt{c^2 d x^2+d}}{49 \sqrt{c^2 x^2+1}}",1,"(d*Sqrt[d + c^2*d*x^2]*(105*a*(1 + c^2*x^2)^3*(-2 + 5*c^2*x^2) - b*c*x*Sqrt[1 + c^2*x^2]*(-210 + 35*c^2*x^2 + 168*c^4*x^4 + 75*c^6*x^6) + 105*b*(1 + c^2*x^2)^3*(-2 + 5*c^2*x^2)*ArcSinh[c*x]))/(3675*c^4*(1 + c^2*x^2))","A",1
129,1,251,254,0.7250288,"\int x^2 \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{-144 a d^{3/2} \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+48 a c d x \sqrt{c^2 x^2+1} \left(8 c^4 x^4+14 c^2 x^2+3\right) \sqrt{c^2 d x^2+d}-18 b d \sqrt{c^2 d x^2+d} \left(8 \sinh ^{-1}(c x)^2-4 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\cosh \left(4 \sinh ^{-1}(c x)\right)\right)+b d \sqrt{c^2 d x^2+d} \left(72 \sinh ^{-1}(c x)^2+12 \left(-3 \sinh \left(2 \sinh ^{-1}(c x)\right)-3 \sinh \left(4 \sinh ^{-1}(c x)\right)+\sinh \left(6 \sinh ^{-1}(c x)\right)\right) \sinh ^{-1}(c x)+18 \cosh \left(2 \sinh ^{-1}(c x)\right)+9 \cosh \left(4 \sinh ^{-1}(c x)\right)-2 \cosh \left(6 \sinh ^{-1}(c x)\right)\right)}{2304 c^3 \sqrt{c^2 x^2+1}}","\frac{d x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{16 c^2}+\frac{1}{6} x^3 \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{8} d x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{32 b c^3 \sqrt{c^2 x^2+1}}-\frac{b d x^2 \sqrt{c^2 d x^2+d}}{32 c \sqrt{c^2 x^2+1}}-\frac{7 b c d x^4 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{b c^3 d x^6 \sqrt{c^2 d x^2+d}}{36 \sqrt{c^2 x^2+1}}",1,"(48*a*c*d*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(3 + 14*c^2*x^2 + 8*c^4*x^4) - 144*a*d^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 18*b*d*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]) + b*d*Sqrt[d + c^2*d*x^2]*(72*ArcSinh[c*x]^2 + 18*Cosh[2*ArcSinh[c*x]] + 9*Cosh[4*ArcSinh[c*x]] - 2*Cosh[6*ArcSinh[c*x]] + 12*ArcSinh[c*x]*(-3*Sinh[2*ArcSinh[c*x]] - 3*Sinh[4*ArcSinh[c*x]] + Sinh[6*ArcSinh[c*x]])))/(2304*c^3*Sqrt[1 + c^2*x^2])","A",1
130,1,102,146,0.1307656,"\int x \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{d \sqrt{c^2 d x^2+d} \left(15 a \left(c^2 x^2+1\right)^3+15 b \left(c^2 x^2+1\right)^3 \sinh ^{-1}(c x)-b c x \left(3 c^4 x^4+10 c^2 x^2+15\right) \sqrt{c^2 x^2+1}\right)}{75 c^2 \left(c^2 x^2+1\right)}","\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 c^2 d}-\frac{b d x \sqrt{c^2 d x^2+d}}{5 c \sqrt{c^2 x^2+1}}-\frac{2 b c d x^3 \sqrt{c^2 d x^2+d}}{15 \sqrt{c^2 x^2+1}}-\frac{b c^3 d x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}",1,"(d*Sqrt[d + c^2*d*x^2]*(15*a*(1 + c^2*x^2)^3 - b*c*x*Sqrt[1 + c^2*x^2]*(15 + 10*c^2*x^2 + 3*c^4*x^4) + 15*b*(1 + c^2*x^2)^3*ArcSinh[c*x]))/(75*c^2*(1 + c^2*x^2))","A",1
131,1,200,180,0.8053737,"\int \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{3 a d^{3/2} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)}{8 c}+\frac{1}{8} a d x \left(2 c^2 x^2+5\right) \sqrt{c^2 d x^2+d}+\frac{b d \sqrt{c^2 d x^2+d} \left(2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)-\cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{8 c \sqrt{c^2 x^2+1}}-\frac{b d \sqrt{c^2 d x^2+d} \left(8 \sinh ^{-1}(c x)^2-4 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{128 c \sqrt{c^2 x^2+1}}","\frac{1}{4} x \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{3}{8} d x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{3 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c \sqrt{c^2 x^2+1}}-\frac{5 b c d x^2 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{b c^3 d x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}",1,"(a*d*x*(5 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2])/8 + (3*a*d^(3/2)*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(8*c) + (b*d*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(8*c*Sqrt[1 + c^2*x^2]) - (b*d*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(128*c*Sqrt[1 + c^2*x^2])","A",1
132,1,248,249,0.7724099,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x,x]","-a d^{3/2} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+\frac{1}{3} a d \left(c^2 x^2+4\right) \sqrt{c^2 d x^2+d}+a d^{3/2} \log (x)+\frac{b d \sqrt{c^2 d x^2+d} \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{b d \sqrt{c^2 d x^2+d} \left(3 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)-c x \left(c^2 x^2+3\right)\right)}{9 \sqrt{c^2 x^2+1}}","\frac{1}{3} \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)+d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{2 d \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{b d \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+\frac{b d \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{4 b c d x \sqrt{c^2 d x^2+d}}{3 \sqrt{c^2 x^2+1}}-\frac{b c^3 d x^3 \sqrt{c^2 d x^2+d}}{9 \sqrt{c^2 x^2+1}}",1,"(a*d*(4 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/3 + (b*d*Sqrt[d + c^2*d*x^2]*(-(c*x*(3 + c^2*x^2)) + 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]))/(9*Sqrt[1 + c^2*x^2]) + 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^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2]","A",0
133,1,200,177,0.7891665,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^2,x]","\frac{1}{8} \left(12 a c d^{3/2} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+\frac{4 a d \left(c^2 x^2-2\right) \sqrt{c^2 d x^2+d}}{x}+\frac{4 b d \sqrt{c^2 d x^2+d} \left(-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right)}{x \sqrt{c^2 x^2+1}}+\frac{b c d \sqrt{c^2 d x^2+d} \left(2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)-\cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}\right)","\frac{3}{2} c^2 d x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{3 c d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b \sqrt{c^2 x^2+1}}-\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x}+\frac{b c d \log (x) \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{b c^3 d x^2 \sqrt{c^2 d x^2+d}}{4 \sqrt{c^2 x^2+1}}",1,"((4*a*d*(-2 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/x + (4*b*d*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]))/(x*Sqrt[1 + c^2*x^2]) + 12*a*c*d^(3/2)*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b*c*d*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/Sqrt[1 + c^2*x^2])/8","A",1
134,1,352,270,4.4628951,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^3,x]","-\frac{3}{2} a c^2 d^{3/2} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+\frac{3}{2} a c^2 d^{3/2} \log (x)+a \left(c^2 d-\frac{d}{2 x^2}\right) \sqrt{c^2 d x^2+d}+\frac{b c^2 d \sqrt{c^2 d x^2+d} \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{b c^2 d \sqrt{c^2 d x^2+d} \left(4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{8 \sqrt{c^2 x^2+1}}","\frac{3}{2} c^2 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{2 x^2}-\frac{3 c^2 d \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{3 b c^2 d \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{c^2 x^2+1}}+\frac{3 b c^2 d \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{c^2 x^2+1}}-\frac{b c d \sqrt{c^2 d x^2+d}}{2 x \sqrt{c^2 x^2+1}}-\frac{b c^3 d x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}",1,"a*(c^2*d - d/(2*x^2))*Sqrt[d + c^2*d*x^2] + (3*a*c^2*d^(3/2)*Log[x])/2 - (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^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (b*c^2*d*Sqrt[d + c^2*d*x^2]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(8*Sqrt[1 + c^2*x^2])","A",0
135,1,214,184,0.765505,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/x^4,x]","\frac{1}{6} d \left(-\frac{2 a \left(4 c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}{x^3}+6 a c^3 \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+\frac{3 b c^2 \sqrt{c^2 d x^2+d} \left(-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right)}{x \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \left(-2 c^3 x^3 \log (c x)+2 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)+c x\right)}{x^3 \sqrt{c^2 x^2+1}}\right)","-\frac{c^2 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{x}-\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^3}+\frac{c^3 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b \sqrt{c^2 x^2+1}}-\frac{b c d \sqrt{c^2 d x^2+d}}{6 x^2 \sqrt{c^2 x^2+1}}+\frac{4 b c^3 d \log (x) \sqrt{c^2 d x^2+d}}{3 \sqrt{c^2 x^2+1}}",1,"(d*((-2*a*(1 + 4*c^2*x^2)*Sqrt[d + c^2*d*x^2])/x^3 + (3*b*c^2*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]))/(x*Sqrt[1 + c^2*x^2]) - (b*Sqrt[d + c^2*d*x^2]*(c*x + 2*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x] - 2*c^3*x^3*Log[c*x]))/(x^3*Sqrt[1 + c^2*x^2]) + 6*a*c^3*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]]))/6","A",1
136,1,140,266,0.1914559,"\int x^3 \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{d^2 \sqrt{c^2 d x^2+d} \left(63 a \left(7 c^2 x^2-2\right) \left(c^2 x^2+1\right)^4+63 b \left(7 c^2 x^2-2\right) \left(c^2 x^2+1\right)^4 \sinh ^{-1}(c x)-b c x \left(49 c^8 x^8+171 c^6 x^6+189 c^4 x^4+21 c^2 x^2-126\right) \sqrt{c^2 x^2+1}\right)}{3969 c^4 \left(c^2 x^2+1\right)}","\frac{\left(c^2 d x^2+d\right)^{9/2} \left(a+b \sinh ^{-1}(c x)\right)}{9 c^4 d^2}-\frac{\left(c^2 d x^2+d\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{7 c^4 d}-\frac{b c d^2 x^5 \sqrt{c^2 d x^2+d}}{21 \sqrt{c^2 x^2+1}}-\frac{b d^2 x^3 \sqrt{c^2 d x^2+d}}{189 c \sqrt{c^2 x^2+1}}-\frac{b c^5 d^2 x^9 \sqrt{c^2 d x^2+d}}{81 \sqrt{c^2 x^2+1}}+\frac{2 b d^2 x \sqrt{c^2 d x^2+d}}{63 c^3 \sqrt{c^2 x^2+1}}-\frac{19 b c^3 d^2 x^7 \sqrt{c^2 d x^2+d}}{441 \sqrt{c^2 x^2+1}}",1,"(d^2*Sqrt[d + c^2*d*x^2]*(63*a*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2) - b*c*x*Sqrt[1 + c^2*x^2]*(-126 + 21*c^2*x^2 + 189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^8) + 63*b*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2)*ArcSinh[c*x]))/(3969*c^4*(1 + c^2*x^2))","A",1
137,1,388,337,0.9372896,"\int x^2 \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{d^2 \left(2880 a c x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-2880 a \sqrt{d} \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+9216 a c^7 x^7 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+26112 a c^5 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+22656 a c^3 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-1440 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2+24 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(-48 \sinh \left(2 \sinh ^{-1}(c x)\right)+24 \sinh \left(4 \sinh ^{-1}(c x)\right)+16 \sinh \left(6 \sinh ^{-1}(c x)\right)+3 \sinh \left(8 \sinh ^{-1}(c x)\right)\right)+576 b \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-144 b \sqrt{c^2 d x^2+d} \cosh \left(4 \sinh ^{-1}(c x)\right)-64 b \sqrt{c^2 d x^2+d} \cosh \left(6 \sinh ^{-1}(c x)\right)-9 b \sqrt{c^2 d x^2+d} \cosh \left(8 \sinh ^{-1}(c x)\right)\right)}{73728 c^3 \sqrt{c^2 x^2+1}}","\frac{5 d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{128 c^2}+\frac{5}{64} d^2 x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{8} x^3 \left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5}{48} d x^3 \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{5 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{256 b c^3 \sqrt{c^2 x^2+1}}-\frac{5 b d^2 x^2 \sqrt{c^2 d x^2+d}}{256 c \sqrt{c^2 x^2+1}}-\frac{59 b c d^2 x^4 \sqrt{c^2 d x^2+d}}{768 \sqrt{c^2 x^2+1}}-\frac{b c^5 d^2 x^8 \sqrt{c^2 d x^2+d}}{64 \sqrt{c^2 x^2+1}}-\frac{17 b c^3 d^2 x^6 \sqrt{c^2 d x^2+d}}{288 \sqrt{c^2 x^2+1}}",1,"(d^2*(2880*a*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 22656*a*c^3*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 26112*a*c^5*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 9216*a*c^7*x^7*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 1440*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2 + 576*b*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 144*b*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 64*b*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 9*b*Sqrt[d + c^2*d*x^2]*Cosh[8*ArcSinh[c*x]] - 2880*a*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 24*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(-48*Sinh[2*ArcSinh[c*x]] + 24*Sinh[4*ArcSinh[c*x]] + 16*Sinh[6*ArcSinh[c*x]] + 3*Sinh[8*ArcSinh[c*x]])))/(73728*c^3*Sqrt[1 + c^2*x^2])","A",1
138,1,112,193,0.151811,"\int x \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{d^2 \sqrt{c^2 d x^2+d} \left(35 a \left(c^2 x^2+1\right)^4+35 b \left(c^2 x^2+1\right)^4 \sinh ^{-1}(c x)-b c x \left(5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right) \sqrt{c^2 x^2+1}\right)}{245 c^2 \left(c^2 x^2+1\right)}","\frac{\left(c^2 d x^2+d\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{7 c^2 d}-\frac{b d^2 x \sqrt{c^2 d x^2+d}}{7 c \sqrt{c^2 x^2+1}}-\frac{b c d^2 x^3 \sqrt{c^2 d x^2+d}}{7 \sqrt{c^2 x^2+1}}-\frac{b c^5 d^2 x^7 \sqrt{c^2 d x^2+d}}{49 \sqrt{c^2 x^2+1}}-\frac{3 b c^3 d^2 x^5 \sqrt{c^2 d x^2+d}}{35 \sqrt{c^2 x^2+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^2*x^2]*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) + 35*b*(1 + c^2*x^2)^4*ArcSinh[c*x]))/(245*c^2*(1 + c^2*x^2))","A",1
139,1,317,254,0.7168792,"\int \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{d^2 \left(1584 a c x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+720 a \sqrt{d} \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+384 a c^5 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+1248 a c^3 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+360 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2+12 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(45 \sinh \left(2 \sinh ^{-1}(c x)\right)+9 \sinh \left(4 \sinh ^{-1}(c x)\right)+\sinh \left(6 \sinh ^{-1}(c x)\right)\right)-270 b \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-27 b \sqrt{c^2 d x^2+d} \cosh \left(4 \sinh ^{-1}(c x)\right)-2 b \sqrt{c^2 d x^2+d} \cosh \left(6 \sinh ^{-1}(c x)\right)\right)}{2304 c \sqrt{c^2 x^2+1}}","\frac{5}{16} d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{32 b c \sqrt{c^2 x^2+1}}+\frac{1}{6} x \left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5}{24} d x \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{25 b c d^2 x^2 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{b d^2 \left(c^2 x^2+1\right)^{5/2} \sqrt{c^2 d x^2+d}}{36 c}-\frac{5 b c^3 d^2 x^4 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}",1,"(d^2*(1584*a*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 1248*a*c^3*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 384*a*c^5*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 360*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2 - 270*b*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 27*b*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 2*b*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] + 720*a*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 12*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(45*Sinh[2*ArcSinh[c*x]] + 9*Sinh[4*ArcSinh[c*x]] + Sinh[6*ArcSinh[c*x]])))/(2304*c*Sqrt[1 + c^2*x^2])","A",1
140,1,361,329,1.2652585,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x,x]","-a d^{5/2} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+\frac{1}{15} a d^2 \left(3 c^4 x^4+11 c^2 x^2+23\right) \sqrt{c^2 d x^2+d}+a d^{5/2} \log (x)+\frac{b d^2 \sqrt{c^2 d x^2+d} \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}-\frac{8 b c d^2 x \left(c^2 x^2+3\right) \sqrt{c^2 d x^2+d}}{45 \sqrt{c^2 x^2+1}}+\frac{2}{3} b d^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)+\frac{1}{15} b d \left(3 c^2 x^2-2\right) \left(c^2 d x^2+d\right)^{3/2} \sinh ^{-1}(c x)-\frac{b c^3 d^2 x^3 \left(3 c^2 x^2+5\right) \sqrt{c^2 d x^2+d}}{75 \sqrt{c^2 x^2+1}}","d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{2 d^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{1}{5} \left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{3} d \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{b d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+\frac{b d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{23 b c d^2 x \sqrt{c^2 d x^2+d}}{15 \sqrt{c^2 x^2+1}}-\frac{b c^5 d^2 x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}-\frac{11 b c^3 d^2 x^3 \sqrt{c^2 d x^2+d}}{45 \sqrt{c^2 x^2+1}}",1,"(-8*b*c*d^2*x*(3 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(45*Sqrt[1 + c^2*x^2]) - (b*c^3*d^2*x^3*(5 + 3*c^2*x^2)*Sqrt[d + c^2*d*x^2])/(75*Sqrt[1 + c^2*x^2]) + (a*d^2*Sqrt[d + c^2*d*x^2]*(23 + 11*c^2*x^2 + 3*c^4*x^4))/15 + (2*b*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/3 + (b*d*(-2 + 3*c^2*x^2)*(d + c^2*d*x^2)^(3/2)*ArcSinh[c*x])/15 + 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^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2]","A",0
141,1,270,257,1.4693109,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^2,x]","\frac{1}{128} d^2 \left(240 a c \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+\frac{16 a \left(2 c^4 x^4+9 c^2 x^2-8\right) \sqrt{c^2 d x^2+d}}{x}+\frac{64 b \sqrt{c^2 d x^2+d} \left(-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right)}{x \sqrt{c^2 x^2+1}}+\frac{32 b c \sqrt{c^2 d x^2+d} \left(2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)-\cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}-\frac{b c \sqrt{c^2 d x^2+d} \left(8 \sinh ^{-1}(c x)^2-4 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}\right)","\frac{15}{8} c^2 d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{15 c d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b \sqrt{c^2 x^2+1}}+\frac{5}{4} c^2 d x \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x}+\frac{b c d^2 \log (x) \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{b c^5 d^2 x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{9 b c^3 d^2 x^2 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+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 + (64*b*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]))/(x*Sqrt[1 + c^2*x^2]) + 240*a*c*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (32*b*c*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/Sqrt[1 + c^2*x^2] - (b*c*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2]))/128","A",1
142,1,467,355,6.7446359,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3,x]","-\frac{5}{2} a c^2 d^{5/2} \log \left(\sqrt{d} \sqrt{d \left(c^2 x^2+1\right)}+d\right)+\frac{5}{2} a c^2 d^{5/2} \log (x)+\sqrt{d \left(c^2 x^2+1\right)} \left(\frac{1}{3} a c^4 d^2 x^2+\frac{7}{3} a c^2 d^2-\frac{a d^2}{2 x^2}\right)+\frac{2 b c^2 d^2 \sqrt{d \left(c^2 x^2+1\right)} \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{b c^2 d^2 \sqrt{d \left(c^2 x^2+1\right)} \left(4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{8 \sqrt{c^2 x^2+1}}+b c^2 d^2 \left(\frac{1}{3} \left(c^2 x^2+1\right) \sqrt{d \left(c^2 x^2+1\right)} \sinh ^{-1}(c x)-\frac{c x \left(c^2 x^2+3\right) \sqrt{d \left(c^2 x^2+1\right)}}{9 \sqrt{c^2 x^2+1}}\right)","\frac{5}{2} c^2 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{5 c^2 d^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{5}{6} c^2 d \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{2 x^2}-\frac{5 b c^2 d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{c^2 x^2+1}}+\frac{5 b c^2 d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{c^2 x^2+1}}-\frac{b c d^2 \sqrt{c^2 d x^2+d}}{2 x \sqrt{c^2 x^2+1}}-\frac{b c^5 d^2 x^3 \sqrt{c^2 d x^2+d}}{9 \sqrt{c^2 x^2+1}}-\frac{7 b c^3 d^2 x \sqrt{c^2 d x^2+d}}{3 \sqrt{c^2 x^2+1}}",1,"Sqrt[d*(1 + c^2*x^2)]*((7*a*c^2*d^2)/3 - (a*d^2)/(2*x^2) + (a*c^4*d^2*x^2)/3) + b*c^2*d^2*(-1/9*(c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/Sqrt[1 + c^2*x^2] + ((1 + c^2*x^2)*Sqrt[d*(1 + c^2*x^2)]*ArcSinh[c*x])/3) + (5*a*c^2*d^(5/2)*Log[x])/2 - (5*a*c^2*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/2 + (2*b*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (b*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(8*Sqrt[1 + c^2*x^2])","A",1
143,1,287,266,0.9709599,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^4,x]","\frac{d^2 \left(4 a \sqrt{c^2 x^2+1} \left(3 c^4 x^4-14 c^2 x^2-2\right) \sqrt{c^2 d x^2+d}+60 a c^3 \sqrt{d} x^3 \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+24 b c^2 x^2 \sqrt{c^2 d x^2+d} \left(-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right)+4 b \sqrt{c^2 d x^2+d} \left(2 c^3 x^3 \log (c x)-2 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)-c x\right)-3 b c^3 x^3 \sqrt{c^2 d x^2+d} \left(\cosh \left(2 \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)\right)\right)}{24 x^3 \sqrt{c^2 x^2+1}}","-\frac{5 c^2 d \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 x}-\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^3}+\frac{5}{2} c^4 d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5 c^3 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b \sqrt{c^2 x^2+1}}-\frac{b c d^2 \sqrt{c^2 d x^2+d}}{6 x^2 \sqrt{c^2 x^2+1}}-\frac{b c^5 d^2 x^2 \sqrt{c^2 d x^2+d}}{4 \sqrt{c^2 x^2+1}}+\frac{7 b c^3 d^2 \log (x) \sqrt{c^2 d x^2+d}}{3 \sqrt{c^2 x^2+1}}",1,"(d^2*(4*a*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(-2 - 14*c^2*x^2 + 3*c^4*x^4) + 24*b*c^2*x^2*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]) + 4*b*Sqrt[d + c^2*d*x^2]*(-(c*x) - 2*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x] + 2*c^3*x^3*Log[c*x]) + 60*a*c^3*Sqrt[d]*x^3*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 3*b*c^3*x^3*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]]))))/(24*x^3*Sqrt[1 + c^2*x^2])","A",1
144,1,28,32,0.0131225,"\int \sqrt{1+x^2} \sinh ^{-1}(x) \, dx","Integrate[Sqrt[1 + x^2]*ArcSinh[x],x]","\frac{1}{4} \left(-x^2+2 \sqrt{x^2+1} x \sinh ^{-1}(x)+\sinh ^{-1}(x)^2\right)","-\frac{x^2}{4}+\frac{1}{2} \sqrt{x^2+1} x \sinh ^{-1}(x)+\frac{1}{4} \sinh ^{-1}(x)^2",1,"(-x^2 + 2*x*Sqrt[1 + x^2]*ArcSinh[x] + ArcSinh[x]^2)/4","A",1
145,1,119,215,0.1683959,"\int \frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^5*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]","\frac{15 a \left(3 c^6 x^6-c^4 x^4+4 c^2 x^2+8\right)+b c x \sqrt{c^2 x^2+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) \sinh ^{-1}(c x)}{225 c^6 \sqrt{c^2 d x^2+d}}","\frac{x^4 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{5 c^2 d}+\frac{8 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{15 c^6 d}-\frac{4 x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{15 c^4 d}-\frac{b x^5 \sqrt{c^2 x^2+1}}{25 c \sqrt{c^2 d x^2+d}}-\frac{8 b x \sqrt{c^2 x^2+1}}{15 c^5 \sqrt{c^2 d x^2+d}}+\frac{4 b x^3 \sqrt{c^2 x^2+1}}{45 c^3 \sqrt{c^2 d x^2+d}}",1,"(b*c*x*Sqrt[1 + c^2*x^2]*(-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)*ArcSinh[c*x])/(225*c^6*Sqrt[d + c^2*d*x^2])","A",1
146,1,151,192,0.6158178,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^4*(a + b*ArcSinh[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{c^2 d x^2+d}}{d}+\frac{48 a \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)}{\sqrt{d}}+\frac{b \sqrt{c^2 x^2+1} \left(4 \sinh ^{-1}(c x) \left(6 \sinh ^{-1}(c x)-8 \sinh \left(2 \sinh ^{-1}(c x)\right)+\sinh \left(4 \sinh ^{-1}(c x)\right)\right)+16 \cosh \left(2 \sinh ^{-1}(c x)\right)-\cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 d x^2+d}}}{128 c^5}","\frac{x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{4 c^2 d}+\frac{3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c^5 \sqrt{c^2 d x^2+d}}-\frac{3 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{8 c^4 d}-\frac{b x^4 \sqrt{c^2 x^2+1}}{16 c \sqrt{c^2 d x^2+d}}+\frac{3 b x^2 \sqrt{c^2 x^2+1}}{16 c^3 \sqrt{c^2 d x^2+d}}",1,"((16*a*c*x*(-3 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2])/d + (48*a*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/Sqrt[d] + (b*Sqrt[1 + c^2*x^2]*(16*Cosh[2*ArcSinh[c*x]] - Cosh[4*ArcSinh[c*x]] + 4*ArcSinh[c*x]*(6*ArcSinh[c*x] - 8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])))/Sqrt[d + c^2*d*x^2])/(128*c^5)","A",1
147,1,93,142,0.1297555,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]","\frac{3 a \left(c^4 x^4-c^2 x^2-2\right)+b c x \sqrt{c^2 x^2+1} \left(6-c^2 x^2\right)+3 b \left(c^4 x^4-c^2 x^2-2\right) \sinh ^{-1}(c x)}{9 c^4 \sqrt{c^2 d x^2+d}}","\frac{x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^2 d}-\frac{2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^4 d}-\frac{b x^3 \sqrt{c^2 x^2+1}}{9 c \sqrt{c^2 d x^2+d}}+\frac{2 b x \sqrt{c^2 x^2+1}}{3 c^3 \sqrt{c^2 d x^2+d}}",1,"(b*c*x*(6 - c^2*x^2)*Sqrt[1 + 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)*ArcSinh[c*x])/(9*c^4*Sqrt[d + c^2*d*x^2])","A",1
148,1,121,119,0.7293239,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]","-\frac{-\frac{4 a c x \sqrt{c^2 d x^2+d}}{d}+\frac{4 a \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)}{\sqrt{d}}+\frac{b \sqrt{c^2 x^2+1} \left(2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-\sinh \left(2 \sinh ^{-1}(c x)\right)\right)+\cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 d x^2+d}}}{8 c^3}","\frac{x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{2 c^2 d}-\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c^3 \sqrt{c^2 d x^2+d}}-\frac{b x^2 \sqrt{c^2 x^2+1}}{4 c \sqrt{c^2 d x^2+d}}",1,"-1/8*((-4*a*c*x*Sqrt[d + c^2*d*x^2])/d + (4*a*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/Sqrt[d] + (b*Sqrt[1 + c^2*x^2]*(Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] - Sinh[2*ArcSinh[c*x]])))/Sqrt[d + c^2*d*x^2])/c^3","A",1
149,1,74,64,0.1035136,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]","\frac{\sqrt{c^2 d x^2+d} \left(a \sqrt{c^2 x^2+1}+b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-b c x\right)}{c^2 d \sqrt{c^2 x^2+1}}","\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d}-\frac{b x \sqrt{c^2 x^2+1}}{c \sqrt{c^2 d x^2+d}}",1,"(Sqrt[d + c^2*d*x^2]*(-(b*c*x) + a*Sqrt[1 + c^2*x^2] + b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]))/(c^2*d*Sqrt[1 + c^2*x^2])","A",1
150,1,48,47,0.0513092,"\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/Sqrt[d + c^2*d*x^2],x]","\frac{\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \left(2 a+b \sinh ^{-1}(c x)\right)}{2 c \sqrt{c^2 d x^2+d}}","\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c \sqrt{c^2 d x^2+d}}",1,"(Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*(2*a + b*ArcSinh[c*x]))/(2*c*Sqrt[d + c^2*d*x^2])","A",1
151,1,129,122,0.3325598,"\int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x*Sqrt[d + c^2*d*x^2]),x]","-\frac{a \log \left(\sqrt{d} \sqrt{d \left(c^2 x^2+1\right)}+d\right)}{\sqrt{d}}+\frac{a \log (x)}{\sqrt{d}}+\frac{b \sqrt{c^2 x^2+1} \left(\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+\sinh ^{-1}(c x) \left(\log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)\right)}{\sqrt{d \left(c^2 x^2+1\right)}}","-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 d x^2+d}}-\frac{b \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 d x^2+d}}+\frac{b \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 d x^2+d}}",1,"(a*Log[x])/Sqrt[d] - (a*Log[d + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/Sqrt[d] + (b*Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]) + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[d*(1 + c^2*x^2)]","A",0
152,1,67,63,0.1632092,"\int \frac{a+b \sinh ^{-1}(c x)}{x^2 \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^2*Sqrt[d + c^2*d*x^2]),x]","\frac{b c \log (x) \sqrt{d \left(c^2 x^2+1\right)}}{d \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{d x}","\frac{b c \sqrt{c^2 x^2+1} \log (x)}{\sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{d x}",1,"-((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(d*x)) + (b*c*Sqrt[d*(1 + c^2*x^2)]*Log[x])/(d*Sqrt[1 + c^2*x^2])","A",1
153,1,229,203,3.0182559,"\int \frac{a+b \sinh ^{-1}(c x)}{x^3 \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^3*Sqrt[d + c^2*d*x^2]),x]","\frac{-\frac{4 a \sqrt{c^2 d x^2+d}}{x^2}+4 a c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)-4 a c^2 \sqrt{d} \log (x)+\frac{b c^2 d^2 \left(c^2 x^2+1\right)^{3/2} \left(-4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{\left(c^2 d x^2+d\right)^{3/2}}}{8 d}","-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{2 d x^2}+\frac{c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 d x^2+d}}+\frac{b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{c^2 d x^2+d}}-\frac{b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{2 \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1}}{2 x \sqrt{c^2 d x^2+d}}",1,"((-4*a*Sqrt[d + c^2*d*x^2])/x^2 - 4*a*c^2*Sqrt[d]*Log[x] + 4*a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b*c^2*d^2*(1 + c^2*x^2)^(3/2)*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 4*PolyLog[2, -E^(-ArcSinh[c*x])] + 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(d + c^2*d*x^2)^(3/2))/(8*d)","A",0
154,1,135,141,0.2039856,"\int \frac{a+b \sinh ^{-1}(c x)}{x^4 \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^4*Sqrt[d + c^2*d*x^2]),x]","\frac{2 a \left(2 c^4 x^4+c^2 x^2-1\right)+b c x \sqrt{c^2 x^2+1} \left(6 c^2 x^2-1\right)+2 b \left(2 c^4 x^4+c^2 x^2-1\right) \sinh ^{-1}(c x)}{6 x^3 \sqrt{c^2 d x^2+d}}-\frac{2 b c^3 \log (x) \sqrt{d \left(c^2 x^2+1\right)}}{3 d \sqrt{c^2 x^2+1}}","\frac{2 c^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 d x}-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 d x^3}-\frac{b c \sqrt{c^2 x^2+1}}{6 x^2 \sqrt{c^2 d x^2+d}}-\frac{2 b c^3 \sqrt{c^2 x^2+1} \log (x)}{3 \sqrt{c^2 d x^2+d}}",1,"(b*c*x*Sqrt[1 + c^2*x^2]*(-1 + 6*c^2*x^2) + 2*a*(-1 + c^2*x^2 + 2*c^4*x^4) + 2*b*(-1 + c^2*x^2 + 2*c^4*x^4)*ArcSinh[c*x])/(6*x^3*Sqrt[d + c^2*d*x^2]) - (2*b*c^3*Sqrt[d*(1 + c^2*x^2)]*Log[x])/(3*d*Sqrt[1 + c^2*x^2])","A",1
155,1,148,212,0.2659283,"\int \frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(3/2),x]","\frac{\sqrt{c^2 d x^2+d} \left(3 a \left(c^4 x^4-4 c^2 x^2-8\right)+b c x \sqrt{c^2 x^2+1} \left(15-c^2 x^2\right)+3 b \left(c^4 x^4-4 c^2 x^2-8\right) \sinh ^{-1}(c x)\right)}{9 c^6 d^2 \left(c^2 x^2+1\right)}+\frac{b \sqrt{d \left(c^2 x^2+1\right)} \tan ^{-1}(c x)}{c^6 d^2 \sqrt{c^2 x^2+1}}","\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^6 d^3}-\frac{2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{c^6 d^2}-\frac{a+b \sinh ^{-1}(c x)}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{b \sqrt{c^2 d x^2+d} \tan ^{-1}(c x)}{c^6 d^2 \sqrt{c^2 x^2+1}}+\frac{5 b x \sqrt{c^2 d x^2+d}}{3 c^5 d^2 \sqrt{c^2 x^2+1}}-\frac{b x^3 \sqrt{c^2 d x^2+d}}{9 c^3 d^2 \sqrt{c^2 x^2+1}}",1,"(Sqrt[d + c^2*d*x^2]*(b*c*x*(15 - c^2*x^2)*Sqrt[1 + c^2*x^2] + 3*a*(-8 - 4*c^2*x^2 + c^4*x^4) + 3*b*(-8 - 4*c^2*x^2 + c^4*x^4)*ArcSinh[c*x]))/(9*c^6*d^2*(1 + c^2*x^2)) + (b*Sqrt[d*(1 + c^2*x^2)]*ArcTan[c*x])/(c^6*d^2*Sqrt[1 + c^2*x^2])","A",1
156,1,161,206,0.4677439,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(3/2),x]","\frac{4 a c \sqrt{d} x \left(c^2 x^2+3\right)-12 a \sqrt{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+b \sqrt{d} \left(8 c x \sinh ^{-1}(c x)-\sqrt{c^2 x^2+1} \left(4 \log \left(c^2 x^2+1\right)+6 \sinh ^{-1}(c x)^2-2 \sinh \left(2 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\cosh \left(2 \sinh ^{-1}(c x)\right)\right)\right)}{8 c^5 d^{3/2} \sqrt{c^2 d x^2+d}}","-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c^5 d \sqrt{c^2 d x^2+d}}+\frac{3 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{2 c^4 d^2}-\frac{b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)}{2 c^5 d \sqrt{c^2 d x^2+d}}-\frac{b x^2 \sqrt{c^2 x^2+1}}{4 c^3 d \sqrt{c^2 d x^2+d}}",1,"(4*a*c*Sqrt[d]*x*(3 + c^2*x^2) - 12*a*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + b*Sqrt[d]*(8*c*x*ArcSinh[c*x] - Sqrt[1 + c^2*x^2]*(6*ArcSinh[c*x]^2 + Cosh[2*ArcSinh[c*x]] + 4*Log[1 + c^2*x^2] - 2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]])))/(8*c^5*d^(3/2)*Sqrt[d + c^2*d*x^2])","A",1
157,1,143,136,0.2380812,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(3/2),x]","\frac{\sqrt{c^2 d x^2+d} \left(a \sqrt{c^2 x^2+1} \left(c^2 x^2+2\right)-b \left(c^3 x^3+c x\right)+b \sqrt{c^2 x^2+1} \left(c^2 x^2+2\right) \sinh ^{-1}(c x)\right)}{c^4 d^2 \left(c^2 x^2+1\right)^{3/2}}-\frac{b \sqrt{d \left(c^2 x^2+1\right)} \tan ^{-1}(c x)}{c^4 d^2 \sqrt{c^2 x^2+1}}","\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{c^4 d^2}+\frac{a+b \sinh ^{-1}(c x)}{c^4 d \sqrt{c^2 d x^2+d}}-\frac{b \sqrt{c^2 d x^2+d} \tan ^{-1}(c x)}{c^4 d^2 \sqrt{c^2 x^2+1}}-\frac{b x \sqrt{c^2 d x^2+d}}{c^3 d^2 \sqrt{c^2 x^2+1}}",1,"(Sqrt[d + c^2*d*x^2]*(a*Sqrt[1 + c^2*x^2]*(2 + c^2*x^2) - b*(c*x + c^3*x^3) + b*Sqrt[1 + c^2*x^2]*(2 + c^2*x^2)*ArcSinh[c*x]))/(c^4*d^2*(1 + c^2*x^2)^(3/2)) - (b*Sqrt[d*(1 + c^2*x^2)]*ArcTan[c*x])/(c^4*d^2*Sqrt[1 + c^2*x^2])","A",1
158,1,146,130,0.2010063,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(3/2),x]","-\frac{a x \sqrt{d \left(c^2 x^2+1\right)}}{c^2 d^2 \left(c^2 x^2+1\right)}+\frac{a \log \left(\sqrt{d} \sqrt{d \left(c^2 x^2+1\right)}+c d x\right)}{c^3 d^{3/2}}+\frac{b \left(\sqrt{c^2 x^2+1} \left(2 \log \left(\sqrt{c^2 x^2+1}\right)+\sinh ^{-1}(c x)^2\right)-2 c x \sinh ^{-1}(c x)\right)}{2 c^3 d \sqrt{d \left(c^2 x^2+1\right)}}","-\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d \sqrt{c^2 d x^2+d}}+\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c^3 d \sqrt{c^2 d x^2+d}}+\frac{b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)}{2 c^3 d \sqrt{c^2 d x^2+d}}",1,"-((a*x*Sqrt[d*(1 + c^2*x^2)])/(c^2*d^2*(1 + c^2*x^2))) + (b*(-2*c*x*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]^2 + 2*Log[Sqrt[1 + c^2*x^2]])))/(2*c^3*d*Sqrt[d*(1 + c^2*x^2)]) + (a*Log[c*d*x + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/(c^3*d^(3/2))","A",1
159,1,82,70,0.1623469,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(3/2),x]","\frac{b \sqrt{d \left(c^2 x^2+1\right)} \tan ^{-1}(c x)}{c^2 d^2 \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d^2 \left(c^2 x^2+1\right)}","\frac{b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{c^2 d \sqrt{c^2 d x^2+d}}",1,"-((Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(c^2*d^2*(1 + c^2*x^2))) + (b*Sqrt[d*(1 + c^2*x^2)]*ArcTan[c*x])/(c^2*d^2*Sqrt[1 + c^2*x^2])","A",1
160,1,100,76,0.1722455,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(3/2),x]","\frac{\sqrt{c^2 d x^2+d} \left(2 a c x \sqrt{c^2 x^2+1}-\left(b c^2 x^2+b\right) \log \left(c^2 x^2+1\right)+2 b c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)\right)}{2 c d^2 \left(c^2 x^2+1\right)^{3/2}}","\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}-\frac{b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)}{2 c d \sqrt{c^2 d x^2+d}}",1,"(Sqrt[d + c^2*d*x^2]*(2*a*c*x*Sqrt[1 + c^2*x^2] + 2*b*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - (b + b*c^2*x^2)*Log[1 + c^2*x^2]))/(2*c*d^2*(1 + c^2*x^2)^(3/2))","A",1
161,1,231,194,0.8504234,"\int \frac{a+b \sinh ^{-1}(c x)}{x \left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)^(3/2)),x]","\frac{\frac{a \sqrt{c^2 d x^2+d}}{c^2 x^2+1}-a \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+a \sqrt{d} \log (x)+\frac{b d \left(\sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-2 \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\sinh ^{-1}(c x)\right)}{\sqrt{c^2 d x^2+d}}}{d^2}","\frac{a+b \sinh ^{-1}(c x)}{d \sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}-\frac{b \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}+\frac{b \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}-\frac{b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{d \sqrt{c^2 d x^2+d}}",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]] + (b*d*(ArcSinh[c*x] - 2*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[d + c^2*d*x^2])/d^2","A",0
162,1,163,143,0.284066,"\int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)^(3/2)),x]","-\frac{\sqrt{c^2 d x^2+d} \left(4 a c^2 x^2 \sqrt{c^2 x^2+1}+2 a \sqrt{c^2 x^2+1}+b c x \left(c^2 x^2+1\right) \log \left(\frac{1}{c^2 x^2}+1\right)-2 b c x \log \left(c^2 x^2+1\right)+2 b \sqrt{c^2 x^2+1} \left(2 c^2 x^2+1\right) \sinh ^{-1}(c x)-2 b c^3 x^3 \log \left(c^2 x^2+1\right)\right)}{2 d^2 x \left(c^2 x^2+1\right)^{3/2}}","-\frac{2 c^2 x \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{d x \sqrt{c^2 d x^2+d}}+\frac{b c \log (x) \sqrt{c^2 d x^2+d}}{d^2 \sqrt{c^2 x^2+1}}+\frac{b c \sqrt{c^2 d x^2+d} \log \left(c^2 x^2+1\right)}{2 d^2 \sqrt{c^2 x^2+1}}",1,"-1/2*(Sqrt[d + c^2*d*x^2]*(2*a*Sqrt[1 + c^2*x^2] + 4*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 2*b*Sqrt[1 + c^2*x^2]*(1 + 2*c^2*x^2)*ArcSinh[c*x] + b*c*x*(1 + c^2*x^2)*Log[1 + 1/(c^2*x^2)] - 2*b*c*x*Log[1 + c^2*x^2] - 2*b*c^3*x^3*Log[1 + c^2*x^2]))/(d^2*x*(1 + c^2*x^2)^(3/2))","A",1
163,1,381,287,6.3737556,"\int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^(3/2)),x]","\frac{3 a c^2 \log \left(\sqrt{d} \sqrt{d \left(c^2 x^2+1\right)}+d\right)}{2 d^{3/2}}-\frac{3 a c^2 \log (x)}{2 d^{3/2}}+\sqrt{d \left(c^2 x^2+1\right)} \left(-\frac{a c^2}{d^2 \left(c^2 x^2+1\right)}-\frac{a}{2 d^2 x^2}\right)+\frac{b c^2 \left(-12 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+12 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-12 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+12 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \sqrt{c^2 x^2+1} \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \sqrt{c^2 x^2+1} \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+16 \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-8 \sinh ^{-1}(c x)\right)}{8 d \sqrt{d \left(c^2 x^2+1\right)}}","-\frac{3 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 d \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \sqrt{c^2 d x^2+d}}+\frac{3 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}+\frac{3 b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{2 d \sqrt{c^2 d x^2+d}}-\frac{3 b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{2 d \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1}}{2 d x \sqrt{c^2 d x^2+d}}+\frac{b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{d \sqrt{c^2 d x^2+d}}",1,"Sqrt[d*(1 + c^2*x^2)]*(-1/2*a/(d^2*x^2) - (a*c^2)/(d^2*(1 + c^2*x^2))) - (3*a*c^2*Log[x])/(2*d^(3/2)) + (3*a*c^2*Log[d + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/(2*d^(3/2)) + (b*c^2*(-8*ArcSinh[c*x] + 16*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] - 2*Sqrt[1 + c^2*x^2]*Coth[ArcSinh[c*x]/2] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 12*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Sqrt[1 + c^2*x^2]*Tanh[ArcSinh[c*x]/2]))/(8*d*Sqrt[d*(1 + c^2*x^2)])","A",0
164,1,216,228,0.3156267,"\int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^(3/2)),x]","\frac{\sqrt{c^2 d x^2+d} \left(8 a c^2 x^2 \sqrt{c^2 x^2+1}-2 a \sqrt{c^2 x^2+1}+16 a c^4 x^4 \sqrt{c^2 x^2+1}-b c^3 x^3-8 b c^5 x^5 \log \left(c^2 x^2+1\right)+2 b \sqrt{c^2 x^2+1} \left(8 c^4 x^4+4 c^2 x^2-1\right) \sinh ^{-1}(c x)+5 b c^3 x^3 \left(c^2 x^2+1\right) \log \left(\frac{1}{c^2 x^2}+1\right)-8 b c^3 x^3 \log \left(c^2 x^2+1\right)-b c x\right)}{6 d^2 x^3 \left(c^2 x^2+1\right)^{3/2}}","\frac{4 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 d x \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left(a+b \sinh ^{-1}(c x)\right)}{3 d \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 d x^2+d}}{6 d^2 x^2 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 \log (x) \sqrt{c^2 d x^2+d}}{3 d^2 \sqrt{c^2 x^2+1}}-\frac{b c^3 \sqrt{c^2 d x^2+d} \log \left(c^2 x^2+1\right)}{2 d^2 \sqrt{c^2 x^2+1}}",1,"(Sqrt[d + c^2*d*x^2]*(-(b*c*x) - b*c^3*x^3 - 2*a*Sqrt[1 + c^2*x^2] + 8*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 16*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 2*b*Sqrt[1 + c^2*x^2]*(-1 + 4*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x] + 5*b*c^3*x^3*(1 + c^2*x^2)*Log[1 + 1/(c^2*x^2)] - 8*b*c^3*x^3*Log[1 + c^2*x^2] - 8*b*c^5*x^5*Log[1 + c^2*x^2]))/(6*d^2*x^3*(1 + c^2*x^2)^(3/2))","A",1
165,1,222,281,1.0044861,"\int \frac{x^6 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^6*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]","\frac{-60 a \sqrt{d} \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+4 a c d x \left(3 c^4 x^4+20 c^2 x^2+15\right)+b d \left(-30 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)^2-\sqrt{c^2 x^2+1} \left(6 c^4 x^4+9 c^2 x^2+28 \left(c^2 x^2+1\right) \log \left(c^2 x^2+1\right)+7\right)+4 c x \left(3 c^4 x^4+20 c^2 x^2+15\right) \sinh ^{-1}(c x)\right)}{24 c^7 d^3 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}","-\frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)}{3 c^2 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{5 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c^7 d^2 \sqrt{c^2 d x^2+d}}+\frac{5 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{2 c^6 d^3}-\frac{5 x^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}-\frac{b}{6 c^7 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{7 b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)}{6 c^7 d^2 \sqrt{c^2 d x^2+d}}-\frac{b x^2 \sqrt{c^2 x^2+1}}{4 c^5 d^2 \sqrt{c^2 d x^2+d}}",1,"(4*a*c*d*x*(15 + 20*c^2*x^2 + 3*c^4*x^4) + b*d*(4*c*x*(15 + 20*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x] - 30*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2 - Sqrt[1 + c^2*x^2]*(7 + 9*c^2*x^2 + 6*c^4*x^4 + 28*(1 + c^2*x^2)*Log[1 + c^2*x^2])) - 60*a*Sqrt[d]*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(24*c^7*d^3*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])","A",1
166,1,154,210,0.27133,"\int \frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]","\frac{\sqrt{c^2 d x^2+d} \left(2 a \left(3 c^4 x^4+12 c^2 x^2+8\right)-b c x \sqrt{c^2 x^2+1} \left(6 c^2 x^2+5\right)+2 b \left(3 c^4 x^4+12 c^2 x^2+8\right) \sinh ^{-1}(c x)\right)}{6 c^6 d^3 \left(c^2 x^2+1\right)^2}-\frac{11 b \sqrt{d \left(c^2 x^2+1\right)} \tan ^{-1}(c x)}{6 c^6 d^3 \sqrt{c^2 x^2+1}}","\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{c^6 d^3}+\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{a+b \sinh ^{-1}(c x)}{3 c^6 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{11 b \sqrt{c^2 d x^2+d} \tan ^{-1}(c x)}{6 c^6 d^3 \sqrt{c^2 x^2+1}}-\frac{b x \sqrt{c^2 d x^2+d}}{c^5 d^3 \sqrt{c^2 x^2+1}}+\frac{b x \sqrt{c^2 d x^2+d}}{6 c^5 d^3 \left(c^2 x^2+1\right)^{3/2}}",1,"(Sqrt[d + c^2*d*x^2]*(-(b*c*x*Sqrt[1 + c^2*x^2]*(5 + 6*c^2*x^2)) + 2*a*(8 + 12*c^2*x^2 + 3*c^4*x^4) + 2*b*(8 + 12*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]))/(6*c^6*d^3*(1 + c^2*x^2)^2) - (11*b*Sqrt[d*(1 + c^2*x^2)]*ArcTan[c*x])/(6*c^6*d^3*Sqrt[1 + c^2*x^2])","A",1
167,1,191,203,0.5308556,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]","\frac{-2 a c \sqrt{d} x \left(4 c^2 x^2+3\right)+6 a \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+b \sqrt{d} \left(\sqrt{c^2 x^2+1}-8 c x \left(c^2 x^2+1\right) \sinh ^{-1}(c x)+\left(c^2 x^2+1\right)^{3/2} \left(4 \log \left(c^2 x^2+1\right)+3 \sinh ^{-1}(c x)^2\right)+2 c x \sinh ^{-1}(c x)\right)}{6 c^5 d^{5/2} \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}","-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 c^2 d \left(c^2 d x^2+d\right)^{3/2}}+\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{b}{6 c^5 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{2 b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}",1,"(-2*a*c*Sqrt[d]*x*(3 + 4*c^2*x^2) + b*Sqrt[d]*(Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x] - 8*c*x*(1 + c^2*x^2)*ArcSinh[c*x] + (1 + c^2*x^2)^(3/2)*(3*ArcSinh[c*x]^2 + 4*Log[1 + c^2*x^2])) + 6*a*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(6*c^5*d^(5/2)*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])","A",1
168,1,151,144,0.2538393,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]","\frac{5 b \sqrt{d \left(c^2 x^2+1\right)} \tan ^{-1}(c x)}{6 c^4 d^3 \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left(2 a \sqrt{c^2 x^2+1} \left(3 c^2 x^2+2\right)+b \left(c^3 x^3+c x\right)+2 b \sqrt{c^2 x^2+1} \left(3 c^2 x^2+2\right) \sinh ^{-1}(c x)\right)}{6 c^4 d^3 \left(c^2 x^2+1\right)^{5/2}}","-\frac{a+b \sinh ^{-1}(c x)}{c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{a+b \sinh ^{-1}(c x)}{3 c^4 d \left(c^2 d x^2+d\right)^{3/2}}+\frac{5 b \sqrt{c^2 d x^2+d} \tan ^{-1}(c x)}{6 c^4 d^3 \sqrt{c^2 x^2+1}}-\frac{b x \sqrt{c^2 d x^2+d}}{6 c^3 d^3 \left(c^2 x^2+1\right)^{3/2}}",1,"-1/6*(Sqrt[d + c^2*d*x^2]*(2*a*Sqrt[1 + c^2*x^2]*(2 + 3*c^2*x^2) + b*(c*x + c^3*x^3) + 2*b*Sqrt[1 + c^2*x^2]*(2 + 3*c^2*x^2)*ArcSinh[c*x]))/(c^4*d^3*(1 + c^2*x^2)^(5/2)) + (5*b*Sqrt[d*(1 + c^2*x^2)]*ArcTan[c*x])/(6*c^4*d^3*Sqrt[1 + c^2*x^2])","A",1
169,1,118,119,0.198921,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]","-\frac{\sqrt{c^2 d x^2+d} \left(-2 a c^3 x^3 \sqrt{c^2 x^2+1}+b c^2 x^2+b \left(c^2 x^2+1\right)^2 \log \left(c^2 x^2+1\right)-2 b c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+b\right)}{6 c^3 d^3 \left(c^2 x^2+1\right)^{5/2}}","\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{b}{6 c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)}{6 c^3 d^2 \sqrt{c^2 d x^2+d}}",1,"-1/6*(Sqrt[d + c^2*d*x^2]*(b + b*c^2*x^2 - 2*a*c^3*x^3*Sqrt[1 + c^2*x^2] - 2*b*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + b*(1 + c^2*x^2)^2*Log[1 + c^2*x^2]))/(c^3*d^3*(1 + c^2*x^2)^(5/2))","A",1
170,1,130,114,0.1882144,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]","\frac{\sqrt{c^2 d x^2+d} \left(-2 a \sqrt{c^2 x^2+1}+b c^3 x^3-2 b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+b c x\right)}{6 c^2 d^3 \left(c^2 x^2+1\right)^{5/2}}+\frac{b \sqrt{d \left(c^2 x^2+1\right)} \tan ^{-1}(c x)}{6 c^2 d^3 \sqrt{c^2 x^2+1}}","-\frac{a+b \sinh ^{-1}(c x)}{3 c^2 d \left(c^2 d x^2+d\right)^{3/2}}+\frac{b x}{6 c d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{6 c^2 d^2 \sqrt{c^2 d x^2+d}}",1,"(Sqrt[d + c^2*d*x^2]*(b*c*x + b*c^3*x^3 - 2*a*Sqrt[1 + c^2*x^2] - 2*b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]))/(6*c^2*d^3*(1 + c^2*x^2)^(5/2)) + (b*Sqrt[d*(1 + c^2*x^2)]*ArcTan[c*x])/(6*c^2*d^3*Sqrt[1 + c^2*x^2])","A",1
171,1,143,147,0.1971763,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + c^2*d*x^2)^(5/2),x]","\frac{\sqrt{c^2 d x^2+d} \left(6 a c x \sqrt{c^2 x^2+1}+4 a c^3 x^3 \sqrt{c^2 x^2+1}+b c^2 x^2-2 b \left(c^2 x^2+1\right)^2 \log \left(c^2 x^2+1\right)+2 b c x \sqrt{c^2 x^2+1} \left(2 c^2 x^2+3\right) \sinh ^{-1}(c x)+b\right)}{6 c d^3 \left(c^2 x^2+1\right)^{5/2}}","\frac{2 x \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{3 d \left(c^2 d x^2+d\right)^{3/2}}+\frac{b}{6 c d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)}{3 c d^2 \sqrt{c^2 d x^2+d}}",1,"(Sqrt[d + c^2*d*x^2]*(b + b*c^2*x^2 + 6*a*c*x*Sqrt[1 + c^2*x^2] + 4*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*b*c*x*Sqrt[1 + c^2*x^2]*(3 + 2*c^2*x^2)*ArcSinh[c*x] - 2*b*(1 + c^2*x^2)^2*Log[1 + c^2*x^2]))/(6*c*d^3*(1 + c^2*x^2)^(5/2))","A",1
172,1,247,262,1.1999282,"\int \frac{a+b \sinh ^{-1}(c x)}{x \left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x*(d + c^2*d*x^2)^(5/2)),x]","\frac{\frac{2 a \left(3 c^2 x^2+4\right) \sqrt{c^2 d x^2+d}}{\left(c^2 x^2+1\right)^2}-6 a \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+6 a \sqrt{d} \log (x)+\frac{b d^2 \left(c^2 x^2+1\right)^{3/2} \left(-\frac{c x}{c^2 x^2+1}+\frac{6 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{2 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}+6 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-6 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+6 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-6 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-14 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{\left(c^2 d x^2+d\right)^{3/2}}}{6 d^3}","\frac{a+b \sinh ^{-1}(c x)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \sqrt{c^2 d x^2+d}}+\frac{a+b \sinh ^{-1}(c x)}{3 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{b \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{d^2 \sqrt{c^2 d x^2+d}}+\frac{b \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{b c x}{6 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{7 b \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{6 d^2 \sqrt{c^2 d x^2+d}}",1,"((2*a*(4 + 3*c^2*x^2)*Sqrt[d + c^2*d*x^2])/(1 + c^2*x^2)^2 + 6*a*Sqrt[d]*Log[x] - 6*a*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b*d^2*(1 + c^2*x^2)^(3/2)*(-((c*x)/(1 + c^2*x^2)) + (2*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + (6*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - 14*ArcTan[Tanh[ArcSinh[c*x]/2]] + 6*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 6*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 6*PolyLog[2, -E^(-ArcSinh[c*x])] - 6*PolyLog[2, E^(-ArcSinh[c*x])]))/(d + c^2*d*x^2)^(3/2))/(6*d^3)","A",0
173,1,227,214,0.3119003,"\int \frac{a+b \sinh ^{-1}(c x)}{x^2 \left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^2*(d + c^2*d*x^2)^(5/2)),x]","-\frac{\sqrt{c^2 d x^2+d} \left(24 a c^2 x^2 \sqrt{c^2 x^2+1}+6 a \sqrt{c^2 x^2+1}+16 a c^4 x^4 \sqrt{c^2 x^2+1}+b c^3 x^3+3 b c x \left(c^2 x^2+1\right)^2 \log \left(\frac{1}{c^2 x^2}+1\right)-8 b c x \log \left(c^2 x^2+1\right)-8 b c^5 x^5 \log \left(c^2 x^2+1\right)+2 b \sqrt{c^2 x^2+1} \left(8 c^4 x^4+12 c^2 x^2+3\right) \sinh ^{-1}(c x)-16 b c^3 x^3 \log \left(c^2 x^2+1\right)+b c x\right)}{6 d^3 x \left(c^2 x^2+1\right)^{5/2}}","-\frac{8 c^2 x \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{4 c^2 x \left(a+b \sinh ^{-1}(c x)\right)}{3 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{d x \left(c^2 d x^2+d\right)^{3/2}}-\frac{b c \sqrt{c^2 d x^2+d}}{6 d^3 \left(c^2 x^2+1\right)^{3/2}}+\frac{b c \log (x) \sqrt{c^2 d x^2+d}}{d^3 \sqrt{c^2 x^2+1}}+\frac{5 b c \sqrt{c^2 d x^2+d} \log \left(c^2 x^2+1\right)}{6 d^3 \sqrt{c^2 x^2+1}}",1,"-1/6*(Sqrt[d + c^2*d*x^2]*(b*c*x + b*c^3*x^3 + 6*a*Sqrt[1 + c^2*x^2] + 24*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 16*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 2*b*Sqrt[1 + c^2*x^2]*(3 + 12*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x] + 3*b*c*x*(1 + c^2*x^2)^2*Log[1 + 1/(c^2*x^2)] - 8*b*c*x*Log[1 + c^2*x^2] - 16*b*c^3*x^3*Log[1 + c^2*x^2] - 8*b*c^5*x^5*Log[1 + c^2*x^2]))/(d^3*x*(1 + c^2*x^2)^(5/2))","A",1
174,1,437,400,6.5604841,"\int \frac{a+b \sinh ^{-1}(c x)}{x^3 \left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[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(-60 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+60 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+\frac{4 c x}{\sqrt{c^2 x^2+1}}-\frac{8 \sinh ^{-1}(c x)}{c^2 x^2+1}-60 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+60 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+6 \sqrt{c^2 x^2+1} \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-6 \sqrt{c^2 x^2+1} \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+104 \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-48 \sinh ^{-1}(c x)\right)}{24 d^2 \sqrt{d \left(c^2 x^2+1\right)}}","-\frac{5 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 d^2 \sqrt{c^2 d x^2+d}}+\frac{5 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{5 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{6 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{2 d x^2 \left(c^2 d x^2+d\right)^{3/2}}+\frac{5 b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{2 d^2 \sqrt{c^2 d x^2+d}}-\frac{5 b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{2 d^2 \sqrt{c^2 d x^2+d}}-\frac{3 b c \sqrt{c^2 x^2+1}}{4 d^2 x \sqrt{c^2 d x^2+d}}+\frac{b c}{4 d^2 x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{13 b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}(c x)}{6 d^2 \sqrt{c^2 d x^2+d}}+\frac{5 b c^3 x}{12 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}",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*((4*c*x)/Sqrt[1 + c^2*x^2] - 48*ArcSinh[c*x] - (8*ArcSinh[c*x])/(1 + c^2*x^2) + 104*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] - 6*Sqrt[1 + c^2*x^2]*Coth[ArcSinh[c*x]/2] - 3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 60*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] + 60*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-ArcSinh[c*x])] - 3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 6*Sqrt[1 + c^2*x^2]*Tanh[ArcSinh[c*x]/2]))/(24*d^2*Sqrt[d*(1 + c^2*x^2)])","A",0
175,1,267,297,0.3612803,"\int \frac{a+b \sinh ^{-1}(c x)}{x^4 \left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(x^4*(d + c^2*d*x^2)^(5/2)),x]","\frac{\sqrt{c^2 d x^2+d} \left(12 a c^2 x^2 \sqrt{c^2 x^2+1}-2 a \sqrt{c^2 x^2+1}+32 a c^6 x^6 \sqrt{c^2 x^2+1}+48 a c^4 x^4 \sqrt{c^2 x^2+1}-b c^3 x^3-16 b c^7 x^7 \log \left(c^2 x^2+1\right)-32 b c^5 x^5 \log \left(c^2 x^2+1\right)+8 b c^3 x^3 \left(c^2 x^2+1\right)^2 \log \left(\frac{1}{c^2 x^2}+1\right)-16 b c^3 x^3 \log \left(c^2 x^2+1\right)+2 b \sqrt{c^2 x^2+1} \left(16 c^6 x^6+24 c^4 x^4+6 c^2 x^2-1\right) \sinh ^{-1}(c x)-b c x\right)}{6 d^3 x^3 \left(c^2 x^2+1\right)^{5/2}}","\frac{2 c^2 \left(a+b \sinh ^{-1}(c x)\right)}{d x \left(c^2 d x^2+d\right)^{3/2}}-\frac{a+b \sinh ^{-1}(c x)}{3 d x^3 \left(c^2 d x^2+d\right)^{3/2}}+\frac{16 c^4 x \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left(a+b \sinh ^{-1}(c x)\right)}{3 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{b c \sqrt{c^2 d x^2+d}}{6 d^3 x^2 \sqrt{c^2 x^2+1}}+\frac{b c^3 \sqrt{c^2 d x^2+d}}{6 d^3 \left(c^2 x^2+1\right)^{3/2}}-\frac{8 b c^3 \log (x) \sqrt{c^2 d x^2+d}}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{4 b c^3 \sqrt{c^2 d x^2+d} \log \left(c^2 x^2+1\right)}{3 d^3 \sqrt{c^2 x^2+1}}",1,"(Sqrt[d + c^2*d*x^2]*(-(b*c*x) - b*c^3*x^3 - 2*a*Sqrt[1 + c^2*x^2] + 12*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 48*a*c^4*x^4*Sqrt[1 + c^2*x^2] + 32*a*c^6*x^6*Sqrt[1 + c^2*x^2] + 2*b*Sqrt[1 + c^2*x^2]*(-1 + 6*c^2*x^2 + 24*c^4*x^4 + 16*c^6*x^6)*ArcSinh[c*x] + 8*b*c^3*x^3*(1 + c^2*x^2)^2*Log[1 + 1/(c^2*x^2)] - 16*b*c^3*x^3*Log[1 + c^2*x^2] - 32*b*c^5*x^5*Log[1 + c^2*x^2] - 16*b*c^7*x^7*Log[1 + c^2*x^2]))/(6*d^3*x^3*(1 + c^2*x^2)^(5/2))","A",1
176,1,121,200,0.1470785,"\int \frac{\sinh ^{-1}(a x)}{\left(c+a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcSinh[a*x]/(c + a^2*c*x^2)^(7/2),x]","\frac{\sqrt{a^2 c x^2+c} \left(4 a x \sqrt{a^2 x^2+1} \left(8 a^4 x^4+20 a^2 x^2+15\right) \sinh ^{-1}(a x)-\left(a^2 x^2+1\right) \left(-8 a^2 x^2+16 \left(a^2 x^2+1\right)^2 \log \left(a^2 x^2+1\right)-11\right)\right)}{60 a c^4 \left(a^2 x^2+1\right)^{7/2}}","\frac{2}{15 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{1}{20 a c^3 \left(a^2 x^2+1\right)^{3/2} \sqrt{a^2 c x^2+c}}-\frac{4 \sqrt{a^2 x^2+1} \log \left(a^2 x^2+1\right)}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{8 x \sinh ^{-1}(a x)}{15 c^3 \sqrt{a^2 c x^2+c}}+\frac{4 x \sinh ^{-1}(a x)}{15 c^2 \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \sinh ^{-1}(a x)}{5 c \left(a^2 c x^2+c\right)^{5/2}}",1,"(Sqrt[c + a^2*c*x^2]*(4*a*x*Sqrt[1 + a^2*x^2]*(15 + 20*a^2*x^2 + 8*a^4*x^4)*ArcSinh[a*x] - (1 + a^2*x^2)*(-11 - 8*a^2*x^2 + 16*(1 + a^2*x^2)^2*Log[1 + a^2*x^2])))/(60*a*c^4*(1 + a^2*x^2)^(7/2))","A",1
177,1,63,86,0.0510066,"\int \frac{x^4 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^4*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{-a^4 x^4+3 a^2 x^2+2 a x \sqrt{a^2 x^2+1} \left(2 a^2 x^2-3\right) \sinh ^{-1}(a x)+3 \sinh ^{-1}(a x)^2}{16 a^5}","\frac{3 \sinh ^{-1}(a x)^2}{16 a^5}+\frac{3 x^2}{16 a^3}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{8 a^4}-\frac{x^4}{16 a}",1,"(3*a^2*x^2 - a^4*x^4 + 2*a*x*Sqrt[1 + a^2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a*x] + 3*ArcSinh[a*x]^2)/(16*a^5)","A",1
178,1,48,70,0.0353389,"\int \frac{x^3 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^3*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{-a^3 x^3+3 \left(a^2 x^2-2\right) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+6 a x}{9 a^4}","\frac{2 x}{3 a^3}+\frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^2}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{3 a^4}-\frac{x^3}{9 a}",1,"(6*a*x - a^3*x^3 + 3*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(9*a^4)","A",1
179,1,42,49,0.0372814,"\int \frac{x^2 \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^2*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]","-\frac{a^2 x^2-2 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+\sinh ^{-1}(a x)^2}{4 a^3}","-\frac{\sinh ^{-1}(a x)^2}{4 a^3}+\frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 a^2}-\frac{x^2}{4 a}",1,"-1/4*(a^2*x^2 - 2*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + ArcSinh[a*x]^2)/a^3","A",1
180,1,28,28,0.0277311,"\int \frac{x \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{x}{a}","\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{x}{a}",1,"-(x/a) + (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/a^2","A",1
181,1,13,13,0.0060737,"\int \frac{\sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]/Sqrt[1 + a^2*x^2],x]","\frac{\sinh ^{-1}(a x)^2}{2 a}","\frac{\sinh ^{-1}(a x)^2}{2 a}",1,"ArcSinh[a*x]^2/(2*a)","A",1
182,1,57,34,0.0438955,"\int \frac{\sinh ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]/(x*Sqrt[1 + a^2*x^2]),x]","\text{Li}_2\left(-e^{-\sinh ^{-1}(a x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(a x)}\right)+\sinh ^{-1}(a x) \left(\log \left(1-e^{-\sinh ^{-1}(a x)}\right)-\log \left(e^{-\sinh ^{-1}(a x)}+1\right)\right)","-\text{Li}_2\left(-e^{\sinh ^{-1}(a x)}\right)+\text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)-2 \sinh ^{-1}(a x) \tanh ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)",1,"ArcSinh[a*x]*(Log[1 - E^(-ArcSinh[a*x])] - Log[1 + E^(-ArcSinh[a*x])]) + PolyLog[2, -E^(-ArcSinh[a*x])] - PolyLog[2, E^(-ArcSinh[a*x])]","A",0
183,1,29,27,0.0387137,"\int \frac{\sinh ^{-1}(a x)}{x^2 \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]/(x^2*Sqrt[1 + a^2*x^2]),x]","a \log (a x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}","a \log (x)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{x}",1,"-((Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/x) + a*Log[a*x]","A",1
184,1,126,80,0.0265449,"\int \frac{\sinh ^{-1}(a x)}{x^3 \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]/(x^3*Sqrt[1 + a^2*x^2]),x]","\frac{1}{8} a^2 \left(-4 \text{Li}_2\left(-e^{-\sinh ^{-1}(a x)}\right)+4 \text{Li}_2\left(e^{-\sinh ^{-1}(a x)}\right)-4 \sinh ^{-1}(a x) \log \left(1-e^{-\sinh ^{-1}(a x)}\right)+4 \sinh ^{-1}(a x) \log \left(e^{-\sinh ^{-1}(a x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(a x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(a x)\right)-\sinh ^{-1}(a x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(a x)\right)-\sinh ^{-1}(a x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(a x)\right)\right)","\frac{1}{2} a^2 \text{Li}_2\left(-e^{\sinh ^{-1}(a x)}\right)-\frac{1}{2} a^2 \text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{2 x^2}+a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)-\frac{a}{2 x}",1,"(a^2*(-2*Coth[ArcSinh[a*x]/2] - ArcSinh[a*x]*Csch[ArcSinh[a*x]/2]^2 - 4*ArcSinh[a*x]*Log[1 - E^(-ArcSinh[a*x])] + 4*ArcSinh[a*x]*Log[1 + E^(-ArcSinh[a*x])] - 4*PolyLog[2, -E^(-ArcSinh[a*x])] + 4*PolyLog[2, E^(-ArcSinh[a*x])] - ArcSinh[a*x]*Sech[ArcSinh[a*x]/2]^2 + 2*Tanh[ArcSinh[a*x]/2]))/8","A",0
185,1,257,313,0.4887599,"\int x^m \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^m*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]),x]","\frac{x^{m+1} \left(\frac{6 d \left(\frac{4 d^2 \left((m+2) \left(c^2 m x^2+c^2 x^2+m+3\right) \left(a+b \sinh ^{-1}(c x)\right)-b c (m+1) x \, _2F_1\left(-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)-2 b c x \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)\right)}{(m+1) (m+2) (m+3)}+\left(c^2 d x^2+d\right)^2 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b c d^2 x \, _2F_1\left(-\frac{3}{2},\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)}{m+2}\right)}{m+5}+\left(c^2 d x^2+d\right)^3 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b c d^3 x \, _2F_1\left(-\frac{5}{2},\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)}{m+2}\right)}{m+7}","\frac{c^6 d^3 x^{m+7} \left(a+b \sinh ^{-1}(c x)\right)}{m+7}+\frac{3 c^4 d^3 x^{m+5} \left(a+b \sinh ^{-1}(c x)\right)}{m+5}+\frac{3 c^2 d^3 x^{m+3} \left(a+b \sinh ^{-1}(c x)\right)}{m+3}+\frac{d^3 x^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{m+1}-\frac{3 b c d^3 \left(35 m^3+455 m^2+1813 m+2161\right) x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{(m+1) (m+2) (m+3)^2 (m+5)^2 (m+7)^2}-\frac{b c d^3 \left(m^4+27 m^3+284 m^2+1329 m+2271\right) \sqrt{c^2 x^2+1} x^{m+2}}{(m+3)^2 (m+5)^2 (m+7)^2}-\frac{b c^5 d^3 \sqrt{c^2 x^2+1} x^{m+6}}{(m+7)^2}-\frac{b c^3 d^3 (m+9) (2 m+13) \sqrt{c^2 x^2+1} x^{m+4}}{(m+5)^2 (m+7)^2}",1,"(x^(1 + m)*((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x]) - (b*c*d^3*x*Hypergeometric2F1[-5/2, 1 + m/2, 2 + m/2, -(c^2*x^2)])/(2 + m) + (6*d*((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]) - (b*c*d^2*x*Hypergeometric2F1[-3/2, 1 + m/2, 2 + m/2, -(c^2*x^2)])/(2 + m) + (4*d^2*((2 + m)*(3 + m + c^2*x^2 + c^2*m*x^2)*(a + b*ArcSinh[c*x]) - b*c*(1 + m)*x*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, -(c^2*x^2)] - 2*b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, -(c^2*x^2)]))/((1 + m)*(2 + m)*(3 + m))))/(5 + m)))/(7 + m)","A",1
186,1,188,217,0.0562139,"\int x^m \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^m*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]),x]","\frac{x^{m+1} \left(\frac{4 d^2 \left((m+2) \left(c^2 m x^2+c^2 x^2+m+3\right) \left(a+b \sinh ^{-1}(c x)\right)-b c (m+1) x \, _2F_1\left(-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)-2 b c x \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)\right)}{(m+1) (m+2) (m+3)}+\left(c^2 d x^2+d\right)^2 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b c d^2 x \, _2F_1\left(-\frac{3}{2},\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)}{m+2}\right)}{m+5}","\frac{c^4 d^2 x^{m+5} \left(a+b \sinh ^{-1}(c x)\right)}{m+5}+\frac{2 c^2 d^2 x^{m+3} \left(a+b \sinh ^{-1}(c x)\right)}{m+3}+\frac{d^2 x^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{m+1}-\frac{b c d^2 \left(15 m^2+100 m+149\right) x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{(m+1) (m+2) (m+3)^2 (m+5)^2}-\frac{b c d^2 \left(m^2+13 m+38\right) \sqrt{c^2 x^2+1} x^{m+2}}{(m+3)^2 (m+5)^2}-\frac{b c^3 d^2 \sqrt{c^2 x^2+1} x^{m+4}}{(m+5)^2}",1,"(x^(1 + m)*((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x]) - (b*c*d^2*x*Hypergeometric2F1[-3/2, 1 + m/2, 2 + m/2, -(c^2*x^2)])/(2 + m) + (4*d^2*((2 + m)*(3 + m + c^2*x^2 + c^2*m*x^2)*(a + b*ArcSinh[c*x]) - b*c*(1 + m)*x*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, -(c^2*x^2)] - 2*b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, -(c^2*x^2)]))/((1 + m)*(2 + m)*(3 + m))))/(5 + m)","A",1
187,1,118,128,0.0448766,"\int x^m \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^m*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x]),x]","\frac{d x^{m+1} \left((m+2) \left(c^2 m x^2+c^2 x^2+m+3\right) \left(a+b \sinh ^{-1}(c x)\right)-b c (m+1) x \, _2F_1\left(-\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)-2 b c x \, _2F_1\left(\frac{1}{2},\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)\right)}{(m+1) (m+2) (m+3)}","\frac{c^2 d x^{m+3} \left(a+b \sinh ^{-1}(c x)\right)}{m+3}+\frac{d x^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{m+1}-\frac{b c d (3 m+7) x^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{(m+1) (m+2) (m+3)^2}-\frac{b c d \sqrt{c^2 x^2+1} x^{m+2}}{(m+3)^2}",1,"(d*x^(1 + m)*((2 + m)*(3 + m + c^2*x^2 + c^2*m*x^2)*(a + b*ArcSinh[c*x]) - b*c*(1 + m)*x*Hypergeometric2F1[-1/2, 1 + m/2, 2 + m/2, -(c^2*x^2)] - 2*b*c*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, -(c^2*x^2)]))/((1 + m)*(2 + m)*(3 + m))","A",1
188,0,0,27,3.8122159,"\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{d+c^2 d x^2} \, dx","Integrate[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2),x]","\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{d+c^2 d x^2} \, dx","\text{Int}\left(\frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d x^2+d},x\right)",0,"Integrate[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2), x]","A",-1
189,0,0,116,5.6333148,"\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2,x]","\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^2} \, dx","\frac{(1-m) \text{Int}\left(\frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d x^2+d},x\right)}{2 d}+\frac{x^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 d^2 \left(c^2 x^2+1\right)}-\frac{b c 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 (m+2)}",0,"Integrate[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^2, x]","A",-1
190,0,0,207,5.9596227,"\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3,x]","\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^3} \, dx","\frac{(1-m) (3-m) \text{Int}\left(\frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d x^2+d},x\right)}{8 d^2}+\frac{(3-m) x^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{8 d^3 \left(c^2 x^2+1\right)}+\frac{x^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{b c (3-m) 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 (m+2)}-\frac{b c 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 (m+2)}",0,"Integrate[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^3, x]","A",-1
191,1,332,618,1.2817075,"\int x^m \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^m*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{d^2 x^{m+1} \sqrt{c^2 d x^2+d} \left(-\frac{5 \left(3 (m+4) \left(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) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right) \left(a+b \sinh ^{-1}(c x)\right)-(m+1) (m+2) \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+b c (m+1) x\right)-\left((m+1) (m+4) (m+2)^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)\right)+b c (m+1) (m+2) x \left(c^2 (m+2) x^2+m+4\right)\right)}{(m+1) (m+2)^2 (m+4)^2 \sqrt{c^2 x^2+1}}+\left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b c x \left(c^4 (m+2) (m+4) x^4+2 c^2 (m+2) (m+6) x^2+(m+4) (m+6)\right)}{(m+2) (m+4) (m+6) \sqrt{c^2 x^2+1}}\right)}{m+6}","-\frac{15 b c d^2 x^{m+2} \sqrt{c^2 d x^2+d} \, _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 (m+4) (m+6) \sqrt{c^2 x^2+1}}+\frac{15 d^2 x^{m+1} \sqrt{c^2 d x^2+d} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right) \left(a+b \sinh ^{-1}(c x)\right)}{(m+6) \left(m^3+7 m^2+14 m+8\right) \sqrt{c^2 x^2+1}}+\frac{15 d^2 x^{m+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{(m+6) \left(m^2+6 m+8\right)}+\frac{x^{m+1} \left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{m+6}+\frac{5 d x^{m+1} \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{(m+4) (m+6)}-\frac{5 b c d^2 x^{m+2} \sqrt{c^2 d x^2+d}}{(m+6) \left(m^2+6 m+8\right) \sqrt{c^2 x^2+1}}-\frac{b c d^2 x^{m+2} \sqrt{c^2 d x^2+d}}{\left(m^2+8 m+12\right) \sqrt{c^2 x^2+1}}-\frac{15 b c d^2 x^{m+2} \sqrt{c^2 d x^2+d}}{(m+2)^2 (m+4) (m+6) \sqrt{c^2 x^2+1}}-\frac{b c^5 d^2 x^{m+6} \sqrt{c^2 d x^2+d}}{(m+6)^2 \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d^2 x^{m+4} \sqrt{c^2 d x^2+d}}{(m+4) (m+6) \sqrt{c^2 x^2+1}}-\frac{5 b c^3 d^2 x^{m+4} \sqrt{c^2 d x^2+d}}{(m+4)^2 (m+6) \sqrt{c^2 x^2+1}}",1,"(d^2*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(-((b*c*x*((4 + m)*(6 + m) + 2*c^2*(2 + m)*(6 + m)*x^2 + c^4*(2 + m)*(4 + m)*x^4))/((2 + m)*(4 + m)*(6 + m)*Sqrt[1 + c^2*x^2])) + (1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]) - (5*(b*c*(1 + m)*(2 + m)*x*(4 + m + c^2*(2 + m)*x^2) - (1 + m)*(2 + m)^2*(4 + m)*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]) + 3*(4 + m)*(b*c*(1 + m)*x - (1 + m)*(2 + m)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) - (2 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)] + 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)^2*Sqrt[1 + c^2*x^2])))/(6 + m)","A",1
192,1,233,390,0.5202027,"\int x^m \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^m*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{d x^{m+1} \sqrt{c^2 d x^2+d} \left(-\frac{3 \left(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) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right) \left(a+b \sinh ^{-1}(c x)\right)-(m+1) (m+2) \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+b c (m+1) x\right)}{(m+1) (m+2)^2 \sqrt{c^2 x^2+1}}+\left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{b c x \left(c^2 (m+2) x^2+m+4\right)}{(m+2) (m+4) \sqrt{c^2 x^2+1}}\right)}{m+4}","-\frac{3 b c d x^{m+2} \sqrt{c^2 d x^2+d} \, _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 (m+4) \sqrt{c^2 x^2+1}}+\frac{3 d x^{m+1} \sqrt{c^2 d x^2+d} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right) \left(a+b \sinh ^{-1}(c x)\right)}{\left(m^3+7 m^2+14 m+8\right) \sqrt{c^2 x^2+1}}+\frac{3 d x^{m+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{m^2+6 m+8}+\frac{x^{m+1} \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{m+4}-\frac{b c d x^{m+2} \sqrt{c^2 d x^2+d}}{\left(m^2+6 m+8\right) \sqrt{c^2 x^2+1}}-\frac{3 b c d x^{m+2} \sqrt{c^2 d x^2+d}}{(m+2)^2 (m+4) \sqrt{c^2 x^2+1}}-\frac{b c^3 d x^{m+4} \sqrt{c^2 d x^2+d}}{(m+4)^2 \sqrt{c^2 x^2+1}}",1,"(d*x^(1 + m)*Sqrt[d + c^2*d*x^2]*(-((b*c*x*(4 + m + c^2*(2 + m)*x^2))/((2 + m)*(4 + m)*Sqrt[1 + c^2*x^2])) + (1 + c^2*x^2)*(a + b*ArcSinh[c*x]) - (3*(b*c*(1 + m)*x - (1 + m)*(2 + m)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) - (2 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)] + 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^2*x^2])))/(4 + m)","A",1
193,1,179,240,0.0694619,"\int x^m \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[x^m*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{x^{m+1} \sqrt{c^2 d x^2+d} \left(-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) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right) \left(a+b \sinh ^{-1}(c x)\right)+(m+1) \left(a (m+2) \sqrt{c^2 x^2+1}+b (m+2) \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-b c x\right)\right)}{(m+1) (m+2)^2 \sqrt{c^2 x^2+1}}","-\frac{b c x^{m+2} \sqrt{c^2 d x^2+d} \, _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^2 x^2+1}}+\frac{x^{m+1} \sqrt{c^2 d x^2+d} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right) \left(a+b \sinh ^{-1}(c x)\right)}{\left(m^2+3 m+2\right) \sqrt{c^2 x^2+1}}+\frac{x^{m+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{m+2}-\frac{b c x^{m+2} \sqrt{c^2 d x^2+d}}{(m+2)^2 \sqrt{c^2 x^2+1}}",1,"(x^(1 + m)*Sqrt[d + c^2*d*x^2]*((1 + m)*(-(b*c*x) + a*(2 + m)*Sqrt[1 + c^2*x^2] + b*(2 + m)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]) + (2 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)] - 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^2*x^2])","A",1
194,1,129,161,0.0544958,"\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^m*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]","\frac{\sqrt{c^2 x^2+1} x^{m+1} \left((m+2) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right) \left(a+b \sinh ^{-1}(c x)\right)-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)\right)}{(m+1) (m+2) \sqrt{c^2 d x^2+d}}","\frac{\sqrt{c^2 x^2+1} 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 \sinh ^{-1}(c x)\right)}{(m+1) \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} 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)}{\left(m^2+3 m+2\right) \sqrt{c^2 d x^2+d}}",1,"(x^(1 + m)*Sqrt[1 + c^2*x^2]*((2 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)] - 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)*Sqrt[d + c^2*d*x^2])","A",1
195,1,206,268,0.2309642,"\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(3/2),x]","\frac{x^{m+1} \left(b c m x \sqrt{c^2 x^2+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{c^2 x^2+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right) \left(a+b \sinh ^{-1}(c x)\right)+(m+1) \left((m+2) \left(a+b \sinh ^{-1}(c x)\right)-b c x \sqrt{c^2 x^2+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{c^2 d x^2+d}}","\frac{b c m \sqrt{c^2 x^2+1} 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 \left(m^2+3 m+2\right) \sqrt{c^2 d x^2+d}}-\frac{m \sqrt{c^2 x^2+1} 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 \sinh ^{-1}(c x)\right)}{d (m+1) \sqrt{c^2 d x^2+d}}+\frac{x^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{d (m+2) \sqrt{c^2 d x^2+d}}",1,"(x^(1 + m)*(-(m*(2 + m)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)]) + (1 + m)*((2 + m)*(a + b*ArcSinh[c*x]) - b*c*x*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, -(c^2*x^2)]) + b*c*m*x*Sqrt[1 + c^2*x^2]*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
196,1,286,402,0.4221987,"\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^m*(a + b*ArcSinh[c*x]))/(d + c^2*d*x^2)^(5/2),x]","\frac{x^{m+1} \left((2-m) \left(c^2 x^2+1\right) \left(-m \sqrt{c^2 x^2+1} \left((m+2) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-c^2 x^2\right) \left(a+b \sinh ^{-1}(c x)\right)-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)\right)+(m+1) (m+2) \left(a+b \sinh ^{-1}(c x)\right)-b c (m+1) x \sqrt{c^2 x^2+1} \, _2F_1\left(1,\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)\right)+(m+1) (m+2) \left(a+b \sinh ^{-1}(c x)\right)-b c (m+1) x \left(c^2 x^2+1\right)^{3/2} \, _2F_1\left(2,\frac{m}{2}+1;\frac{m}{2}+2;-c^2 x^2\right)\right)}{3 d^2 (m+1) (m+2) \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}","\frac{b c (2-m) m \sqrt{c^2 x^2+1} 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 \left(m^2+3 m+2\right) \sqrt{c^2 d x^2+d}}-\frac{(2-m) m \sqrt{c^2 x^2+1} 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 \sinh ^{-1}(c x)\right)}{3 d^2 (m+1) \sqrt{c^2 d x^2+d}}+\frac{(2-m) x^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{x^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{b c (2-m) \sqrt{c^2 x^2+1} x^{m+2} \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{3 d^2 (m+2) \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} x^{m+2} \, _2F_1\left(2,\frac{m+2}{2};\frac{m+4}{2};-c^2 x^2\right)}{3 d^2 (m+2) \sqrt{c^2 d x^2+d}}",1,"(x^(1 + m)*((1 + m)*(2 + m)*(a + b*ArcSinh[c*x]) - b*c*(1 + m)*x*(1 + c^2*x^2)^(3/2)*Hypergeometric2F1[2, 1 + m/2, 2 + m/2, -(c^2*x^2)] + (2 - m)*(1 + c^2*x^2)*((1 + m)*(2 + m)*(a + b*ArcSinh[c*x]) - b*c*(1 + m)*x*Sqrt[1 + c^2*x^2]*Hypergeometric2F1[1, 1 + m/2, 2 + m/2, -(c^2*x^2)] - m*Sqrt[1 + c^2*x^2]*((2 + m)*(a + b*ArcSinh[c*x])*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(c^2*x^2)] - b*c*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(c^2*x^2)]))))/(3*d^2*(1 + m)*(2 + m)*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])","A",1
197,1,97,102,0.0344629,"\int \frac{x^m \sinh ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^m*ArcSinh[a*x])/Sqrt[1 + a^2*x^2],x]","\frac{x^{m+1} \left((m+2) \sinh ^{-1}(a x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right)-a x \, _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)\right)}{(m+1) (m+2)}","\frac{x^{m+1} \sinh ^{-1}(a x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};-a^2 x^2\right)}{m+1}-\frac{a 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)}{m^2+3 m+2}",1,"(x^(1 + m)*((2 + m)*ArcSinh[a*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, -(a^2*x^2)] - a*x*HypergeometricPFQ[{1, 1 + m/2, 1 + m/2}, {3/2 + m/2, 2 + m/2}, -(a^2*x^2)]))/((1 + m)*(2 + m))","A",1
198,1,201,283,0.2913345,"\int x^4 \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^4*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d \left(11025 a^2 c^5 x^5 \left(5 c^2 x^2+7\right)-210 a b \sqrt{c^2 x^2+1} \left(75 c^6 x^6+57 c^4 x^4-76 c^2 x^2+152\right)-210 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+1} \left(75 c^6 x^6+57 c^4 x^4-76 c^2 x^2+152\right)-105 a c^5 x^5 \left(5 c^2 x^2+7\right)\right)+11025 b^2 c^5 x^5 \left(5 c^2 x^2+7\right) \sinh ^{-1}(c x)^2+b^2 \left(2250 c^7 x^7+2394 c^5 x^5-5320 c^3 x^3+31920 c x\right)\right)}{385875 c^5}","\frac{1}{7} d x^5 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{4 b d x^4 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{175 c}-\frac{2 b d \left(c^2 x^2+1\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{49 c^5}+\frac{4 b d \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{35 c^5}-\frac{2 b d \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{21 c^5}-\frac{32 b d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{525 c^5}+\frac{16 b d x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{525 c^3}+\frac{2}{35} d x^5 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{304 b^2 d x}{3675 c^4}+\frac{2}{343} b^2 c^2 d x^7-\frac{152 b^2 d x^3}{11025 c^2}+\frac{38 b^2 d x^5}{6125}",1,"(d*(11025*a^2*c^5*x^5*(7 + 5*c^2*x^2) - 210*a*b*Sqrt[1 + c^2*x^2]*(152 - 76*c^2*x^2 + 57*c^4*x^4 + 75*c^6*x^6) + b^2*(31920*c*x - 5320*c^3*x^3 + 2394*c^5*x^5 + 2250*c^7*x^7) - 210*b*(-105*a*c^5*x^5*(7 + 5*c^2*x^2) + b*Sqrt[1 + c^2*x^2]*(152 - 76*c^2*x^2 + 57*c^4*x^4 + 75*c^6*x^6))*ArcSinh[c*x] + 11025*b^2*c^5*x^5*(7 + 5*c^2*x^2)*ArcSinh[c*x]^2))/(385875*c^5)","A",1
199,1,186,198,0.2662803,"\int x^3 \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d \left(c x \left(18 a^2 c^3 x^3 \left(2 c^2 x^2+3\right)-6 a b \sqrt{c^2 x^2+1} \left(2 c^4 x^4+2 c^2 x^2-3\right)+b^2 c x \left(2 c^4 x^4+3 c^2 x^2-9\right)\right)+6 b \sinh ^{-1}(c x) \left(3 a \left(4 c^6 x^6+6 c^4 x^4-1\right)+b c x \sqrt{c^2 x^2+1} \left(-2 c^4 x^4-2 c^2 x^2+3\right)\right)+9 b^2 \left(4 c^6 x^6+6 c^4 x^4-1\right) \sinh ^{-1}(c x)^2\right)}{216 c^4}","-\frac{d \left(a+b \sinh ^{-1}(c x)\right)^2}{24 c^4}-\frac{1}{18} b c d x^5 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{6} d x^4 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b d x^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{18 c}+\frac{b d x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{12 c^3}+\frac{1}{12} d x^4 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{108} b^2 c^2 d x^6-\frac{b^2 d x^2}{24 c^2}+\frac{1}{72} b^2 d x^4",1,"(d*(c*x*(18*a^2*c^3*x^3*(3 + 2*c^2*x^2) - 6*a*b*Sqrt[1 + c^2*x^2]*(-3 + 2*c^2*x^2 + 2*c^4*x^4) + b^2*c*x*(-9 + 3*c^2*x^2 + 2*c^4*x^4)) + 6*b*(b*c*x*Sqrt[1 + c^2*x^2]*(3 - 2*c^2*x^2 - 2*c^4*x^4) + 3*a*(-1 + 6*c^4*x^4 + 4*c^6*x^6))*ArcSinh[c*x] + 9*b^2*(-1 + 6*c^4*x^4 + 4*c^6*x^6)*ArcSinh[c*x]^2))/(216*c^4)","A",1
200,1,177,206,0.2477244,"\int x^2 \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d \left(225 a^2 c^3 x^3 \left(3 c^2 x^2+5\right)-30 a b \sqrt{c^2 x^2+1} \left(9 c^4 x^4+13 c^2 x^2-26\right)-30 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+1} \left(9 c^4 x^4+13 c^2 x^2-26\right)-15 a c^3 x^3 \left(3 c^2 x^2+5\right)\right)+2 b^2 c x \left(27 c^4 x^4+65 c^2 x^2-390\right)+225 b^2 c^3 x^3 \left(3 c^2 x^2+5\right) \sinh ^{-1}(c x)^2\right)}{3375 c^3}","-\frac{4 b d x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{45 c}+\frac{1}{5} d x^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b d \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{25 c^3}+\frac{2 b d \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{15 c^3}+\frac{8 b d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{45 c^3}+\frac{2}{15} d x^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{125} b^2 c^2 d x^5-\frac{52 b^2 d x}{225 c^2}+\frac{26}{675} b^2 d x^3",1,"(d*(225*a^2*c^3*x^3*(5 + 3*c^2*x^2) - 30*a*b*Sqrt[1 + c^2*x^2]*(-26 + 13*c^2*x^2 + 9*c^4*x^4) + 2*b^2*c*x*(-390 + 65*c^2*x^2 + 27*c^4*x^4) - 30*b*(-15*a*c^3*x^3*(5 + 3*c^2*x^2) + b*Sqrt[1 + c^2*x^2]*(-26 + 13*c^2*x^2 + 9*c^4*x^4))*ArcSinh[c*x] + 225*b^2*c^3*x^3*(5 + 3*c^2*x^2)*ArcSinh[c*x]^2))/(3375*c^3)","A",1
201,1,155,135,0.2609574,"\int x \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d \left(c x \left(8 a^2 c x \left(c^2 x^2+2\right)-2 a b \sqrt{c^2 x^2+1} \left(2 c^2 x^2+5\right)+b^2 c x \left(c^2 x^2+5\right)\right)+2 b \sinh ^{-1}(c x) \left(a \left(8 c^4 x^4+16 c^2 x^2+5\right)-b c x \sqrt{c^2 x^2+1} \left(2 c^2 x^2+5\right)\right)+b^2 \left(8 c^4 x^4+16 c^2 x^2+5\right) \sinh ^{-1}(c x)^2\right)}{32 c^2}","-\frac{b d x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 c}-\frac{3 b d x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{16 c}+\frac{d \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^2}-\frac{3 d \left(a+b \sinh ^{-1}(c x)\right)^2}{32 c^2}+\frac{1}{32} b^2 c^2 d x^4+\frac{5}{32} b^2 d x^2",1,"(d*(c*x*(8*a^2*c*x*(2 + c^2*x^2) + b^2*c*x*(5 + c^2*x^2) - 2*a*b*Sqrt[1 + c^2*x^2]*(5 + 2*c^2*x^2)) + 2*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(5 + 2*c^2*x^2)) + a*(5 + 16*c^2*x^2 + 8*c^4*x^4))*ArcSinh[c*x] + b^2*(5 + 16*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x]^2))/(32*c^2)","A",1
202,1,135,125,0.2061486,"\int \left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d \left(9 a^2 c x \left(c^2 x^2+3\right)-6 a b \sqrt{c^2 x^2+1} \left(c^2 x^2+7\right)-6 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+1} \left(c^2 x^2+7\right)-3 a c x \left(c^2 x^2+3\right)\right)+2 b^2 c x \left(c^2 x^2+21\right)+9 b^2 c x \left(c^2 x^2+3\right) \sinh ^{-1}(c x)^2\right)}{27 c}","\frac{1}{3} d x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b d \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{9 c}-\frac{4 b d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 c}+\frac{2}{3} d x \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{27} b^2 c^2 d x^3+\frac{14}{9} b^2 d x",1,"(d*(9*a^2*c*x*(3 + c^2*x^2) - 6*a*b*Sqrt[1 + c^2*x^2]*(7 + c^2*x^2) + 2*b^2*c*x*(21 + c^2*x^2) - 6*b*(-3*a*c*x*(3 + c^2*x^2) + b*Sqrt[1 + c^2*x^2]*(7 + c^2*x^2))*ArcSinh[c*x] + 9*b^2*c*x*(3 + c^2*x^2)*ArcSinh[c*x]^2))/(27*c)","A",1
203,1,209,166,0.3868506,"\int \frac{\left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x,x]","\frac{1}{8} d \left(4 a^2 c^2 x^2+8 a^2 \log (x)-4 a b \left(c x \sqrt{c^2 x^2+1}-\sinh ^{-1}(c x)\right)+8 a b c^2 x^2 \sinh ^{-1}(c x)-8 a b \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+8 a b \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+2 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)\right)+8 b^2 \left(\sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)-\frac{1}{3} \sinh ^{-1}(c x)^3+\sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)\right)-2 b^2 \sinh ^{-1}(c x) \sinh \left(2 \sinh ^{-1}(c x)\right)+b^2 \left(2 \sinh ^{-1}(c x)^2+1\right) \cosh \left(2 \sinh ^{-1}(c x)\right)\right)","\frac{1}{2} d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{1}{2} b c d x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)-b d \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{d \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b}-\frac{1}{4} d \left(a+b \sinh ^{-1}(c x)\right)^2+d \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{4} b^2 c^2 d x^2-\frac{1}{2} b^2 d \text{Li}_3\left(e^{-2 \sinh ^{-1}(c x)}\right)",1,"(d*(4*a^2*c^2*x^2 - 4*a*b*(c*x*Sqrt[1 + c^2*x^2] - ArcSinh[c*x]) + 8*a*b*c^2*x^2*ArcSinh[c*x] + b^2*(1 + 2*ArcSinh[c*x]^2)*Cosh[2*ArcSinh[c*x]] + 8*a*b*ArcSinh[c*x]*(ArcSinh[c*x] + 2*Log[1 - E^(-2*ArcSinh[c*x])]) + 8*a^2*Log[x] - 8*a*b*PolyLog[2, E^(-2*ArcSinh[c*x])] + 8*b^2*(-1/3*ArcSinh[c*x]^3 + ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - PolyLog[3, E^(2*ArcSinh[c*x])]/2) - 2*b^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))/8","A",0
204,1,192,131,0.4325684,"\int \frac{\left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x^2,x]","\frac{d \left(a^2 c^2 x^2-a^2+2 a b c x \left(c x \sinh ^{-1}(c x)-\sqrt{c^2 x^2+1}\right)-2 a b \left(c x \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+\sinh ^{-1}(c x)\right)+b^2 c x \left(-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 c x+c x \sinh ^{-1}(c x)^2\right)-b^2 \left(-2 c x \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+2 c x \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+2 c x \left(\log \left(e^{-\sinh ^{-1}(c x)}+1\right)-\log \left(1-e^{-\sinh ^{-1}(c x)}\right)\right)\right)\right)\right)}{x}","-2 b c d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)-\frac{d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{x}+2 c^2 d x \left(a+b \sinh ^{-1}(c x)\right)^2-4 b c d \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+2 b^2 c^2 d x-2 b^2 c d \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+2 b^2 c d \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)",1,"(d*(-a^2 + a^2*c^2*x^2 + 2*a*b*c*x*(-Sqrt[1 + c^2*x^2] + c*x*ArcSinh[c*x]) + b^2*c*x*(2*c*x - 2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2) - 2*a*b*(ArcSinh[c*x] + c*x*ArcTanh[Sqrt[1 + c^2*x^2]]) - b^2*(ArcSinh[c*x]*(ArcSinh[c*x] + 2*c*x*(-Log[1 - E^(-ArcSinh[c*x])] + Log[1 + E^(-ArcSinh[c*x])])) - 2*c*x*PolyLog[2, -E^(-ArcSinh[c*x])] + 2*c*x*PolyLog[2, E^(-ArcSinh[c*x])])))/x","A",0
205,1,212,180,0.3780353,"\int \frac{\left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x^3,x]","\frac{1}{2} d \left(2 a^2 c^2 \log (x)-\frac{a^2}{x^2}+2 a b c^2 \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+2 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)\right)-\text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)\right)-\frac{2 a b \left(c x \sqrt{c^2 x^2+1}+\sinh ^{-1}(c x)\right)}{x^2}-\frac{1}{3} b^2 c^2 \left(-6 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+3 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)+2 \sinh ^{-1}(c x)^2 \left(\sinh ^{-1}(c x)-3 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)\right)\right)-\frac{b^2 \left(-2 c^2 x^2 \log (c x)+2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2\right)}{x^2}\right)","-b c^2 d \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{2 x^2}-\frac{b c d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{x}+\frac{c^2 d \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b}+\frac{1}{2} c^2 d \left(a+b \sinh ^{-1}(c x)\right)^2+c^2 d \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{1}{2} b^2 c^2 d \text{Li}_3\left(e^{-2 \sinh ^{-1}(c x)}\right)+b^2 c^2 d \log (x)",1,"(d*(-(a^2/x^2) - (2*a*b*(c*x*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/x^2 + 2*a^2*c^2*Log[x] - (b^2*(2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]^2 - 2*c^2*x^2*Log[c*x]))/x^2 + 2*a*b*c^2*(ArcSinh[c*x]*(ArcSinh[c*x] + 2*Log[1 - E^(-2*ArcSinh[c*x])]) - PolyLog[2, E^(-2*ArcSinh[c*x])]) - (b^2*c^2*(2*ArcSinh[c*x]^2*(ArcSinh[c*x] - 3*Log[1 - E^(2*ArcSinh[c*x])]) - 6*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] + 3*PolyLog[3, E^(2*ArcSinh[c*x])]))/3))/2","A",0
206,1,245,158,0.7975649,"\int \frac{\left(d+c^2 d x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + c^2*d*x^2)*(a + b*ArcSinh[c*x])^2)/x^4,x]","-\frac{d \left(3 a^2 c^2 x^2+a^2+a b c x \sqrt{c^2 x^2+1}+6 a b c^2 x^2 \sinh ^{-1}(c x)+5 a b c^3 x^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+2 a b \sinh ^{-1}(c x)-5 b^2 c^3 x^3 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+5 b^2 c^3 x^3 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-5 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+5 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+b^2 c^2 x^2+3 b^2 c^2 x^2 \sinh ^{-1}(c x)^2+b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+b^2 \sinh ^{-1}(c x)^2\right)}{3 x^3}","-\frac{10}{3} b c^3 d \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{b c d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^2}-\frac{d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 x^3}-\frac{2 c^2 d \left(a+b \sinh ^{-1}(c x)\right)^2}{3 x}-\frac{5}{3} b^2 c^3 d \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+\frac{5}{3} b^2 c^3 d \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)-\frac{b^2 c^2 d}{3 x}",1,"-1/3*(d*(a^2 + 3*a^2*c^2*x^2 + b^2*c^2*x^2 + a*b*c*x*Sqrt[1 + c^2*x^2] + 2*a*b*ArcSinh[c*x] + 6*a*b*c^2*x^2*ArcSinh[c*x] + b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + b^2*ArcSinh[c*x]^2 + 3*b^2*c^2*x^2*ArcSinh[c*x]^2 + 5*a*b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] - 5*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 5*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 5*b^2*c^3*x^3*PolyLog[2, -E^(-ArcSinh[c*x])] + 5*b^2*c^3*x^3*PolyLog[2, E^(-ArcSinh[c*x])]))/x^3","A",0
207,1,251,386,0.4071504,"\int x^4 \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^4*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]","\frac{d^2 \left(99225 a^2 c^5 x^5 \left(35 c^4 x^4+90 c^2 x^2+63\right)-630 a b \sqrt{c^2 x^2+1} \left(1225 c^8 x^8+2650 c^6 x^6+789 c^4 x^4-1052 c^2 x^2+2104\right)-630 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+1} \left(1225 c^8 x^8+2650 c^6 x^6+789 c^4 x^4-1052 c^2 x^2+2104\right)-315 a c^5 x^5 \left(35 c^4 x^4+90 c^2 x^2+63\right)\right)+99225 b^2 c^5 x^5 \left(35 c^4 x^4+90 c^2 x^2+63\right) \sinh ^{-1}(c x)^2+2 b^2 c x \left(42875 c^8 x^8+119250 c^6 x^6+49707 c^4 x^4-110460 c^2 x^2+662760\right)\right)}{31255875 c^5}","\frac{1}{9} d^2 x^5 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{4}{63} d^2 x^5 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{16 b d^2 x^4 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{1575 c}-\frac{2 b d^2 \left(c^2 x^2+1\right)^{9/2} \left(a+b \sinh ^{-1}(c x)\right)}{81 c^5}+\frac{20 b d^2 \left(c^2 x^2+1\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{441 c^5}+\frac{2 b d^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{315 c^5}-\frac{8 b d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{189 c^5}-\frac{128 b d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{4725 c^5}+\frac{64 b d^2 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{4725 c^3}+\frac{8}{315} d^2 x^5 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{729} b^2 c^4 d^2 x^9+\frac{4208 b^2 d^2 x}{99225 c^4}+\frac{212 b^2 c^2 d^2 x^7}{27783}-\frac{2104 b^2 d^2 x^3}{297675 c^2}+\frac{526 b^2 d^2 x^5}{165375}",1,"(d^2*(99225*a^2*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4) - 630*a*b*Sqrt[1 + c^2*x^2]*(2104 - 1052*c^2*x^2 + 789*c^4*x^4 + 2650*c^6*x^6 + 1225*c^8*x^8) + 2*b^2*c*x*(662760 - 110460*c^2*x^2 + 49707*c^4*x^4 + 119250*c^6*x^6 + 42875*c^8*x^8) - 630*b*(-315*a*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4) + b*Sqrt[1 + c^2*x^2]*(2104 - 1052*c^2*x^2 + 789*c^4*x^4 + 2650*c^6*x^6 + 1225*c^8*x^8))*ArcSinh[c*x] + 99225*b^2*c^5*x^5*(63 + 90*c^2*x^2 + 35*c^4*x^4)*ArcSinh[c*x]^2))/(31255875*c^5)","A",1
208,1,237,296,0.3635574,"\int x^3 \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]","\frac{d^2 \left(c x \left(1152 a^2 c^3 x^3 \left(3 c^4 x^4+8 c^2 x^2+6\right)-6 a b \sqrt{c^2 x^2+1} \left(144 c^6 x^6+344 c^4 x^4+146 c^2 x^2-219\right)+b^2 c x \left(108 c^6 x^6+344 c^4 x^4+219 c^2 x^2-657\right)\right)+6 b \sinh ^{-1}(c x) \left(3 a \left(384 c^8 x^8+1024 c^6 x^6+768 c^4 x^4-73\right)-b c x \sqrt{c^2 x^2+1} \left(144 c^6 x^6+344 c^4 x^4+146 c^2 x^2-219\right)\right)+9 b^2 \left(384 c^8 x^8+1024 c^6 x^6+768 c^4 x^4-73\right) \sinh ^{-1}(c x)^2\right)}{27648 c^4}","-\frac{73 d^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3072 c^4}-\frac{1}{32} b c d^2 x^5 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{25}{576} b c d^2 x^5 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{8} d^2 x^4 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{12} d^2 x^4 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{73 b d^2 x^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2304 c}+\frac{73 b d^2 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{1536 c^3}+\frac{1}{24} d^2 x^4 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{256} b^2 c^4 d^2 x^8+\frac{43 b^2 c^2 d^2 x^6}{3456}-\frac{73 b^2 d^2 x^2}{3072 c^2}+\frac{73 b^2 d^2 x^4}{9216}",1,"(d^2*(c*x*(1152*a^2*c^3*x^3*(6 + 8*c^2*x^2 + 3*c^4*x^4) + b^2*c*x*(-657 + 219*c^2*x^2 + 344*c^4*x^4 + 108*c^6*x^6) - 6*a*b*Sqrt[1 + c^2*x^2]*(-219 + 146*c^2*x^2 + 344*c^4*x^4 + 144*c^6*x^6)) + 6*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(-219 + 146*c^2*x^2 + 344*c^4*x^4 + 144*c^6*x^6)) + 3*a*(-73 + 768*c^4*x^4 + 1024*c^6*x^6 + 384*c^8*x^8))*ArcSinh[c*x] + 9*b^2*(-73 + 768*c^4*x^4 + 1024*c^6*x^6 + 384*c^8*x^8)*ArcSinh[c*x]^2))/(27648*c^4)","A",1
209,1,227,303,0.4042848,"\int x^2 \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]","\frac{d^2 \left(11025 a^2 c^3 x^3 \left(15 c^4 x^4+42 c^2 x^2+35\right)-210 a b \sqrt{c^2 x^2+1} \left(225 c^6 x^6+612 c^4 x^4+409 c^2 x^2-818\right)-210 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+1} \left(225 c^6 x^6+612 c^4 x^4+409 c^2 x^2-818\right)-105 a c^3 x^3 \left(15 c^4 x^4+42 c^2 x^2+35\right)\right)+2 b^2 c x \left(3375 c^6 x^6+12852 c^4 x^4+14315 c^2 x^2-85890\right)+11025 b^2 c^3 x^3 \left(15 c^4 x^4+42 c^2 x^2+35\right) \sinh ^{-1}(c x)^2\right)}{1157625 c^3}","-\frac{16 b d^2 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{315 c}+\frac{1}{7} d^2 x^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{4}{35} d^2 x^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b d^2 \left(c^2 x^2+1\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{49 c^3}+\frac{2 b d^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{175 c^3}+\frac{8 b d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{105 c^3}+\frac{32 b d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{315 c^3}+\frac{8}{105} d^2 x^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{343} b^2 c^4 d^2 x^7+\frac{136 b^2 c^2 d^2 x^5}{6125}-\frac{1636 b^2 d^2 x}{11025 c^2}+\frac{818 b^2 d^2 x^3}{33075}",1,"(d^2*(11025*a^2*c^3*x^3*(35 + 42*c^2*x^2 + 15*c^4*x^4) - 210*a*b*Sqrt[1 + c^2*x^2]*(-818 + 409*c^2*x^2 + 612*c^4*x^4 + 225*c^6*x^6) + 2*b^2*c*x*(-85890 + 14315*c^2*x^2 + 12852*c^4*x^4 + 3375*c^6*x^6) - 210*b*(-105*a*c^3*x^3*(35 + 42*c^2*x^2 + 15*c^4*x^4) + b*Sqrt[1 + c^2*x^2]*(-818 + 409*c^2*x^2 + 612*c^4*x^4 + 225*c^6*x^6))*ArcSinh[c*x] + 11025*b^2*c^3*x^3*(35 + 42*c^2*x^2 + 15*c^4*x^4)*ArcSinh[c*x]^2))/(1157625*c^3)","A",1
210,1,208,204,0.5000167,"\int x \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]","\frac{d^2 \left(c x \left(144 a^2 c x \left(c^4 x^4+3 c^2 x^2+3\right)-6 a b \sqrt{c^2 x^2+1} \left(8 c^4 x^4+26 c^2 x^2+33\right)+b^2 c x \left(8 c^4 x^4+39 c^2 x^2+99\right)\right)+6 b \sinh ^{-1}(c x) \left(3 a \left(16 c^6 x^6+48 c^4 x^4+48 c^2 x^2+11\right)-b c x \sqrt{c^2 x^2+1} \left(8 c^4 x^4+26 c^2 x^2+33\right)\right)+9 b^2 \left(16 c^6 x^6+48 c^4 x^4+48 c^2 x^2+11\right) \sinh ^{-1}(c x)^2\right)}{864 c^2}","-\frac{b d^2 x \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{18 c}-\frac{5 b d^2 x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{72 c}-\frac{5 b d^2 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{48 c}+\frac{d^2 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{6 c^2}-\frac{5 d^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{96 c^2}+\frac{5}{288} b^2 c^2 d^2 x^4+\frac{b^2 d^2 \left(c^2 x^2+1\right)^3}{108 c^2}+\frac{25}{288} b^2 d^2 x^2",1,"(d^2*(c*x*(144*a^2*c*x*(3 + 3*c^2*x^2 + c^4*x^4) - 6*a*b*Sqrt[1 + c^2*x^2]*(33 + 26*c^2*x^2 + 8*c^4*x^4) + b^2*c*x*(99 + 39*c^2*x^2 + 8*c^4*x^4)) + 6*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(33 + 26*c^2*x^2 + 8*c^4*x^4)) + 3*a*(11 + 48*c^2*x^2 + 48*c^4*x^4 + 16*c^6*x^6))*ArcSinh[c*x] + 9*b^2*(11 + 48*c^2*x^2 + 48*c^4*x^4 + 16*c^6*x^6)*ArcSinh[c*x]^2))/(864*c^2)","A",1
211,1,191,214,0.4063413,"\int \left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]","\frac{d^2 \left(225 a^2 c x \left(3 c^4 x^4+10 c^2 x^2+15\right)-30 a b \sqrt{c^2 x^2+1} \left(9 c^4 x^4+38 c^2 x^2+149\right)-30 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+1} \left(9 c^4 x^4+38 c^2 x^2+149\right)-15 a c x \left(3 c^4 x^4+10 c^2 x^2+15\right)\right)+2 b^2 c x \left(27 c^4 x^4+190 c^2 x^2+2235\right)+225 b^2 c x \left(3 c^4 x^4+10 c^2 x^2+15\right) \sinh ^{-1}(c x)^2\right)}{3375 c}","\frac{1}{5} d^2 x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{4}{15} d^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b d^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{25 c}-\frac{8 b d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{45 c}-\frac{16 b d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{15 c}+\frac{8}{15} d^2 x \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{125} b^2 c^4 d^2 x^5+\frac{76}{675} b^2 c^2 d^2 x^3+\frac{298}{225} b^2 d^2 x",1,"(d^2*(225*a^2*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) - 30*a*b*Sqrt[1 + c^2*x^2]*(149 + 38*c^2*x^2 + 9*c^4*x^4) + 2*b^2*c*x*(2235 + 190*c^2*x^2 + 27*c^4*x^4) - 30*b*(-15*a*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 + c^2*x^2]*(149 + 38*c^2*x^2 + 9*c^4*x^4))*ArcSinh[c*x] + 225*b^2*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]^2))/(3375*c)","A",1
212,1,323,257,0.3955598,"\int \frac{\left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x,x]","\frac{1}{768} d^2 \left(192 a^2 c^4 x^4+768 a^2 c^2 x^2+768 a^2 \log (c x)+384 a b c^4 x^4 \sinh ^{-1}(c x)-624 a b c x \sqrt{c^2 x^2+1}+1536 a b c^2 x^2 \sinh ^{-1}(c x)-96 a b c^3 x^3 \sqrt{c^2 x^2+1}-768 a b \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+768 a b \sinh ^{-1}(c x)^2+624 a b \sinh ^{-1}(c x)+1536 a b \sinh ^{-1}(c x) \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)+768 b^2 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-384 b^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)-256 b^2 \sinh ^{-1}(c x)^3-288 b^2 \sinh ^{-1}(c x) \sinh \left(2 \sinh ^{-1}(c x)\right)-12 b^2 \sinh ^{-1}(c x) \sinh \left(4 \sinh ^{-1}(c x)\right)+768 b^2 \sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+144 b^2 \cosh \left(2 \sinh ^{-1}(c x)\right)+288 b^2 \sinh ^{-1}(c x)^2 \cosh \left(2 \sinh ^{-1}(c x)\right)+3 b^2 \cosh \left(4 \sinh ^{-1}(c x)\right)+24 b^2 \sinh ^{-1}(c x)^2 \cosh \left(4 \sinh ^{-1}(c x)\right)\right)","-\frac{1}{8} b c d^2 x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{11}{16} b c d^2 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{4} d^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{2} d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-b d^2 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{d^2 \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b}-\frac{11}{32} d^2 \left(a+b \sinh ^{-1}(c x)\right)^2+d^2 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{32} b^2 c^4 d^2 x^4+\frac{13}{32} b^2 c^2 d^2 x^2-\frac{1}{2} b^2 d^2 \text{Li}_3\left(e^{-2 \sinh ^{-1}(c x)}\right)",1,"(d^2*(768*a^2*c^2*x^2 + 192*a^2*c^4*x^4 - 624*a*b*c*x*Sqrt[1 + c^2*x^2] - 96*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] + 624*a*b*ArcSinh[c*x] + 1536*a*b*c^2*x^2*ArcSinh[c*x] + 384*a*b*c^4*x^4*ArcSinh[c*x] + 768*a*b*ArcSinh[c*x]^2 - 256*b^2*ArcSinh[c*x]^3 + 144*b^2*Cosh[2*ArcSinh[c*x]] + 288*b^2*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] + 3*b^2*Cosh[4*ArcSinh[c*x]] + 24*b^2*ArcSinh[c*x]^2*Cosh[4*ArcSinh[c*x]] + 1536*a*b*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] + 768*b^2*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 768*a^2*Log[c*x] - 768*a*b*PolyLog[2, E^(-2*ArcSinh[c*x])] + 768*b^2*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - 384*b^2*PolyLog[3, E^(2*ArcSinh[c*x])] - 288*b^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 12*b^2*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/768","A",0
213,1,306,229,1.1698678,"\int \frac{\left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^2,x]","\frac{1}{54} d^2 \left(18 a^2 c^4 x^3+108 a^2 c^2 x-\frac{54 a^2}{x}+36 a b c^4 x^3 \sinh ^{-1}(c x)-12 a b c \left(c^2 x^2-2\right) \sqrt{c^2 x^2+1}+216 a b c \left(c x \sinh ^{-1}(c x)-\sqrt{c^2 x^2+1}\right)-\frac{108 a b \left(c x \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+\sinh ^{-1}(c x)\right)}{x}+2 b^2 c^2 x \left(2 c^2 x^2+9 c^2 x^2 \sinh ^{-1}(c x)^2-12\right)-189 b^2 c \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+108 b^2 c^2 x \left(\sinh ^{-1}(c x)^2+2\right)+108 b^2 c \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-108 b^2 c \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-\frac{54 b^2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+2 c x \left(\log \left(e^{-\sinh ^{-1}(c x)}+1\right)-\log \left(1-e^{-\sinh ^{-1}(c x)}\right)\right)\right)}{x}-3 b^2 c \sinh ^{-1}(c x) \cosh \left(3 \sinh ^{-1}(c x)\right)\right)","\frac{4}{3} c^2 d^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2}{9} b c d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{10}{3} b c d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)-\frac{d^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{x}+\frac{8}{3} c^2 d^2 x \left(a+b \sinh ^{-1}(c x)\right)^2-4 b c d^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{2}{27} b^2 c^4 d^2 x^3+\frac{32}{9} b^2 c^2 d^2 x-2 b^2 c d^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+2 b^2 c d^2 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)",1,"(d^2*((-54*a^2)/x + 108*a^2*c^2*x + 18*a^2*c^4*x^3 - 12*a*b*c*(-2 + c^2*x^2)*Sqrt[1 + c^2*x^2] + 36*a*b*c^4*x^3*ArcSinh[c*x] - 189*b^2*c*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 216*a*b*c*(-Sqrt[1 + c^2*x^2] + c*x*ArcSinh[c*x]) + 108*b^2*c^2*x*(2 + ArcSinh[c*x]^2) + 2*b^2*c^2*x*(-12 + 2*c^2*x^2 + 9*c^2*x^2*ArcSinh[c*x]^2) - (108*a*b*(ArcSinh[c*x] + c*x*ArcTanh[Sqrt[1 + c^2*x^2]]))/x - 3*b^2*c*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]] - (54*b^2*ArcSinh[c*x]*(ArcSinh[c*x] + 2*c*x*(-Log[1 - E^(-ArcSinh[c*x])] + Log[1 + E^(-ArcSinh[c*x])])))/x + 108*b^2*c*PolyLog[2, -E^(-ArcSinh[c*x])] - 108*b^2*c*PolyLog[2, E^(-ArcSinh[c*x])]))/54","A",0
214,1,305,272,0.8818818,"\int \frac{\left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^3,x]","\frac{1}{2} d^2 \left(a^2 c^4 x^2+4 a^2 c^2 \log (x)-\frac{a^2}{x^2}+4 a b c^2 \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+2 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)\right)-\text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)\right)+a b c^2 \left(\left(2 c^2 x^2+1\right) \sinh ^{-1}(c x)-c x \sqrt{c^2 x^2+1}\right)-\frac{2 a b \left(c x \sqrt{c^2 x^2+1}+\sinh ^{-1}(c x)\right)}{x^2}-\frac{2}{3} b^2 c^2 \left(-6 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+3 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)+2 \sinh ^{-1}(c x)^2 \left(\sinh ^{-1}(c x)-3 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)\right)\right)-\frac{b^2 \left(-2 c^2 x^2 \log (c x)+2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2\right)}{x^2}+\frac{1}{4} b^2 c^2 \left(\left(2 \sinh ^{-1}(c x)^2+1\right) \cosh \left(2 \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \sinh \left(2 \sinh ^{-1}(c x)\right)\right)\right)","-2 b c^2 d^2 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+c^2 d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b c d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{x}-\frac{d^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 x^2}+\frac{2 c^2 d^2 \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b}+\frac{1}{4} c^2 d^2 \left(a+b \sinh ^{-1}(c x)\right)^2+2 c^2 d^2 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{2} b c^3 d^2 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{4} b^2 c^4 d^2 x^2-b^2 c^2 d^2 \text{Li}_3\left(e^{-2 \sinh ^{-1}(c x)}\right)+b^2 c^2 d^2 \log (x)",1,"(d^2*(-(a^2/x^2) + a^2*c^4*x^2 - (2*a*b*(c*x*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/x^2 + a*b*c^2*(-(c*x*Sqrt[1 + c^2*x^2]) + (1 + 2*c^2*x^2)*ArcSinh[c*x]) + 4*a^2*c^2*Log[x] - (b^2*(2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]^2 - 2*c^2*x^2*Log[c*x]))/x^2 + 4*a*b*c^2*(ArcSinh[c*x]*(ArcSinh[c*x] + 2*Log[1 - E^(-2*ArcSinh[c*x])]) - PolyLog[2, E^(-2*ArcSinh[c*x])]) - (2*b^2*c^2*(2*ArcSinh[c*x]^2*(ArcSinh[c*x] - 3*Log[1 - E^(2*ArcSinh[c*x])]) - 6*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] + 3*PolyLog[3, E^(2*ArcSinh[c*x])]))/3 + (b^2*c^2*((1 + 2*ArcSinh[c*x]^2)*Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))/4))/2","A",0
215,1,357,248,0.9119277,"\int \frac{\left(d+c^2 d x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2)/x^4,x]","\frac{d^2 \left(3 a^2 c^4 x^4-6 a^2 c^2 x^2-a^2+6 a b c^4 x^4 \sinh ^{-1}(c x)-a b c x \sqrt{c^2 x^2+1}-12 a b c^2 x^2 \sinh ^{-1}(c x)-6 a b c^3 x^3 \sqrt{c^2 x^2+1}-11 a b c^3 x^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-2 a b \sinh ^{-1}(c x)+6 b^2 c^4 x^4+3 b^2 c^4 x^4 \sinh ^{-1}(c x)^2+11 b^2 c^3 x^3 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-11 b^2 c^3 x^3 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+11 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-11 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-b^2 c^2 x^2-6 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-6 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-b^2 \sinh ^{-1}(c x)^2\right)}{3 x^3}","\frac{8}{3} c^4 d^2 x \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{22}{3} b c^3 d^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{4 c^2 d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 x}-\frac{b c d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^2}-\frac{d^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 x^3}-\frac{5}{3} b c^3 d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+2 b^2 c^4 d^2 x-\frac{11}{3} b^2 c^3 d^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+\frac{11}{3} b^2 c^3 d^2 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)-\frac{b^2 c^2 d^2}{3 x}",1,"(d^2*(-a^2 - 6*a^2*c^2*x^2 - b^2*c^2*x^2 + 3*a^2*c^4*x^4 + 6*b^2*c^4*x^4 - a*b*c*x*Sqrt[1 + c^2*x^2] - 6*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 2*a*b*ArcSinh[c*x] - 12*a*b*c^2*x^2*ArcSinh[c*x] + 6*a*b*c^4*x^4*ArcSinh[c*x] - b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - b^2*ArcSinh[c*x]^2 - 6*b^2*c^2*x^2*ArcSinh[c*x]^2 + 3*b^2*c^4*x^4*ArcSinh[c*x]^2 - 11*a*b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] + 11*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 11*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 11*b^2*c^3*x^3*PolyLog[2, -E^(-ArcSinh[c*x])] - 11*b^2*c^3*x^3*PolyLog[2, E^(-ArcSinh[c*x])]))/(3*x^3)","A",0
216,1,299,465,0.4774325,"\int x^4 \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^4*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]","\frac{d^3 \left(12006225 a^2 c^5 x^5 \left(105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right)-6930 a b \sqrt{c^2 x^2+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)-6930 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+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)-3465 a c^5 x^5 \left(105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right)\right)+12006225 b^2 c^5 x^5 \left(105 c^6 x^6+385 c^4 x^4+495 c^2 x^2+231\right) \sinh ^{-1}(c x)^2+2 b^2 c x \left(10418625 c^{10} x^{10}+42917875 c^8 x^8+58224375 c^6 x^6+13120569 c^4 x^4-29156820 c^2 x^2+174940920\right)\right)}{13867189875 c^5}","\frac{1}{11} d^3 x^5 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{33} d^3 x^5 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{8}{231} d^3 x^5 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{32 b d^3 x^4 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{5775 c}-\frac{2 b d^3 \left(c^2 x^2+1\right)^{11/2} \left(a+b \sinh ^{-1}(c x)\right)}{121 c^5}+\frac{8 b d^3 \left(c^2 x^2+1\right)^{9/2} \left(a+b \sinh ^{-1}(c x)\right)}{297 c^5}-\frac{2 b d^3 \left(c^2 x^2+1\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{1617 c^5}+\frac{4 b d^3 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{1155 c^5}-\frac{16 b d^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{693 c^5}-\frac{256 b d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{17325 c^5}+\frac{128 b d^3 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{17325 c^3}+\frac{16 d^3 x^5 \left(a+b \sinh ^{-1}(c x)\right)^2}{1155}+\frac{2 b^2 c^6 d^3 x^{11}}{1331}+\frac{182 b^2 c^4 d^3 x^9}{29403}+\frac{100976 b^2 d^3 x}{4002075 c^4}+\frac{9410 b^2 c^2 d^3 x^7}{1120581}-\frac{50488 b^2 d^3 x^3}{12006225 c^2}+\frac{12622 b^2 d^3 x^5}{6670125}",1,"(d^3*(12006225*a^2*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6) - 6930*a*b*Sqrt[1 + c^2*x^2]*(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) + 2*b^2*c*x*(174940920 - 29156820*c^2*x^2 + 13120569*c^4*x^4 + 58224375*c^6*x^6 + 42917875*c^8*x^8 + 10418625*c^10*x^10) - 6930*b*(-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^2*x^2]*(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))*ArcSinh[c*x] + 12006225*b^2*c^5*x^5*(231 + 495*c^2*x^2 + 385*c^4*x^4 + 105*c^6*x^6)*ArcSinh[c*x]^2))/(13867189875*c^5)","A",1
217,1,285,376,0.4569999,"\int x^3 \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]","\frac{d^3 \left(c x \left(28800 a^2 c^3 x^3 \left(4 c^6 x^6+15 c^4 x^4+20 c^2 x^2+10\right)-30 a b \sqrt{c^2 x^2+1} \left(768 c^8 x^8+2736 c^6 x^6+3208 c^4 x^4+790 c^2 x^2-1185\right)+b^2 c x \left(2304 c^8 x^8+10260 c^6 x^6+16040 c^4 x^4+5925 c^2 x^2-17775\right)\right)+30 b \sinh ^{-1}(c x) \left(15 a \left(512 c^{10} x^{10}+1920 c^8 x^8+2560 c^6 x^6+1280 c^4 x^4-79\right)-b c x \sqrt{c^2 x^2+1} \left(768 c^8 x^8+2736 c^6 x^6+3208 c^4 x^4+790 c^2 x^2-1185\right)\right)+225 b^2 \left(512 c^{10} x^{10}+1920 c^8 x^8+2560 c^6 x^6+1280 c^4 x^4-79\right) \sinh ^{-1}(c x)^2\right)}{1152000 c^4}","-\frac{79 d^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{5120 c^4}-\frac{1}{50} b c d^3 x^5 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{1}{32} b c d^3 x^5 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{31}{960} b c d^3 x^5 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{10} d^3 x^4 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{3}{40} d^3 x^4 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{20} d^3 x^4 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{79 b d^3 x^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3840 c}+\frac{79 b d^3 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2560 c^3}+\frac{1}{40} d^3 x^4 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{500} b^2 c^6 d^3 x^{10}+\frac{57 b^2 c^4 d^3 x^8}{6400}+\frac{401 b^2 c^2 d^3 x^6}{28800}-\frac{79 b^2 d^3 x^2}{5120 c^2}+\frac{79 b^2 d^3 x^4}{15360}",1,"(d^3*(c*x*(28800*a^2*c^3*x^3*(10 + 20*c^2*x^2 + 15*c^4*x^4 + 4*c^6*x^6) - 30*a*b*Sqrt[1 + c^2*x^2]*(-1185 + 790*c^2*x^2 + 3208*c^4*x^4 + 2736*c^6*x^6 + 768*c^8*x^8) + b^2*c*x*(-17775 + 5925*c^2*x^2 + 16040*c^4*x^4 + 10260*c^6*x^6 + 2304*c^8*x^8)) + 30*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(-1185 + 790*c^2*x^2 + 3208*c^4*x^4 + 2736*c^6*x^6 + 768*c^8*x^8)) + 15*a*(-79 + 1280*c^4*x^4 + 2560*c^6*x^6 + 1920*c^8*x^8 + 512*c^10*x^10))*ArcSinh[c*x] + 225*b^2*(-79 + 1280*c^4*x^4 + 2560*c^6*x^6 + 1920*c^8*x^8 + 512*c^10*x^10)*ArcSinh[c*x]^2))/(1152000*c^4)","A",1
218,1,275,382,0.5099375,"\int x^2 \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]","\frac{d^3 \left(99225 a^2 c^3 x^3 \left(35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right)-630 a b \sqrt{c^2 x^2+1} \left(1225 c^8 x^8+4675 c^6 x^6+6297 c^4 x^4+2629 c^2 x^2-5258\right)-630 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+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 a c^3 x^3 \left(35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right)\right)+b^2 \left(85750 c^9 x^9+420750 c^7 x^7+793422 c^5 x^5+552090 c^3 x^3-3312540 c x\right)+99225 b^2 c^3 x^3 \left(35 c^6 x^6+135 c^4 x^4+189 c^2 x^2+105\right) \sinh ^{-1}(c x)^2\right)}{31255875 c^3}","-\frac{32 b d^3 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{945 c}+\frac{1}{9} d^3 x^3 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{21} d^3 x^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{8}{105} d^3 x^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b d^3 \left(c^2 x^2+1\right)^{9/2} \left(a+b \sinh ^{-1}(c x)\right)}{81 c^3}+\frac{2 b d^3 \left(c^2 x^2+1\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{441 c^3}+\frac{4 b d^3 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{525 c^3}+\frac{16 b d^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{315 c^3}+\frac{64 b d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{945 c^3}+\frac{16}{315} d^3 x^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{729} b^2 c^6 d^3 x^9+\frac{374 b^2 c^4 d^3 x^7}{27783}+\frac{4198 b^2 c^2 d^3 x^5}{165375}-\frac{10516 b^2 d^3 x}{99225 c^2}+\frac{5258 b^2 d^3 x^3}{297675}",1,"(d^3*(99225*a^2*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6) - 630*a*b*Sqrt[1 + c^2*x^2]*(-5258 + 2629*c^2*x^2 + 6297*c^4*x^4 + 4675*c^6*x^6 + 1225*c^8*x^8) + b^2*(-3312540*c*x + 552090*c^3*x^3 + 793422*c^5*x^5 + 420750*c^7*x^7 + 85750*c^9*x^9) - 630*b*(-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^2*x^2]*(-5258 + 2629*c^2*x^2 + 6297*c^4*x^4 + 4675*c^6*x^6 + 1225*c^8*x^8))*ArcSinh[c*x] + 99225*b^2*c^3*x^3*(105 + 189*c^2*x^2 + 135*c^4*x^4 + 35*c^6*x^6)*ArcSinh[c*x]^2))/(31255875*c^3)","A",1
219,1,256,261,0.6647551,"\int x \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]","\frac{d^3 \left(c x \left(1152 a^2 c x \left(c^6 x^6+4 c^4 x^4+6 c^2 x^2+4\right)-6 a b \sqrt{c^2 x^2+1} \left(48 c^6 x^6+200 c^4 x^4+326 c^2 x^2+279\right)+b^2 c x \left(36 c^6 x^6+200 c^4 x^4+489 c^2 x^2+837\right)\right)+6 b \sinh ^{-1}(c x) \left(3 a \left(128 c^8 x^8+512 c^6 x^6+768 c^4 x^4+512 c^2 x^2+93\right)-b c x \sqrt{c^2 x^2+1} \left(48 c^6 x^6+200 c^4 x^4+326 c^2 x^2+279\right)\right)+9 b^2 \left(128 c^8 x^8+512 c^6 x^6+768 c^4 x^4+512 c^2 x^2+93\right) \sinh ^{-1}(c x)^2\right)}{9216 c^2}","-\frac{b d^3 x \left(c^2 x^2+1\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{32 c}-\frac{7 b d^3 x \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{192 c}-\frac{35 b d^3 x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{768 c}-\frac{35 b d^3 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{512 c}+\frac{d^3 \left(c^2 x^2+1\right)^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{8 c^2}-\frac{35 d^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{1024 c^2}+\frac{35 b^2 c^2 d^3 x^4}{3072}+\frac{b^2 d^3 \left(c^2 x^2+1\right)^4}{256 c^2}+\frac{7 b^2 d^3 \left(c^2 x^2+1\right)^3}{1152 c^2}+\frac{175 b^2 d^3 x^2}{3072}",1,"(d^3*(c*x*(1152*a^2*c*x*(4 + 6*c^2*x^2 + 4*c^4*x^4 + c^6*x^6) + b^2*c*x*(837 + 489*c^2*x^2 + 200*c^4*x^4 + 36*c^6*x^6) - 6*a*b*Sqrt[1 + c^2*x^2]*(279 + 326*c^2*x^2 + 200*c^4*x^4 + 48*c^6*x^6)) + 6*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(279 + 326*c^2*x^2 + 200*c^4*x^4 + 48*c^6*x^6)) + 3*a*(93 + 512*c^2*x^2 + 768*c^4*x^4 + 512*c^6*x^6 + 128*c^8*x^8))*ArcSinh[c*x] + 9*b^2*(93 + 512*c^2*x^2 + 768*c^4*x^4 + 512*c^6*x^6 + 128*c^8*x^8)*ArcSinh[c*x]^2))/(9216*c^2)","A",1
220,1,239,291,0.5615831,"\int \left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]","\frac{d^3 \left(11025 a^2 c x \left(5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right)-210 a b \sqrt{c^2 x^2+1} \left(75 c^6 x^6+351 c^4 x^4+757 c^2 x^2+2161\right)-210 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+1} \left(75 c^6 x^6+351 c^4 x^4+757 c^2 x^2+2161\right)-105 a c x \left(5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right)\right)+2 b^2 c x \left(1125 c^6 x^6+7371 c^4 x^4+26495 c^2 x^2+226905\right)+11025 b^2 c x \left(5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right) \sinh ^{-1}(c x)^2\right)}{385875 c}","\frac{1}{7} d^3 x \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{6}{35} d^3 x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{8}{35} d^3 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b d^3 \left(c^2 x^2+1\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)}{49 c}-\frac{12 b d^3 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{175 c}-\frac{16 b d^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{105 c}-\frac{32 b d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{35 c}+\frac{16}{35} d^3 x \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{343} b^2 c^6 d^3 x^7+\frac{234 b^2 c^4 d^3 x^5}{6125}+\frac{1514 b^2 c^2 d^3 x^3}{11025}+\frac{4322 b^2 d^3 x}{3675}",1,"(d^3*(11025*a^2*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) - 210*a*b*Sqrt[1 + c^2*x^2]*(2161 + 757*c^2*x^2 + 351*c^4*x^4 + 75*c^6*x^6) + 2*b^2*c*x*(226905 + 26495*c^2*x^2 + 7371*c^4*x^4 + 1125*c^6*x^6) - 210*b*(-105*a*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 + c^2*x^2]*(2161 + 757*c^2*x^2 + 351*c^4*x^4 + 75*c^6*x^6))*ArcSinh[c*x] + 11025*b^2*c*x*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6)*ArcSinh[c*x]^2))/(385875*c)","A",1
221,1,416,337,0.6928584,"\int \frac{\left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x,x]","\frac{d^3 \left(576 a^2 c^6 x^6+2592 a^2 c^4 x^4+5184 a^2 c^2 x^2+3456 a^2 \log (c x)+1152 a b c^6 x^6 \sinh ^{-1}(c x)+5184 a b c^4 x^4 \sinh ^{-1}(c x)-3600 a b c x \sqrt{c^2 x^2+1}+10368 a b c^2 x^2 \sinh ^{-1}(c x)-192 a b c^5 x^5 \sqrt{c^2 x^2+1}-1056 a b c^3 x^3 \sqrt{c^2 x^2+1}-3456 a b \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+3456 a b \sinh ^{-1}(c x)^2+3600 a b \sinh ^{-1}(c x)+6912 a b \sinh ^{-1}(c x) \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)+3456 b^2 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-1728 b^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)-1152 b^2 \sinh ^{-1}(c x)^3-1566 b^2 \sinh ^{-1}(c x) \sinh \left(2 \sinh ^{-1}(c x)\right)-108 b^2 \sinh ^{-1}(c x) \sinh \left(4 \sinh ^{-1}(c x)\right)-6 b^2 \sinh ^{-1}(c x) \sinh \left(6 \sinh ^{-1}(c x)\right)+3456 b^2 \sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+783 b^2 \cosh \left(2 \sinh ^{-1}(c x)\right)+1566 b^2 \sinh ^{-1}(c x)^2 \cosh \left(2 \sinh ^{-1}(c x)\right)+27 b^2 \cosh \left(4 \sinh ^{-1}(c x)\right)+216 b^2 \sinh ^{-1}(c x)^2 \cosh \left(4 \sinh ^{-1}(c x)\right)+b^2 \cosh \left(6 \sinh ^{-1}(c x)\right)+18 b^2 \sinh ^{-1}(c x)^2 \cosh \left(6 \sinh ^{-1}(c x)\right)\right)}{3456}","-\frac{1}{18} b c d^3 x \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{7}{36} b c d^3 x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{19}{24} b c d^3 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{6} d^3 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{4} d^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{2} d^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-b d^3 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{d^3 \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b}-\frac{19}{48} d^3 \left(a+b \sinh ^{-1}(c x)\right)^2+d^3 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{7}{144} b^2 c^4 d^3 x^4+\frac{71}{144} b^2 c^2 d^3 x^2+\frac{1}{108} b^2 d^3 \left(c^2 x^2+1\right)^3-\frac{1}{2} b^2 d^3 \text{Li}_3\left(e^{-2 \sinh ^{-1}(c x)}\right)",1,"(d^3*(5184*a^2*c^2*x^2 + 2592*a^2*c^4*x^4 + 576*a^2*c^6*x^6 - 3600*a*b*c*x*Sqrt[1 + c^2*x^2] - 1056*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 192*a*b*c^5*x^5*Sqrt[1 + c^2*x^2] + 3600*a*b*ArcSinh[c*x] + 10368*a*b*c^2*x^2*ArcSinh[c*x] + 5184*a*b*c^4*x^4*ArcSinh[c*x] + 1152*a*b*c^6*x^6*ArcSinh[c*x] + 3456*a*b*ArcSinh[c*x]^2 - 1152*b^2*ArcSinh[c*x]^3 + 783*b^2*Cosh[2*ArcSinh[c*x]] + 1566*b^2*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] + 27*b^2*Cosh[4*ArcSinh[c*x]] + 216*b^2*ArcSinh[c*x]^2*Cosh[4*ArcSinh[c*x]] + b^2*Cosh[6*ArcSinh[c*x]] + 18*b^2*ArcSinh[c*x]^2*Cosh[6*ArcSinh[c*x]] + 6912*a*b*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] + 3456*b^2*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 3456*a^2*Log[c*x] - 3456*a*b*PolyLog[2, E^(-2*ArcSinh[c*x])] + 3456*b^2*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - 1728*b^2*PolyLog[3, E^(2*ArcSinh[c*x])] - 1566*b^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 108*b^2*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]] - 6*b^2*ArcSinh[c*x]*Sinh[6*ArcSinh[c*x]]))/3456","A",0
222,1,466,307,1.4267256,"\int \frac{\left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^2,x]","\frac{1}{720} d^3 \left(144 a^2 c^6 x^5+720 a^2 c^4 x^3+2160 a^2 c^2 x-\frac{720 a^2}{x}+288 a b c^6 x^5 \sinh ^{-1}(c x)+1440 a b c^4 x^3 \sinh ^{-1}(c x)-\frac{17568}{5} a b c \sqrt{c^2 x^2+1}-1440 a b c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+4320 a b c^2 x \sinh ^{-1}(c x)-\frac{288}{5} a b c^5 x^4 \sqrt{c^2 x^2+1}-\frac{2016}{5} a b c^3 x^2 \sqrt{c^2 x^2+1}-\frac{1440 a b \sinh ^{-1}(c x)}{x}-3420 b^2 c \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+3460 b^2 c^2 x+1890 b^2 c^2 x \sinh ^{-1}(c x)^2+360 b^2 c^2 x \sinh ^{-1}(c x)^2 \cosh \left(2 \sinh ^{-1}(c x)\right)+80 b^2 c^2 x \cosh \left(2 \sinh ^{-1}(c x)\right)+1440 b^2 c \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-1440 b^2 c \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-10 b^2 c \sinh \left(3 \sinh ^{-1}(c x)\right)-45 b^2 c \sinh ^{-1}(c x)^2 \sinh \left(3 \sinh ^{-1}(c x)\right)+\frac{18}{25} b^2 c \sinh \left(5 \sinh ^{-1}(c x)\right)+9 b^2 c \sinh ^{-1}(c x)^2 \sinh \left(5 \sinh ^{-1}(c x)\right)-\frac{720 b^2 \sinh ^{-1}(c x)^2}{x}+1440 b^2 c \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-1440 b^2 c \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-90 b^2 c \sinh ^{-1}(c x) \cosh \left(3 \sinh ^{-1}(c x)\right)-\frac{18}{5} b^2 c \sinh ^{-1}(c x) \cosh \left(5 \sinh ^{-1}(c x)\right)\right)","\frac{6}{5} c^2 d^3 x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{8}{5} c^2 d^3 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2}{25} b c d^3 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{2}{5} b c d^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{22}{5} b c d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)-\frac{d^3 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{x}+\frac{16}{5} c^2 d^3 x \left(a+b \sinh ^{-1}(c x)\right)^2-4 b c d^3 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{2}{125} b^2 c^6 d^3 x^5+\frac{14}{75} b^2 c^4 d^3 x^3+\frac{122}{25} b^2 c^2 d^3 x-2 b^2 c d^3 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+2 b^2 c d^3 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)",1,"(d^3*((-720*a^2)/x + 2160*a^2*c^2*x + 3460*b^2*c^2*x + 720*a^2*c^4*x^3 + 144*a^2*c^6*x^5 - (17568*a*b*c*Sqrt[1 + c^2*x^2])/5 - (2016*a*b*c^3*x^2*Sqrt[1 + c^2*x^2])/5 - (288*a*b*c^5*x^4*Sqrt[1 + c^2*x^2])/5 - (1440*a*b*ArcSinh[c*x])/x + 4320*a*b*c^2*x*ArcSinh[c*x] + 1440*a*b*c^4*x^3*ArcSinh[c*x] + 288*a*b*c^6*x^5*ArcSinh[c*x] - 3420*b^2*c*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - (720*b^2*ArcSinh[c*x]^2)/x + 1890*b^2*c^2*x*ArcSinh[c*x]^2 - 1440*a*b*c*ArcTanh[Sqrt[1 + c^2*x^2]] + 80*b^2*c^2*x*Cosh[2*ArcSinh[c*x]] + 360*b^2*c^2*x*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] - 90*b^2*c*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]] - (18*b^2*c*ArcSinh[c*x]*Cosh[5*ArcSinh[c*x]])/5 + 1440*b^2*c*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 1440*b^2*c*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 1440*b^2*c*PolyLog[2, -E^(-ArcSinh[c*x])] - 1440*b^2*c*PolyLog[2, E^(-ArcSinh[c*x])] - 10*b^2*c*Sinh[3*ArcSinh[c*x]] - 45*b^2*c*ArcSinh[c*x]^2*Sinh[3*ArcSinh[c*x]] + (18*b^2*c*Sinh[5*ArcSinh[c*x]])/25 + 9*b^2*c*ArcSinh[c*x]^2*Sinh[5*ArcSinh[c*x]]))/720","A",0
223,1,459,354,1.2024003,"\int \frac{\left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^3,x]","\frac{1}{256} d^3 \left(64 a^2 c^6 x^4+384 a^2 c^4 x^2+768 a^2 c^2 \log (x)-\frac{128 a^2}{x^2}+128 a b c^6 x^4 \sinh ^{-1}(c x)+768 a b c^4 x^2 \sinh ^{-1}(c x)-768 a b c^2 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-\frac{256 a b c \sqrt{c^2 x^2+1}}{x}+768 a b c^2 \sinh ^{-1}(c x)^2+336 a b c^2 \sinh ^{-1}(c x)+1536 a b c^2 \sinh ^{-1}(c x) \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)-32 a b c^5 x^3 \sqrt{c^2 x^2+1}-336 a b c^3 x \sqrt{c^2 x^2+1}-\frac{256 a b \sinh ^{-1}(c x)}{x^2}+768 b^2 c^2 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-384 b^2 c^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)-\frac{256 b^2 c \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{x}+256 b^2 c^2 \log (c x)-256 b^2 c^2 \sinh ^{-1}(c x)^3-160 b^2 c^2 \sinh ^{-1}(c x) \sinh \left(2 \sinh ^{-1}(c x)\right)-4 b^2 c^2 \sinh ^{-1}(c x) \sinh \left(4 \sinh ^{-1}(c x)\right)+768 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+80 b^2 c^2 \cosh \left(2 \sinh ^{-1}(c x)\right)+160 b^2 c^2 \sinh ^{-1}(c x)^2 \cosh \left(2 \sinh ^{-1}(c x)\right)+b^2 c^2 \cosh \left(4 \sinh ^{-1}(c x)\right)+8 b^2 c^2 \sinh ^{-1}(c x)^2 \cosh \left(4 \sinh ^{-1}(c x)\right)-\frac{128 b^2 \sinh ^{-1}(c x)^2}{x^2}\right)","-3 b c^2 d^3 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)+\frac{3}{4} c^2 d^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{3}{2} c^2 d^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b c d^3 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{x}-\frac{d^3 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 x^2}+\frac{c^2 d^3 \left(a+b \sinh ^{-1}(c x)\right)^3}{b}-\frac{3}{32} c^2 d^3 \left(a+b \sinh ^{-1}(c x)\right)^2+3 c^2 d^3 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{7}{8} b c^3 d^3 x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{3}{16} b c^3 d^3 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{32} b^2 c^6 d^3 x^4+\frac{21}{32} b^2 c^4 d^3 x^2-\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left(e^{-2 \sinh ^{-1}(c x)}\right)+b^2 c^2 d^3 \log (x)",1,"(d^3*((-128*a^2)/x^2 + 384*a^2*c^4*x^2 + 64*a^2*c^6*x^4 - (256*a*b*c*Sqrt[1 + c^2*x^2])/x - 336*a*b*c^3*x*Sqrt[1 + c^2*x^2] - 32*a*b*c^5*x^3*Sqrt[1 + c^2*x^2] + 336*a*b*c^2*ArcSinh[c*x] - (256*a*b*ArcSinh[c*x])/x^2 + 768*a*b*c^4*x^2*ArcSinh[c*x] + 128*a*b*c^6*x^4*ArcSinh[c*x] - (256*b^2*c*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/x + 768*a*b*c^2*ArcSinh[c*x]^2 - (128*b^2*ArcSinh[c*x]^2)/x^2 - 256*b^2*c^2*ArcSinh[c*x]^3 + 80*b^2*c^2*Cosh[2*ArcSinh[c*x]] + 160*b^2*c^2*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] + b^2*c^2*Cosh[4*ArcSinh[c*x]] + 8*b^2*c^2*ArcSinh[c*x]^2*Cosh[4*ArcSinh[c*x]] + 1536*a*b*c^2*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] + 768*b^2*c^2*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 768*a^2*c^2*Log[x] + 256*b^2*c^2*Log[c*x] - 768*a*b*c^2*PolyLog[2, E^(-2*ArcSinh[c*x])] + 768*b^2*c^2*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - 384*b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])] - 160*b^2*c^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 4*b^2*c^2*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/256","A",0
224,1,461,326,1.0990223,"\int \frac{\left(d+c^2 d x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2)/x^4,x]","\frac{d^3 \left(9 a^2 c^6 x^6+81 a^2 c^4 x^4-81 a^2 c^2 x^2-9 a^2+18 a b c^6 x^6 \sinh ^{-1}(c x)+162 a b c^4 x^4 \sinh ^{-1}(c x)-9 a b c x \sqrt{c^2 x^2+1}-162 a b c^2 x^2 \sinh ^{-1}(c x)-6 a b c^5 x^5 \sqrt{c^2 x^2+1}-150 a b c^3 x^3 \sqrt{c^2 x^2+1}-153 a b c^3 x^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-18 a b \sinh ^{-1}(c x)+2 b^2 c^6 x^6+9 b^2 c^6 x^6 \sinh ^{-1}(c x)^2+150 b^2 c^4 x^4+81 b^2 c^4 x^4 \sinh ^{-1}(c x)^2+153 b^2 c^3 x^3 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-153 b^2 c^3 x^3 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+153 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-153 b^2 c^3 x^3 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-9 b^2 c^2 x^2-81 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-9 b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-6 b^2 c^5 x^5 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-150 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-9 b^2 \sinh ^{-1}(c x)^2\right)}{27 x^3}","\frac{16}{3} c^4 d^3 x \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{34}{3} b c^3 d^3 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{2 c^2 d^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{x}-\frac{b c d^3 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^2}-\frac{d^3 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 x^3}+\frac{8}{3} c^4 d^3 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{9} b c^3 d^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-5 b c^3 d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)+\frac{2}{27} b^2 c^6 d^3 x^3+\frac{50}{9} b^2 c^4 d^3 x-\frac{17}{3} b^2 c^3 d^3 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)+\frac{17}{3} b^2 c^3 d^3 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)-\frac{b^2 c^2 d^3}{3 x}",1,"(d^3*(-9*a^2 - 81*a^2*c^2*x^2 - 9*b^2*c^2*x^2 + 81*a^2*c^4*x^4 + 150*b^2*c^4*x^4 + 9*a^2*c^6*x^6 + 2*b^2*c^6*x^6 - 9*a*b*c*x*Sqrt[1 + c^2*x^2] - 150*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 6*a*b*c^5*x^5*Sqrt[1 + c^2*x^2] - 18*a*b*ArcSinh[c*x] - 162*a*b*c^2*x^2*ArcSinh[c*x] + 162*a*b*c^4*x^4*ArcSinh[c*x] + 18*a*b*c^6*x^6*ArcSinh[c*x] - 9*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 150*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^5*x^5*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 9*b^2*ArcSinh[c*x]^2 - 81*b^2*c^2*x^2*ArcSinh[c*x]^2 + 81*b^2*c^4*x^4*ArcSinh[c*x]^2 + 9*b^2*c^6*x^6*ArcSinh[c*x]^2 - 153*a*b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] + 153*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 153*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 153*b^2*c^3*x^3*PolyLog[2, -E^(-ArcSinh[c*x])] - 153*b^2*c^3*x^3*PolyLog[2, E^(-ArcSinh[c*x])]))/(27*x^3)","A",0
225,1,365,277,1.3238247,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{d+c^2 d x^2} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2),x]","\frac{a^2 c^3 x^3-3 a^2 c x+3 a^2 \tan ^{-1}(c x)-\frac{2}{3} a b \left(-3 c^3 x^3 \sinh ^{-1}(c x)+c^2 x^2 \sqrt{c^2 x^2+1}-11 \sqrt{c^2 x^2+1}+9 i \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)-9 i \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+9 c x \sinh ^{-1}(c x)-9 i \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)+9 i \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)+3 b^2 \left(\frac{5}{2} \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+i \left(-2 \sinh ^{-1}(c x) \left(\text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)\right)-2 \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)+2 \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)-\left(\sinh ^{-1}(c x)^2 \left(\log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-\log \left(1+i e^{-\sinh ^{-1}(c x)}\right)\right)\right)\right)-\frac{5}{4} c x \left(\sinh ^{-1}(c x)^2+2\right)+\frac{1}{108} \left(9 \sinh ^{-1}(c x)^2+2\right) \sinh \left(3 \sinh ^{-1}(c x)\right)-\frac{1}{18} \sinh ^{-1}(c x) \cosh \left(3 \sinh ^{-1}(c x)\right)\right)}{3 c^5 d}","-\frac{2 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d}+\frac{2 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d}+\frac{2 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c^5 d}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{c^4 d}+\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^2 d}+\frac{22 b \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c^5 d}-\frac{2 b x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c^3 d}+\frac{2 i b^2 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{c^5 d}-\frac{2 i b^2 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{c^5 d}-\frac{22 b^2 x}{9 c^4 d}+\frac{2 b^2 x^3}{27 c^2 d}",1,"(-3*a^2*c*x + a^2*c^3*x^3 + 3*a^2*ArcTan[c*x] - (2*a*b*(-11*Sqrt[1 + c^2*x^2] + c^2*x^2*Sqrt[1 + c^2*x^2] + 9*c*x*ArcSinh[c*x] - 3*c^3*x^3*ArcSinh[c*x] - (9*I)*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (9*I)*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + (9*I)*PolyLog[2, (-I)*E^ArcSinh[c*x]] - (9*I)*PolyLog[2, I*E^ArcSinh[c*x]]))/3 + 3*b^2*((5*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/2 - (5*c*x*(2 + ArcSinh[c*x]^2))/4 - (ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]])/18 + I*(-(ArcSinh[c*x]^2*(Log[1 - I/E^ArcSinh[c*x]] - Log[1 + I/E^ArcSinh[c*x]])) - 2*ArcSinh[c*x]*(PolyLog[2, (-I)/E^ArcSinh[c*x]] - PolyLog[2, I/E^ArcSinh[c*x]]) - 2*PolyLog[3, (-I)/E^ArcSinh[c*x]] + 2*PolyLog[3, I/E^ArcSinh[c*x]]) + ((2 + 9*ArcSinh[c*x]^2)*Sinh[3*ArcSinh[c*x]])/108))/(3*c^5*d)","A",0
226,1,279,199,0.4554696,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{d+c^2 d x^2} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2),x]","\frac{12 a^2 c^2 x^2-12 a^2 \log \left(c^2 x^2+1\right)-12 a b c x \sqrt{c^2 x^2+1}+24 a b c^2 x^2 \sinh ^{-1}(c x)-48 a b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)-48 a b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+24 a b \sinh ^{-1}(c x)^2+12 a b \sinh ^{-1}(c x)-48 a b \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-48 a b \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+24 b^2 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)+12 b^2 \text{Li}_3\left(-e^{-2 \sinh ^{-1}(c x)}\right)-8 b^2 \sinh ^{-1}(c x)^3-6 b^2 \sinh \left(2 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-24 b^2 \sinh ^{-1}(c x)^2 \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)+6 b^2 \sinh ^{-1}(c x)^2 \cosh \left(2 \sinh ^{-1}(c x)\right)+3 b^2 \cosh \left(2 \sinh ^{-1}(c x)\right)}{24 c^4 d}","-\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^4 d}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c^4 d}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^4 d}-\frac{\log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c^4 d}+\frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^2 d}-\frac{b x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 c^3 d}+\frac{b^2 \text{Li}_3\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 c^4 d}+\frac{b^2 x^2}{4 c^2 d}",1,"(12*a^2*c^2*x^2 - 12*a*b*c*x*Sqrt[1 + c^2*x^2] + 12*a*b*ArcSinh[c*x] + 24*a*b*c^2*x^2*ArcSinh[c*x] + 24*a*b*ArcSinh[c*x]^2 - 8*b^2*ArcSinh[c*x]^3 + 3*b^2*Cosh[2*ArcSinh[c*x]] + 6*b^2*ArcSinh[c*x]^2*Cosh[2*ArcSinh[c*x]] - 24*b^2*ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] - 48*a*b*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - 48*a*b*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - 12*a^2*Log[1 + c^2*x^2] + 24*b^2*ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] - 48*a*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] - 48*a*b*PolyLog[2, I*E^ArcSinh[c*x]] + 12*b^2*PolyLog[3, -E^(-2*ArcSinh[c*x])] - 6*b^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]])/(24*c^4*d)","C",0
227,1,293,198,0.593831,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{d+c^2 d x^2} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2),x]","-\frac{a^2 \tan ^{-1}(c x)}{c^3 d}+\frac{a^2 x}{c^2 d}+\frac{2 a b \left(-\sqrt{c^2 x^2+1}+i \left(\text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)\right)+c x \sinh ^{-1}(c x)+i \sinh ^{-1}(c x) \left(\log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-\log \left(1+i e^{-\sinh ^{-1}(c x)}\right)\right)\right)}{c^3 d}+\frac{b^2 \left(-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-i \left(-2 \sinh ^{-1}(c x) \left(\text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)\right)-2 \left(\text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)\right)-\left(\sinh ^{-1}(c x)^2 \left(\log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-\log \left(1+i e^{-\sinh ^{-1}(c x)}\right)\right)\right)\right)+c x \left(\sinh ^{-1}(c x)^2+2\right)\right)}{c^3 d}","\frac{2 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d}-\frac{2 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d}-\frac{2 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c^3 d}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{c^2 d}-\frac{2 b \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d}-\frac{2 i b^2 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{c^3 d}+\frac{2 i b^2 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{c^3 d}+\frac{2 b^2 x}{c^2 d}",1,"(a^2*x)/(c^2*d) - (a^2*ArcTan[c*x])/(c^3*d) + (2*a*b*(-Sqrt[1 + c^2*x^2] + c*x*ArcSinh[c*x] + I*ArcSinh[c*x]*(Log[1 - I/E^ArcSinh[c*x]] - Log[1 + I/E^ArcSinh[c*x]]) + I*(PolyLog[2, (-I)/E^ArcSinh[c*x]] - PolyLog[2, I/E^ArcSinh[c*x]])))/(c^3*d) + (b^2*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*(2 + ArcSinh[c*x]^2) - I*(-(ArcSinh[c*x]^2*(Log[1 - I/E^ArcSinh[c*x]] - Log[1 + I/E^ArcSinh[c*x]])) - 2*ArcSinh[c*x]*(PolyLog[2, (-I)/E^ArcSinh[c*x]] - PolyLog[2, I/E^ArcSinh[c*x]]) - 2*(PolyLog[3, (-I)/E^ArcSinh[c*x]] - PolyLog[3, I/E^ArcSinh[c*x]]))))/(c^3*d)","A",0
228,1,281,105,0.2255023,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{d+c^2 d x^2} \, dx","Integrate[(x*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2),x]","\frac{3 a^2 \log \left(c^2 x^2+1\right)+12 b \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right) \left(a+b \sinh ^{-1}(c x)\right)+12 b \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right) \left(a+b \sinh ^{-1}(c x)\right)+12 a b \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+12 a b \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-6 a b \sinh ^{-1}(c x)^2-12 b^2 \text{Li}_3\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)-12 b^2 \text{Li}_3\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)+6 b^2 \sinh ^{-1}(c x)^2 \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+6 b^2 \sinh ^{-1}(c x)^2 \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-2 b^2 \sinh ^{-1}(c x)^3}{6 c^2 d}","\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c^2 d}+\frac{\log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c^2 d}-\frac{b^2 \text{Li}_3\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 c^2 d}",1,"(-6*a*b*ArcSinh[c*x]^2 - 2*b^2*ArcSinh[c*x]^3 + 12*a*b*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 6*b^2*ArcSinh[c*x]^2*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 12*a*b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 6*b^2*ArcSinh[c*x]^2*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 3*a^2*Log[1 + c^2*x^2] + 12*b*(a + b*ArcSinh[c*x])*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 12*b*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 12*b^2*PolyLog[3, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 12*b^2*PolyLog[3, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(6*c^2*d)","B",0
229,1,274,138,0.2422997,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d+c^2 d x^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2),x]","-\frac{c \left(a^2 \sqrt{-c^2} \tan ^{-1}(c x)+2 b c \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right) \left(a+b \sinh ^{-1}(c x)\right)-2 b c \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right) \left(a+b \sinh ^{-1}(c x)\right)-2 a b c \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+2 a b c \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-2 b^2 c \text{Li}_3\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)+2 b^2 c \text{Li}_3\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)-b^2 c \sinh ^{-1}(c x)^2 \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+b^2 c \sinh ^{-1}(c x)^2 \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)\right)}{\left(-c^2\right)^{3/2} d}","-\frac{2 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c d}+\frac{2 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c d}+\frac{2 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c d}+\frac{2 i b^2 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{c d}-\frac{2 i b^2 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{c d}",1,"-((c*(a^2*Sqrt[-c^2]*ArcTan[c*x] - 2*a*b*c*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - b^2*c*ArcSinh[c*x]^2*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 2*a*b*c*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + b^2*c*ArcSinh[c*x]^2*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 2*b*c*(a + b*ArcSinh[c*x])*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 2*b*c*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 2*b^2*c*PolyLog[3, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 2*b^2*c*PolyLog[3, (Sqrt[-c^2]*E^ArcSinh[c*x])/c]))/((-c^2)^(3/2)*d))","A",0
230,1,400,116,0.3600983,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x \left(d+c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)),x]","-\frac{2 a^3+3 a^2 b \log \left(c^2 x^2+1\right)+6 a^2 b \sinh ^{-1}(c x)-6 a^2 b \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+12 b^2 \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right) \left(a+b \sinh ^{-1}(c x)\right)+12 b^2 \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right) \left(a+b \sinh ^{-1}(c x)\right)+12 a b^2 \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+12 a b^2 \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-6 a b^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-12 a b^2 \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)-12 b^3 \text{Li}_3\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)-12 b^3 \text{Li}_3\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)+6 b^3 \sinh ^{-1}(c x)^2 \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+6 b^3 \sinh ^{-1}(c x)^2 \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-6 b^3 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+3 b^3 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)-6 b^3 \sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)}{6 b d}","-\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}+\frac{b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}-\frac{2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d}+\frac{b^2 \text{Li}_3\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d}-\frac{b^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d}",1,"-1/6*(2*a^3 + 6*a^2*b*ArcSinh[c*x] + 12*a*b^2*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 6*b^3*ArcSinh[c*x]^2*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 12*a*b^2*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 6*b^3*ArcSinh[c*x]^2*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 6*a^2*b*Log[1 - E^(2*ArcSinh[c*x])] - 12*a*b^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] - 6*b^3*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 3*a^2*b*Log[1 + c^2*x^2] + 12*b^2*(a + b*ArcSinh[c*x])*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 12*b^2*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 6*a*b^2*PolyLog[2, E^(2*ArcSinh[c*x])] - 6*b^3*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - 12*b^3*PolyLog[3, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 12*b^3*PolyLog[3, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 3*b^3*PolyLog[3, E^(2*ArcSinh[c*x])])/(b*d)","B",0
231,1,363,204,1.063974,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^2 \left(d+c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)),x]","-\frac{a^2 c \tan ^{-1}(c x)+\frac{a^2}{x}+2 a b c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+\frac{1}{2} i a b c \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)\right)-\frac{1}{2} i a b c \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)\right)+\frac{2 a b \sinh ^{-1}(c x)}{x}-b^2 c \left(2 i \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-2 i \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+2 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-2 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+2 i \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)-2 i \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)-\frac{\sinh ^{-1}(c x)^2}{c x}+i \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-i \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+2 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-2 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{d}","\frac{2 i b c \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}-\frac{2 i b c \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d x}-\frac{2 c \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d}-\frac{4 b c \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}-\frac{2 b^2 c \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{d}+\frac{2 b^2 c \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{d}-\frac{2 i b^2 c \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{d}+\frac{2 i b^2 c \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{d}",1,"-((a^2/x + (2*a*b*ArcSinh[c*x])/x + a^2*c*ArcTan[c*x] + 2*a*b*c*ArcTanh[Sqrt[1 + c^2*x^2]] + (I/2)*a*b*c*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - (I/2)*a*b*c*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]]) - b^2*c*(-(ArcSinh[c*x]^2/(c*x)) + 2*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + I*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - I*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 2*PolyLog[2, -E^(-ArcSinh[c*x])] + (2*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (2*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - 2*PolyLog[2, E^(-ArcSinh[c*x])] + (2*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] - (2*I)*PolyLog[3, I/E^ArcSinh[c*x]]))/d)","A",0
232,1,419,194,1.0513955,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^3 \left(d+c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)),x]","-\frac{-a^2 c^2 \log \left(c^2 x^2+1\right)+2 a^2 c^2 \log (x)+\frac{a^2}{x^2}-4 a b c^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)-4 a b c^2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+2 a b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+\frac{2 a b c \sqrt{c^2 x^2+1}}{x}-4 a b c^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-4 a b c^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+4 a b c^2 \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+\frac{2 a b \sinh ^{-1}(c x)}{x^2}+2 b^2 c^2 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)+2 b^2 c^2 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+b^2 c^2 \text{Li}_3\left(-e^{-2 \sinh ^{-1}(c x)}\right)-b^2 c^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)+\frac{2 b^2 c \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{x}-2 b^2 c^2 \log (c x)-\frac{4}{3} b^2 c^2 \sinh ^{-1}(c x)^3-2 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)+2 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+\frac{1}{12} i \pi ^3 b^2 c^2+\frac{b^2 \sinh ^{-1}(c x)^2}{x^2}}{2 d}","\frac{b c^2 \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}-\frac{b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}-\frac{b c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{d x}+\frac{2 c^2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 d x^2}-\frac{b^2 c^2 \text{Li}_3\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d}+\frac{b^2 c^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d}+\frac{b^2 c^2 \log (x)}{d}",1,"-1/2*((I/12)*b^2*c^2*Pi^3 + a^2/x^2 + (2*a*b*c*Sqrt[1 + c^2*x^2])/x + (2*a*b*ArcSinh[c*x])/x^2 + (2*b^2*c*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/x + (b^2*ArcSinh[c*x]^2)/x^2 - (4*b^2*c^2*ArcSinh[c*x]^3)/3 - 2*b^2*c^2*ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] - 4*a*b*c^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - 4*a*b*c^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + 4*a*b*c^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 2*b^2*c^2*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 2*a^2*c^2*Log[x] - 2*b^2*c^2*Log[c*x] - a^2*c^2*Log[1 + c^2*x^2] + 2*b^2*c^2*ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] - 4*a*b*c^2*PolyLog[2, (-I)*E^ArcSinh[c*x]] - 4*a*b*c^2*PolyLog[2, I*E^ArcSinh[c*x]] + 2*a*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])] + 2*b^2*c^2*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] + b^2*c^2*PolyLog[3, -E^(-2*ArcSinh[c*x])] - b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/d","C",0
233,1,602,297,7.9600678,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^4 \left(d+c^2 d x^2\right)} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)),x]","\frac{a^2 c^3 \tan ^{-1}(c x)}{d}+\frac{a^2 c^2}{d x}-\frac{a^2}{3 d x^3}+\frac{2 a b \left(-\frac{1}{2} i c^4 \left(\frac{2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{c}\right)+\frac{1}{2} i c^4 \left(\frac{2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)}{c}\right)-\frac{c \sqrt{c^2 x^2+1}}{6 x^2}-c^2 \left(-c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-\frac{\sinh ^{-1}(c x)}{x}\right)+\frac{1}{6} c^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-\frac{\sinh ^{-1}(c x)}{3 x^3}\right)}{d}+\frac{b^2 c^3 \left(-\frac{8 \sinh ^{-1}(c x)^2 \sinh ^4\left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{c^3 x^3}-56 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-48 i \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+48 i \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+56 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-48 i \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)+48 i \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)-56 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-24 i \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+24 i \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+56 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-14 \sinh ^{-1}(c x)^2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+4 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+14 \sinh ^{-1}(c x)^2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\frac{1}{2} c x \sinh ^{-1}(c x)^2 \text{csch}^4\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{24 d}","-\frac{2 i b c^3 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}+\frac{2 i b c^3 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d}+\frac{2 c^3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d}+\frac{14 b c^3 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d}-\frac{b c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 d x^2}+\frac{c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{d x}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{3 d x^3}+\frac{7 b^2 c^3 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{3 d}-\frac{7 b^2 c^3 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{3 d}+\frac{2 i b^2 c^3 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{d}-\frac{2 i b^2 c^3 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{d}-\frac{b^2 c^2}{3 d x}",1,"-1/3*a^2/(d*x^3) + (a^2*c^2)/(d*x) + (a^2*c^3*ArcTan[c*x])/d + (2*a*b*(-1/6*(c*Sqrt[1 + c^2*x^2])/x^2 - ArcSinh[c*x]/(3*x^3) + (c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/6 - c^2*(-(ArcSinh[c*x]/x) - c*ArcTanh[Sqrt[1 + c^2*x^2]]) - (I/2)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/c) + (I/2)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, I*E^ArcSinh[c*x]])/c)))/d + (b^2*c^3*(-4*Coth[ArcSinh[c*x]/2] + 14*ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] - 2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - (c*x*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^4)/2 - 56*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - (24*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] + (24*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 56*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 56*PolyLog[2, -E^(-ArcSinh[c*x])] - (48*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (48*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] + 56*PolyLog[2, E^(-ArcSinh[c*x])] - (48*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] + (48*I)*PolyLog[3, I/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 - (8*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2]^4)/(c^3*x^3) + 4*Tanh[ArcSinh[c*x]/2] - 14*ArcSinh[c*x]^2*Tanh[ArcSinh[c*x]/2]))/(24*d)","B",0
234,1,482,279,2.2287742,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^2,x]","\frac{-\frac{3 a^2 \tan ^{-1}(c x)}{c^5}+\frac{2 a^2 x}{c^4}+\frac{a^2 x}{c^6 x^2+c^4}-\frac{2 a b \left(-2 c^3 x^3 \sinh ^{-1}(c x)-3 i \left(c^2 x^2+1\right) \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)+3 i \left(c^2 x^2+1\right) \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+2 c^2 x^2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1}+3 i c^2 x^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-3 i c^2 x^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)-3 c x \sinh ^{-1}(c x)+3 i \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-3 i \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)}{c^7 x^2+c^5}+\frac{2 b^2 \left(\frac{c x \sinh ^{-1}(c x)^2}{2 c^2 x^2+2}-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\frac{\sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{1}{2} i \left(6 \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-6 \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+6 \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)-6 \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)+3 \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-3 \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+c x \left(\sinh ^{-1}(c x)^2+2\right)\right)}{c^5}}{2 d^2}","\frac{3 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d^2}-\frac{3 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d^2}-\frac{3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c^5 d^2}+\frac{3 x \left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^4 d^2}-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^2 d^2 \left(c^2 x^2+1\right)}-\frac{2 b \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d^2}+\frac{b \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d^2 \sqrt{c^2 x^2+1}}-\frac{3 i b^2 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{c^5 d^2}+\frac{3 i b^2 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{c^5 d^2}-\frac{b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac{2 b^2 x}{c^4 d^2}",1,"((2*a^2*x)/c^4 + (a^2*x)/(c^4 + c^6*x^2) - (3*a^2*ArcTan[c*x])/c^5 - (2*a*b*(Sqrt[1 + c^2*x^2] + 2*c^2*x^2*Sqrt[1 + c^2*x^2] - 3*c*x*ArcSinh[c*x] - 2*c^3*x^3*ArcSinh[c*x] + (3*I)*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + (3*I)*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - (3*I)*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (3*I)*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - (3*I)*(1 + c^2*x^2)*PolyLog[2, (-I)*E^ArcSinh[c*x]] + (3*I)*(1 + c^2*x^2)*PolyLog[2, I*E^ArcSinh[c*x]]))/(c^5 + c^7*x^2) + (2*b^2*(ArcSinh[c*x]/Sqrt[1 + c^2*x^2] - 2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + (c*x*ArcSinh[c*x]^2)/(2 + 2*c^2*x^2) + c*x*(2 + ArcSinh[c*x]^2) + (I/2)*((4*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 3*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - 3*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 6*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - 6*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] + 6*PolyLog[3, (-I)/E^ArcSinh[c*x]] - 6*PolyLog[3, I/E^ArcSinh[c*x]])))/c^5)/(2*d^2)","A",0
235,1,320,213,1.098554,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^2,x]","\frac{\frac{a^2}{c^2 x^2+1}+a^2 \log \left(c^2 x^2+1\right)-\frac{a b \left(\sqrt{c^2 x^2+1}-i \sinh ^{-1}(c x)\right)}{c x+i}-\frac{a b \left(\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right)}{c x-i}+4 a b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)+4 a b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)-a b \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-a b \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)+2 b^2 \left(\frac{1}{2} \log \left(c^2 x^2+1\right)+\frac{\sinh ^{-1}(c x)^2}{2 c^2 x^2+2}-\frac{c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\sinh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(-e^{-2 \sinh ^{-1}(c x)}\right)+\frac{1}{3} \sinh ^{-1}(c x)^3+\sinh ^{-1}(c x)^2 \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)\right)}{2 c^4 d^2}","\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^4 d^2}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c^4 d^2}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^4 d^2}+\frac{\log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c^4 d^2}-\frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^2 d^2 \left(c^2 x^2+1\right)}-\frac{b x \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d^2 \sqrt{c^2 x^2+1}}-\frac{b^2 \text{Li}_3\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 c^4 d^2}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^4 d^2}",1,"(a^2/(1 + c^2*x^2) - (a*b*(Sqrt[1 + c^2*x^2] - I*ArcSinh[c*x]))/(I + c*x) - (a*b*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(-I + c*x) - a*b*ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - a*b*ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) + a^2*Log[1 + c^2*x^2] + 4*a*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + 4*a*b*PolyLog[2, I*E^ArcSinh[c*x]] + 2*b^2*(-((c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2]) + ArcSinh[c*x]^2/(2 + 2*c^2*x^2) + ArcSinh[c*x]^3/3 + ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] + Log[1 + c^2*x^2]/2 - ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] - PolyLog[3, -E^(-2*ArcSinh[c*x])]/2))/(2*c^4*d^2)","C",0
236,1,385,213,1.8635263,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^2,x]","-\frac{\frac{a^2 c x}{c^2 x^2+1}+a^2 \left(-\tan ^{-1}(c x)\right)+\frac{a b \left(\sinh ^{-1}(c x)-i \sqrt{c^2 x^2+1}\right)}{c x-i}+\frac{a b \left(\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right)}{c x+i}-\frac{1}{2} i a b \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)\right)+\frac{1}{2} i a b \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)\right)+\frac{b^2 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}+\frac{2 b^2 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+i b^2 \left(2 \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-2 \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+2 \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)-2 \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)+\sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{2 c^3 d^2}","-\frac{i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d^2}+\frac{i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d^2}+\frac{\tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c^3 d^2}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^2 d^2 \left(c^2 x^2+1\right)}-\frac{b \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d^2 \sqrt{c^2 x^2+1}}+\frac{i b^2 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{c^3 d^2}-\frac{i b^2 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{c^3 d^2}+\frac{b^2 \tan ^{-1}(c x)}{c^3 d^2}",1,"-1/2*((a^2*c*x)/(1 + c^2*x^2) + (2*b^2*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + (b^2*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) + (a*b*((-I)*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(-I + c*x) + (a*b*(I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(I + c*x) - a^2*ArcTan[c*x] - (I/2)*a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) + (I/2)*a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]]) + I*b^2*((4*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 2*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] + 2*PolyLog[3, (-I)/E^ArcSinh[c*x]] - 2*PolyLog[3, I/E^ArcSinh[c*x]]))/(c^3*d^2)","A",0
237,1,145,85,0.2197479,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^2,x]","-\frac{a^2}{2 c^2 d^2 \left(c^2 x^2+1\right)}+\frac{a b x}{c d^2 \sqrt{c^2 x^2+1}}+\frac{b \sinh ^{-1}(c x) \left(b c x \sqrt{c^2 x^2+1}-a\right)}{c^2 d^2 \left(c^2 x^2+1\right)}-\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^2 d^2}-\frac{b^2 \sinh ^{-1}(c x)^2}{2 c^2 d^2 \left(c^2 x^2+1\right)}","\frac{b x \left(a+b \sinh ^{-1}(c x)\right)}{c d^2 \sqrt{c^2 x^2+1}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^2 d^2 \left(c^2 x^2+1\right)}-\frac{b^2 \log \left(c^2 x^2+1\right)}{2 c^2 d^2}",1,"-1/2*a^2/(c^2*d^2*(1 + c^2*x^2)) + (a*b*x)/(c*d^2*Sqrt[1 + c^2*x^2]) + (b*(-a + b*c*x*Sqrt[1 + c^2*x^2])*ArcSinh[c*x])/(c^2*d^2*(1 + c^2*x^2)) - (b^2*ArcSinh[c*x]^2)/(2*c^2*d^2*(1 + c^2*x^2)) - (b^2*Log[1 + c^2*x^2])/(2*c^2*d^2)","A",1
238,1,403,210,1.4169789,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^2,x]","\frac{\frac{a^2 x}{c^2 x^2+1}+\frac{a^2 \tan ^{-1}(c x)}{c}+\frac{2 a b \left(-i \left(c^2 x^2+1\right) \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)+i \left(c^2 x^2+1\right) \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)+\sqrt{c^2 x^2+1}+i c^2 x^2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-i c^2 x^2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)+c x \sinh ^{-1}(c x)+i \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)-i \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)}{c^3 x^2+c}+\frac{2 b^2 \left(\frac{c x \sinh ^{-1}(c x)^2}{2 c^2 x^2+2}+\frac{\sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{1}{2} i \left(2 \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-2 \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+2 \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)-2 \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)+\sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{c}}{2 d^2}","\frac{b \left(a+b \sinh ^{-1}(c x)\right)}{c d^2 \sqrt{c^2 x^2+1}}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{2 d^2 \left(c^2 x^2+1\right)}-\frac{i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c d^2}+\frac{i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c d^2}+\frac{\tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c d^2}+\frac{i b^2 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{c d^2}-\frac{i b^2 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{c d^2}-\frac{b^2 \tan ^{-1}(c x)}{c d^2}",1,"((a^2*x)/(1 + c^2*x^2) + (a^2*ArcTan[c*x])/c + (2*a*b*(Sqrt[1 + c^2*x^2] + c*x*ArcSinh[c*x] + I*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + I*c^2*x^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - I*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - I*c^2*x^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] - I*(1 + c^2*x^2)*PolyLog[2, (-I)*E^ArcSinh[c*x]] + I*(1 + c^2*x^2)*PolyLog[2, I*E^ArcSinh[c*x]]))/(c + c^3*x^2) + (2*b^2*(ArcSinh[c*x]/Sqrt[1 + c^2*x^2] + (c*x*ArcSinh[c*x]^2)/(2 + 2*c^2*x^2) - (I/2)*((-4*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 2*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] + 2*PolyLog[3, (-I)/E^ArcSinh[c*x]] - 2*PolyLog[3, I/E^ArcSinh[c*x]])))/c)/(2*d^2)","A",0
239,1,428,193,1.8567852,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x \left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^2),x]","\frac{\frac{a^2}{c^2 x^2+1}-a^2 \log \left(c^2 x^2+1\right)+2 a^2 \log (c x)-\frac{a b \left(\sqrt{c^2 x^2+1}-i \sinh ^{-1}(c x)\right)}{c x+i}-\frac{a b \left(\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right)}{c x-i}+2 a b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+a b \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)\right)+a b \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)\right)-2 a b \sinh ^{-1}(c x)^2+4 a b \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+2 b^2 \left(\frac{1}{2} \log \left(c^2 x^2+1\right)+\frac{\sinh ^{-1}(c x)^2}{2 c^2 x^2+2}-\frac{c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\sinh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)+\sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+\frac{1}{2} \text{Li}_3\left(-e^{-2 \sinh ^{-1}(c x)}\right)-\frac{1}{2} \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)-\frac{2}{3} \sinh ^{-1}(c x)^3-\sinh ^{-1}(c x)^2 \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)+\sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+\frac{i \pi ^3}{24}\right)}{2 d^2}","-\frac{b c x \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \sqrt{c^2 x^2+1}}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 d^2 \left(c^2 x^2+1\right)}-\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}+\frac{b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}-\frac{2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d^2}+\frac{b^2 \log \left(c^2 x^2+1\right)}{2 d^2}+\frac{b^2 \text{Li}_3\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d^2}-\frac{b^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d^2}",1,"(a^2/(1 + c^2*x^2) - (a*b*(Sqrt[1 + c^2*x^2] - I*ArcSinh[c*x]))/(I + c*x) - (a*b*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(-I + c*x) - 2*a*b*ArcSinh[c*x]^2 + 4*a*b*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 2*a^2*Log[c*x] - a^2*Log[1 + c^2*x^2] + a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) + a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]]) + 2*a*b*PolyLog[2, E^(2*ArcSinh[c*x])] + 2*b^2*((I/24)*Pi^3 - (c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ArcSinh[c*x]^2/(2 + 2*c^2*x^2) - (2*ArcSinh[c*x]^3)/3 - ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] + ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + Log[1 + c^2*x^2]/2 + ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] + ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] + PolyLog[3, -E^(-2*ArcSinh[c*x])]/2 - PolyLog[3, E^(2*ArcSinh[c*x])]/2))/(2*d^2)","C",0
240,1,549,287,7.2108839,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^2 \left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^2),x]","-\frac{a^2 c^2 x}{2 d^2 \left(c^2 x^2+1\right)}-\frac{3 a^2 c \tan ^{-1}(c x)}{2 d^2}-\frac{a^2}{d^2 x}+\frac{2 a b c \left(\frac{\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)}{4 (-1-i c x)}-\frac{\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}}{4 (c x+i)}-\tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+\frac{3}{4} i \left(2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)-\frac{1}{2} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)-\frac{3}{4} i \left(2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)-\frac{1}{2} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-\frac{\sinh ^{-1}(c x)}{c x}\right)}{d^2}+\frac{b^2 c \left(-\frac{c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}-\frac{2 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+6 i \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-6 i \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+6 i \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)-6 i \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)+3 i \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-3 i \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+\sinh ^{-1}(c x)^2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+\sinh ^{-1}(c x)^2 \left(-\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{2 d^2}","-\frac{b c \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \sqrt{c^2 x^2+1}}-\frac{3 c^2 x \left(a+b \sinh ^{-1}(c x)\right)^2}{2 d^2 \left(c^2 x^2+1\right)}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d^2 x \left(c^2 x^2+1\right)}+\frac{3 i b c \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}-\frac{3 i b c \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}-\frac{3 c \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d^2}-\frac{4 b c \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}-\frac{2 b^2 c \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{d^2}+\frac{2 b^2 c \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{d^2}-\frac{3 i b^2 c \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{d^2}+\frac{3 i b^2 c \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{d^2}+\frac{b^2 c \tan ^{-1}(c x)}{d^2}",1,"-(a^2/(d^2*x)) - (a^2*c^2*x)/(2*d^2*(1 + c^2*x^2)) - (3*a^2*c*ArcTan[c*x])/(2*d^2) + (2*a*b*c*((Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x])/(4*(-1 - I*c*x)) - ArcSinh[c*x]/(c*x) - (I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x])/(4*(I + c*x)) - ArcTanh[Sqrt[1 + c^2*x^2]] + ((3*I)/4)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + 2*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - ((3*I)/4)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 2*PolyLog[2, I*E^ArcSinh[c*x]])))/d^2 + (b^2*c*((-2*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) + 4*ArcTan[Tanh[ArcSinh[c*x]/2]] - ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + (3*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - (3*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] + (6*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (6*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - 4*PolyLog[2, E^(-ArcSinh[c*x])] + (6*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] - (6*I)*PolyLog[3, I/E^ArcSinh[c*x]] + ArcSinh[c*x]^2*Tanh[ArcSinh[c*x]/2]))/(2*d^2)","A",0
241,1,594,253,0.9101457,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^3 \left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^2),x]","\frac{2 a^2 c^2 \log \left(c^2 x^2+1\right)+\frac{a^2}{c^2 x^4+x^2}+4 a^2 c^2 \sinh ^{-1}(c x)-4 a^2 c^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)-\frac{2 a^2}{x^2}-4 a b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+8 b c^2 \text{Li}_2\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right) \left(a+b \sinh ^{-1}(c x)\right)+8 b c^2 \text{Li}_2\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right) \left(a+b \sinh ^{-1}(c x)\right)-\frac{2 a b c}{x \sqrt{c^2 x^2+1}}+\frac{2 a b \sinh ^{-1}(c x)}{c^2 x^4+x^2}+8 a b c^2 \sinh ^{-1}(c x) \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+8 a b c^2 \sinh ^{-1}(c x) \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-8 a b c^2 \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)-\frac{4 a b \sinh ^{-1}(c x)}{x^2}-4 b^2 c^2 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-8 b^2 c^2 \text{Li}_3\left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}\right)-8 b^2 c^2 \text{Li}_3\left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}\right)+2 b^2 c^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)-b^2 c^2 \log \left(c^2 x^2+1\right)-\frac{2 b^2 c \sinh ^{-1}(c x)}{x \sqrt{c^2 x^2+1}}+\frac{b^2 \sinh ^{-1}(c x)^2}{c^2 x^4+x^2}+2 b^2 c^2 \log (x)+4 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left(\frac{c e^{\sinh ^{-1}(c x)}}{\sqrt{-c^2}}+1\right)+4 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left(\frac{\sqrt{-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right)-4 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)-\frac{2 b^2 \sinh ^{-1}(c x)^2}{x^2}}{2 d^2}","\frac{2 b c^2 \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}-\frac{2 b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}-\frac{c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{d^2 \left(c^2 x^2+1\right)}-\frac{b c \left(a+b \sinh ^{-1}(c x)\right)}{d^2 x \sqrt{c^2 x^2+1}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 d^2 x^2 \left(c^2 x^2+1\right)}+\frac{4 c^2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d^2}-\frac{b^2 c^2 \text{Li}_3\left(-e^{2 \sinh ^{-1}(c x)}\right)}{d^2}+\frac{b^2 c^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)}{d^2}-\frac{b^2 c^2 \log \left(c^2 x^2+1\right)}{2 d^2}+\frac{b^2 c^2 \log (x)}{d^2}",1,"((-2*a^2)/x^2 - (2*a*b*c)/(x*Sqrt[1 + c^2*x^2]) + a^2/(x^2 + c^2*x^4) + 4*a^2*c^2*ArcSinh[c*x] - (4*a*b*ArcSinh[c*x])/x^2 - (2*b^2*c*ArcSinh[c*x])/(x*Sqrt[1 + c^2*x^2]) + (2*a*b*ArcSinh[c*x])/(x^2 + c^2*x^4) - (2*b^2*ArcSinh[c*x]^2)/x^2 + (b^2*ArcSinh[c*x]^2)/(x^2 + c^2*x^4) + 8*a*b*c^2*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 4*b^2*c^2*ArcSinh[c*x]^2*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 8*a*b*c^2*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 4*b^2*c^2*ArcSinh[c*x]^2*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 4*a^2*c^2*Log[1 - E^(2*ArcSinh[c*x])] - 8*a*b*c^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] - 4*b^2*c^2*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 2*b^2*c^2*Log[x] + 2*a^2*c^2*Log[1 + c^2*x^2] - b^2*c^2*Log[1 + c^2*x^2] + 8*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 8*b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 4*a*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])] - 4*b^2*c^2*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - 8*b^2*c^2*PolyLog[3, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 8*b^2*c^2*PolyLog[3, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 2*b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d^2)","B",0
242,1,764,401,9.0374427,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^4 \left(d+c^2 d x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^2),x]","\frac{5 a^2 c^3 \tan ^{-1}(c x)}{2 d^2}+\frac{2 a^2 c^2}{d^2 x}+\frac{a^2 c^4 x}{2 d^2 \left(c^2 x^2+1\right)}-\frac{a^2}{3 d^2 x^3}+\frac{2 a b \left(-\frac{5}{4} i c^4 \left(\frac{2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{c}\right)+\frac{5}{4} i c^4 \left(\frac{2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)}{c}\right)-\frac{c \sqrt{c^2 x^2+1}}{6 x^2}-2 c^2 \left(-c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-\frac{\sinh ^{-1}(c x)}{x}\right)+\frac{c^4 \left(\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right)}{4 \left(c^2 x+i c\right)}-\frac{c^3 \left(\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right)}{4 (-1-i c x)}+\frac{1}{6} c^3 \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)-\frac{\sinh ^{-1}(c x)}{3 x^3}\right)}{d^2}+\frac{b^2 c^3 \left(-\frac{8 \sinh ^{-1}(c x)^2 \sinh ^4\left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{c^3 x^3}+\frac{12 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}+\frac{24 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-104 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-120 i \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+120 i \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+104 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-120 i \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)+120 i \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)-104 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-60 i \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+60 i \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+104 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-26 \sinh ^{-1}(c x)^2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+4 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+26 \sinh ^{-1}(c x)^2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\frac{1}{2} c x \sinh ^{-1}(c x)^2 \text{csch}^4\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-48 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{24 d^2}","-\frac{5 i b c^3 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}+\frac{5 i b c^3 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2}+\frac{5 c^3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d^2}+\frac{26 b c^3 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2}+\frac{5 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d^2 x \left(c^2 x^2+1\right)}-\frac{b c \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 x^2 \sqrt{c^2 x^2+1}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{3 d^2 x^3 \left(c^2 x^2+1\right)}+\frac{5 c^4 x \left(a+b \sinh ^{-1}(c x)\right)^2}{2 d^2 \left(c^2 x^2+1\right)}+\frac{2 b c^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 x^2+1}}+\frac{13 b^2 c^3 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{3 d^2}-\frac{13 b^2 c^3 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{3 d^2}+\frac{5 i b^2 c^3 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{d^2}-\frac{5 i b^2 c^3 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{d^2}-\frac{b^2 c^3 \tan ^{-1}(c x)}{d^2}-\frac{b^2 c^2}{3 d^2 x}",1,"-1/3*a^2/(d^2*x^3) + (2*a^2*c^2)/(d^2*x) + (a^2*c^4*x)/(2*d^2*(1 + c^2*x^2)) + (5*a^2*c^3*ArcTan[c*x])/(2*d^2) + (2*a*b*(-1/6*(c*Sqrt[1 + c^2*x^2])/x^2 - (c^3*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(4*(-1 - I*c*x)) - ArcSinh[c*x]/(3*x^3) + (c^4*(I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(4*(I*c + c^2*x)) + (c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/6 - 2*c^2*(-(ArcSinh[c*x]/x) - c*ArcTanh[Sqrt[1 + c^2*x^2]]) - ((5*I)/4)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/c) + ((5*I)/4)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, I*E^ArcSinh[c*x]])/c)))/d^2 + (b^2*c^3*((24*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + (12*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) - 48*ArcTan[Tanh[ArcSinh[c*x]/2]] - 4*Coth[ArcSinh[c*x]/2] + 26*ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] - 2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - (c*x*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^4)/2 - 104*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - (60*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] + (60*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 104*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 104*PolyLog[2, -E^(-ArcSinh[c*x])] - (120*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (120*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] + 104*PolyLog[2, E^(-ArcSinh[c*x])] - (120*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] + (120*I)*PolyLog[3, I/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 - (8*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2]^4)/(c^3*x^3) + 4*Tanh[ArcSinh[c*x]/2] - 26*ArcSinh[c*x]^2*Tanh[ArcSinh[c*x]/2]))/(24*d^2)","A",0
243,1,552,320,3.3965718,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^3,x]","\frac{-\frac{15 a^2 c x}{c^2 x^2+1}+\frac{6 a^2 c x}{\left(c^2 x^2+1\right)^2}+9 a^2 \tan ^{-1}(c x)+\frac{15 a b \left(\sqrt{c^2 x^2+1}-i \sinh ^{-1}(c x)\right)}{-1+i c x}+\frac{15 a b \left(\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right)}{-1-i c x}-\frac{i a b \left(3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x-2 i)\right)}{(c x-i)^2}+\frac{i a b \left(3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x+2 i)\right)}{(c x+i)^2}+\frac{9}{2} i a b \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)\right)-\frac{9}{2} i a b \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)\right)+b^2 \left(-\frac{2 c x}{c^2 x^2+1}-\frac{15 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}+\frac{6 c x \sinh ^{-1}(c x)^2}{\left(c^2 x^2+1\right)^2}-\frac{30 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{4 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}-18 i \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+18 i \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-18 i \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)+18 i \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)-9 i \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+9 i \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+56 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{24 c^5 d^3}","-\frac{3 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 c^5 d^3}+\frac{3 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 c^5 d^3}+\frac{3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^5 d^3}-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^2 d^3 \left(c^2 x^2+1\right)^2}-\frac{5 b \left(a+b \sinh ^{-1}(c x)\right)}{4 c^5 d^3 \sqrt{c^2 x^2+1}}+\frac{b \left(a+b \sinh ^{-1}(c x)\right)}{6 c^5 d^3 \left(c^2 x^2+1\right)^{3/2}}-\frac{3 x \left(a+b \sinh ^{-1}(c x)\right)^2}{8 c^4 d^3 \left(c^2 x^2+1\right)}+\frac{3 i b^2 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{4 c^5 d^3}-\frac{3 i b^2 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{4 c^5 d^3}+\frac{7 b^2 \tan ^{-1}(c x)}{6 c^5 d^3}-\frac{b^2 x}{12 c^4 d^3 \left(c^2 x^2+1\right)}",1,"((6*a^2*c*x)/(1 + c^2*x^2)^2 - (15*a^2*c*x)/(1 + c^2*x^2) + (15*a*b*(Sqrt[1 + c^2*x^2] - I*ArcSinh[c*x]))/(-1 + I*c*x) + (15*a*b*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(-1 - I*c*x) - (I*a*b*((-2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(-I + c*x)^2 + (I*a*b*((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(I + c*x)^2 + 9*a^2*ArcTan[c*x] + ((9*I)/2)*a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - ((9*I)/2)*a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]]) + b^2*((-2*c*x)/(1 + c^2*x^2) + (4*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) - (30*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + (6*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 - (15*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) + 56*ArcTan[Tanh[ArcSinh[c*x]/2]] - (9*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] + (9*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] - (18*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (18*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - (18*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] + (18*I)*PolyLog[3, I/E^ArcSinh[c*x]]))/(24*c^5*d^3)","A",0
244,1,186,167,0.397588,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^3,x]","-\frac{6 a^2 c^2 x^2+3 a^2-6 a b c x \sqrt{c^2 x^2+1}+2 b \sinh ^{-1}(c x) \left(a \left(6 c^2 x^2+3\right)-b c x \sqrt{c^2 x^2+1} \left(4 c^2 x^2+3\right)\right)-8 a b c^3 x^3 \sqrt{c^2 x^2+1}+b^2 c^2 x^2+3 b^2 \left(2 c^2 x^2+1\right) \sinh ^{-1}(c x)^2+b^2+4 \left(b c^2 x^2+b\right)^2 \log \left(c^2 x^2+1\right)}{12 c^4 d^3 \left(c^2 x^2+1\right)^2}","-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^4 d^3}+\frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{4 d^3 \left(c^2 x^2+1\right)^2}+\frac{b x^3 \left(a+b \sinh ^{-1}(c x)\right)}{6 c d^3 \left(c^2 x^2+1\right)^{3/2}}+\frac{b x \left(a+b \sinh ^{-1}(c x)\right)}{2 c^3 d^3 \sqrt{c^2 x^2+1}}-\frac{b^2}{12 c^4 d^3 \left(c^2 x^2+1\right)}-\frac{b^2 \log \left(c^2 x^2+1\right)}{3 c^4 d^3}",1,"-1/12*(3*a^2 + b^2 + 6*a^2*c^2*x^2 + b^2*c^2*x^2 - 6*a*b*c*x*Sqrt[1 + c^2*x^2] - 8*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(3 + 4*c^2*x^2)) + a*(3 + 6*c^2*x^2))*ArcSinh[c*x] + 3*b^2*(1 + 2*c^2*x^2)*ArcSinh[c*x]^2 + 4*(b + b*c^2*x^2)^2*Log[1 + c^2*x^2])/(c^4*d^3*(1 + c^2*x^2)^2)","A",1
245,1,550,318,2.5119044,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^3,x]","\frac{\frac{3 a^2 c x}{c^2 x^2+1}-\frac{6 a^2 c x}{\left(c^2 x^2+1\right)^2}+3 a^2 \tan ^{-1}(c x)+\frac{a b \left(\sqrt{c^2 x^2+1} (2+i c x)+3 i \sinh ^{-1}(c x)\right)}{(c x-i)^2}+\frac{3 a b \left(\sinh ^{-1}(c x)-i \sqrt{c^2 x^2+1}\right)}{c x-i}+\frac{3 a b \left(\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right)}{c x+i}-\frac{i a b \left(3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x+2 i)\right)}{(c x+i)^2}+\frac{3}{2} i a b \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)\right)-\frac{3}{2} i a b \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)\right)+b^2 \left(\frac{2 c x}{c^2 x^2+1}+\frac{3 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}-\frac{6 c x \sinh ^{-1}(c x)^2}{\left(c^2 x^2+1\right)^2}+\frac{6 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{4 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}-6 i \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+6 i \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-6 i \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)+6 i \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)-3 i \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+3 i \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-8 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{24 c^3 d^3}","-\frac{i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 c^3 d^3}+\frac{i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 c^3 d^3}+\frac{\tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^3 d^3}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{8 c^2 d^3 \left(c^2 x^2+1\right)}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^2 d^3 \left(c^2 x^2+1\right)^2}+\frac{b \left(a+b \sinh ^{-1}(c x)\right)}{4 c^3 d^3 \sqrt{c^2 x^2+1}}-\frac{b \left(a+b \sinh ^{-1}(c x)\right)}{6 c^3 d^3 \left(c^2 x^2+1\right)^{3/2}}+\frac{i b^2 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{4 c^3 d^3}-\frac{i b^2 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{4 c^3 d^3}-\frac{b^2 \tan ^{-1}(c x)}{6 c^3 d^3}+\frac{b^2 x}{12 c^2 d^3 \left(c^2 x^2+1\right)}",1,"((-6*a^2*c*x)/(1 + c^2*x^2)^2 + (3*a^2*c*x)/(1 + c^2*x^2) + (a*b*((2 + I*c*x)*Sqrt[1 + c^2*x^2] + (3*I)*ArcSinh[c*x]))/(-I + c*x)^2 + (3*a*b*((-I)*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(-I + c*x) + (3*a*b*(I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(I + c*x) - (I*a*b*((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(I + c*x)^2 + 3*a^2*ArcTan[c*x] + ((3*I)/2)*a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - ((3*I)/2)*a*b*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]]) + b^2*((2*c*x)/(1 + c^2*x^2) - (4*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + (6*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (6*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 + (3*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) - 8*ArcTan[Tanh[ArcSinh[c*x]/2]] - (3*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] + (3*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] - (6*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (6*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - (6*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] + (6*I)*PolyLog[3, I/E^ArcSinh[c*x]]))/(24*c^3*d^3)","A",0
246,1,152,145,0.2241609,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^3,x]","\frac{-3 a^2+6 a b c x \sqrt{c^2 x^2+1}+2 b \sinh ^{-1}(c x) \left(b c x \sqrt{c^2 x^2+1} \left(2 c^2 x^2+3\right)-3 a\right)+4 a b c^3 x^3 \sqrt{c^2 x^2+1}+b^2 c^2 x^2-3 b^2 \sinh ^{-1}(c x)^2+b^2-2 \left(b c^2 x^2+b\right)^2 \log \left(c^2 x^2+1\right)}{12 d^3 \left(c^3 x^2+c\right)^2}","\frac{b x \left(a+b \sinh ^{-1}(c x)\right)}{3 c d^3 \sqrt{c^2 x^2+1}}+\frac{b x \left(a+b \sinh ^{-1}(c x)\right)}{6 c d^3 \left(c^2 x^2+1\right)^{3/2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^2 d^3 \left(c^2 x^2+1\right)^2}+\frac{b^2}{12 c^2 d^3 \left(c^2 x^2+1\right)}-\frac{b^2 \log \left(c^2 x^2+1\right)}{6 c^2 d^3}",1,"(-3*a^2 + b^2 + b^2*c^2*x^2 + 6*a*b*c*x*Sqrt[1 + c^2*x^2] + 4*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*b*(-3*a + b*c*x*Sqrt[1 + c^2*x^2]*(3 + 2*c^2*x^2))*ArcSinh[c*x] - 3*b^2*ArcSinh[c*x]^2 - 2*(b + b*c^2*x^2)^2*Log[1 + c^2*x^2])/(12*d^3*(c + c^3*x^2)^2)","A",1
247,1,546,309,2.3364052,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^3,x]","\frac{\frac{9 a^2 x}{c^2 x^2+1}+\frac{6 a^2 x}{\left(c^2 x^2+1\right)^2}+\frac{9 a^2 \tan ^{-1}(c x)}{c}+\frac{a b \left(\frac{9 \left(\sinh ^{-1}(c x)-i \sqrt{c^2 x^2+1}\right)}{c x-i}+\frac{9 \left(\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right)}{c x+i}-\frac{i \left(3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x-2 i)\right)}{(c x-i)^2}+\frac{i \left(3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x+2 i)\right)}{(c x+i)^2}+\frac{9}{2} i \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)\right)-\frac{9}{2} i \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)\right)\right)}{c}+\frac{b^2 \left(-\frac{2 c x}{c^2 x^2+1}+\frac{9 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}+\frac{6 c x \sinh ^{-1}(c x)^2}{\left(c^2 x^2+1\right)^2}+\frac{18 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{4 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}-18 i \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+18 i \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-18 i \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)+18 i \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)-9 i \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+9 i \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-40 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{c}}{24 d^3}","\frac{3 b \left(a+b \sinh ^{-1}(c x)\right)}{4 c d^3 \sqrt{c^2 x^2+1}}+\frac{b \left(a+b \sinh ^{-1}(c x)\right)}{6 c d^3 \left(c^2 x^2+1\right)^{3/2}}+\frac{3 x \left(a+b \sinh ^{-1}(c x)\right)^2}{8 d^3 \left(c^2 x^2+1\right)}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{3 i b \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 c d^3}+\frac{3 i b \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 c d^3}+\frac{3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{4 c d^3}-\frac{b^2 x}{12 d^3 \left(c^2 x^2+1\right)}+\frac{3 i b^2 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{4 c d^3}-\frac{3 i b^2 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{4 c d^3}-\frac{5 b^2 \tan ^{-1}(c x)}{6 c d^3}",1,"((6*a^2*x)/(1 + c^2*x^2)^2 + (9*a^2*x)/(1 + c^2*x^2) + (9*a^2*ArcTan[c*x])/c + (a*b*((9*((-I)*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(-I + c*x) + (9*(I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(I + c*x) - (I*((-2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(-I + c*x)^2 + (I*((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(I + c*x)^2 + ((9*I)/2)*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - ((9*I)/2)*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]])))/c + (b^2*((-2*c*x)/(1 + c^2*x^2) + (4*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + (18*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + (6*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 + (9*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) - 40*ArcTan[Tanh[ArcSinh[c*x]/2]] - (9*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] + (9*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] - (18*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (18*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - (18*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] + (18*I)*PolyLog[3, I/E^ArcSinh[c*x]]))/c)/(24*d^3)","A",0
248,1,560,275,3.7766791,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x \left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^3),x]","\frac{\frac{12 a^2}{c^2 x^2+1}+\frac{6 a^2}{\left(c^2 x^2+1\right)^2}-12 a^2 \log \left(c^2 x^2+1\right)+24 a^2 \log (c x)+a b \left(-\frac{15 \left(\sqrt{c^2 x^2+1}-i \sinh ^{-1}(c x)\right)}{c x+i}-\frac{15 \left(\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right)}{c x-i}-\frac{3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x-2 i)}{(c x-i)^2}-\frac{3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x+2 i)}{(c x+i)^2}+24 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+12 \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)\right)+12 \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-4 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)\right)-24 \sinh ^{-1}(c x)^2+48 \sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)\right)+b^2 \left(-\frac{2}{c^2 x^2+1}+16 \log \left(c^2 x^2+1\right)+\frac{12 \sinh ^{-1}(c x)^2}{c^2 x^2+1}+\frac{6 \sinh ^{-1}(c x)^2}{\left(c^2 x^2+1\right)^2}-\frac{32 c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{4 c x \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}+24 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)+24 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)+12 \text{Li}_3\left(-e^{-2 \sinh ^{-1}(c x)}\right)-12 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)-16 \sinh ^{-1}(c x)^3-24 \sinh ^{-1}(c x)^2 \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)+24 \sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)+i \pi ^3\right)}{24 d^3}","-\frac{4 b c x \left(a+b \sinh ^{-1}(c x)\right)}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{b c x \left(a+b \sinh ^{-1}(c x)\right)}{6 d^3 \left(c^2 x^2+1\right)^{3/2}}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 d^3 \left(c^2 x^2+1\right)}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{b \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^3}+\frac{b \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^3}-\frac{2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d^3}-\frac{b^2}{12 d^3 \left(c^2 x^2+1\right)}+\frac{2 b^2 \log \left(c^2 x^2+1\right)}{3 d^3}+\frac{b^2 \text{Li}_3\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d^3}-\frac{b^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d^3}",1,"((6*a^2)/(1 + c^2*x^2)^2 + (12*a^2)/(1 + c^2*x^2) + 24*a^2*Log[c*x] - 12*a^2*Log[1 + c^2*x^2] + a*b*((-15*(Sqrt[1 + c^2*x^2] - I*ArcSinh[c*x]))/(I + c*x) - (15*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(-I + c*x) - 24*ArcSinh[c*x]^2 - ((-2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x])/(-I + c*x)^2 - ((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x])/(I + c*x)^2 + 48*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 12*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) + 12*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]]) + 24*PolyLog[2, E^(2*ArcSinh[c*x])]) + b^2*(I*Pi^3 - 2/(1 + c^2*x^2) - (4*c*x*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) - (32*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + (6*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 + (12*ArcSinh[c*x]^2)/(1 + c^2*x^2) - 16*ArcSinh[c*x]^3 - 24*ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] + 24*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 16*Log[1 + c^2*x^2] + 24*ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] + 24*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] + 12*PolyLog[3, -E^(-2*ArcSinh[c*x])] - 12*PolyLog[3, E^(2*ArcSinh[c*x])]))/(24*d^3)","C",0
249,1,716,389,7.6974498,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^2 \left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^3),x]","-\frac{7 a^2 c^2 x}{8 d^3 \left(c^2 x^2+1\right)}-\frac{a^2 c^2 x}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{15 a^2 c \tan ^{-1}(c x)}{8 d^3}-\frac{a^2}{d^3 x}+\frac{2 a b c \left(\frac{7 \left(\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right)}{16 (-1-i c x)}-\frac{7 \left(\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right)}{16 (c x+i)}+\frac{i \left(3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x-2 i)\right)}{48 (c x-i)^2}-\frac{i \left(3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x+2 i)\right)}{48 (c x+i)^2}-\tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)+\frac{15}{16} i \left(2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)-\frac{1}{2} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)\right)-\frac{15}{16} i \left(2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)-\frac{1}{2} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)\right)-\frac{\sinh ^{-1}(c x)}{c x}\right)}{d^3}+\frac{b^2 c \left(\frac{2 c x}{c^2 x^2+1}-\frac{21 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}-\frac{6 c x \sinh ^{-1}(c x)^2}{\left(c^2 x^2+1\right)^2}-\frac{42 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{4 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}+90 i \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-90 i \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+48 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-48 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+90 i \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)-90 i \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)+45 i \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-45 i \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+48 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-48 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+12 \sinh ^{-1}(c x)^2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-12 \sinh ^{-1}(c x)^2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+88 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{24 d^3}","-\frac{7 b c \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3 \sqrt{c^2 x^2+1}}-\frac{b c \left(a+b \sinh ^{-1}(c x)\right)}{6 d^3 \left(c^2 x^2+1\right)^{3/2}}-\frac{15 c^2 x \left(a+b \sinh ^{-1}(c x)\right)^2}{8 d^3 \left(c^2 x^2+1\right)}-\frac{5 c^2 x \left(a+b \sinh ^{-1}(c x)\right)^2}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d^3 x \left(c^2 x^2+1\right)^2}+\frac{15 i b c \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3}-\frac{15 i b c \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3}-\frac{15 c \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{4 d^3}-\frac{4 b c \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^3}+\frac{b^2 c^2 x}{12 d^3 \left(c^2 x^2+1\right)}-\frac{2 b^2 c \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{d^3}+\frac{2 b^2 c \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{d^3}-\frac{15 i b^2 c \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{4 d^3}+\frac{15 i b^2 c \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{4 d^3}+\frac{11 b^2 c \tan ^{-1}(c x)}{6 d^3}",1,"-(a^2/(d^3*x)) - (a^2*c^2*x)/(4*d^3*(1 + c^2*x^2)^2) - (7*a^2*c^2*x)/(8*d^3*(1 + c^2*x^2)) - (15*a^2*c*ArcTan[c*x])/(8*d^3) + (2*a*b*c*((7*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(16*(-1 - I*c*x)) - ArcSinh[c*x]/(c*x) - (7*(I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(16*(I + c*x)) + ((I/48)*((-2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(-I + c*x)^2 - ((I/48)*((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(I + c*x)^2 - ArcTanh[Sqrt[1 + c^2*x^2]] + ((15*I)/16)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]] + 2*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - ((15*I)/16)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 2*PolyLog[2, I*E^ArcSinh[c*x]])))/d^3 + (b^2*c*((2*c*x)/(1 + c^2*x^2) - (4*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) - (42*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (6*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 - (21*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) + 88*ArcTan[Tanh[ArcSinh[c*x]/2]] - 12*ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] + 48*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + (45*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - (45*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] - 48*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 48*PolyLog[2, -E^(-ArcSinh[c*x])] + (90*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (90*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - 48*PolyLog[2, E^(-ArcSinh[c*x])] + (90*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] - (90*I)*PolyLog[3, I/E^ArcSinh[c*x]] + 12*ArcSinh[c*x]^2*Tanh[ArcSinh[c*x]/2]))/(24*d^3)","A",0
250,1,759,381,9.4625908,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^3 \left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^3),x]","-\frac{a^2 c^2}{d^3 \left(c^2 x^2+1\right)}-\frac{a^2 c^2}{4 d^3 \left(c^2 x^2+1\right)^2}+\frac{3 a^2 c^2 \log \left(c^2 x^2+1\right)}{2 d^3}-\frac{3 a^2 c^2 \log (x)}{d^3}-\frac{a^2}{2 d^3 x^2}+\frac{2 a b \left(\frac{3}{2} c^3 \left(\frac{2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left(1+i e^{\sinh ^{-1}(c x)}\right)}{c}\right)+\frac{3}{2} c^3 \left(\frac{2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c}-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \sinh ^{-1}(c x) \log \left(1-i e^{\sinh ^{-1}(c x)}\right)}{c}\right)-3 c^2 \left(\frac{1}{2} \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)-\frac{1}{2} \sinh ^{-1}(c x)^2+\sinh ^{-1}(c x) \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)\right)-\frac{c^2 \left(-3 \sinh ^{-1}(c x)+(-c x+2 i) \sqrt{c^2 x^2+1}\right)}{48 (c x-i)^2}-\frac{9 i c^2 \left(\sqrt{c^2 x^2+1}+i \sinh ^{-1}(c x)\right)}{16 (-1-i c x)}+\frac{c^2 \left(3 \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} (c x+2 i)\right)}{48 (c x+i)^2}-\frac{c x \sqrt{c^2 x^2+1}+\sinh ^{-1}(c x)}{2 x^2}-\frac{9 i c^3 \left(\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right)}{16 \left(c^2 x+i c\right)}\right)}{d^3}+\frac{b^2 c^2 \left(\frac{1}{24} \left(\frac{2}{c^2 x^2+1}-56 \log \left(\sqrt{c^2 x^2+1}\right)-\frac{24 \sinh ^{-1}(c x)^2}{c^2 x^2+1}-\frac{12 \sinh ^{-1}(c x)^2}{c^2 x^2}-\frac{6 \sinh ^{-1}(c x)^2}{\left(c^2 x^2+1\right)^2}-\frac{24 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{c x}+\frac{56 c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{4 c x \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}-36 \text{Li}_3\left(-e^{-2 \sinh ^{-1}(c x)}\right)+36 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)+24 \log (c x)+48 \sinh ^{-1}(c x)^3+72 \sinh ^{-1}(c x)^2 \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)-72 \sinh ^{-1}(c x)^2 \log \left(1-e^{2 \sinh ^{-1}(c x)}\right)-3 i \pi ^3\right)-3 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)-3 \sinh ^{-1}(c x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)\right)}{d^3}","\frac{3 b c^2 \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^3}-\frac{3 b c^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^3}-\frac{3 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 d^3 \left(c^2 x^2+1\right)}-\frac{3 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{4 d^3 \left(c^2 x^2+1\right)^2}-\frac{b c \left(a+b \sinh ^{-1}(c x)\right)}{d^3 x \left(c^2 x^2+1\right)^{3/2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 d^3 x^2 \left(c^2 x^2+1\right)^2}+\frac{6 c^2 \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d^3}+\frac{4 b c^3 x \left(a+b \sinh ^{-1}(c x)\right)}{3 d^3 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 x \left(a+b \sinh ^{-1}(c x)\right)}{6 d^3 \left(c^2 x^2+1\right)^{3/2}}-\frac{3 b^2 c^2 \text{Li}_3\left(-e^{2 \sinh ^{-1}(c x)}\right)}{2 d^3}+\frac{3 b^2 c^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c x)}\right)}{2 d^3}+\frac{b^2 c^2}{12 d^3 \left(c^2 x^2+1\right)}-\frac{7 b^2 c^2 \log \left(c^2 x^2+1\right)}{6 d^3}+\frac{b^2 c^2 \log (x)}{d^3}",1,"-1/2*a^2/(d^3*x^2) - (a^2*c^2)/(4*d^3*(1 + c^2*x^2)^2) - (a^2*c^2)/(d^3*(1 + c^2*x^2)) - (3*a^2*c^2*Log[x])/d^3 + (3*a^2*c^2*Log[1 + c^2*x^2])/(2*d^3) + (2*a*b*(-1/48*(c^2*((2*I - c*x)*Sqrt[1 + c^2*x^2] - 3*ArcSinh[c*x]))/(-I + c*x)^2 - (((9*I)/16)*c^2*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(-1 - I*c*x) - (((9*I)/16)*c^3*(I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(I*c + c^2*x) - (c*x*Sqrt[1 + c^2*x^2] + ArcSinh[c*x])/(2*x^2) + (c^2*((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(48*(I + c*x)^2) + (3*c^3*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/c))/2 + (3*c^3*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, I*E^ArcSinh[c*x]])/c))/2 - 3*c^2*(-1/2*ArcSinh[c*x]^2 + ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + PolyLog[2, E^(2*ArcSinh[c*x])]/2)))/d^3 + (b^2*c^2*(-3*ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] - 3*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] + ((-3*I)*Pi^3 + 2/(1 + c^2*x^2) + (4*c*x*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + (56*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (24*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(c*x) - (12*ArcSinh[c*x]^2)/(c^2*x^2) - (6*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 - (24*ArcSinh[c*x]^2)/(1 + c^2*x^2) + 48*ArcSinh[c*x]^3 + 72*ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] - 72*ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + 24*Log[c*x] - 56*Log[Sqrt[1 + c^2*x^2]] - 36*PolyLog[3, -E^(-2*ArcSinh[c*x])] + 36*PolyLog[3, E^(2*ArcSinh[c*x])])/24))/d^3","C",0
251,1,937,529,10.2945032,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^4 \left(d+c^2 d x^2\right)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^3),x]","\frac{11 a^2 x c^4}{8 d^3 \left(c^2 x^2+1\right)}+\frac{a^2 x c^4}{4 d^3 \left(c^2 x^2+1\right)^2}+\frac{35 a^2 \tan ^{-1}(c x) c^3}{8 d^3}+\frac{b^2 \left(-\frac{1}{2} c x \sinh ^{-1}(c x)^2 \text{csch}^4\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\frac{8 \sinh ^{-1}(c x)^2 \sinh ^4\left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{c^3 x^3}+\frac{33 c x \sinh ^{-1}(c x)^2}{c^2 x^2+1}+\frac{6 c x \sinh ^{-1}(c x)^2}{\left(c^2 x^2+1\right)^2}-2 \sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+\frac{66 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{4 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}-136 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+38 \sinh ^{-1}(c x)^2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-152 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-105 i \sinh ^{-1}(c x)^2 \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+105 i \sinh ^{-1}(c x)^2 \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+152 \sinh ^{-1}(c x) \log \left(1+e^{-\sinh ^{-1}(c x)}\right)-152 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-210 i \sinh ^{-1}(c x) \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+210 i \sinh ^{-1}(c x) \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+152 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-210 i \text{Li}_3\left(-i e^{-\sinh ^{-1}(c x)}\right)+210 i \text{Li}_3\left(i e^{-\sinh ^{-1}(c x)}\right)-38 \sinh ^{-1}(c x)^2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+4 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\frac{2 c x}{c^2 x^2+1}\right) c^3}{24 d^3}+\frac{3 a^2 c^2}{d^3 x}+\frac{2 a b \left(\frac{11 \left(\sinh ^{-1}(c x)+i \sqrt{c^2 x^2+1}\right) c^4}{16 \left(x c^2+i c\right)}-\frac{35}{16} i \left(-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \log \left(1+i e^{\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)}{c}+\frac{2 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c}\right) c^4+\frac{35}{16} i \left(-\frac{\sinh ^{-1}(c x)^2}{2 c}+\frac{2 \log \left(1-i e^{\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)}{c}+\frac{2 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c}\right) c^4+\frac{i \left((2 i-c x) \sqrt{c^2 x^2+1}-3 \sinh ^{-1}(c x)\right) c^3}{48 (c x-i)^2}-\frac{11 \left(i \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1}\right) c^3}{16 (-i c x-1)}+\frac{i \left(\sqrt{c^2 x^2+1} (c x+2 i)+3 \sinh ^{-1}(c x)\right) c^3}{48 (c x+i)^2}+\frac{1}{6} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right) c^3-3 \left(-\frac{\sinh ^{-1}(c x)}{x}-c \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)\right) c^2-\frac{\sqrt{c^2 x^2+1} c}{6 x^2}-\frac{\sinh ^{-1}(c x)}{3 x^3}\right)}{d^3}-\frac{a^2}{3 d^3 x^3}","-\frac{35 i b c^3 \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3}+\frac{35 i b c^3 \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{4 d^3}+\frac{35 c^3 \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{4 d^3}+\frac{38 b c^3 \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d^3}+\frac{7 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d^3 x \left(c^2 x^2+1\right)^2}-\frac{b c \left(a+b \sinh ^{-1}(c x)\right)}{3 d^3 x^2 \left(c^2 x^2+1\right)^{3/2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{3 d^3 x^3 \left(c^2 x^2+1\right)^2}+\frac{35 c^4 x \left(a+b \sinh ^{-1}(c x)\right)^2}{8 d^3 \left(c^2 x^2+1\right)}+\frac{35 c^4 x \left(a+b \sinh ^{-1}(c x)\right)^2}{12 d^3 \left(c^2 x^2+1\right)^2}+\frac{29 b c^3 \left(a+b \sinh ^{-1}(c x)\right)}{12 d^3 \sqrt{c^2 x^2+1}}-\frac{b c^3 \left(a+b \sinh ^{-1}(c x)\right)}{6 d^3 \left(c^2 x^2+1\right)^{3/2}}+\frac{19 b^2 c^3 \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right)}{3 d^3}-\frac{19 b^2 c^3 \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right)}{3 d^3}+\frac{35 i b^2 c^3 \text{Li}_3\left(-i e^{\sinh ^{-1}(c x)}\right)}{4 d^3}-\frac{35 i b^2 c^3 \text{Li}_3\left(i e^{\sinh ^{-1}(c x)}\right)}{4 d^3}-\frac{17 b^2 c^3 \tan ^{-1}(c x)}{6 d^3}+\frac{b^2 c^2}{6 d^3 x \left(c^2 x^2+1\right)}-\frac{b^2 c^2}{2 d^3 x}+\frac{b^2 c^4 x}{12 d^3 \left(c^2 x^2+1\right)}",1,"-1/3*a^2/(d^3*x^3) + (3*a^2*c^2)/(d^3*x) + (a^2*c^4*x)/(4*d^3*(1 + c^2*x^2)^2) + (11*a^2*c^4*x)/(8*d^3*(1 + c^2*x^2)) + (35*a^2*c^3*ArcTan[c*x])/(8*d^3) + (2*a*b*(-1/6*(c*Sqrt[1 + c^2*x^2])/x^2 + ((I/48)*c^3*((2*I - c*x)*Sqrt[1 + c^2*x^2] - 3*ArcSinh[c*x]))/(-I + c*x)^2 - (11*c^3*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(16*(-1 - I*c*x)) - ArcSinh[c*x]/(3*x^3) + (11*c^4*(I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(16*(I*c + c^2*x)) + ((I/48)*c^3*((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(I + c*x)^2 + (c^3*ArcTanh[Sqrt[1 + c^2*x^2]])/6 - 3*c^2*(-(ArcSinh[c*x]/x) - c*ArcTanh[Sqrt[1 + c^2*x^2]]) - ((35*I)/16)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/c) + ((35*I)/16)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, I*E^ArcSinh[c*x]])/c)))/d^3 + (b^2*c^3*((-2*c*x)/(1 + c^2*x^2) + (4*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + (66*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + (6*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 + (33*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) - 136*ArcTan[Tanh[ArcSinh[c*x]/2]] - 4*Coth[ArcSinh[c*x]/2] + 38*ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] - 2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - (c*x*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^4)/2 - 152*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - (105*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] + (105*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 152*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 152*PolyLog[2, -E^(-ArcSinh[c*x])] - (210*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (210*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] + 152*PolyLog[2, E^(-ArcSinh[c*x])] - (210*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] + (210*I)*PolyLog[3, I/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 - (8*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2]^4)/(c^3*x^3) + 4*Tanh[ArcSinh[c*x]/2] - 38*ArcSinh[c*x]^2*Tanh[ArcSinh[c*x]/2]))/(24*d^3)","A",0
252,1,284,300,0.9309716,"\int \left(\pi +c^2 \pi  x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{\pi ^{5/2} \left(12 \sinh ^{-1}(c x) \left(360 a^2+540 a b \sinh \left(2 \sinh ^{-1}(c x)\right)+108 a b \sinh \left(4 \sinh ^{-1}(c x)\right)+12 a b \sinh \left(6 \sinh ^{-1}(c x)\right)-270 b^2 \cosh \left(2 \sinh ^{-1}(c x)\right)-27 b^2 \cosh \left(4 \sinh ^{-1}(c x)\right)-2 b^2 \cosh \left(6 \sinh ^{-1}(c x)\right)\right)+9504 a^2 c x \sqrt{c^2 x^2+1}+2304 a^2 c^5 x^5 \sqrt{c^2 x^2+1}+7488 a^2 c^3 x^3 \sqrt{c^2 x^2+1}+72 b \sinh ^{-1}(c x)^2 \left(60 a+45 b \sinh \left(2 \sinh ^{-1}(c x)\right)+9 b \sinh \left(4 \sinh ^{-1}(c x)\right)+b \sinh \left(6 \sinh ^{-1}(c x)\right)\right)-3240 a b \cosh \left(2 \sinh ^{-1}(c x)\right)-324 a b \cosh \left(4 \sinh ^{-1}(c x)\right)-24 a b \cosh \left(6 \sinh ^{-1}(c x)\right)+1440 b^2 \sinh ^{-1}(c x)^3+1620 b^2 \sinh \left(2 \sinh ^{-1}(c x)\right)+81 b^2 \sinh \left(4 \sinh ^{-1}(c x)\right)+4 b^2 \sinh \left(6 \sinh ^{-1}(c x)\right)\right)}{13824 c}","-\frac{\pi ^{5/2} b \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)}{18 c}-\frac{5 \pi ^{5/2} b \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{48 c}+\frac{1}{6} x \left(\pi  c^2 x^2+\pi \right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{5}{24} \pi  x \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{5}{16} \pi ^2 x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{5}{16} \pi ^{5/2} b c x^2 \left(a+b \sinh ^{-1}(c x)\right)+\frac{5 \pi ^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{48 b c}+\frac{1}{108} \pi ^{5/2} b^2 x \left(c^2 x^2+1\right)^{5/2}+\frac{65 \pi ^{5/2} b^2 x \left(c^2 x^2+1\right)^{3/2}}{1728}+\frac{245 \pi ^{5/2} b^2 x \sqrt{c^2 x^2+1}}{1152}-\frac{115 \pi ^{5/2} b^2 \sinh ^{-1}(c x)}{1152 c}",1,"(Pi^(5/2)*(9504*a^2*c*x*Sqrt[1 + c^2*x^2] + 7488*a^2*c^3*x^3*Sqrt[1 + c^2*x^2] + 2304*a^2*c^5*x^5*Sqrt[1 + c^2*x^2] + 1440*b^2*ArcSinh[c*x]^3 - 3240*a*b*Cosh[2*ArcSinh[c*x]] - 324*a*b*Cosh[4*ArcSinh[c*x]] - 24*a*b*Cosh[6*ArcSinh[c*x]] + 1620*b^2*Sinh[2*ArcSinh[c*x]] + 81*b^2*Sinh[4*ArcSinh[c*x]] + 4*b^2*Sinh[6*ArcSinh[c*x]] + 72*b*ArcSinh[c*x]^2*(60*a + 45*b*Sinh[2*ArcSinh[c*x]] + 9*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]]) + 12*ArcSinh[c*x]*(360*a^2 - 270*b^2*Cosh[2*ArcSinh[c*x]] - 27*b^2*Cosh[4*ArcSinh[c*x]] - 2*b^2*Cosh[6*ArcSinh[c*x]] + 540*a*b*Sinh[2*ArcSinh[c*x]] + 108*a*b*Sinh[4*ArcSinh[c*x]] + 12*a*b*Sinh[6*ArcSinh[c*x]])))/(13824*c)","A",1
253,1,202,210,0.5691422,"\int \left(\pi +c^2 \pi  x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{\pi ^{3/2} \left(160 a^2 c x \sqrt{c^2 x^2+1}+64 a^2 c^3 x^3 \sqrt{c^2 x^2+1}+4 \sinh ^{-1}(c x) \left(4 a \left(6 a+8 b \sinh \left(2 \sinh ^{-1}(c x)\right)+b \sinh \left(4 \sinh ^{-1}(c x)\right)\right)-16 b^2 \cosh \left(2 \sinh ^{-1}(c x)\right)-b^2 \cosh \left(4 \sinh ^{-1}(c x)\right)\right)+8 b \sinh ^{-1}(c x)^2 \left(12 a+8 b \sinh \left(2 \sinh ^{-1}(c x)\right)+b \sinh \left(4 \sinh ^{-1}(c x)\right)\right)-64 a b \cosh \left(2 \sinh ^{-1}(c x)\right)-4 a b \cosh \left(4 \sinh ^{-1}(c x)\right)+32 b^2 \sinh ^{-1}(c x)^3+32 b^2 \sinh \left(2 \sinh ^{-1}(c x)\right)+b^2 \sinh \left(4 \sinh ^{-1}(c x)\right)\right)}{256 c}","\frac{1}{4} x \left(\pi  c^2 x^2+\pi \right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{3}{8} \pi  x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{\pi ^{3/2} b \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{8 c}-\frac{3}{8} \pi ^{3/2} b c x^2 \left(a+b \sinh ^{-1}(c x)\right)+\frac{\pi ^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{8 b c}+\frac{1}{32} \pi ^{3/2} b^2 x \left(c^2 x^2+1\right)^{3/2}+\frac{15}{64} \pi ^{3/2} b^2 x \sqrt{c^2 x^2+1}-\frac{9 \pi ^{3/2} b^2 \sinh ^{-1}(c x)}{64 c}",1,"(Pi^(3/2)*(160*a^2*c*x*Sqrt[1 + c^2*x^2] + 64*a^2*c^3*x^3*Sqrt[1 + c^2*x^2] + 32*b^2*ArcSinh[c*x]^3 - 64*a*b*Cosh[2*ArcSinh[c*x]] - 4*a*b*Cosh[4*ArcSinh[c*x]] + 32*b^2*Sinh[2*ArcSinh[c*x]] + b^2*Sinh[4*ArcSinh[c*x]] + 8*b*ArcSinh[c*x]^2*(12*a + 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]) + 4*ArcSinh[c*x]*(-16*b^2*Cosh[2*ArcSinh[c*x]] - b^2*Cosh[4*ArcSinh[c*x]] + 4*a*(6*a + 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]))))/(256*c)","A",1
254,1,124,122,0.3532413,"\int \sqrt{\pi +c^2 \pi  x^2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])^2,x]","\frac{\sqrt{\pi } \left(3 \left(4 a^2 c x \sqrt{c^2 x^2+1}-2 a b \cosh \left(2 \sinh ^{-1}(c x)\right)+b^2 \sinh \left(2 \sinh ^{-1}(c x)\right)\right)+6 \sinh ^{-1}(c x) \left(2 a \left(a+b \sinh \left(2 \sinh ^{-1}(c x)\right)\right)-b^2 \cosh \left(2 \sinh ^{-1}(c x)\right)\right)+6 b \sinh ^{-1}(c x)^2 \left(2 a+b \sinh \left(2 \sinh ^{-1}(c x)\right)\right)+4 b^2 \sinh ^{-1}(c x)^3\right)}{24 c}","\frac{1}{2} x \sqrt{\pi  c^2 x^2+\pi } \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{1}{2} \sqrt{\pi } b c x^2 \left(a+b \sinh ^{-1}(c x)\right)+\frac{\sqrt{\pi } \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b c}+\frac{1}{4} \sqrt{\pi } b^2 x \sqrt{c^2 x^2+1}-\frac{\sqrt{\pi } b^2 \sinh ^{-1}(c x)}{4 c}",1,"(Sqrt[Pi]*(4*b^2*ArcSinh[c*x]^3 + 6*b*ArcSinh[c*x]^2*(2*a + b*Sinh[2*ArcSinh[c*x]]) + 3*(4*a^2*c*x*Sqrt[1 + c^2*x^2] - 2*a*b*Cosh[2*ArcSinh[c*x]] + b^2*Sinh[2*ArcSinh[c*x]]) + 6*ArcSinh[c*x]*(-(b^2*Cosh[2*ArcSinh[c*x]]) + 2*a*(a + b*Sinh[2*ArcSinh[c*x]]))))/(24*c)","A",1
255,1,25,25,0.0240582,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{\pi +c^2 \pi  x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/Sqrt[Pi + c^2*Pi*x^2],x]","\frac{\left(a+b \sinh ^{-1}(c x)\right)^3}{3 \sqrt{\pi } b c}","\frac{\left(a+b \sinh ^{-1}(c x)\right)^3}{3 \sqrt{\pi } b c}",1,"(a + b*ArcSinh[c*x])^3/(3*b*c*Sqrt[Pi])","A",1
256,1,153,104,0.3856978,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(\pi +c^2 \pi  x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(Pi + c^2*Pi*x^2)^(3/2),x]","\frac{a \left(a c x-b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)\right)+2 b \sinh ^{-1}(c x) \left(a c x-b \sqrt{c^2 x^2+1} \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)\right)+b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)-\left(b^2 \left(\sqrt{c^2 x^2+1}-c x\right) \sinh ^{-1}(c x)^2\right)}{\pi ^{3/2} c \sqrt{c^2 x^2+1}}","\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{\pi  \sqrt{\pi  c^2 x^2+\pi }}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\pi ^{3/2} c}-\frac{2 b \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{\pi ^{3/2} c}-\frac{b^2 \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{\pi ^{3/2} c}",1,"(-(b^2*(-(c*x) + Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2) + 2*b*ArcSinh[c*x]*(a*c*x - b*Sqrt[1 + c^2*x^2]*Log[1 + E^(-2*ArcSinh[c*x])]) + a*(a*c*x - b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2]) + b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[c*x])])/(c*Pi^(3/2)*Sqrt[1 + c^2*x^2])","A",0
257,1,293,204,0.6300563,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(\pi +c^2 \pi  x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(Pi + c^2*Pi*x^2)^(5/2),x]","\frac{2 a^2 c^3 x^3+3 a^2 c x+a b \sqrt{c^2 x^2+1}-2 a b c^2 x^2 \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)-2 a b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)-b \sinh ^{-1}(c x) \left(-4 a c^3 x^3-6 a c x-b \sqrt{c^2 x^2+1}+4 b \left(c^2 x^2+1\right)^{3/2} \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)\right)-b^2 c^3 x^3+2 b^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)-b^2 \left(-2 c^3 x^3+2 c^2 x^2 \sqrt{c^2 x^2+1}+2 \sqrt{c^2 x^2+1}-3 c x\right) \sinh ^{-1}(c x)^2-b^2 c x}{3 \pi ^{5/2} c \left(c^2 x^2+1\right)^{3/2}}","\frac{b \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi ^{5/2} c \left(c^2 x^2+1\right)}+\frac{2 x \left(a+b \sinh ^{-1}(c x)\right)^2}{3 \pi ^2 \sqrt{\pi  c^2 x^2+\pi }}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{3 \pi  \left(\pi  c^2 x^2+\pi \right)^{3/2}}+\frac{2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 \pi ^{5/2} c}-\frac{4 b \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 \pi ^{5/2} c}-\frac{b^2 x}{3 \pi ^{5/2} \sqrt{c^2 x^2+1}}-\frac{2 b^2 \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 \pi ^{5/2} c}",1,"(3*a^2*c*x - b^2*c*x + 2*a^2*c^3*x^3 - b^2*c^3*x^3 + a*b*Sqrt[1 + c^2*x^2] - b^2*(-3*c*x - 2*c^3*x^3 + 2*Sqrt[1 + c^2*x^2] + 2*c^2*x^2*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 - b*ArcSinh[c*x]*(-6*a*c*x - 4*a*c^3*x^3 - b*Sqrt[1 + c^2*x^2] + 4*b*(1 + c^2*x^2)^(3/2)*Log[1 + E^(-2*ArcSinh[c*x])]) - 2*a*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] - 2*a*b*c^2*x^2*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + 2*b^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(-2*ArcSinh[c*x])])/(3*c*Pi^(5/2)*(1 + c^2*x^2)^(3/2))","A",0
258,1,222,358,0.2740575,"\int x^3 \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]","\frac{\sqrt{c^2 d x^2+d} \left(-30 a b c x \sqrt{c^2 x^2+1} \left(9 c^4 x^4+5 c^2 x^2-30\right)-30 b \sinh ^{-1}(c x) \left(b c x \sqrt{c^2 x^2+1} \left(9 c^4 x^4+5 c^2 x^2-30\right)-15 a \left(c^2 x^2+1\right)^2 \left(3 c^2 x^2-2\right)\right)+225 \left(3 c^2 x^2-2\right) \left(a c^2 x^2+a\right)^2+2 b^2 \left(27 c^6 x^6+16 c^4 x^4-439 c^2 x^2-428\right)+225 \left(3 c^2 x^2-2\right) \left(b c^2 x^2+b\right)^2 \sinh ^{-1}(c x)^2\right)}{3375 c^4 \left(c^2 x^2+1\right)}","\frac{x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{15 c^2}-\frac{2 b c x^5 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{25 \sqrt{c^2 x^2+1}}+\frac{1}{5} x^4 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{45 c \sqrt{c^2 x^2+1}}-\frac{2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{15 c^4}+\frac{4 a b x \sqrt{c^2 d x^2+d}}{15 c^3 \sqrt{c^2 x^2+1}}+\frac{2 b^2 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d}}{125 c^4}-\frac{26 b^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}{675 c^4}-\frac{52 b^2 \sqrt{c^2 d x^2+d}}{225 c^4}+\frac{4 b^2 x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{15 c^3 \sqrt{c^2 x^2+1}}",1,"(Sqrt[d + c^2*d*x^2]*(225*(-2 + 3*c^2*x^2)*(a + a*c^2*x^2)^2 - 30*a*b*c*x*Sqrt[1 + c^2*x^2]*(-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^2*x^2]*(-30 + 5*c^2*x^2 + 9*c^4*x^4))*ArcSinh[c*x] + 225*(-2 + 3*c^2*x^2)*(b + b*c^2*x^2)^2*ArcSinh[c*x]^2))/(3375*c^4*(1 + c^2*x^2))","A",1
259,1,207,291,1.324299,"\int x^2 \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]","-\frac{-96 a^2 c x \left(2 c^2 x^2+1\right) \sqrt{c^2 d x^2+d}+96 a^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+\frac{12 a b \sqrt{c^2 d x^2+d} \left(8 \sinh ^{-1}(c x)^2-4 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{b^2 \sqrt{c^2 d x^2+d} \left(32 \sinh ^{-1}(c x)^3-3 \left(8 \sinh ^{-1}(c x)^2+1\right) \sinh \left(4 \sinh ^{-1}(c x)\right)+12 \sinh ^{-1}(c x) \cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}}{768 c^3}","-\frac{b x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{8 c \sqrt{c^2 x^2+1}}+\frac{x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{8 c^2}-\frac{b c x^4 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{8 \sqrt{c^2 x^2+1}}+\frac{1}{4} x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{24 b c^3 \sqrt{c^2 x^2+1}}+\frac{b^2 x \sqrt{c^2 d x^2+d}}{64 c^2}+\frac{1}{32} b^2 x^3 \sqrt{c^2 d x^2+d}-\frac{b^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{64 c^3 \sqrt{c^2 x^2+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]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (12*a*b*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2] + (b^2*Sqrt[d + c^2*d*x^2]*(32*ArcSinh[c*x]^3 + 12*ArcSinh[c*x]*Cosh[4*ArcSinh[c*x]] - 3*(1 + 8*ArcSinh[c*x]^2)*Sinh[4*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2])/c^3","A",1
260,1,166,180,0.2715438,"\int x \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]","\frac{\sqrt{c^2 d x^2+d} \left(-6 a b c x \sqrt{c^2 x^2+1} \left(c^2 x^2+3\right)+6 b \sinh ^{-1}(c x) \left(3 a \left(c^2 x^2+1\right)^2-b c x \sqrt{c^2 x^2+1} \left(c^2 x^2+3\right)\right)+9 \left(a c^2 x^2+a\right)^2+2 b^2 \left(c^4 x^4+8 c^2 x^2+7\right)+9 \left(b c^2 x^2+b\right)^2 \sinh ^{-1}(c x)^2\right)}{27 c^2 \left(c^2 x^2+1\right)}","-\frac{2 b x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 c \sqrt{c^2 x^2+1}}+\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^2 d}-\frac{2 b c x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{9 \sqrt{c^2 x^2+1}}+\frac{2 b^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}{27 c^2}+\frac{4 b^2 \sqrt{c^2 d x^2+d}}{9 c^2}",1,"(Sqrt[d + c^2*d*x^2]*(-6*a*b*c*x*Sqrt[1 + c^2*x^2]*(3 + c^2*x^2) + 9*(a + a*c^2*x^2)^2 + 2*b^2*(7 + 8*c^2*x^2 + c^4*x^4) + 6*b*(3*a*(1 + c^2*x^2)^2 - b*c*x*Sqrt[1 + c^2*x^2]*(3 + c^2*x^2))*ArcSinh[c*x] + 9*(b + b*c^2*x^2)^2*ArcSinh[c*x]^2))/(27*c^2*(1 + c^2*x^2))","A",1
261,1,200,184,1.0682161,"\int \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]","\frac{1}{24} \left(12 a^2 x \sqrt{c^2 d x^2+d}+\frac{12 a^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)}{c}+\frac{6 a b \sqrt{c^2 d x^2+d} \left(2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)-\cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{c \sqrt{c^2 x^2+1}}+\frac{b^2 \sqrt{c^2 d x^2+d} \left(4 \sinh ^{-1}(c x)^3+\left(6 \sinh ^{-1}(c x)^2+3\right) \sinh \left(2 \sinh ^{-1}(c x)\right)-6 \sinh ^{-1}(c x) \cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{c \sqrt{c^2 x^2+1}}\right)","\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b c \sqrt{c^2 x^2+1}}+\frac{1}{2} x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b c x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{c^2 x^2+1}}+\frac{1}{4} b^2 x \sqrt{c^2 d x^2+d}-\frac{b^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{4 c \sqrt{c^2 x^2+1}}",1,"(12*a^2*x*Sqrt[d + c^2*d*x^2] + (12*a^2*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/c + (b^2*Sqrt[d + c^2*d*x^2]*(4*ArcSinh[c*x]^3 - 6*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] + (3 + 6*ArcSinh[c*x]^2)*Sinh[2*ArcSinh[c*x]]))/(c*Sqrt[1 + c^2*x^2]) + (6*a*b*Sqrt[d + c^2*d*x^2]*(-Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(c*Sqrt[1 + c^2*x^2]))/24","A",1
262,1,352,338,1.1664178,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x,x]","a^2 \sqrt{c^2 d x^2+d}-a^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+a^2 \sqrt{d} \log (c x)+\frac{2 a b \sqrt{c^2 d x^2+d} \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{b^2 \sqrt{c^2 d x^2+d} \left(2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-2 \sinh ^{-1}(c x) \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+2 \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-2 \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)-2 c x \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2 \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x)^2 \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}","-\frac{2 b \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{2 b \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{2 a b c x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}+\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 x^2+1}}+\frac{2 b^2 \sqrt{c^2 d x^2+d} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{2 b^2 \sqrt{c^2 d x^2+d} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+2 b^2 \sqrt{c^2 d x^2+d}-\frac{2 b^2 c x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+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^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (b^2*Sqrt[d + c^2*d*x^2]*(2*Sqrt[1 + c^2*x^2] - 2*c*x*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 + ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 2*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - 2*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 2*PolyLog[3, -E^(-ArcSinh[c*x])] - 2*PolyLog[3, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2]","A",0
263,1,232,209,1.1662774,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^2,x]","-\frac{a^2 \sqrt{c^2 d x^2+d}}{x}+a^2 c \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+\frac{a b \sqrt{c^2 d x^2+d} \left(-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right)}{x \sqrt{c^2 x^2+1}}+\frac{b^2 c \sqrt{c^2 d x^2+d} \left(\sinh ^{-1}(c x) \left(\left(3-\frac{3 \sqrt{c^2 x^2+1}}{c x}\right) \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2+6 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)\right)-3 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)\right)}{3 \sqrt{c^2 x^2+1}}","\frac{c \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b \sqrt{c^2 x^2+1}}+\frac{c \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{x}+\frac{2 b c \sqrt{c^2 d x^2+d} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{b^2 c \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}",1,"-((a^2*Sqrt[d + c^2*d*x^2])/x) + (a*b*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]))/(x*Sqrt[1 + c^2*x^2]) + a^2*c*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (b^2*c*Sqrt[d + c^2*d*x^2]*(ArcSinh[c*x]*((3 - (3*Sqrt[1 + c^2*x^2])/(c*x))*ArcSinh[c*x] + ArcSinh[c*x]^2 + 6*Log[1 - E^(-2*ArcSinh[c*x])]) - 3*PolyLog[2, E^(-2*ArcSinh[c*x])]))/(3*Sqrt[1 + c^2*x^2])","A",0
264,1,446,358,4.8826391,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^3,x]","\frac{1}{8} \left(-\frac{4 a^2 \sqrt{c^2 d x^2+d}}{x^2}-4 a^2 c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+4 a^2 c^2 \sqrt{d} \log (x)+\frac{2 a b c^2 \sqrt{c^2 d x^2+d} \left(4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \left(8 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-8 \sinh ^{-1}(c x) \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+8 \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-8 \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x)^2 \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x)^2 \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+4 \sinh ^{-1}(c x) \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 \sinh ^{-1}(c x) \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+\sinh ^{-1}(c x)^2 \left(-\text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-\sinh ^{-1}(c x)^2 \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+8 \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{\sqrt{c^2 x^2+1}}\right)","-\frac{b c^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{b c^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{b c \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{x \sqrt{c^2 x^2+1}}-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 x^2}-\frac{c^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 x^2+1}}+\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{b^2 c^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{\sqrt{c^2 x^2+1}}",1,"((-4*a^2*Sqrt[d + c^2*d*x^2])/x^2 + 4*a^2*c^2*Sqrt[d]*Log[x] - 4*a^2*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (2*a*b*c^2*Sqrt[d + c^2*d*x^2]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/Sqrt[1 + c^2*x^2] + (b^2*c^2*Sqrt[d + c^2*d*x^2]*(-4*ArcSinh[c*x]*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 8*Log[Tanh[ArcSinh[c*x]/2]] + 8*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - 8*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 8*PolyLog[3, -E^(-ArcSinh[c*x])] - 8*PolyLog[3, E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/Sqrt[1 + c^2*x^2])/8","A",0
265,1,240,294,0.8008452,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^4,x]","-\frac{\sqrt{c^2 d x^2+d} \left(a^2 c^2 x^2 \sqrt{c^2 x^2+1}+a^2 \sqrt{c^2 x^2+1}-2 a b c^3 x^3 \log (c x)-b \sinh ^{-1}(c x) \left(-2 a \left(c^2 x^2+1\right)^{3/2}+2 b c^3 x^3 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)-b c x\right)+a b c x+b^2 c^3 x^3 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+b^2 c^2 x^2 \sqrt{c^2 x^2+1}+b^2 \left(-c^3 x^3+c^2 x^2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1}\right) \sinh ^{-1}(c x)^2\right)}{3 x^3 \sqrt{c^2 x^2+1}}","-\frac{b c \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^2}-\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d x^3}+\frac{c^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 \sqrt{c^2 x^2+1}}+\frac{2 b c^3 \sqrt{c^2 d x^2+d} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 x^2+1}}-\frac{b^2 c^2 \sqrt{c^2 d x^2+d}}{3 x}-\frac{b^2 c^3 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)}{3 \sqrt{c^2 x^2+1}}+\frac{b^2 c^3 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{3 \sqrt{c^2 x^2+1}}",1,"-1/3*(Sqrt[d + c^2*d*x^2]*(a*b*c*x + a^2*Sqrt[1 + c^2*x^2] + a^2*c^2*x^2*Sqrt[1 + c^2*x^2] + b^2*c^2*x^2*Sqrt[1 + c^2*x^2] + b^2*(-(c^3*x^3) + Sqrt[1 + c^2*x^2] + c^2*x^2*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 - b*ArcSinh[c*x]*(-(b*c*x) - 2*a*(1 + c^2*x^2)^(3/2) + 2*b*c^3*x^3*Log[1 - E^(-2*ArcSinh[c*x])]) - 2*a*b*c^3*x^3*Log[c*x] + b^2*c^3*x^3*PolyLog[2, E^(-2*ArcSinh[c*x])]))/(x^3*Sqrt[1 + c^2*x^2])","A",0
266,1,251,482,0.3853734,"\int x^3 \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d \sqrt{c^2 d x^2+d} \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 \left(75 c^6 x^6+168 c^4 x^4+35 c^2 x^2-210\right) \sqrt{c^2 x^2+1}-210 b \sinh ^{-1}(c x) \left(b c x \sqrt{c^2 x^2+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 \sinh ^{-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{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{35 c^2}-\frac{16 b c d x^5 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{175 \sqrt{c^2 x^2+1}}+\frac{1}{7} x^4 \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{3}{35} d x^4 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b d x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{105 c \sqrt{c^2 x^2+1}}-\frac{2 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{35 c^4}+\frac{4 a b d x \sqrt{c^2 d x^2+d}}{35 c^3 \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d x^7 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{49 \sqrt{c^2 x^2+1}}+\frac{2 b^2 d \left(c^2 x^2+1\right)^3 \sqrt{c^2 d x^2+d}}{343 c^4}-\frac{38 b^2 d \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d}}{6125 c^4}-\frac{304 b^2 d \sqrt{c^2 d x^2+d}}{3675 c^4}-\frac{152 b^2 d \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}{11025 c^4}+\frac{4 b^2 d x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{35 c^3 \sqrt{c^2 x^2+1}}",1,"(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^2*x^2]*(-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^2*x^2]*(-210 + 35*c^2*x^2 + 168*c^4*x^4 + 75*c^6*x^6))*ArcSinh[c*x] + 11025*b^2*(1 + c^2*x^2)^3*(-2 + 5*c^2*x^2)*ArcSinh[c*x]^2))/(385875*c^4*(1 + c^2*x^2))","A",1
267,1,508,405,1.1907781,"\int x^2 \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{-864 a^2 d^{3/2} \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+864 a^2 c d x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+2304 a^2 c^5 d x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+4032 a^2 c^3 d x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+72 b d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left(-12 a-3 b \sinh \left(2 \sinh ^{-1}(c x)\right)+3 b \sinh \left(4 \sinh ^{-1}(c x)\right)+b \sinh \left(6 \sinh ^{-1}(c x)\right)\right)+216 a b d \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-108 a b d \sqrt{c^2 d x^2+d} \cosh \left(4 \sinh ^{-1}(c x)\right)-24 a b d \sqrt{c^2 d x^2+d} \cosh \left(6 \sinh ^{-1}(c x)\right)+12 b d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(-36 a \sinh \left(2 \sinh ^{-1}(c x)\right)+36 a \sinh \left(4 \sinh ^{-1}(c x)\right)+12 a \sinh \left(6 \sinh ^{-1}(c x)\right)+18 b \cosh \left(2 \sinh ^{-1}(c x)\right)-9 b \cosh \left(4 \sinh ^{-1}(c x)\right)-2 b \cosh \left(6 \sinh ^{-1}(c x)\right)\right)-288 b^2 d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3-108 b^2 d \sqrt{c^2 d x^2+d} \sinh \left(2 \sinh ^{-1}(c x)\right)+27 b^2 d \sqrt{c^2 d x^2+d} \sinh \left(4 \sinh ^{-1}(c x)\right)+4 b^2 d \sqrt{c^2 d x^2+d} \sinh \left(6 \sinh ^{-1}(c x)\right)}{13824 c^3 \sqrt{c^2 x^2+1}}","-\frac{b d x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{16 c \sqrt{c^2 x^2+1}}+\frac{d x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 c^2}-\frac{7 b c d x^4 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{48 \sqrt{c^2 x^2+1}}+\frac{1}{6} x^3 \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{8} d x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{48 b c^3 \sqrt{c^2 x^2+1}}-\frac{b c^3 d x^6 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{18 \sqrt{c^2 x^2+1}}-\frac{7 b^2 d x \sqrt{c^2 d x^2+d}}{1152 c^2}+\frac{1}{108} b^2 c^2 d x^5 \sqrt{c^2 d x^2+d}+\frac{43 b^2 d x^3 \sqrt{c^2 d x^2+d}}{1728}+\frac{7 b^2 d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c^3 \sqrt{c^2 x^2+1}}",1,"(864*a^2*c*d*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 4032*a^2*c^3*d*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 2304*a^2*c^5*d*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 288*b^2*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 + 216*a*b*d*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 108*a*b*d*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 24*a*b*d*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 864*a^2*d^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 108*b^2*d*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + 27*b^2*d*Sqrt[d + c^2*d*x^2]*Sinh[4*ArcSinh[c*x]] + 4*b^2*d*Sqrt[d + c^2*d*x^2]*Sinh[6*ArcSinh[c*x]] + 12*b*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(18*b*Cosh[2*ArcSinh[c*x]] - 9*b*Cosh[4*ArcSinh[c*x]] - 2*b*Cosh[6*ArcSinh[c*x]] - 36*a*Sinh[2*ArcSinh[c*x]] + 36*a*Sinh[4*ArcSinh[c*x]] + 12*a*Sinh[6*ArcSinh[c*x]]) + 72*b*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(-12*a - 3*b*Sinh[2*ArcSinh[c*x]] + 3*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]]))/(13824*c^3*Sqrt[1 + c^2*x^2])","A",1
268,1,198,267,0.3636962,"\int x \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d \sqrt{c^2 d x^2+d} \left(225 a^2 \left(c^2 x^2+1\right)^3-30 a b c x \left(3 c^4 x^4+10 c^2 x^2+15\right) \sqrt{c^2 x^2+1}+30 b \sinh ^{-1}(c x) \left(15 a \left(c^2 x^2+1\right)^3-b c x \sqrt{c^2 x^2+1} \left(3 c^4 x^4+10 c^2 x^2+15\right)\right)+225 b^2 \left(c^2 x^2+1\right)^3 \sinh ^{-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{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{5 c \sqrt{c^2 x^2+1}}+\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{5 c^2 d}-\frac{4 b c d x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{15 \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d x^5 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{25 \sqrt{c^2 x^2+1}}+\frac{2 b^2 d \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d}}{125 c^2}+\frac{16 b^2 d \sqrt{c^2 d x^2+d}}{75 c^2}+\frac{8 b^2 d \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}{225 c^2}",1,"(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^2*x^2]*(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^2*x^2]*(15 + 10*c^2*x^2 + 3*c^4*x^4))*ArcSinh[c*x] + 225*b^2*(1 + c^2*x^2)^3*ArcSinh[c*x]^2))/(1125*c^2*(1 + c^2*x^2))","A",1
269,1,329,294,1.9096224,"\int \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{288 a^2 d^{3/2} \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+96 a^2 c d x \sqrt{c^2 x^2+1} \left(2 c^2 x^2+5\right) \sqrt{c^2 d x^2+d}-192 a b d \sqrt{c^2 d x^2+d} \left(\cosh \left(2 \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)\right)-12 a b d \sqrt{c^2 d x^2+d} \left(8 \sinh ^{-1}(c x)^2-4 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\cosh \left(4 \sinh ^{-1}(c x)\right)\right)+32 b^2 d \sqrt{c^2 d x^2+d} \left(4 \sinh ^{-1}(c x)^3+\left(6 \sinh ^{-1}(c x)^2+3\right) \sinh \left(2 \sinh ^{-1}(c x)\right)-6 \sinh ^{-1}(c x) \cosh \left(2 \sinh ^{-1}(c x)\right)\right)-b^2 d \sqrt{c^2 d x^2+d} \left(32 \sinh ^{-1}(c x)^3-3 \left(8 \sinh ^{-1}(c x)^2+1\right) \sinh \left(4 \sinh ^{-1}(c x)\right)+12 \sinh ^{-1}(c x) \cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{768 c \sqrt{c^2 x^2+1}}","\frac{d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{8 b c \sqrt{c^2 x^2+1}}+\frac{1}{4} x \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{3}{8} d x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b d \left(c^2 x^2+1\right)^{3/2} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{8 c}-\frac{3 b c d x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{8 \sqrt{c^2 x^2+1}}+\frac{15}{64} b^2 d x \sqrt{c^2 d x^2+d}+\frac{1}{32} b^2 d x \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}-\frac{9 b^2 d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{64 c \sqrt{c^2 x^2+1}}",1,"(96*a^2*c*d*x*Sqrt[1 + c^2*x^2]*(5 + 2*c^2*x^2)*Sqrt[d + c^2*d*x^2] + 288*a^2*d^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 32*b^2*d*Sqrt[d + c^2*d*x^2]*(4*ArcSinh[c*x]^3 - 6*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] + (3 + 6*ArcSinh[c*x]^2)*Sinh[2*ArcSinh[c*x]]) - 192*a*b*d*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])) - 12*a*b*d*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]) - b^2*d*Sqrt[d + c^2*d*x^2]*(32*ArcSinh[c*x]^3 + 12*ArcSinh[c*x]*Cosh[4*ArcSinh[c*x]] - 3*(1 + 8*ArcSinh[c*x]^2)*Sinh[4*ArcSinh[c*x]]))/(768*c*Sqrt[1 + c^2*x^2])","A",1
270,1,520,498,2.2860405,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x,x]","-a^2 d^{3/2} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+\frac{1}{3} a^2 d \left(c^2 x^2+4\right) \sqrt{c^2 d x^2+d}+a^2 d^{3/2} \log (c x)+\frac{2 a b d \sqrt{c^2 d x^2+d} \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}-\frac{2 a b d \sqrt{c^2 d x^2+d} \left(c^3 x^3-3 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)+3 c x\right)}{9 \sqrt{c^2 x^2+1}}+\frac{b^2 d \sqrt{c^2 d x^2+d} \left(2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \left(\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)\right)+2 \left(\text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)\right)-2 c x \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2 \left(\log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{b^2 d \sqrt{c^2 d x^2+d} \left(27 \sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x)^2+2\right)-6 \sinh ^{-1}(c x) \left(9 c x+\sinh \left(3 \sinh ^{-1}(c x)\right)\right)+\left(9 \sinh ^{-1}(c x)^2+2\right) \cosh \left(3 \sinh ^{-1}(c x)\right)\right)}{108 \sqrt{c^2 x^2+1}}","-\frac{2 b d \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{2 b d \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{2 a b c d x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{2 b c d x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 x^2+1}}+\frac{1}{3} \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 d \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 x^2+1}}-\frac{2 b c^3 d x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{9 \sqrt{c^2 x^2+1}}+\frac{2 b^2 d \sqrt{c^2 d x^2+d} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{2 b^2 d \sqrt{c^2 d x^2+d} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+\frac{22}{9} b^2 d \sqrt{c^2 d x^2+d}+\frac{2}{27} b^2 d \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}-\frac{2 b^2 c d x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}",1,"(a^2*d*(4 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/3 - (2*a*b*d*Sqrt[d + c^2*d*x^2]*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]))/(9*Sqrt[1 + c^2*x^2]) + 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^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (b^2*d*Sqrt[d + c^2*d*x^2]*(2*Sqrt[1 + c^2*x^2] - 2*c*x*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 + ArcSinh[c*x]^2*(Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]) + 2*ArcSinh[c*x]*(PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]) + 2*(PolyLog[3, -E^(-ArcSinh[c*x])] - PolyLog[3, E^(-ArcSinh[c*x])])))/Sqrt[1 + c^2*x^2] + (b^2*d*Sqrt[d + c^2*d*x^2]*(27*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + (2 + 9*ArcSinh[c*x]^2)*Cosh[3*ArcSinh[c*x]] - 6*ArcSinh[c*x]*(9*c*x + Sinh[3*ArcSinh[c*x]])))/(108*Sqrt[1 + c^2*x^2])","A",0
271,1,369,398,2.9692618,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x^2,x]","\frac{36 a^2 c d^{3/2} x \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+12 a^2 d \left(c^2 x^2-2\right) \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+24 a b d \sqrt{c^2 d x^2+d} \left(-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 c x \log (c x)+c x \sinh ^{-1}(c x)^2\right)-6 a b c d x \sqrt{c^2 d x^2+d} \left(\cosh \left(2 \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)\right)-8 b^2 d \sqrt{c^2 d x^2+d} \left(\sinh ^{-1}(c x) \left(3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-c x \left(\sinh ^{-1}(c x)+3\right) \sinh ^{-1}(c x)-6 c x \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)\right)+3 c x \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)\right)+b^2 c d x \sqrt{c^2 d x^2+d} \left(4 \sinh ^{-1}(c x)^3+\left(6 \sinh ^{-1}(c x)^2+3\right) \sinh \left(2 \sinh ^{-1}(c x)\right)-6 \sinh ^{-1}(c x) \cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{24 x \sqrt{c^2 x^2+1}}","\frac{3}{2} c^2 d x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{c d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{2 b \sqrt{c^2 x^2+1}}+\frac{c d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 x^2+1}}+b c d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x}+\frac{2 b c d \sqrt{c^2 d x^2+d} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{3 b c^3 d x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{c^2 x^2+1}}-\frac{b^2 c d \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+\frac{1}{4} b^2 c^2 d x \sqrt{c^2 d x^2+d}-\frac{5 b^2 c d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{4 \sqrt{c^2 x^2+1}}",1,"(12*a^2*d*(-2 + c^2*x^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 24*a*b*d*Sqrt[d + c^2*d*x^2]*(-2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + c*x*ArcSinh[c*x]^2 + 2*c*x*Log[c*x]) + 36*a^2*c*d^(3/2)*x*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 8*b^2*d*Sqrt[d + c^2*d*x^2]*(ArcSinh[c*x]*(3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - c*x*ArcSinh[c*x]*(3 + ArcSinh[c*x]) - 6*c*x*Log[1 - E^(-2*ArcSinh[c*x])]) + 3*c*x*PolyLog[2, E^(-2*ArcSinh[c*x])]) + b^2*c*d*x*Sqrt[d + c^2*d*x^2]*(4*ArcSinh[c*x]^3 - 6*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] + (3 + 6*ArcSinh[c*x]^2)*Sinh[2*ArcSinh[c*x]]) - 6*a*b*c*d*x*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(24*x*Sqrt[1 + c^2*x^2])","A",0
272,1,771,541,7.8780683,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x^3,x]","-\frac{3}{2} a^2 c^2 d^{3/2} \log \left(\sqrt{d} \sqrt{d \left(c^2 x^2+1\right)}+d\right)+\frac{3}{2} a^2 c^2 d^{3/2} \log (x)+\sqrt{d \left(c^2 x^2+1\right)} \left(a^2 c^2 d-\frac{a^2 d}{2 x^2}\right)+\frac{2 a b c^2 d \sqrt{d \left(c^2 x^2+1\right)} \left(\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-c x+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{a b c^2 d \sqrt{d \left(c^2 x^2+1\right)} \left(4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{4 \sqrt{c^2 x^2+1}}+b^2 c^2 d \sqrt{d \left(c^2 x^2+1\right)} \left(\frac{2 \sinh ^{-1}(c x) \left(\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{2 \left(\text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)\right)}{\sqrt{c^2 x^2+1}}-\frac{2 c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{\sinh ^{-1}(c x)^2 \left(\log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 x^2+1}}+\sinh ^{-1}(c x)^2+2\right)+\frac{b^2 c^2 d \sqrt{d \left(c^2 x^2+1\right)} \left(8 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-8 \sinh ^{-1}(c x) \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+8 \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-8 \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x)^2 \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x)^2 \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+4 \sinh ^{-1}(c x) \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 \sinh ^{-1}(c x) \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+\sinh ^{-1}(c x)^2 \left(-\text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-\sinh ^{-1}(c x)^2 \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+8 \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{8 \sqrt{c^2 x^2+1}}","-\frac{3 b c^2 d \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{3 b c^2 d \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{3}{2} c^2 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b c d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{x \sqrt{c^2 x^2+1}}-\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 x^2}-\frac{3 c^2 d \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 x^2+1}}-\frac{3 a b c^3 d x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}+\frac{b c^3 d x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{3 b^2 c^2 d \sqrt{c^2 d x^2+d} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{3 b^2 c^2 d \sqrt{c^2 d x^2+d} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+2 b^2 c^2 d \sqrt{c^2 d x^2+d}-\frac{b^2 c^2 d \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{\sqrt{c^2 x^2+1}}-\frac{3 b^2 c^3 d x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+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^2*x^2)]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + b^2*c^2*d*Sqrt[d*(1 + c^2*x^2)]*(2 - (2*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ArcSinh[c*x]^2 + (ArcSinh[c*x]^2*(Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (2*ArcSinh[c*x]*(PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (2*(PolyLog[3, -E^(-ArcSinh[c*x])] - PolyLog[3, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2]) + (a*b*c^2*d*Sqrt[d*(1 + c^2*x^2)]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(4*Sqrt[1 + c^2*x^2]) + (b^2*c^2*d*Sqrt[d*(1 + c^2*x^2)]*(-4*ArcSinh[c*x]*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 8*Log[Tanh[ArcSinh[c*x]/2]] + 8*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - 8*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 8*PolyLog[3, -E^(-ArcSinh[c*x])] - 8*PolyLog[3, E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/(8*Sqrt[1 + c^2*x^2])","A",0
273,1,458,378,1.4152482,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/x^4,x]","\frac{-4 a^2 c^2 d x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-a^2 d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+3 a^2 c^3 d^{3/2} x^3 \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)-a b c d x \sqrt{c^2 d x^2+d}+8 a b c^3 d x^3 \sqrt{c^2 d x^2+d} \log (c x)+b d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left(3 a c^3 x^3-b \left(-4 c^3 x^3+4 c^2 x^2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1}\right)\right)+b d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(-2 a \sqrt{c^2 x^2+1} \left(4 c^2 x^2+1\right)+8 b c^3 x^3 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)-b c x\right)-b^2 c^2 d x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-4 b^2 c^3 d x^3 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+b^2 c^3 d x^3 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3}{3 x^3 \sqrt{c^2 x^2+1}}","-\frac{c^2 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{x}-\frac{b c d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^2}-\frac{\left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 x^3}+\frac{c^3 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b \sqrt{c^2 x^2+1}}+\frac{4 c^3 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 \sqrt{c^2 x^2+1}}+\frac{8 b c^3 d \sqrt{c^2 d x^2+d} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 x^2+1}}-\frac{b^2 c^2 d \sqrt{c^2 d x^2+d}}{3 x}-\frac{4 b^2 c^3 d \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)}{3 \sqrt{c^2 x^2+1}}+\frac{b^2 c^3 d \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{3 \sqrt{c^2 x^2+1}}",1,"(-(a*b*c*d*x*Sqrt[d + c^2*d*x^2]) - a^2*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 4*a^2*c^2*d*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - b^2*c^2*d*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + b*d*Sqrt[d + c^2*d*x^2]*(3*a*c^3*x^3 - b*(-4*c^3*x^3 + Sqrt[1 + c^2*x^2] + 4*c^2*x^2*Sqrt[1 + c^2*x^2]))*ArcSinh[c*x]^2 + b^2*c^3*d*x^3*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 + b*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(-(b*c*x) - 2*a*Sqrt[1 + c^2*x^2]*(1 + 4*c^2*x^2) + 8*b*c^3*x^3*Log[1 - E^(-2*ArcSinh[c*x])]) + 8*a*b*c^3*d*x^3*Sqrt[d + c^2*d*x^2]*Log[c*x] + 3*a^2*c^3*d^(3/2)*x^3*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 4*b^2*c^3*d*x^3*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*x^3*Sqrt[1 + c^2*x^2])","A",0
274,1,277,625,0.4584595,"\int x^3 \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d^2 \sqrt{c^2 d x^2+d} \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 \left(49 c^8 x^8+171 c^6 x^6+189 c^4 x^4+21 c^2 x^2-126\right) \sqrt{c^2 x^2+1}-126 b \sinh ^{-1}(c x) \left(b c x \sqrt{c^2 x^2+1} \left(49 c^8 x^8+171 c^6 x^6+189 c^4 x^4+21 c^2 x^2-126\right)-63 a \left(c^2 x^2+1\right)^4 \left(7 c^2 x^2-2\right)\right)+3969 b^2 \left(7 c^2 x^2-2\right) \left(c^2 x^2+1\right)^4 \sinh ^{-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{d^2 x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{63 c^2}-\frac{2 b c d^2 x^5 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{21 \sqrt{c^2 x^2+1}}+\frac{1}{21} d^2 x^4 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b d^2 x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{189 c \sqrt{c^2 x^2+1}}+\frac{1}{9} x^4 \left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{5}{63} d x^4 \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b c^5 d^2 x^9 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{81 \sqrt{c^2 x^2+1}}-\frac{2 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{63 c^4}+\frac{4 a b d^2 x \sqrt{c^2 d x^2+d}}{63 c^3 \sqrt{c^2 x^2+1}}-\frac{38 b c^3 d^2 x^7 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{441 \sqrt{c^2 x^2+1}}+\frac{2 b^2 d^2 \left(c^2 x^2+1\right)^4 \sqrt{c^2 d x^2+d}}{729 c^4}-\frac{50 b^2 d^2 \left(c^2 x^2+1\right)^3 \sqrt{c^2 d x^2+d}}{27783 c^4}-\frac{160 b^2 d^2 \sqrt{c^2 d x^2+d}}{3969 c^4}-\frac{4 b^2 d^2 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d}}{1323 c^4}-\frac{80 b^2 d^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}{11907 c^4}+\frac{4 b^2 d^2 x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{63 c^3 \sqrt{c^2 x^2+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^2*x^2]*(-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^2*x^2]*(-126 + 21*c^2*x^2 + 189*c^4*x^4 + 171*c^6*x^6 + 49*c^8*x^8))*ArcSinh[c*x] + 3969*b^2*(1 + c^2*x^2)^4*(-2 + 7*c^2*x^2)*ArcSinh[c*x]^2))/(250047*c^4*(1 + c^2*x^2))","A",1
275,1,619,536,2.0632356,"\int x^2 \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d^2 \left(34560 a^2 c x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-34560 a^2 \sqrt{d} \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+110592 a^2 c^7 x^7 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+313344 a^2 c^5 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+271872 a^2 c^3 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+288 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left(-120 a-48 b \sinh \left(2 \sinh ^{-1}(c x)\right)+24 b \sinh \left(4 \sinh ^{-1}(c x)\right)+16 b \sinh \left(6 \sinh ^{-1}(c x)\right)+3 b \sinh \left(8 \sinh ^{-1}(c x)\right)\right)+13824 a b \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-3456 a b \sqrt{c^2 d x^2+d} \cosh \left(4 \sinh ^{-1}(c x)\right)-1536 a b \sqrt{c^2 d x^2+d} \cosh \left(6 \sinh ^{-1}(c x)\right)-216 a b \sqrt{c^2 d x^2+d} \cosh \left(8 \sinh ^{-1}(c x)\right)+24 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(-1152 a \sinh \left(2 \sinh ^{-1}(c x)\right)+576 a \sinh \left(4 \sinh ^{-1}(c x)\right)+384 a \sinh \left(6 \sinh ^{-1}(c x)\right)+72 a \sinh \left(8 \sinh ^{-1}(c x)\right)+576 b \cosh \left(2 \sinh ^{-1}(c x)\right)-144 b \cosh \left(4 \sinh ^{-1}(c x)\right)-64 b \cosh \left(6 \sinh ^{-1}(c x)\right)-9 b \cosh \left(8 \sinh ^{-1}(c x)\right)\right)-11520 b^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3-6912 b^2 \sqrt{c^2 d x^2+d} \sinh \left(2 \sinh ^{-1}(c x)\right)+864 b^2 \sqrt{c^2 d x^2+d} \sinh \left(4 \sinh ^{-1}(c x)\right)+256 b^2 \sqrt{c^2 d x^2+d} \sinh \left(6 \sinh ^{-1}(c x)\right)+27 b^2 \sqrt{c^2 d x^2+d} \sinh \left(8 \sinh ^{-1}(c x)\right)\right)}{884736 c^3 \sqrt{c^2 x^2+1}}","-\frac{5 b d^2 x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{128 c \sqrt{c^2 x^2+1}}+\frac{5 d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{128 c^2}-\frac{59 b c d^2 x^4 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{384 \sqrt{c^2 x^2+1}}+\frac{5}{64} d^2 x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{8} x^3 \left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{5}{48} d x^3 \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b c^5 d^2 x^8 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{32 \sqrt{c^2 x^2+1}}-\frac{5 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{384 b c^3 \sqrt{c^2 x^2+1}}-\frac{17 b c^3 d^2 x^6 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{144 \sqrt{c^2 x^2+1}}-\frac{359 b^2 d^2 x \sqrt{c^2 d x^2+d}}{36864 c^2}+\frac{209 b^2 c^2 d^2 x^5 \sqrt{c^2 d x^2+d}}{13824}+\frac{1079 b^2 d^2 x^3 \sqrt{c^2 d x^2+d}}{55296}+\frac{1}{256} b^2 c^4 d^2 x^7 \sqrt{c^2 d x^2+d}+\frac{359 b^2 d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{36864 c^3 \sqrt{c^2 x^2+1}}",1,"(d^2*(34560*a^2*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 271872*a^2*c^3*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 313344*a^2*c^5*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 110592*a^2*c^7*x^7*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 11520*b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 + 13824*a*b*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 3456*a*b*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 1536*a*b*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 216*a*b*Sqrt[d + c^2*d*x^2]*Cosh[8*ArcSinh[c*x]] - 34560*a^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 6912*b^2*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + 864*b^2*Sqrt[d + c^2*d*x^2]*Sinh[4*ArcSinh[c*x]] + 256*b^2*Sqrt[d + c^2*d*x^2]*Sinh[6*ArcSinh[c*x]] + 27*b^2*Sqrt[d + c^2*d*x^2]*Sinh[8*ArcSinh[c*x]] + 24*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(576*b*Cosh[2*ArcSinh[c*x]] - 144*b*Cosh[4*ArcSinh[c*x]] - 64*b*Cosh[6*ArcSinh[c*x]] - 9*b*Cosh[8*ArcSinh[c*x]] - 1152*a*Sinh[2*ArcSinh[c*x]] + 576*a*Sinh[4*ArcSinh[c*x]] + 384*a*Sinh[6*ArcSinh[c*x]] + 72*a*Sinh[8*ArcSinh[c*x]]) + 288*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(-120*a - 48*b*Sinh[2*ArcSinh[c*x]] + 24*b*Sinh[4*ArcSinh[c*x]] + 16*b*Sinh[6*ArcSinh[c*x]] + 3*b*Sinh[8*ArcSinh[c*x]])))/(884736*c^3*Sqrt[1 + c^2*x^2])","A",1
276,1,224,366,0.4255178,"\int x \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d^2 \sqrt{c^2 d x^2+d} \left(3675 a^2 \left(c^2 x^2+1\right)^4-210 a b c x \left(5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right) \sqrt{c^2 x^2+1}+210 b \sinh ^{-1}(c x) \left(35 a \left(c^2 x^2+1\right)^4-b c x \sqrt{c^2 x^2+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 \sinh ^{-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{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{7 c \sqrt{c^2 x^2+1}}-\frac{2 b c d^2 x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{7 \sqrt{c^2 x^2+1}}+\frac{\left(c^2 d x^2+d\right)^{7/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{7 c^2 d}-\frac{2 b c^5 d^2 x^7 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{49 \sqrt{c^2 x^2+1}}-\frac{6 b c^3 d^2 x^5 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{35 \sqrt{c^2 x^2+1}}+\frac{2 b^2 d^2 \left(c^2 x^2+1\right)^3 \sqrt{c^2 d x^2+d}}{343 c^2}+\frac{32 b^2 d^2 \sqrt{c^2 d x^2+d}}{245 c^2}+\frac{12 b^2 d^2 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d}}{1225 c^2}+\frac{16 b^2 d^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}{735 c^2}",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^2*x^2]*(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^2*x^2]*(35 + 35*c^2*x^2 + 21*c^4*x^4 + 5*c^6*x^6))*ArcSinh[c*x] + 3675*b^2*(1 + c^2*x^2)^4*ArcSinh[c*x]^2))/(25725*c^2*(1 + c^2*x^2))","A",1
277,1,499,420,1.517041,"\int \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{d^2 \left(9504 a^2 c x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+4320 a^2 \sqrt{d} \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+2304 a^2 c^5 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+7488 a^2 c^3 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+72 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left(60 a+45 b \sinh \left(2 \sinh ^{-1}(c x)\right)+9 b \sinh \left(4 \sinh ^{-1}(c x)\right)+b \sinh \left(6 \sinh ^{-1}(c x)\right)\right)-3240 a b \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-324 a b \sqrt{c^2 d x^2+d} \cosh \left(4 \sinh ^{-1}(c x)\right)-24 a b \sqrt{c^2 d x^2+d} \cosh \left(6 \sinh ^{-1}(c x)\right)-12 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(-540 a \sinh \left(2 \sinh ^{-1}(c x)\right)-108 a \sinh \left(4 \sinh ^{-1}(c x)\right)-12 a \sinh \left(6 \sinh ^{-1}(c x)\right)+270 b \cosh \left(2 \sinh ^{-1}(c x)\right)+27 b \cosh \left(4 \sinh ^{-1}(c x)\right)+2 b \cosh \left(6 \sinh ^{-1}(c x)\right)\right)+1440 b^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3+1620 b^2 \sqrt{c^2 d x^2+d} \sinh \left(2 \sinh ^{-1}(c x)\right)+81 b^2 \sqrt{c^2 d x^2+d} \sinh \left(4 \sinh ^{-1}(c x)\right)+4 b^2 \sqrt{c^2 d x^2+d} \sinh \left(6 \sinh ^{-1}(c x)\right)\right)}{13824 c \sqrt{c^2 x^2+1}}","\frac{5 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{48 b c \sqrt{c^2 x^2+1}}+\frac{5}{16} d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b d^2 \left(c^2 x^2+1\right)^{5/2} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{18 c}-\frac{5 b d^2 \left(c^2 x^2+1\right)^{3/2} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{48 c}-\frac{5 b c d^2 x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{16 \sqrt{c^2 x^2+1}}+\frac{1}{6} x \left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{5}{24} d x \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{108} b^2 d^2 x \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d}+\frac{245 b^2 d^2 x \sqrt{c^2 d x^2+d}}{1152}+\frac{65 b^2 d^2 x \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}{1728}-\frac{115 b^2 d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{1152 c \sqrt{c^2 x^2+1}}",1,"(d^2*(9504*a^2*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 7488*a^2*c^3*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 2304*a^2*c^5*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 1440*b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 - 3240*a*b*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 324*a*b*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 24*a*b*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] + 4320*a^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 1620*b^2*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + 81*b^2*Sqrt[d + c^2*d*x^2]*Sinh[4*ArcSinh[c*x]] + 4*b^2*Sqrt[d + c^2*d*x^2]*Sinh[6*ArcSinh[c*x]] - 12*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(270*b*Cosh[2*ArcSinh[c*x]] + 27*b*Cosh[4*ArcSinh[c*x]] + 2*b*Cosh[6*ArcSinh[c*x]] - 540*a*Sinh[2*ArcSinh[c*x]] - 108*a*Sinh[4*ArcSinh[c*x]] - 12*a*Sinh[6*ArcSinh[c*x]]) + 72*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(60*a + 45*b*Sinh[2*ArcSinh[c*x]] + 9*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]])))/(13824*c*Sqrt[1 + c^2*x^2])","A",1
278,1,710,635,4.4533568,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x,x]","\frac{d^2 \left(54000 a^2 \sqrt{d} \sqrt{c^2 x^2+1} \log (c x)-54000 a^2 \sqrt{d} \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+3600 a^2 \sqrt{c^2 x^2+1} \left(3 c^4 x^4+11 c^2 x^2+23\right) \sqrt{c^2 d x^2+d}-108000 a b \sqrt{c^2 d x^2+d} \left(-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+c x-\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+\sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)-480 a b \sqrt{c^2 d x^2+d} \left(c x \left(9 c^4 x^4+5 c^2 x^2-30\right)-15 \sqrt{c^2 x^2+1} \left(3 c^4 x^4+c^2 x^2-2\right) \sinh ^{-1}(c x)\right)-24000 a b \sqrt{c^2 d x^2+d} \left(c^3 x^3-3 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)+3 c x\right)+54000 b^2 \sqrt{c^2 d x^2+d} \left(2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2+2 \sinh ^{-1}(c x) \left(\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)\right)+2 \left(\text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)\right)-2 c x \sinh ^{-1}(c x)+\sinh ^{-1}(c x)^2 \left(\log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)\right)+1000 b^2 \sqrt{c^2 d x^2+d} \left(27 \sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x)^2+2\right)-6 \sinh ^{-1}(c x) \left(9 c x+\sinh \left(3 \sinh ^{-1}(c x)\right)\right)+\left(9 \sinh ^{-1}(c x)^2+2\right) \cosh \left(3 \sinh ^{-1}(c x)\right)\right)-b^2 \sqrt{c^2 d x^2+d} \left(6750 \sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x)^2+2\right)+480 c x \left(9 c^4 x^4+5 c^2 x^2-30\right) \sinh ^{-1}(c x)+125 \left(9 \sinh ^{-1}(c x)^2+2\right) \cosh \left(3 \sinh ^{-1}(c x)\right)-27 \left(25 \sinh ^{-1}(c x)^2+2\right) \cosh \left(5 \sinh ^{-1}(c x)\right)\right)\right)}{54000 \sqrt{c^2 x^2+1}}","-\frac{2 b d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{2 b d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}-\frac{2 a b c d^2 x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{16 b c d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{15 \sqrt{c^2 x^2+1}}+d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 d^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 x^2+1}}+\frac{1}{5} \left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{3} d \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 b c^5 d^2 x^5 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{25 \sqrt{c^2 x^2+1}}-\frac{22 b c^3 d^2 x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{45 \sqrt{c^2 x^2+1}}+\frac{2 b^2 d^2 \sqrt{c^2 d x^2+d} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{2 b^2 d^2 \sqrt{c^2 d x^2+d} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+\frac{598}{225} b^2 d^2 \sqrt{c^2 d x^2+d}+\frac{2}{125} b^2 d^2 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d}+\frac{74}{675} b^2 d^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}-\frac{2 b^2 c d^2 x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}",1,"(d^2*(3600*a^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(23 + 11*c^2*x^2 + 3*c^4*x^4) - 24000*a*b*Sqrt[d + c^2*d*x^2]*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]) - 480*a*b*Sqrt[d + c^2*d*x^2]*(c*x*(-30 + 5*c^2*x^2 + 9*c^4*x^4) - 15*Sqrt[1 + c^2*x^2]*(-2 + c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]) - b^2*Sqrt[d + c^2*d*x^2]*(480*c*x*(-30 + 5*c^2*x^2 + 9*c^4*x^4)*ArcSinh[c*x] + 6750*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + 125*(2 + 9*ArcSinh[c*x]^2)*Cosh[3*ArcSinh[c*x]] - 27*(2 + 25*ArcSinh[c*x]^2)*Cosh[5*ArcSinh[c*x]]) + 54000*a^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*x] - 54000*a^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 108000*a*b*Sqrt[d + c^2*d*x^2]*(c*x - Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - PolyLog[2, -E^(-ArcSinh[c*x])] + PolyLog[2, E^(-ArcSinh[c*x])]) + 54000*b^2*Sqrt[d + c^2*d*x^2]*(2*Sqrt[1 + c^2*x^2] - 2*c*x*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 + ArcSinh[c*x]^2*(Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]) + 2*ArcSinh[c*x]*(PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]) + 2*(PolyLog[3, -E^(-ArcSinh[c*x])] - PolyLog[3, E^(-ArcSinh[c*x])])) + 1000*b^2*Sqrt[d + c^2*d*x^2]*(27*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + (2 + 9*ArcSinh[c*x]^2)*Cosh[3*ArcSinh[c*x]] - 6*ArcSinh[c*x]*(9*c*x + Sinh[3*ArcSinh[c*x]]))))/(54000*Sqrt[1 + c^2*x^2])","A",0
279,1,550,530,2.1757613,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^2,x]","\frac{d^2 \left(288 a^2 c^2 x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-256 a^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+480 a^2 c \sqrt{d} x \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+64 a^2 c^4 x^4 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+512 a b c x \sqrt{c^2 d x^2+d} \log (c x)+8 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left(60 a c x-32 b \sqrt{c^2 x^2+1}+32 b c x+16 b c x \sinh \left(2 \sinh ^{-1}(c x)\right)+b c x \sinh \left(4 \sinh ^{-1}(c x)\right)\right)-128 a b c x \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-4 a b c x \sqrt{c^2 d x^2+d} \cosh \left(4 \sinh ^{-1}(c x)\right)-4 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(128 a \sqrt{c^2 x^2+1}-64 a c x \sinh \left(2 \sinh ^{-1}(c x)\right)-4 a c x \sinh \left(4 \sinh ^{-1}(c x)\right)-128 b c x \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)+32 b c x \cosh \left(2 \sinh ^{-1}(c x)\right)+b c x \cosh \left(4 \sinh ^{-1}(c x)\right)\right)-256 b^2 c x \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+160 b^2 c x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3+64 b^2 c x \sqrt{c^2 d x^2+d} \sinh \left(2 \sinh ^{-1}(c x)\right)+b^2 c x \sqrt{c^2 d x^2+d} \sinh \left(4 \sinh ^{-1}(c x)\right)\right)}{256 x \sqrt{c^2 x^2+1}}","\frac{15}{8} c^2 d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{5 c d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{8 b \sqrt{c^2 x^2+1}}+\frac{c d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 x^2+1}}-\frac{1}{8} b c d^2 \left(c^2 x^2+1\right)^{3/2} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+b c d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{2 b c d^2 \sqrt{c^2 d x^2+d} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{5}{4} c^2 d x \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x}-\frac{15 b c^3 d^2 x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{8 \sqrt{c^2 x^2+1}}-\frac{b^2 c d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+\frac{31}{64} b^2 c^2 d^2 x \sqrt{c^2 d x^2+d}+\frac{1}{32} b^2 c^2 d^2 x \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}-\frac{89 b^2 c d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{64 \sqrt{c^2 x^2+1}}",1,"(d^2*(-256*a^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 288*a^2*c^2*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 64*a^2*c^4*x^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 160*b^2*c*x*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 - 128*a*b*c*x*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 4*a*b*c*x*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] + 512*a*b*c*x*Sqrt[d + c^2*d*x^2]*Log[c*x] + 480*a^2*c*Sqrt[d]*x*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 256*b^2*c*x*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])] + 64*b^2*c*x*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + b^2*c*x*Sqrt[d + c^2*d*x^2]*Sinh[4*ArcSinh[c*x]] - 4*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(128*a*Sqrt[1 + c^2*x^2] + 32*b*c*x*Cosh[2*ArcSinh[c*x]] + b*c*x*Cosh[4*ArcSinh[c*x]] - 128*b*c*x*Log[1 - E^(-2*ArcSinh[c*x])] - 64*a*c*x*Sinh[2*ArcSinh[c*x]] - 4*a*c*x*Sinh[4*ArcSinh[c*x]]) + 8*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(60*a*c*x + 32*b*c*x - 32*b*Sqrt[1 + c^2*x^2] + 16*b*c*x*Sinh[2*ArcSinh[c*x]] + b*c*x*Sinh[4*ArcSinh[c*x]])))/(256*x*Sqrt[1 + c^2*x^2])","A",0
280,1,990,687,8.0234555,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^3,x]","\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}+2 a b c^2 \left(\frac{1}{3} \left(c^2 x^2+1\right) \sqrt{d \left(c^2 x^2+1\right)} \sinh ^{-1}(c x)-\frac{c x \sqrt{d \left(c^2 x^2+1\right)} \left(c^2 x^2+3\right)}{9 \sqrt{c^2 x^2+1}}\right) d^2+\frac{4 a b c^2 \sqrt{d \left(c^2 x^2+1\right)} \left(-c x+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x) \log \left(1+e^{-\sinh ^{-1}(c x)}\right)+\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)\right) d^2}{\sqrt{c^2 x^2+1}}+2 b^2 c^2 \sqrt{d \left(c^2 x^2+1\right)} \left(\frac{\left(\log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\log \left(1+e^{-\sinh ^{-1}(c x)}\right)\right) \sinh ^{-1}(c x)^2}{\sqrt{c^2 x^2+1}}+\sinh ^{-1}(c x)^2+\frac{2 \left(\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)\right) \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}-\frac{2 c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{2 \left(\text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)\right)}{\sqrt{c^2 x^2+1}}+2\right) d^2+\frac{b^2 c^2 \sqrt{d \left(c^2 x^2+1\right)} \left(27 \sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x)^2+2\right)+\left(9 \sinh ^{-1}(c x)^2+2\right) \cosh \left(3 \sinh ^{-1}(c x)\right)-6 \sinh ^{-1}(c x) \left(9 c x+\sinh \left(3 \sinh ^{-1}(c x)\right)\right)\right) d^2}{108 \sqrt{c^2 x^2+1}}+\frac{a b c^2 \sqrt{d \left(c^2 x^2+1\right)} \left(-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(1+e^{-\sinh ^{-1}(c x)}\right)+4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) d^2}{4 \sqrt{c^2 x^2+1}}+\frac{b^2 c^2 \sqrt{d \left(c^2 x^2+1\right)} \left(-\text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2-\text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2+4 \log \left(1-e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)^2-4 \log \left(1+e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)^2-4 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+8 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)-8 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)+4 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+8 \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+8 \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-8 \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)\right) d^2}{8 \sqrt{c^2 x^2+1}}+\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{5 b c^2 d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{5 b c^2 d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}}+\frac{5}{2} c^2 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b c d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{x \sqrt{c^2 x^2+1}}-\frac{5 c^2 d^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 x^2+1}}+\frac{5}{6} c^2 d \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 x^2}-\frac{2 b c^5 d^2 x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{9 \sqrt{c^2 x^2+1}}-\frac{5 a b c^3 d^2 x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}+\frac{b c^3 d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 x^2+1}}+\frac{5 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}-\frac{5 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 x^2+1}}+\frac{40}{9} b^2 c^2 d^2 \sqrt{c^2 d x^2+d}+\frac{2}{27} b^2 c^2 d^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}-\frac{b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{\sqrt{c^2 x^2+1}}-\frac{5 b^2 c^3 d^2 x \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}",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) + 2*a*b*c^2*d^2*(-1/9*(c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/Sqrt[1 + c^2*x^2] + ((1 + c^2*x^2)*Sqrt[d*(1 + c^2*x^2)]*ArcSinh[c*x])/3) + (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^2*x^2)]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + 2*b^2*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(2 - (2*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ArcSinh[c*x]^2 + (ArcSinh[c*x]^2*(Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (2*ArcSinh[c*x]*(PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (2*(PolyLog[3, -E^(-ArcSinh[c*x])] - PolyLog[3, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2]) + (b^2*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(27*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + (2 + 9*ArcSinh[c*x]^2)*Cosh[3*ArcSinh[c*x]] - 6*ArcSinh[c*x]*(9*c*x + Sinh[3*ArcSinh[c*x]])))/(108*Sqrt[1 + c^2*x^2]) + (a*b*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(4*Sqrt[1 + c^2*x^2]) + (b^2*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(-4*ArcSinh[c*x]*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 8*Log[Tanh[ArcSinh[c*x]/2]] + 8*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - 8*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 8*PolyLog[3, -E^(-ArcSinh[c*x])] - 8*PolyLog[3, E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/(8*Sqrt[1 + c^2*x^2])","A",0
281,1,616,561,2.3558462,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^4,x]","\frac{d^2 \left(-56 a^2 c^2 x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-8 a^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+12 a^2 c^4 x^4 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+60 a^2 c^3 \sqrt{d} x^3 \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)-8 a b c x \sqrt{c^2 d x^2+d}+112 a b c^3 x^3 \sqrt{c^2 d x^2+d} \log (c x)+2 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2 \left(30 a c^3 x^3+3 b c^3 x^3 \sinh \left(2 \sinh ^{-1}(c x)\right)-4 b \left(-7 c^3 x^3+7 c^2 x^2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1}\right)\right)-6 a b c^3 x^3 \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-2 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(-6 a c^3 x^3 \sinh \left(2 \sinh ^{-1}(c x)\right)+56 a c^2 x^2 \sqrt{c^2 x^2+1}+8 a \sqrt{c^2 x^2+1}-56 b c^3 x^3 \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)+3 b c^3 x^3 \cosh \left(2 \sinh ^{-1}(c x)\right)+4 b c x\right)-8 b^2 c^2 x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}-56 b^2 c^3 x^3 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+20 b^2 c^3 x^3 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^3+3 b^2 c^3 x^3 \sqrt{c^2 d x^2+d} \sinh \left(2 \sinh ^{-1}(c x)\right)\right)}{24 x^3 \sqrt{c^2 x^2+1}}","-\frac{b c d^2 \left(c^2 x^2+1\right)^{3/2} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^2}-\frac{5 c^2 d \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 x}-\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 x^3}-\frac{5 b c^5 d^2 x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{c^2 x^2+1}}+\frac{5}{2} c^4 d^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{5 c^3 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b \sqrt{c^2 x^2+1}}+\frac{7 c^3 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 \sqrt{c^2 x^2+1}}+\frac{7}{3} b c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{14 b c^3 d^2 \sqrt{c^2 d x^2+d} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 x^2+1}}-\frac{b^2 c^2 d^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}{3 x}+\frac{7}{12} b^2 c^4 d^2 x \sqrt{c^2 d x^2+d}-\frac{7 b^2 c^3 d^2 \sqrt{c^2 d x^2+d} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)}{3 \sqrt{c^2 x^2+1}}-\frac{23 b^2 c^3 d^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{12 \sqrt{c^2 x^2+1}}",1,"(d^2*(-8*a*b*c*x*Sqrt[d + c^2*d*x^2] - 8*a^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 56*a^2*c^2*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 8*b^2*c^2*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 12*a^2*c^4*x^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 20*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 - 6*a*b*c^3*x^3*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] + 112*a*b*c^3*x^3*Sqrt[d + c^2*d*x^2]*Log[c*x] + 60*a^2*c^3*Sqrt[d]*x^3*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 56*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])] + 3*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] - 2*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(4*b*c*x + 8*a*Sqrt[1 + c^2*x^2] + 56*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 3*b*c^3*x^3*Cosh[2*ArcSinh[c*x]] - 56*b*c^3*x^3*Log[1 - E^(-2*ArcSinh[c*x])] - 6*a*c^3*x^3*Sinh[2*ArcSinh[c*x]]) + 2*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(30*a*c^3*x^3 - 4*b*(-7*c^3*x^3 + Sqrt[1 + c^2*x^2] + 7*c^2*x^2*Sqrt[1 + c^2*x^2]) + 3*b*c^3*x^3*Sinh[2*ArcSinh[c*x]])))/(24*x^3*Sqrt[1 + c^2*x^2])","A",0
282,1,98,153,0.0772127,"\int \frac{x^4 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^4*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2],x]","\frac{a x \sqrt{a^2 x^2+1} \left(2 a^2 x^2-15\right)+8 a x \sqrt{a^2 x^2+1} \left(2 a^2 x^2-3\right) \sinh ^{-1}(a x)^2+\left(-8 a^4 x^4+24 a^2 x^2+15\right) \sinh ^{-1}(a x)+8 \sinh ^{-1}(a x)^3}{64 a^5}","\frac{\sinh ^{-1}(a x)^3}{8 a^5}+\frac{15 \sinh ^{-1}(a x)}{64 a^5}+\frac{3 x^2 \sinh ^{-1}(a x)}{8 a^3}+\frac{x^3 \sqrt{a^2 x^2+1}}{32 a^2}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{4 a^2}-\frac{15 x \sqrt{a^2 x^2+1}}{64 a^4}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{8 a^4}-\frac{x^4 \sinh ^{-1}(a x)}{8 a}",1,"(a*x*Sqrt[1 + a^2*x^2]*(-15 + 2*a^2*x^2) + (15 + 24*a^2*x^2 - 8*a^4*x^4)*ArcSinh[a*x] + 8*a*x*Sqrt[1 + a^2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a*x]^2 + 8*ArcSinh[a*x]^3)/(64*a^5)","A",1
283,1,79,122,0.0621925,"\int \frac{x^3 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^3*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2],x]","\frac{2 \left(a^2 x^2-20\right) \sqrt{a^2 x^2+1}+9 \left(a^2 x^2-2\right) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2-6 a x \left(a^2 x^2-6\right) \sinh ^{-1}(a x)}{27 a^4}","\frac{4 x \sinh ^{-1}(a x)}{3 a^3}+\frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^2}+\frac{2 \left(a^2 x^2+1\right)^{3/2}}{27 a^4}-\frac{14 \sqrt{a^2 x^2+1}}{9 a^4}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{3 a^4}-\frac{2 x^3 \sinh ^{-1}(a x)}{9 a}",1,"(2*(-20 + a^2*x^2)*Sqrt[1 + a^2*x^2] - 6*a*x*(-6 + a^2*x^2)*ArcSinh[a*x] + 9*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/(27*a^4)","A",1
284,1,72,87,0.049608,"\int \frac{x^2 \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^2*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2],x]","\frac{3 a x \sqrt{a^2 x^2+1}+6 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2-3 \left(2 a^2 x^2+1\right) \sinh ^{-1}(a x)-2 \sinh ^{-1}(a x)^3}{12 a^3}","-\frac{\sinh ^{-1}(a x)^3}{6 a^3}-\frac{\sinh ^{-1}(a x)}{4 a^3}+\frac{x \sqrt{a^2 x^2+1}}{4 a^2}+\frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 a^2}-\frac{x^2 \sinh ^{-1}(a x)}{2 a}",1,"(3*a*x*Sqrt[1 + a^2*x^2] - 3*(1 + 2*a^2*x^2)*ArcSinh[a*x] + 6*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2 - 2*ArcSinh[a*x]^3)/(12*a^3)","A",1
285,1,48,52,0.0326974,"\int \frac{x \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2],x]","\frac{2 \sqrt{a^2 x^2+1}+\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2-2 a x \sinh ^{-1}(a x)}{a^2}","\frac{2 \sqrt{a^2 x^2+1}}{a^2}+\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a^2}-\frac{2 x \sinh ^{-1}(a x)}{a}",1,"(2*Sqrt[1 + a^2*x^2] - 2*a*x*ArcSinh[a*x] + Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/a^2","A",1
286,1,13,13,0.0073017,"\int \frac{\sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^2/Sqrt[1 + a^2*x^2],x]","\frac{\sinh ^{-1}(a x)^3}{3 a}","\frac{\sinh ^{-1}(a x)^3}{3 a}",1,"ArcSinh[a*x]^3/(3*a)","A",1
287,1,100,68,0.1125576,"\int \frac{\sinh ^{-1}(a x)^2}{x \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^2/(x*Sqrt[1 + a^2*x^2]),x]","2 \sinh ^{-1}(a x) \text{Li}_2\left(-e^{-\sinh ^{-1}(a x)}\right)-2 \sinh ^{-1}(a x) \text{Li}_2\left(e^{-\sinh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-e^{-\sinh ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{-\sinh ^{-1}(a x)}\right)+\sinh ^{-1}(a x)^2 \log \left(1-e^{-\sinh ^{-1}(a x)}\right)-\sinh ^{-1}(a x)^2 \log \left(e^{-\sinh ^{-1}(a x)}+1\right)","-2 \sinh ^{-1}(a x) \text{Li}_2\left(-e^{\sinh ^{-1}(a x)}\right)+2 \sinh ^{-1}(a x) \text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)+2 \text{Li}_3\left(-e^{\sinh ^{-1}(a x)}\right)-2 \text{Li}_3\left(e^{\sinh ^{-1}(a x)}\right)-2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)",1,"ArcSinh[a*x]^2*Log[1 - E^(-ArcSinh[a*x])] - ArcSinh[a*x]^2*Log[1 + E^(-ArcSinh[a*x])] + 2*ArcSinh[a*x]*PolyLog[2, -E^(-ArcSinh[a*x])] - 2*ArcSinh[a*x]*PolyLog[2, E^(-ArcSinh[a*x])] + 2*PolyLog[3, -E^(-ArcSinh[a*x])] - 2*PolyLog[3, E^(-ArcSinh[a*x])]","A",0
288,1,65,66,0.3302423,"\int \frac{\sinh ^{-1}(a x)^2}{x^2 \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^2/(x^2*Sqrt[1 + a^2*x^2]),x]","a \left(\sinh ^{-1}(a x) \left(-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a x}+\sinh ^{-1}(a x)+2 \log \left(1-e^{-2 \sinh ^{-1}(a x)}\right)\right)-\text{Li}_2\left(e^{-2 \sinh ^{-1}(a x)}\right)\right)","-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{x}+a \text{Li}_2\left(e^{2 \sinh ^{-1}(a x)}\right)-a \sinh ^{-1}(a x)^2+2 a \sinh ^{-1}(a x) \log \left(1-e^{2 \sinh ^{-1}(a x)}\right)",1,"a*(ArcSinh[a*x]*(ArcSinh[a*x] - (Sqrt[1 + a^2*x^2]*ArcSinh[a*x])/(a*x) + 2*Log[1 - E^(-2*ArcSinh[a*x])]) - PolyLog[2, E^(-2*ArcSinh[a*x])])","A",0
289,1,188,135,1.2007356,"\int \frac{\sinh ^{-1}(a x)^2}{x^3 \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^2/(x^3*Sqrt[1 + a^2*x^2]),x]","\frac{1}{8} a^2 \left(-8 \sinh ^{-1}(a x) \text{Li}_2\left(-e^{-\sinh ^{-1}(a x)}\right)+8 \sinh ^{-1}(a x) \text{Li}_2\left(e^{-\sinh ^{-1}(a x)}\right)-8 \text{Li}_3\left(-e^{-\sinh ^{-1}(a x)}\right)+8 \text{Li}_3\left(e^{-\sinh ^{-1}(a x)}\right)-4 \sinh ^{-1}(a x)^2 \log \left(1-e^{-\sinh ^{-1}(a x)}\right)+4 \sinh ^{-1}(a x)^2 \log \left(e^{-\sinh ^{-1}(a x)}+1\right)+4 \sinh ^{-1}(a x) \tanh \left(\frac{1}{2} \sinh ^{-1}(a x)\right)-4 \sinh ^{-1}(a x) \coth \left(\frac{1}{2} \sinh ^{-1}(a x)\right)+\sinh ^{-1}(a x)^2 \left(-\text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(a x)\right)\right)-\sinh ^{-1}(a x)^2 \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(a x)\right)+8 \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(a x)\right)\right)\right)","a^2 \sinh ^{-1}(a x) \text{Li}_2\left(-e^{\sinh ^{-1}(a x)}\right)-a^2 \sinh ^{-1}(a x) \text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)-a^2 \text{Li}_3\left(-e^{\sinh ^{-1}(a x)}\right)+a^2 \text{Li}_3\left(e^{\sinh ^{-1}(a x)}\right)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{2 x^2}-a^2 \tanh ^{-1}\left(\sqrt{a^2 x^2+1}\right)+a^2 \sinh ^{-1}(a x)^2 \tanh ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)-\frac{a \sinh ^{-1}(a x)}{x}",1,"(a^2*(-4*ArcSinh[a*x]*Coth[ArcSinh[a*x]/2] - ArcSinh[a*x]^2*Csch[ArcSinh[a*x]/2]^2 - 4*ArcSinh[a*x]^2*Log[1 - E^(-ArcSinh[a*x])] + 4*ArcSinh[a*x]^2*Log[1 + E^(-ArcSinh[a*x])] + 8*Log[Tanh[ArcSinh[a*x]/2]] - 8*ArcSinh[a*x]*PolyLog[2, -E^(-ArcSinh[a*x])] + 8*ArcSinh[a*x]*PolyLog[2, E^(-ArcSinh[a*x])] - 8*PolyLog[3, -E^(-ArcSinh[a*x])] + 8*PolyLog[3, E^(-ArcSinh[a*x])] - ArcSinh[a*x]^2*Sech[ArcSinh[a*x]/2]^2 + 4*ArcSinh[a*x]*Tanh[ArcSinh[a*x]/2]))/8","A",0
290,1,230,383,0.3665422,"\int \frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^5*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]","\frac{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^2 x^2+1} \left(9 c^4 x^4-20 c^2 x^2+120\right)+30 b \sinh ^{-1}(c x) \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^2 x^2+1} \left(-9 c^4 x^4+20 c^2 x^2-120\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) \sinh ^{-1}(c x)^2}{3375 c^6 \sqrt{c^2 d x^2+d}}","-\frac{2 b x^5 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{25 c \sqrt{c^2 d x^2+d}}+\frac{x^4 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{5 c^2 d}+\frac{8 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{15 c^6 d}-\frac{16 a b x \sqrt{c^2 x^2+1}}{15 c^5 \sqrt{c^2 d x^2+d}}-\frac{4 x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{15 c^4 d}+\frac{8 b x^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{45 c^3 \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \left(c^2 x^2+1\right)^3}{125 c^6 \sqrt{c^2 d x^2+d}}-\frac{76 b^2 \left(c^2 x^2+1\right)^2}{675 c^6 \sqrt{c^2 d x^2+d}}+\frac{298 b^2 \left(c^2 x^2+1\right)}{225 c^6 \sqrt{c^2 d x^2+d}}-\frac{16 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{15 c^5 \sqrt{c^2 d x^2+d}}",1,"(-30*a*b*c*x*Sqrt[1 + c^2*x^2]*(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^2*x^2]*(-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))*ArcSinh[c*x] + 225*b^2*(8 + 4*c^2*x^2 - c^4*x^4 + 3*c^6*x^6)*ArcSinh[c*x]^2)/(3375*c^6*Sqrt[d + c^2*d*x^2])","A",1
291,1,268,323,0.8240248,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^4*(a + b*ArcSinh[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{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+4 a b \sqrt{d} \sqrt{c^2 x^2+1} \left(4 \sinh ^{-1}(c x) \left(6 \sinh ^{-1}(c x)-8 \sinh \left(2 \sinh ^{-1}(c x)\right)+\sinh \left(4 \sinh ^{-1}(c x)\right)\right)+16 \cosh \left(2 \sinh ^{-1}(c x)\right)-\cosh \left(4 \sinh ^{-1}(c x)\right)\right)+b^2 \sqrt{d} \sqrt{c^2 x^2+1} \left(32 \sinh ^{-1}(c x)^3+8 \left(\sinh \left(4 \sinh ^{-1}(c x)\right)-8 \sinh \left(2 \sinh ^{-1}(c x)\right)\right) \sinh ^{-1}(c x)^2-32 \sinh \left(2 \sinh ^{-1}(c x)\right)+\sinh \left(4 \sinh ^{-1}(c x)\right)-4 \sinh ^{-1}(c x) \left(\cosh \left(4 \sinh ^{-1}(c x)\right)-16 \cosh \left(2 \sinh ^{-1}(c x)\right)\right)\right)}{256 c^5 \sqrt{d} \sqrt{c^2 d x^2+d}}","-\frac{b x^4 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{8 c \sqrt{c^2 d x^2+d}}+\frac{x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^2 d}+\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{8 b c^5 \sqrt{c^2 d x^2+d}}-\frac{3 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{8 c^4 d}+\frac{3 b x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{8 c^3 \sqrt{c^2 d x^2+d}}+\frac{b^2 x^3 \left(c^2 x^2+1\right)}{32 c^2 \sqrt{c^2 d x^2+d}}+\frac{15 b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{64 c^5 \sqrt{c^2 d x^2+d}}-\frac{15 b^2 x \left(c^2 x^2+1\right)}{64 c^4 \sqrt{c^2 d x^2+d}}",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]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + b^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*(32*ArcSinh[c*x]^3 - 4*ArcSinh[c*x]*(-16*Cosh[2*ArcSinh[c*x]] + Cosh[4*ArcSinh[c*x]]) - 32*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]] + 8*ArcSinh[c*x]^2*(-8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])) + 4*a*b*Sqrt[d]*Sqrt[1 + c^2*x^2]*(16*Cosh[2*ArcSinh[c*x]] - Cosh[4*ArcSinh[c*x]] + 4*ArcSinh[c*x]*(6*ArcSinh[c*x] - 8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])))/(256*c^5*Sqrt[d]*Sqrt[d + c^2*d*x^2])","A",1
292,1,176,265,0.2794758,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]","\frac{9 a^2 \left(c^4 x^4-c^2 x^2-2\right)-6 a b c x \left(c^2 x^2-6\right) \sqrt{c^2 x^2+1}-6 b \sinh ^{-1}(c x) \left(a \left(-3 c^4 x^4+3 c^2 x^2+6\right)+b c x \sqrt{c^2 x^2+1} \left(c^2 x^2-6\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) \sinh ^{-1}(c x)^2}{27 c^4 \sqrt{c^2 d x^2+d}}","\frac{x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^2 d}-\frac{2 b x^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c \sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^4 d}+\frac{4 a b x \sqrt{c^2 x^2+1}}{3 c^3 \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \left(c^2 x^2+1\right)^2}{27 c^4 \sqrt{c^2 d x^2+d}}-\frac{14 b^2 \left(c^2 x^2+1\right)}{9 c^4 \sqrt{c^2 d x^2+d}}+\frac{4 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^3 \sqrt{c^2 d x^2+d}}",1,"(-6*a*b*c*x*(-6 + c^2*x^2)*Sqrt[1 + c^2*x^2] + 2*b^2*(-20 - 19*c^2*x^2 + c^4*x^4) + 9*a^2*(-2 - c^2*x^2 + c^4*x^4) - 6*b*(b*c*x*(-6 + c^2*x^2)*Sqrt[1 + c^2*x^2] + a*(6 + 3*c^2*x^2 - 3*c^4*x^4))*ArcSinh[c*x] + 9*b^2*(-2 - c^2*x^2 + c^4*x^4)*ArcSinh[c*x]^2)/(27*c^4*Sqrt[d + c^2*d*x^2])","A",1
293,1,198,204,0.8906111,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]","\frac{12 a^2 c x \left(c^2 d x^2+d\right)-12 a^2 \sqrt{d} \sqrt{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)-6 a b d \sqrt{c^2 x^2+1} \left(2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-\sinh \left(2 \sinh ^{-1}(c x)\right)\right)+\cosh \left(2 \sinh ^{-1}(c x)\right)\right)-b^2 d \sqrt{c^2 x^2+1} \left(4 \sinh ^{-1}(c x)^3-3 \left(2 \sinh ^{-1}(c x)^2+1\right) \sinh \left(2 \sinh ^{-1}(c x)\right)+6 \sinh ^{-1}(c x) \cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{24 c^3 d \sqrt{c^2 d x^2+d}}","\frac{x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^2 d}-\frac{b x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 c \sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b c^3 \sqrt{c^2 d x^2+d}}+\frac{b^2 x \left(c^2 x^2+1\right)}{4 c^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^3 \sqrt{c^2 d x^2+d}}",1,"(12*a^2*c*x*(d + c^2*d*x^2) - 12*a^2*Sqrt[d]*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 6*a*b*d*Sqrt[1 + c^2*x^2]*(Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*(ArcSinh[c*x] - Sinh[2*ArcSinh[c*x]])) - b^2*d*Sqrt[1 + c^2*x^2]*(4*ArcSinh[c*x]^3 + 6*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] - 3*(1 + 2*ArcSinh[c*x]^2)*Sinh[2*ArcSinh[c*x]]))/(24*c^3*d*Sqrt[d + c^2*d*x^2])","A",1
294,1,127,138,0.280364,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]","\frac{\sqrt{c^2 d x^2+d} \left(a^2 \sqrt{c^2 x^2+1}-2 b \sinh ^{-1}(c x) \left(b c x-a \sqrt{c^2 x^2+1}\right)-2 a b c x+2 b^2 \sqrt{c^2 x^2+1}+b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2\right)}{c^2 d \sqrt{c^2 x^2+1}}","-\frac{2 a b x \sqrt{c^2 x^2+1}}{c \sqrt{c^2 d x^2+d}}+\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{c^2 d}+\frac{2 b^2 \left(c^2 x^2+1\right)}{c^2 \sqrt{c^2 d x^2+d}}-\frac{2 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{c \sqrt{c^2 d x^2+d}}",1,"(Sqrt[d + c^2*d*x^2]*(-2*a*b*c*x + a^2*Sqrt[1 + c^2*x^2] + 2*b^2*Sqrt[1 + c^2*x^2] - 2*b*(b*c*x - a*Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2))/(c^2*d*Sqrt[1 + c^2*x^2])","A",1
295,1,62,47,0.1330465,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/Sqrt[d + c^2*d*x^2],x]","\frac{\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \left(3 a^2+3 a b \sinh ^{-1}(c x)+b^2 \sinh ^{-1}(c x)^2\right)}{3 c \sqrt{c^2 d x^2+d}}","\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c \sqrt{c^2 d x^2+d}}",1,"(Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*(3*a^2 + 3*a*b*ArcSinh[c*x] + b^2*ArcSinh[c*x]^2))/(3*c*Sqrt[d + c^2*d*x^2])","A",1
296,1,266,223,0.8658469,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x*Sqrt[d + c^2*d*x^2]),x]","-\frac{a^2 \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)}{\sqrt{d}}+\frac{a^2 \log (c x)}{\sqrt{d}}+\frac{2 a b \sqrt{c^2 x^2+1} \left(\text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+\sinh ^{-1}(c x) \left(\log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)\right)}{\sqrt{c^2 d x^2+d}}+\frac{b^2 \sqrt{c^2 x^2+1} \left(2 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-2 \sinh ^{-1}(c x) \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+2 \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-2 \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)+\sinh ^{-1}(c x)^2 \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x)^2 \log \left(e^{-\sinh ^{-1}(c x)}+1\right)\right)}{\sqrt{c^2 d x^2+d}}","-\frac{2 b \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 d x^2+d}}+\frac{2 b \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 d x^2+d}}+\frac{2 b^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 d x^2+d}}-\frac{2 b^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 d x^2+d}}",1,"(a^2*Log[c*x])/Sqrt[d] - (a^2*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/Sqrt[d] + (2*a*b*Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])]) + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2, E^(-ArcSinh[c*x])]))/Sqrt[d + c^2*d*x^2] + (b^2*Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 2*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - 2*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 2*PolyLog[3, -E^(-ArcSinh[c*x])] - 2*PolyLog[3, E^(-ArcSinh[c*x])]))/Sqrt[d + c^2*d*x^2]","A",0
297,1,168,167,0.4583735,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^2 \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^2*Sqrt[d + c^2*d*x^2]),x]","\frac{-a \left(a c^2 x^2+a-2 b c x \sqrt{c^2 x^2+1} \log (c x)\right)-2 b \sinh ^{-1}(c x) \left(a c^2 x^2+a-b c x \sqrt{c^2 x^2+1} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)\right)-b^2 c x \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+b^2 \left(-c^2 x^2+c x \sqrt{c^2 x^2+1}-1\right) \sinh ^{-1}(c x)^2}{x \sqrt{c^2 d x^2+d}}","-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{d x}+\frac{c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 d x^2+d}}+\frac{2 b c \sqrt{c^2 x^2+1} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 d x^2+d}}-\frac{b^2 c \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)}{\sqrt{c^2 d x^2+d}}",1,"(b^2*(-1 - c^2*x^2 + c*x*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 - 2*b*ArcSinh[c*x]*(a + a*c^2*x^2 - b*c*x*Sqrt[1 + c^2*x^2]*Log[1 - E^(-2*ArcSinh[c*x])]) - a*(a + a*c^2*x^2 - 2*b*c*x*Sqrt[1 + c^2*x^2]*Log[c*x]) - b^2*c*x*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(x*Sqrt[d + c^2*d*x^2])","A",0
298,1,455,360,5.473622,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^3 \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^3*Sqrt[d + c^2*d*x^2]),x]","\frac{-\frac{4 a^2 \sqrt{c^2 d x^2+d}}{x^2}+4 a^2 c^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)-4 a^2 c^2 \sqrt{d} \log (x)+\frac{2 a b c^2 d^2 \left(c^2 x^2+1\right)^{3/2} \left(-4 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+4 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-2 \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{\left(c^2 d x^2+d\right)^{3/2}}+\frac{b^2 c^2 d^2 \left(c^2 x^2+1\right)^{3/2} \left(-8 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+8 \sinh ^{-1}(c x) \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)-8 \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)+8 \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)-4 \sinh ^{-1}(c x)^2 \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+4 \sinh ^{-1}(c x)^2 \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+4 \sinh ^{-1}(c x) \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 \sinh ^{-1}(c x) \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+\sinh ^{-1}(c x)^2 \left(-\text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-\sinh ^{-1}(c x)^2 \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)+8 \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{\left(c^2 d x^2+d\right)^{3/2}}}{8 d}","\frac{b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 d x^2+d}}-\frac{b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{x \sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 d x^2}+\frac{c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 d x^2+d}}+\frac{b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{\sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{\sqrt{c^2 d x^2+d}}",1,"((-4*a^2*Sqrt[d + c^2*d*x^2])/x^2 - 4*a^2*c^2*Sqrt[d]*Log[x] + 4*a^2*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (2*a*b*c^2*d^2*(1 + c^2*x^2)^(3/2)*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 4*PolyLog[2, -E^(-ArcSinh[c*x])] + 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(d + c^2*d*x^2)^(3/2) + (b^2*c^2*d^2*(1 + c^2*x^2)^(3/2)*(-4*ArcSinh[c*x]*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 - 4*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] + 4*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 8*Log[Tanh[ArcSinh[c*x]/2]] - 8*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] + 8*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] - 8*PolyLog[3, -E^(-ArcSinh[c*x])] + 8*PolyLog[3, E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/(d + c^2*d*x^2)^(3/2))/(8*d)","A",0
299,1,278,299,0.6467691,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^4 \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^4*Sqrt[d + c^2*d*x^2]),x]","\frac{2 a^2 c^4 x^4+a^2 c^2 x^2-a^2-a b c x \sqrt{c^2 x^2+1}-4 a b c^3 x^3 \sqrt{c^2 x^2+1} \log (c x)-b \sinh ^{-1}(c x) \left(-2 a \left(2 c^4 x^4+c^2 x^2-1\right)+b c x \sqrt{c^2 x^2+1}+4 b c^3 x^3 \sqrt{c^2 x^2+1} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)\right)-b^2 c^4 x^4-b^2 c^2 x^2+2 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+b^2 \left(2 c^4 x^4+c^2 x^2-2 c^3 x^3 \sqrt{c^2 x^2+1}-1\right) \sinh ^{-1}(c x)^2}{3 x^3 \sqrt{c^2 d x^2+d}}","\frac{2 c^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d x}-\frac{b c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 x^2 \sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d x^3}-\frac{2 c^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 \sqrt{c^2 d x^2+d}}-\frac{4 b c^3 \sqrt{c^2 x^2+1} \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \left(c^2 x^2+1\right)}{3 x \sqrt{c^2 d x^2+d}}+\frac{2 b^2 c^3 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)}{3 \sqrt{c^2 d x^2+d}}",1,"(-a^2 + a^2*c^2*x^2 - b^2*c^2*x^2 + 2*a^2*c^4*x^4 - b^2*c^4*x^4 - a*b*c*x*Sqrt[1 + c^2*x^2] + b^2*(-1 + c^2*x^2 + 2*c^4*x^4 - 2*c^3*x^3*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 - b*ArcSinh[c*x]*(b*c*x*Sqrt[1 + c^2*x^2] - 2*a*(-1 + c^2*x^2 + 2*c^4*x^4) + 4*b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[1 - E^(-2*ArcSinh[c*x])]) - 4*a*b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[c*x] + 2*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*x^3*Sqrt[d + c^2*d*x^2])","A",0
300,1,427,515,0.6331485,"\int \frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2),x]","\frac{9 a^2 c^4 x^4-36 a^2 c^2 x^2-72 a^2+18 a b c^4 x^4 \sinh ^{-1}(c x)+90 a b c x \sqrt{c^2 x^2+1}-72 a b c^2 x^2 \sinh ^{-1}(c x)+108 a b \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-6 a b c^3 x^3 \sqrt{c^2 x^2+1}-144 a b \sinh ^{-1}(c x)+2 b^2 c^4 x^4+9 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-54 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+54 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-92 b^2 c^2 x^2-36 b^2 c^2 x^2 \sinh ^{-1}(c x)^2+90 b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-54 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+54 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-6 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-72 b^2 \sinh ^{-1}(c x)^2-94 b^2}{27 c^6 d \sqrt{c^2 d x^2+d}}","-\frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{8 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^6 d^2}+\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{16 a b x \sqrt{c^2 x^2+1}}{3 c^5 d \sqrt{c^2 d x^2+d}}-\frac{2 b x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d \sqrt{c^2 d x^2+d}}+\frac{4 x^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^4 d^2}-\frac{2 b x^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c^3 d \sqrt{c^2 d x^2+d}}-\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c^6 d \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \left(c^2 x^2+1\right)^2}{27 c^6 d \sqrt{c^2 d x^2+d}}-\frac{32 b^2 \left(c^2 x^2+1\right)}{9 c^6 d \sqrt{c^2 d x^2+d}}+\frac{16 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d \sqrt{c^2 d x^2+d}}",1,"(-72*a^2 - 94*b^2 - 36*a^2*c^2*x^2 - 92*b^2*c^2*x^2 + 9*a^2*c^4*x^4 + 2*b^2*c^4*x^4 + 90*a*b*c*x*Sqrt[1 + c^2*x^2] - 6*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 144*a*b*ArcSinh[c*x] - 72*a*b*c^2*x^2*ArcSinh[c*x] + 18*a*b*c^4*x^4*ArcSinh[c*x] + 90*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 72*b^2*ArcSinh[c*x]^2 - 36*b^2*c^2*x^2*ArcSinh[c*x]^2 + 9*b^2*c^4*x^4*ArcSinh[c*x]^2 + 108*a*b*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] - (54*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + (54*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] - (54*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (54*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(27*c^6*d*Sqrt[d + c^2*d*x^2])","A",0
301,1,288,400,1.9048964,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcSinh[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{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+2 a b \sqrt{d} \left(8 c x \sinh ^{-1}(c x)-\sqrt{c^2 x^2+1} \left(4 \log \left(c^2 x^2+1\right)+6 \sinh ^{-1}(c x)^2-2 \sinh \left(2 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\cosh \left(2 \sinh ^{-1}(c x)\right)\right)\right)+b^2 \sqrt{d} \left(8 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)+\sqrt{c^2 x^2+1} \left(-4 \sinh ^{-1}(c x)^3+2 \left(\sinh \left(2 \sinh ^{-1}(c x)\right)-4\right) \sinh ^{-1}(c x)^2+\sinh \left(2 \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \left(8 \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)+\cosh \left(2 \sinh ^{-1}(c x)\right)\right)\right)+8 c x \sinh ^{-1}(c x)^2\right)}{8 c^5 d^{3/2} \sqrt{c^2 d x^2+d}}","-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{2 b c^5 d \sqrt{c^2 d x^2+d}}+\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{c^5 d \sqrt{c^2 d x^2+d}}-\frac{2 b \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^5 d \sqrt{c^2 d x^2+d}}+\frac{3 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^4 d^2}-\frac{b x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 c^3 d \sqrt{c^2 d x^2+d}}-\frac{b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c^5 d \sqrt{c^2 d x^2+d}}-\frac{b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c^5 d \sqrt{c^2 d x^2+d}}+\frac{b^2 x \left(c^2 x^2+1\right)}{4 c^4 d \sqrt{c^2 d x^2+d}}",1,"(4*a^2*c*Sqrt[d]*x*(3 + c^2*x^2) - 12*a^2*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + b^2*Sqrt[d]*(8*c*x*ArcSinh[c*x]^2 + 8*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[c*x])] + Sqrt[1 + c^2*x^2]*(-4*ArcSinh[c*x]^3 - 2*ArcSinh[c*x]*(Cosh[2*ArcSinh[c*x]] + 8*Log[1 + E^(-2*ArcSinh[c*x])]) + 2*ArcSinh[c*x]^2*(-4 + Sinh[2*ArcSinh[c*x]]) + Sinh[2*ArcSinh[c*x]])) + 2*a*b*Sqrt[d]*(8*c*x*ArcSinh[c*x] - Sqrt[1 + c^2*x^2]*(6*ArcSinh[c*x]^2 + Cosh[2*ArcSinh[c*x]] + 4*Log[1 + c^2*x^2] - 2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]])))/(8*c^5*d^(3/2)*Sqrt[d + c^2*d*x^2])","A",0
302,1,318,383,0.4677179,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2),x]","\frac{a^2 c^2 x^2+2 a^2-2 a b c x \sqrt{c^2 x^2+1}+2 a b c^2 x^2 \sinh ^{-1}(c x)-4 a b \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4 a b \sinh ^{-1}(c x)+2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+2 b^2 c^2 x^2+b^2 c^2 x^2 \sinh ^{-1}(c x)^2-2 b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-2 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+2 b^2 \sinh ^{-1}(c x)^2+2 b^2}{c^4 d \sqrt{c^2 d x^2+d}}","-\frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{c^2 d \sqrt{c^2 d x^2+d}}+\frac{2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{c^4 d^2}-\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^4 d \sqrt{c^2 d x^2+d}}-\frac{4 a b x \sqrt{c^2 x^2+1}}{c^3 d \sqrt{c^2 d x^2+d}}+\frac{2 b x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c^4 d \sqrt{c^2 d x^2+d}}-\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c^4 d \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \left(c^2 x^2+1\right)}{c^4 d \sqrt{c^2 d x^2+d}}-\frac{4 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{c^3 d \sqrt{c^2 d x^2+d}}",1,"(2*a^2 + 2*b^2 + a^2*c^2*x^2 + 2*b^2*c^2*x^2 - 2*a*b*c*x*Sqrt[1 + c^2*x^2] + 4*a*b*ArcSinh[c*x] + 2*a*b*c^2*x^2*ArcSinh[c*x] - 2*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 2*b^2*ArcSinh[c*x]^2 + b^2*c^2*x^2*ArcSinh[c*x]^2 - 4*a*b*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] + (2*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + (2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(c^4*d*Sqrt[d + c^2*d*x^2])","A",0
303,1,215,233,1.0104643,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2),x]","\frac{3 a^2 \sqrt{d} \sqrt{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)-3 a^2 c d x-3 a b d \left(2 c x \sinh ^{-1}(c x)-\sqrt{c^2 x^2+1} \left(\log \left(c^2 x^2+1\right)+\sinh ^{-1}(c x)^2\right)\right)+b^2 d \left(\sinh ^{-1}(c x) \left(\sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+3\right)+6 \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)\right)-3 c x \sinh ^{-1}(c x)\right)-3 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)\right)}{3 c^3 d^2 \sqrt{c^2 d x^2+d}}","-\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{c^2 d \sqrt{c^2 d x^2+d}}+\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c^3 d \sqrt{c^2 d x^2+d}}-\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{c^3 d \sqrt{c^2 d x^2+d}}+\frac{2 b \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^3 d \sqrt{c^2 d x^2+d}}+\frac{b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c^3 d \sqrt{c^2 d x^2+d}}",1,"(-3*a^2*c*d*x - 3*a*b*d*(2*c*x*ArcSinh[c*x] - Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]^2 + Log[1 + c^2*x^2])) + 3*a^2*Sqrt[d]*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + b^2*d*(ArcSinh[c*x]*(-3*c*x*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(3 + ArcSinh[c*x]) + 6*Log[1 + E^(-2*ArcSinh[c*x])])) - 3*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[c*x])]))/(3*c^3*d^2*Sqrt[d + c^2*d*x^2])","A",0
304,1,217,188,0.4594305,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2),x]","-\frac{a^2-4 a b \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+2 a b \sinh ^{-1}(c x)+2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+2 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-2 i b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+b^2 \sinh ^{-1}(c x)^2}{c^2 d \sqrt{c^2 d x^2+d}}","-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{c^2 d \sqrt{c^2 d x^2+d}}+\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^2 d \sqrt{c^2 d x^2+d}}-\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c^2 d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c^2 d \sqrt{c^2 d x^2+d}}",1,"-((a^2 + 2*a*b*ArcSinh[c*x] + b^2*ArcSinh[c*x]^2 - 4*a*b*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] + (2*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + (2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(c^2*d*Sqrt[d + c^2*d*x^2]))","A",0
305,1,152,179,0.439236,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^(3/2),x]","\frac{a \left(a c x-b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)\right)+2 b \sinh ^{-1}(c x) \left(a c x-b \sqrt{c^2 x^2+1} \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)\right)+b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)-\left(b^2 \left(\sqrt{c^2 x^2+1}-c x\right) \sinh ^{-1}(c x)^2\right)}{c d \sqrt{c^2 d x^2+d}}","\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{d \sqrt{c^2 d x^2+d}}+\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{c d \sqrt{c^2 d x^2+d}}-\frac{2 b \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c d \sqrt{c^2 d x^2+d}}-\frac{b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c d \sqrt{c^2 d x^2+d}}",1,"(-(b^2*(-(c*x) + Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2) + 2*b*ArcSinh[c*x]*(a*c*x - b*Sqrt[1 + c^2*x^2]*Log[1 + E^(-2*ArcSinh[c*x])]) + a*(a*c*x - b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2]) + b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[c*x])])/(c*d*Sqrt[d + c^2*d*x^2])","A",0
306,1,568,412,1.5445826,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x \left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^(3/2)),x]","\frac{a^2 \sqrt{d} \sqrt{c^2 d x^2+d} \log (c x)-a^2 \sqrt{d} \sqrt{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+a^2 d+2 a b d \left(\sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-\sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-2 \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\sinh ^{-1}(c x)\right)+b^2 d \left(2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+2 i \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-2 i \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+2 \sqrt{c^2 x^2+1} \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-2 \sqrt{c^2 x^2+1} \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2 \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2 \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+2 i \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-2 i \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+\sinh ^{-1}(c x)^2\right)}{d^2 \sqrt{c^2 d x^2+d}}","-\frac{2 b \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}+\frac{2 b \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d \sqrt{c^2 d x^2+d}}-\frac{4 b \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}-\frac{2 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}-\frac{2 b^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}",1,"(a^2*d + a^2*Sqrt[d]*Sqrt[d + c^2*d*x^2]*Log[c*x] - a^2*Sqrt[d]*Sqrt[d + c^2*d*x^2]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 2*a*b*d*(ArcSinh[c*x] - 2*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-ArcSinh[c*x])]) + b^2*d*(ArcSinh[c*x]^2 + Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] + (2*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - (2*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] + (2*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (2*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]] - 2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 2*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^(-ArcSinh[c*x])] - 2*Sqrt[1 + c^2*x^2]*PolyLog[3, E^(-ArcSinh[c*x])]))/(d^2*Sqrt[d + c^2*d*x^2])","A",0
307,1,296,305,0.9260249,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^2 \left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^(3/2)),x]","-\frac{2 a^2 c^2 x^2+a^2-2 a b c x \sqrt{c^2 x^2+1} \log (c x)-a b c x \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)+4 a b c^2 x^2 \sinh ^{-1}(c x)+2 a b \sinh ^{-1}(c x)+b^2 c x \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)+b^2 c x \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)+2 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-2 b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2-2 b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)-2 b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)+b^2 \sinh ^{-1}(c x)^2}{d x \sqrt{c^2 d x^2+d}}","-\frac{2 c^2 x \left(a+b \sinh ^{-1}(c x)\right)^2}{d \sqrt{c^2 d x^2+d}}-\frac{2 c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{d \sqrt{c^2 d x^2+d}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d x \sqrt{c^2 d x^2+d}}+\frac{4 b c \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}-\frac{4 b c \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}+\frac{b^2 c \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}+\frac{b^2 c \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}",1,"-((a^2 + 2*a^2*c^2*x^2 + 2*a*b*ArcSinh[c*x] + 4*a*b*c^2*x^2*ArcSinh[c*x] + b^2*ArcSinh[c*x]^2 + 2*b^2*c^2*x^2*ArcSinh[c*x]^2 - 2*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 - 2*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] - 2*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] - 2*a*b*c*x*Sqrt[1 + c^2*x^2]*Log[c*x] - a*b*c*x*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + b^2*c*x*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[c*x])] + b^2*c*x*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(d*x*Sqrt[d + c^2*d*x^2]))","A",0
308,1,884,573,7.5340785,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^3 \left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^(3/2)),x]","-\frac{3 a^2 \log (x) c^2}{2 d^{3/2}}+\frac{3 a^2 \log \left(d+\sqrt{d \left(c^2 x^2+1\right)} \sqrt{d}\right) c^2}{2 d^{3/2}}+\frac{a b \left(-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-8 \sinh ^{-1}(c x)+16 \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-2 \sqrt{c^2 x^2+1} \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-12 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+12 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1+e^{-\sinh ^{-1}(c x)}\right)-12 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+12 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+2 \sqrt{c^2 x^2+1} \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) c^2}{4 d \sqrt{d \left(c^2 x^2+1\right)}}+\frac{b^2 \left(-\sqrt{c^2 x^2+1} \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2-\sqrt{c^2 x^2+1} \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2-12 \sqrt{c^2 x^2+1} \log \left(1-e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)^2+12 \sqrt{c^2 x^2+1} \log \left(1+e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)^2-8 \sinh ^{-1}(c x)^2-4 \sqrt{c^2 x^2+1} \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-16 i \sqrt{c^2 x^2+1} \log \left(1-i e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)+16 i \sqrt{c^2 x^2+1} \log \left(1+i e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)-24 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)+24 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)+4 \sqrt{c^2 x^2+1} \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+8 \sqrt{c^2 x^2+1} \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-16 i \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+16 i \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-24 \sqrt{c^2 x^2+1} \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)+24 \sqrt{c^2 x^2+1} \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)\right) c^2}{8 d \sqrt{d \left(c^2 x^2+1\right)}}+\sqrt{d \left(c^2 x^2+1\right)} \left(-\frac{c^2 a^2}{d^2 \left(c^2 x^2+1\right)}-\frac{a^2}{2 d^2 x^2}\right)","\frac{3 b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}-\frac{3 b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}-\frac{3 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 d \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{d x \sqrt{c^2 d x^2+d}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 d x^2 \sqrt{c^2 d x^2+d}}+\frac{4 b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{c^2 d x^2+d}}+\frac{3 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d \sqrt{c^2 d x^2+d}}-\frac{2 i b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}+\frac{2 i b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}-\frac{3 b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}+\frac{3 b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{d \sqrt{c^2 d x^2+d}}",1,"Sqrt[d*(1 + c^2*x^2)]*(-1/2*a^2/(d^2*x^2) - (a^2*c^2)/(d^2*(1 + c^2*x^2))) - (3*a^2*c^2*Log[x])/(2*d^(3/2)) + (3*a^2*c^2*Log[d + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/(2*d^(3/2)) + (a*b*c^2*(-8*ArcSinh[c*x] + 16*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] - 2*Sqrt[1 + c^2*x^2]*Coth[ArcSinh[c*x]/2] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 12*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] + 12*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Sqrt[1 + c^2*x^2]*Tanh[ArcSinh[c*x]/2]))/(4*d*Sqrt[d*(1 + c^2*x^2)]) + (b^2*c^2*(-8*ArcSinh[c*x]^2 - 4*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Coth[ArcSinh[c*x]/2] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 - 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - (16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + (16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 8*Sqrt[1 + c^2*x^2]*Log[Tanh[ArcSinh[c*x]/2]] - 24*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - (16*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (16*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]] + 24*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] - 24*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^(-ArcSinh[c*x])] + 24*Sqrt[1 + c^2*x^2]*PolyLog[3, E^(-ArcSinh[c*x])] - Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 + 4*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/(8*d*Sqrt[d*(1 + c^2*x^2)])","A",0
309,1,438,452,0.8876121,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^4 \left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^4*(d + c^2*d*x^2)^(3/2)),x]","\frac{8 a^2 c^4 x^4+4 a^2 c^2 x^2-a^2+16 a b c^4 x^4 \sinh ^{-1}(c x)-a b c x \sqrt{c^2 x^2+1}+8 a b c^2 x^2 \sinh ^{-1}(c x)-10 a b c^3 x^3 \sqrt{c^2 x^2+1} \log (c x)-3 a b c^3 x^3 \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)-2 a b \sinh ^{-1}(c x)-b^2 c^4 x^4+8 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-b^2 c^2 x^2+4 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+3 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)+5 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-8 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2-10 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)-6 b^2 c^3 x^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)-b^2 \sinh ^{-1}(c x)^2}{3 d x^3 \sqrt{c^2 d x^2+d}}","\frac{4 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d x \sqrt{c^2 d x^2+d}}-\frac{b c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 d x^2 \sqrt{c^2 d x^2+d}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{3 d x^3 \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d \sqrt{c^2 d x^2+d}}+\frac{8 c^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d \sqrt{c^2 d x^2+d}}-\frac{16 b c^3 \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d \sqrt{c^2 d x^2+d}}+\frac{20 b c^3 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \left(c^2 x^2+1\right)}{3 d x \sqrt{c^2 d x^2+d}}-\frac{b^2 c^3 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{d \sqrt{c^2 d x^2+d}}-\frac{5 b^2 c^3 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{3 d \sqrt{c^2 d x^2+d}}",1,"(-a^2 + 4*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 - b^2*c^4*x^4 - a*b*c*x*Sqrt[1 + c^2*x^2] - 2*a*b*ArcSinh[c*x] + 8*a*b*c^2*x^2*ArcSinh[c*x] + 16*a*b*c^4*x^4*ArcSinh[c*x] - b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - b^2*ArcSinh[c*x]^2 + 4*b^2*c^2*x^2*ArcSinh[c*x]^2 + 8*b^2*c^4*x^4*ArcSinh[c*x]^2 - 8*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 - 10*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] - 6*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] - 10*a*b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[c*x] - 3*a*b*c^3*x^3*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + 3*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[c*x])] + 5*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*d*x^3*Sqrt[d + c^2*d*x^2])","A",0
310,1,333,512,1.7696447,"\int \frac{x^5 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]","\frac{\sqrt{c^2 d x^2+d} \left(a^2 \left(3 c^4 x^4+12 c^2 x^2+8\right)+a b \left(2 \left(3 c^4 x^4+12 c^2 x^2+8\right) \sinh ^{-1}(c x)-\sqrt{c^2 x^2+1} \left(c x \left(6 c^2 x^2+5\right)+22 \left(c^2 x^2+1\right) \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)+b^2 \left(11 i \left(c^2 x^2+1\right)^{3/2} \left(\text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-\text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)\right)+3 \left(c^2 x^2+1\right)^2 \left(\sinh ^{-1}(c x)^2+2\right)-6 c x \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)+\left(c^2 x^2+1\right) \left(6 \sinh ^{-1}(c x)^2+1\right)+c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+11 i \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \left(\log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-\log \left(1+i e^{-\sinh ^{-1}(c x)}\right)\right)-\sinh ^{-1}(c x)^2\right)\right)}{3 c^6 d^3 \left(c^2 x^2+1\right)^2}","-\frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^2 d \left(c^2 d x^2+d\right)^{3/2}}+\frac{8 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^6 d^3}-\frac{22 b \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{16 a b x \sqrt{c^2 x^2+1}}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}+\frac{11 b x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{4 x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}-\frac{b x^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{11 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{11 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \left(c^2 x^2+1\right)}{c^6 d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2}{3 c^6 d^2 \sqrt{c^2 d x^2+d}}-\frac{16 b^2 x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}",1,"(Sqrt[d + c^2*d*x^2]*(a^2*(8 + 12*c^2*x^2 + 3*c^4*x^4) + a*b*(2*(8 + 12*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x] - Sqrt[1 + c^2*x^2]*(c*x*(5 + 6*c^2*x^2) + 22*(1 + c^2*x^2)*ArcTan[Tanh[ArcSinh[c*x]/2]])) + b^2*(c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*c*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x] - ArcSinh[c*x]^2 + 3*(1 + c^2*x^2)^2*(2 + ArcSinh[c*x]^2) + (1 + c^2*x^2)*(1 + 6*ArcSinh[c*x]^2) + (11*I)*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*(Log[1 - I/E^ArcSinh[c*x]] - Log[1 + I/E^ArcSinh[c*x]]) + (11*I)*(1 + c^2*x^2)^(3/2)*(PolyLog[2, (-I)/E^ArcSinh[c*x]] - PolyLog[2, I/E^ArcSinh[c*x]]))))/(3*c^6*d^3*(1 + c^2*x^2)^2)","A",0
311,1,359,398,1.2518677,"\int \frac{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]","\frac{a^2 (-c) \sqrt{d} x \left(4 c^2 x^2+3\right)+3 a^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+a b \sqrt{d} \left(\sqrt{c^2 x^2+1}-8 c x \left(c^2 x^2+1\right) \sinh ^{-1}(c x)+\left(c^2 x^2+1\right)^{3/2} \left(4 \log \left(c^2 x^2+1\right)+3 \sinh ^{-1}(c x)^2\right)+2 c x \sinh ^{-1}(c x)\right)-b^2 \sqrt{d} \left(c^3 x^3+4 c^3 x^3 \sinh ^{-1}(c x)^2+4 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)-\left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)^3-4 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)^2-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-8 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)+c x+3 c x \sinh ^{-1}(c x)^2\right)}{3 c^5 d^{5/2} \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}","-\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^2 d \left(c^2 d x^2+d\right)^{3/2}}+\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{4 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 b \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{c^4 d^2 \sqrt{c^2 d x^2+d}}-\frac{b x^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{4 b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 x}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}",1,"(-(a^2*c*Sqrt[d]*x*(3 + 4*c^2*x^2)) + a*b*Sqrt[d]*(Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x] - 8*c*x*(1 + c^2*x^2)*ArcSinh[c*x] + (1 + c^2*x^2)^(3/2)*(3*ArcSinh[c*x]^2 + 4*Log[1 + c^2*x^2])) + 3*a^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - b^2*Sqrt[d]*(c*x + c^3*x^3 - Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 3*c*x*ArcSinh[c*x]^2 + 4*c^3*x^3*ArcSinh[c*x]^2 - 4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2 - (1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^3 - 8*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] + 4*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(-2*ArcSinh[c*x])]))/(3*c^5*d^(5/2)*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])","A",0
312,1,301,307,1.1075241,"\int \frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]","\frac{-\left(a^2 \left(3 c^2 x^2+2\right)\right)+a b \left(\sqrt{c^2 x^2+1} \left(10 \left(c^2 x^2+1\right) \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-c x\right)-2 \left(3 c^2 x^2+2\right) \sinh ^{-1}(c x)\right)-b^2 \left(5 i \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-5 i \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+c^2 x^2+3 c^2 x^2 \sinh ^{-1}(c x)^2+c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+5 i \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-5 i \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+2 \sinh ^{-1}(c x)^2+1\right)}{3 c^4 d^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}","-\frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^2 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{10 b \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}-\frac{b x \left(a+b \sinh ^{-1}(c x)\right)}{3 c^3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{5 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}+\frac{5 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2}{3 c^4 d^2 \sqrt{c^2 d x^2+d}}",1,"(-(a^2*(2 + 3*c^2*x^2)) + a*b*(-2*(2 + 3*c^2*x^2)*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*(-(c*x) + 10*(1 + c^2*x^2)*ArcTan[Tanh[ArcSinh[c*x]/2]])) - b^2*(1 + c^2*x^2 + c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 2*ArcSinh[c*x]^2 + 3*c^2*x^2*ArcSinh[c*x]^2 + (5*I)*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - (5*I)*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + (5*I)*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (5*I)*(1 + c^2*x^2)^(3/2)*PolyLog[2, I/E^ArcSinh[c*x]]))/(3*c^4*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])","A",0
313,1,280,312,0.9622806,"\int \frac{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]","\frac{a^2 c^3 x^3-a b \sqrt{c^2 x^2+1}-a b c^2 x^2 \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)-a b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)-b \sinh ^{-1}(c x) \left(-2 a c^3 x^3+b \sqrt{c^2 x^2+1}+2 b \left(c^2 x^2+1\right)^{3/2} \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)\right)+b^2 c^3 x^3+b^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)-b^2 \left(-c^3 x^3+c^2 x^2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1}\right) \sinh ^{-1}(c x)^2+b^2 c x}{3 c^3 d^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d}}","\frac{b x^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 c d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d \left(c^2 d x^2+d\right)^{3/2}}+\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^3 d^2 \sqrt{c^2 d x^2+d}}-\frac{2 b \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c^3 d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2 x}{3 c^2 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 c^3 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^3 d^2 \sqrt{c^2 d x^2+d}}",1,"(b^2*c*x + a^2*c^3*x^3 + b^2*c^3*x^3 - a*b*Sqrt[1 + c^2*x^2] - b^2*(-(c^3*x^3) + Sqrt[1 + c^2*x^2] + c^2*x^2*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 - b*ArcSinh[c*x]*(-2*a*c^3*x^3 + b*Sqrt[1 + c^2*x^2] + 2*b*(1 + c^2*x^2)^(3/2)*Log[1 + E^(-2*ArcSinh[c*x])]) - a*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] - a*b*c^2*x^2*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + b^2*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(-2*ArcSinh[c*x])])/(3*c^3*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])","A",0
314,1,254,270,1.0164528,"\int \frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]","\frac{-a^2+a b \left(\sqrt{c^2 x^2+1} \left(2 \left(c^2 x^2+1\right) \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+c x\right)-2 \sinh ^{-1}(c x)\right)+b^2 \left(-i \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+i \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+c^2 x^2+c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-i \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+i \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-\sinh ^{-1}(c x)^2+1\right)}{3 c^2 d \left(c^2 d x^2+d\right)^{3/2}}","\frac{b x \left(a+b \sinh ^{-1}(c x)\right)}{3 c d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{2 b \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c^2 d^2 \sqrt{c^2 d x^2+d}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{3 c^2 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c^2 d^2 \sqrt{c^2 d x^2+d}}+\frac{i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{3 c^2 d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2}{3 c^2 d^2 \sqrt{c^2 d x^2+d}}",1,"(-a^2 + a*b*(-2*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*(c*x + 2*(1 + c^2*x^2)*ArcTan[Tanh[ArcSinh[c*x]/2]])) + b^2*(1 + c^2*x^2 + c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - ArcSinh[c*x]^2 - I*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + I*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] - I*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)/E^ArcSinh[c*x]] + I*(1 + c^2*x^2)^(3/2)*PolyLog[2, I/E^ArcSinh[c*x]]))/(3*c^2*d*(d + c^2*d*x^2)^(3/2))","A",0
315,1,236,292,1.2680002,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + c^2*d*x^2)^(5/2),x]","\frac{a^2 c x \left(2 c^2 x^2+3\right)+a b \left(\left(4 c^3 x^3+6 c x\right) \sinh ^{-1}(c x)+\sqrt{c^2 x^2+1} \left(1-2 \left(c^2 x^2+1\right) \log \left(c^2 x^2+1\right)\right)\right)-b^2 \left(c^3 x^3-2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)-2 c x \left(c^2 x^2+1\right) \sinh ^{-1}(c x)^2-\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+2 \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)\right)+c x-c x \sinh ^{-1}(c x)^2\right)}{3 d^2 \left(c^3 x^2+c\right) \sqrt{c^2 d x^2+d}}","\frac{b \left(a+b \sinh ^{-1}(c x)\right)}{3 c d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{2 x \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c d^2 \sqrt{c^2 d x^2+d}}-\frac{4 b \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c d^2 \sqrt{c^2 d x^2+d}}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{2 b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 c d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 x}{3 d^2 \sqrt{c^2 d x^2+d}}",1,"(a^2*c*x*(3 + 2*c^2*x^2) + a*b*((6*c*x + 4*c^3*x^3)*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*(1 - 2*(1 + c^2*x^2)*Log[1 + c^2*x^2])) - b^2*(c*x + c^3*x^3 - Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - c*x*ArcSinh[c*x]^2 - 2*c*x*(1 + c^2*x^2)*ArcSinh[c*x]^2 + 2*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*(ArcSinh[c*x] + 2*Log[1 + E^(-2*ArcSinh[c*x])]) - 2*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(-2*ArcSinh[c*x])]))/(3*d^2*(c + c^3*x^2)*Sqrt[d + c^2*d*x^2])","A",0
316,1,547,518,3.8076203,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x \left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x*(d + c^2*d*x^2)^(5/2)),x]","\frac{\frac{a^2 \left(3 c^2 x^2+4\right) \sqrt{c^2 d x^2+d}}{\left(c^2 x^2+1\right)^2}-3 a^2 \sqrt{d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+d\right)+3 a^2 \sqrt{d} \log (c x)+\frac{a b d^2 \left(c^2 x^2+1\right)^{3/2} \left(-\frac{c x}{c^2 x^2+1}+\frac{6 \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{2 \sinh ^{-1}(c x)}{\left(c^2 x^2+1\right)^{3/2}}+6 \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-6 \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+6 \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-6 \sinh ^{-1}(c x) \log \left(e^{-\sinh ^{-1}(c x)}+1\right)-14 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{\left(c^2 d x^2+d\right)^{3/2}}+\frac{b^2 d^2 \left(c^2 x^2+1\right)^{3/2} \left(-\frac{1}{\sqrt{c^2 x^2+1}}+\frac{3 \sinh ^{-1}(c x)^2}{\sqrt{c^2 x^2+1}}+\frac{\sinh ^{-1}(c x)^2}{\left(c^2 x^2+1\right)^{3/2}}-\frac{c x \sinh ^{-1}(c x)}{c^2 x^2+1}+6 \sinh ^{-1}(c x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)-6 \sinh ^{-1}(c x) \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+7 i \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-7 i \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+6 \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)-6 \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)+3 \sinh ^{-1}(c x)^2 \log \left(1-e^{-\sinh ^{-1}(c x)}\right)-3 \sinh ^{-1}(c x)^2 \log \left(e^{-\sinh ^{-1}(c x)}+1\right)+7 i \sinh ^{-1}(c x) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-7 i \sinh ^{-1}(c x) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)\right)}{\left(c^2 d x^2+d\right)^{3/2}}}{3 d^3}","-\frac{2 b \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \sqrt{c^2 d x^2+d}}+\frac{2 b \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{b c x \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d^2 \sqrt{c^2 d x^2+d}}-\frac{14 b \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d^2 \sqrt{c^2 d x^2+d}}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{3 d \left(c^2 d x^2+d\right)^{3/2}}+\frac{7 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{7 i b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{2 b^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{2 b^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2}{3 d^2 \sqrt{c^2 d x^2+d}}",1,"((a^2*(4 + 3*c^2*x^2)*Sqrt[d + c^2*d*x^2])/(1 + c^2*x^2)^2 + 3*a^2*Sqrt[d]*Log[c*x] - 3*a^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (a*b*d^2*(1 + c^2*x^2)^(3/2)*(-((c*x)/(1 + c^2*x^2)) + (2*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + (6*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - 14*ArcTan[Tanh[ArcSinh[c*x]/2]] + 6*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 6*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 6*PolyLog[2, -E^(-ArcSinh[c*x])] - 6*PolyLog[2, E^(-ArcSinh[c*x])]))/(d + c^2*d*x^2)^(3/2) + (b^2*d^2*(1 + c^2*x^2)^(3/2)*(-(1/Sqrt[1 + c^2*x^2]) - (c*x*ArcSinh[c*x])/(1 + c^2*x^2) + ArcSinh[c*x]^2/(1 + c^2*x^2)^(3/2) + (3*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] + (7*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - (7*I)*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] - 3*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 6*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] + (7*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (7*I)*PolyLog[2, I/E^ArcSinh[c*x]] - 6*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 6*PolyLog[3, -E^(-ArcSinh[c*x])] - 6*PolyLog[3, E^(-ArcSinh[c*x])]))/(d + c^2*d*x^2)^(3/2))/(3*d^3)","A",0
317,1,408,421,1.8835212,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^2 \left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^(5/2)),x]","-\frac{8 a^2 c^4 x^4+12 a^2 c^2 x^2+3 a^2+16 a b c^4 x^4 \sinh ^{-1}(c x)+a b c x \sqrt{c^2 x^2+1}-6 a b c x \left(c^2 x^2+1\right)^{3/2} \log (c x)-5 a b c x \left(c^2 x^2+1\right)^{3/2} \log \left(c^2 x^2+1\right)+24 a b c^2 x^2 \sinh ^{-1}(c x)+6 a b \sinh ^{-1}(c x)-b^2 c^4 x^4+8 b^2 c^4 x^4 \sinh ^{-1}(c x)^2+5 b^2 c x \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)+3 b^2 c x \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-b^2 c^2 x^2+12 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-8 b^2 c x \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)^2+b^2 c x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-6 b^2 c x \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)-10 b^2 c x \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)+3 b^2 \sinh ^{-1}(c x)^2}{3 d x \left(c^2 d x^2+d\right)^{3/2}}","-\frac{b c \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{8 c^2 x \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{8 c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{16 b c \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{4 b c \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{4 c^2 x \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d x \left(c^2 d x^2+d\right)^{3/2}}+\frac{5 b^2 c \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2 c \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2 c^2 x}{3 d^2 \sqrt{c^2 d x^2+d}}",1,"-1/3*(3*a^2 + 12*a^2*c^2*x^2 - b^2*c^2*x^2 + 8*a^2*c^4*x^4 - b^2*c^4*x^4 + a*b*c*x*Sqrt[1 + c^2*x^2] + 6*a*b*ArcSinh[c*x] + 24*a*b*c^2*x^2*ArcSinh[c*x] + 16*a*b*c^4*x^4*ArcSinh[c*x] + b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 3*b^2*ArcSinh[c*x]^2 + 12*b^2*c^2*x^2*ArcSinh[c*x]^2 + 8*b^2*c^4*x^4*ArcSinh[c*x]^2 - 8*b^2*c*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2 - 6*b^2*c*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] - 10*b^2*c*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] - 6*a*b*c*x*(1 + c^2*x^2)^(3/2)*Log[c*x] - 5*a*b*c*x*(1 + c^2*x^2)^(3/2)*Log[1 + c^2*x^2] + 5*b^2*c*x*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(-2*ArcSinh[c*x])] + 3*b^2*c*x*(1 + c^2*x^2)^(3/2)*PolyLog[2, E^(-2*ArcSinh[c*x])])/(d*x*(d + c^2*d*x^2)^(3/2))","A",0
318,1,983,687,8.0194861,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^3 \left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[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(-3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\frac{8 \sinh ^{-1}(c x)}{c^2 x^2+1}-48 \sinh ^{-1}(c x)+104 \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-6 \sqrt{c^2 x^2+1} \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-60 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-e^{-\sinh ^{-1}(c x)}\right)+60 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1+e^{-\sinh ^{-1}(c x)}\right)-60 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right)+60 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right)+6 \sqrt{c^2 x^2+1} \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+\frac{4 c x}{\sqrt{c^2 x^2+1}}\right) c^2}{12 d^2 \sqrt{d \left(c^2 x^2+1\right)}}+\frac{b^2 \left(-3 \sqrt{c^2 x^2+1} \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2-3 \sqrt{c^2 x^2+1} \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2-60 \sqrt{c^2 x^2+1} \log \left(1-e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)^2+60 \sqrt{c^2 x^2+1} \log \left(1+e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)^2-\frac{8 \sinh ^{-1}(c x)^2}{c^2 x^2+1}-48 \sinh ^{-1}(c x)^2-12 \sqrt{c^2 x^2+1} \coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-104 i \sqrt{c^2 x^2+1} \log \left(1-i e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)+104 i \sqrt{c^2 x^2+1} \log \left(1+i e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)-120 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)+120 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)+12 \sqrt{c^2 x^2+1} \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\frac{8 c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+24 \sqrt{c^2 x^2+1} \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-104 i \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+104 i \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-120 \sqrt{c^2 x^2+1} \text{Li}_3\left(-e^{-\sinh ^{-1}(c x)}\right)+120 \sqrt{c^2 x^2+1} \text{Li}_3\left(e^{-\sinh ^{-1}(c x)}\right)+8\right) c^2}{24 d^2 \sqrt{d \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{5 b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{5 b c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{d^2 \sqrt{c^2 d x^2+d}}-\frac{5 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 d^2 \sqrt{c^2 d x^2+d}}-\frac{b c \left(a+b \sinh ^{-1}(c x)\right)}{d^2 x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{26 b c^2 \sqrt{c^2 x^2+1} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{5 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{d^2 \sqrt{c^2 d x^2+d}}-\frac{5 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{6 d \left(c^2 d x^2+d\right)^{3/2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 d x^2 \left(c^2 d x^2+d\right)^{3/2}}-\frac{2 b c^3 x \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}-\frac{13 i b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{13 i b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{5 b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(-e^{\sinh ^{-1}(c x)}\right)}{d^2 \sqrt{c^2 d x^2+d}}+\frac{5 b^2 c^2 \sqrt{c^2 x^2+1} \text{Li}_3\left(e^{\sinh ^{-1}(c x)}\right)}{d^2 \sqrt{c^2 d x^2+d}}+\frac{b^2 c^2}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{d^2 \sqrt{c^2 d x^2+d}}",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*((4*c*x)/Sqrt[1 + c^2*x^2] - 48*ArcSinh[c*x] - (8*ArcSinh[c*x])/(1 + c^2*x^2) + 104*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] - 6*Sqrt[1 + c^2*x^2]*Coth[ArcSinh[c*x]/2] - 3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 - 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] + 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 60*Sqrt[1 + c^2*x^2]*PolyLog[2, -E^(-ArcSinh[c*x])] + 60*Sqrt[1 + c^2*x^2]*PolyLog[2, E^(-ArcSinh[c*x])] - 3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 6*Sqrt[1 + c^2*x^2]*Tanh[ArcSinh[c*x]/2]))/(12*d^2*Sqrt[d*(1 + c^2*x^2)]) + (b^2*c^2*(8 + (8*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - 48*ArcSinh[c*x]^2 - (8*ArcSinh[c*x]^2)/(1 + c^2*x^2) - 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Coth[ArcSinh[c*x]/2] - 3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 - 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 - E^(-ArcSinh[c*x])] - (104*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + (104*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + 60*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 24*Sqrt[1 + c^2*x^2]*Log[Tanh[ArcSinh[c*x]/2]] - 120*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - (104*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (104*I)*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]] + 120*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] - 120*Sqrt[1 + c^2*x^2]*PolyLog[3, -E^(-ArcSinh[c*x])] + 120*Sqrt[1 + c^2*x^2]*PolyLog[3, E^(-ArcSinh[c*x])] - 3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 + 12*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/(24*d^2*Sqrt[d*(1 + c^2*x^2)])","A",0
319,1,417,506,3.0510866,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{x^4 \left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[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}-\frac{a b \left(c x \sqrt{c^2 x^2+1} \left(16 \left(c^4 x^4+c^2 x^2\right) \log (c x)+8 \left(c^4 x^4+c^2 x^2\right) \log \left(c^2 x^2+1\right)+1\right)-2 \left(16 c^6 x^6+24 c^4 x^4+6 c^2 x^2-1\right) \sinh ^{-1}(c x)\right)}{x^3}+b^2 c^3 \left(c^2 x^2+1\right)^{3/2} \left(-\frac{\sqrt{c^2 x^2+1}}{c x}-\frac{c x}{\sqrt{c^2 x^2+1}}+\frac{8 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2}{c x}+\frac{8 c x \sinh ^{-1}(c x)^2}{\sqrt{c^2 x^2+1}}+\frac{c x \sinh ^{-1}(c x)^2}{\left(c^2 x^2+1\right)^{3/2}}+\frac{\sinh ^{-1}(c x)}{c^2 x^2+1}-\frac{\sinh ^{-1}(c x)}{c^2 x^2}-\frac{\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2}{c^3 x^3}+8 \text{Li}_2\left(-e^{-2 \sinh ^{-1}(c x)}\right)+8 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c x)}\right)-16 \sinh ^{-1}(c x)^2-16 \sinh ^{-1}(c x) \log \left(1-e^{-2 \sinh ^{-1}(c x)}\right)-16 \sinh ^{-1}(c x) \log \left(e^{-2 \sinh ^{-1}(c x)}+1\right)\right)}{3 d \left(c^2 d x^2+d\right)^{3/2}}","-\frac{b c \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}}+\frac{2 c^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{d x \left(c^2 d x^2+d\right)^{3/2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{3 d x^3 \left(c^2 d x^2+d\right)^{3/2}}+\frac{16 c^4 x \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{8 c^4 x \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d \left(c^2 d x^2+d\right)^{3/2}}+\frac{16 c^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{32 b c^3 \sqrt{c^2 x^2+1} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 d x^2+d}}+\frac{32 b c^3 \sqrt{c^2 x^2+1} \tanh ^{-1}\left(e^{2 \sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{b^2 c^2}{3 d^2 x \sqrt{c^2 d x^2+d}}-\frac{2 b^2 c^4 x}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{8 b^2 c^3 \sqrt{c^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 d^2 \sqrt{c^2 d x^2+d}}-\frac{8 b^2 c^3 \sqrt{c^2 x^2+1} \text{Li}_2\left(e^{2 \sinh ^{-1}(c x)}\right)}{3 d^2 \sqrt{c^2 d x^2+d}}",1,"((a^2*(-1 + 6*c^2*x^2 + 24*c^4*x^4 + 16*c^6*x^6))/x^3 - (a*b*(-2*(-1 + 6*c^2*x^2 + 24*c^4*x^4 + 16*c^6*x^6)*ArcSinh[c*x] + c*x*Sqrt[1 + c^2*x^2]*(1 + 16*(c^2*x^2 + c^4*x^4)*Log[c*x] + 8*(c^2*x^2 + c^4*x^4)*Log[1 + c^2*x^2])))/x^3 + b^2*c^3*(1 + c^2*x^2)^(3/2)*(-((c*x)/Sqrt[1 + c^2*x^2]) - Sqrt[1 + c^2*x^2]/(c*x) - ArcSinh[c*x]/(c^2*x^2) + ArcSinh[c*x]/(1 + c^2*x^2) - 16*ArcSinh[c*x]^2 + (c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^(3/2) + (8*c*x*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] - (Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2)/(c^3*x^3) + (8*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2)/(c*x) - 16*ArcSinh[c*x]*Log[1 - E^(-2*ArcSinh[c*x])] - 16*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[c*x])] + 8*PolyLog[2, -E^(-2*ArcSinh[c*x])] + 8*PolyLog[2, E^(-2*ArcSinh[c*x])]))/(3*d*(d + c^2*d*x^2)^(3/2))","A",0
320,1,178,366,0.7965564,"\int \frac{\sinh ^{-1}(a x)^2}{\left(c+a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcSinh[a*x]^2/(c + a^2*c*x^2)^(7/2),x]","\frac{16 \sqrt{a^2 x^2+1} \text{Li}_2\left(-e^{-2 \sinh ^{-1}(a x)}\right)+a x \left(-\frac{1}{a^2 x^2+1}-10\right)+\frac{\sinh ^{-1}(a x) \left(8 a^2 x^2-32 \left(a^2 x^2+1\right)^2 \log \left(e^{-2 \sinh ^{-1}(a x)}+1\right)+11\right)}{\left(a^2 x^2+1\right)^{3/2}}+\left(\frac{2 a x \left(8 a^4 x^4+20 a^2 x^2+15\right)}{\left(a^2 x^2+1\right)^2}-16 \sqrt{a^2 x^2+1}\right) \sinh ^{-1}(a x)^2}{30 a c^3 \sqrt{a^2 c x^2+c}}","-\frac{8 \sqrt{a^2 x^2+1} \text{Li}_2\left(-e^{2 \sinh ^{-1}(a x)}\right)}{15 a c^3 \sqrt{a^2 c x^2+c}}-\frac{x}{3 c^3 \sqrt{a^2 c x^2+c}}-\frac{x}{30 c^3 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c}}+\frac{8 x \sinh ^{-1}(a x)^2}{15 c^3 \sqrt{a^2 c x^2+c}}+\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 \sinh ^{-1}(a x)}{15 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{\sinh ^{-1}(a x)}{10 a c^3 \left(a^2 x^2+1\right)^{3/2} \sqrt{a^2 c x^2+c}}-\frac{16 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \log \left(e^{2 \sinh ^{-1}(a x)}+1\right)}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 x \sinh ^{-1}(a x)^2}{15 c^2 \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \sinh ^{-1}(a x)^2}{5 c \left(a^2 c x^2+c\right)^{5/2}}",1,"(a*x*(-10 - (1 + a^2*x^2)^(-1)) + (-16*Sqrt[1 + a^2*x^2] + (2*a*x*(15 + 20*a^2*x^2 + 8*a^4*x^4))/(1 + a^2*x^2)^2)*ArcSinh[a*x]^2 + (ArcSinh[a*x]*(11 + 8*a^2*x^2 - 32*(1 + a^2*x^2)^2*Log[1 + E^(-2*ArcSinh[a*x])]))/(1 + a^2*x^2)^(3/2) + 16*Sqrt[1 + a^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[a*x])])/(30*a*c^3*Sqrt[c + a^2*c*x^2])","A",0
321,0,0,936,2.8032573,"\int x^m \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^m*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2,x]","\int x^m \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","\frac{\left(c^2 d x^2+d\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 x^{m+1}}{m+6}+\frac{5 d \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 x^{m+1}}{(m+4) (m+6)}+\frac{15 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2 x^{m+1}}{(m+6) \left(m^2+6 m+8\right)}-\frac{30 b c d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^{m+2}}{(m+2)^2 (m+4) (m+6) \sqrt{c^2 x^2+1}}-\frac{10 b c d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^{m+2}}{(m+6) \left(m^2+6 m+8\right) \sqrt{c^2 x^2+1}}-\frac{2 b c d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^{m+2}}{\left(m^2+8 m+12\right) \sqrt{c^2 x^2+1}}+\frac{10 b^2 c^2 d^2 (3 m+10) \sqrt{c^2 d x^2+d} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};-c^2 x^2\right) x^{m+3}}{(m+2) (m+3) (m+4)^3 (m+6) \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^2 d^2 \left(15 m^2+130 m+264\right) \sqrt{c^2 d x^2+d} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};-c^2 x^2\right) x^{m+3}}{(m+2) (m+3) (m+4)^2 (m+6)^3 \sqrt{c^2 x^2+1}}+\frac{30 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};-c^2 x^2\right) x^{m+3}}{(m+2)^2 (m+3) (m+4) (m+6) \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^2 d^2 \left(m^2+15 m+52\right) \sqrt{c^2 d x^2+d} x^{m+3}}{(m+4)^2 (m+6)^3}+\frac{10 b^2 c^2 d^2 \sqrt{c^2 d x^2+d} x^{m+3}}{(m+4)^3 (m+6)}-\frac{4 b c^3 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^{m+4}}{(m+4) (m+6) \sqrt{c^2 x^2+1}}-\frac{10 b c^3 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^{m+4}}{(m+4)^2 (m+6) \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^4 d^2 \sqrt{c^2 d x^2+d} x^{m+5}}{(m+6)^3}-\frac{2 b c^5 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^{m+6}}{(m+6)^2 \sqrt{c^2 x^2+1}}+\frac{15 d^3 \text{Int}\left(\frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 d x^2+d}},x\right)}{(m+6) \left(m^2+6 m+8\right)}",0,"Integrate[x^m*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2, x]","A",-1
322,0,0,488,0.4314347,"\int x^m \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^m*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2,x]","\int x^m \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","\frac{3 d^2 \text{Int}\left(\frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 d x^2+d}},x\right)}{m^2+6 m+8}+\frac{3 d x^{m+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{m^2+6 m+8}-\frac{2 b c d x^{m+2} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{\left(m^2+6 m+8\right) \sqrt{c^2 x^2+1}}+\frac{x^{m+1} \left(c^2 d x^2+d\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{m+4}-\frac{6 b c d x^{m+2} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{(m+2)^2 (m+4) \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d x^{m+4} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{(m+4)^2 \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^2 d (3 m+10) x^{m+3} \sqrt{c^2 d x^2+d} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};-c^2 x^2\right)}{(m+2) (m+3) (m+4)^3 \sqrt{c^2 x^2+1}}+\frac{6 b^2 c^2 d x^{m+3} \sqrt{c^2 d x^2+d} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};-c^2 x^2\right)}{(m+2)^2 (m+3) (m+4) \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^2 d x^{m+3} \sqrt{c^2 d x^2+d}}{(m+4)^3}",0,"Integrate[x^m*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2, x]","A",-1
323,0,0,199,0.2932877,"\int x^m \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[x^m*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]","\int x^m \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","\frac{d \text{Int}\left(\frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 d x^2+d}},x\right)}{m+2}+\frac{x^{m+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{m+2}-\frac{2 b c x^{m+2} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{(m+2)^2 \sqrt{c^2 x^2+1}}+\frac{2 b^2 c^2 x^{m+3} \sqrt{c^2 d x^2+d} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};-c^2 x^2\right)}{(m+2)^2 (m+3) \sqrt{c^2 x^2+1}}",0,"Integrate[x^m*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2, x]","A",-1
324,0,0,31,2.860071,"\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]","\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+c^2 d x^2}} \, dx","\text{Int}\left(\frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{c^2 d x^2+d}},x\right)",0,"Integrate[(x^m*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2], x]","A",-1
325,0,0,31,4.3506941,"\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","Integrate[(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2),x]","\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(c^2 d x^2+d\right)^{3/2}},x\right)",0,"Integrate[(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2), x]","A",-1
326,0,0,31,4.4691063,"\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","Integrate[(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]","\int \frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+c^2 d x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^m \left(a+b \sinh ^{-1}(c x)\right)^2}{\left(c^2 d x^2+d\right)^{5/2}},x\right)",0,"Integrate[(x^m*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2), x]","A",-1
327,0,0,26,0.4990856,"\int \frac{x^m \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^m*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2],x]","\int \frac{x^m \sinh ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx","\text{Int}\left(\frac{x^m \sinh ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}},x\right)",0,"Integrate[(x^m*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2], x]","A",-1
328,1,169,359,0.2634424,"\int \left(c+a^2 c x^2\right)^3 \sinh ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)^3*ArcSinh[a*x]^3,x]","\frac{c^3 \left(-2 \sqrt{a^2 x^2+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) \sinh ^{-1}(a x)^3-11025 \sqrt{a^2 x^2+1} \left(75 a^6 x^6+351 a^4 x^4+757 a^2 x^2+2161\right) \sinh ^{-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) \sinh ^{-1}(a x)\right)}{13505625 a}","\frac{6}{343} a^6 c^3 x^7 \sinh ^{-1}(a x)+\frac{702 a^4 c^3 x^5 \sinh ^{-1}(a x)}{6125}+\frac{1514 a^2 c^3 x^3 \sinh ^{-1}(a x)}{3675}-\frac{6 c^3 \left(a^2 x^2+1\right)^{7/2}}{2401 a}-\frac{2664 c^3 \left(a^2 x^2+1\right)^{5/2}}{214375 a}-\frac{30256 c^3 \left(a^2 x^2+1\right)^{3/2}}{385875 a}-\frac{413312 c^3 \sqrt{a^2 x^2+1}}{128625 a}+\frac{1}{7} c^3 x \left(a^2 x^2+1\right)^3 \sinh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left(a^2 x^2+1\right)^2 \sinh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left(a^2 x^2+1\right) \sinh ^{-1}(a x)^3-\frac{3 c^3 \left(a^2 x^2+1\right)^{7/2} \sinh ^{-1}(a x)^2}{49 a}-\frac{18 c^3 \left(a^2 x^2+1\right)^{5/2} \sinh ^{-1}(a x)^2}{175 a}-\frac{8 c^3 \left(a^2 x^2+1\right)^{3/2} \sinh ^{-1}(a x)^2}{35 a}-\frac{48 c^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{35 a}+\frac{16}{35} c^3 x \sinh ^{-1}(a x)^3+\frac{4322 c^3 x \sinh ^{-1}(a x)}{1225}",1,"(c^3*(-2*Sqrt[1 + a^2*x^2]*(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)*ArcSinh[a*x] - 11025*Sqrt[1 + a^2*x^2]*(2161 + 757*a^2*x^2 + 351*a^4*x^4 + 75*a^6*x^6)*ArcSinh[a*x]^2 + 385875*a*x*(35 + 35*a^2*x^2 + 21*a^4*x^4 + 5*a^6*x^6)*ArcSinh[a*x]^3))/(13505625*a)","A",1
329,1,137,265,0.1501506,"\int \left(c+a^2 c x^2\right)^2 \sinh ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)^2*ArcSinh[a*x]^3,x]","\frac{c^2 \left(-2 \sqrt{a^2 x^2+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) \sinh ^{-1}(a x)^3-225 \sqrt{a^2 x^2+1} \left(9 a^4 x^4+38 a^2 x^2+149\right) \sinh ^{-1}(a x)^2+30 a x \left(27 a^4 x^4+190 a^2 x^2+2235\right) \sinh ^{-1}(a x)\right)}{16875 a}","\frac{6}{125} a^4 c^2 x^5 \sinh ^{-1}(a x)+\frac{76}{225} a^2 c^2 x^3 \sinh ^{-1}(a x)-\frac{6 c^2 \left(a^2 x^2+1\right)^{5/2}}{625 a}-\frac{272 c^2 \left(a^2 x^2+1\right)^{3/2}}{3375 a}-\frac{4144 c^2 \sqrt{a^2 x^2+1}}{1125 a}+\frac{1}{5} c^2 x \left(a^2 x^2+1\right)^2 \sinh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left(a^2 x^2+1\right) \sinh ^{-1}(a x)^3-\frac{3 c^2 \left(a^2 x^2+1\right)^{5/2} \sinh ^{-1}(a x)^2}{25 a}-\frac{4 c^2 \left(a^2 x^2+1\right)^{3/2} \sinh ^{-1}(a x)^2}{15 a}-\frac{8 c^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{5 a}+\frac{8}{15} c^2 x \sinh ^{-1}(a x)^3+\frac{298}{75} c^2 x \sinh ^{-1}(a x)",1,"(c^2*(-2*Sqrt[1 + a^2*x^2]*(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)*ArcSinh[a*x] - 225*Sqrt[1 + a^2*x^2]*(149 + 38*a^2*x^2 + 9*a^4*x^4)*ArcSinh[a*x]^2 + 1125*a*x*(15 + 10*a^2*x^2 + 3*a^4*x^4)*ArcSinh[a*x]^3))/(16875*a)","A",1
330,1,99,153,0.0824403,"\int \left(c+a^2 c x^2\right) \sinh ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)*ArcSinh[a*x]^3,x]","\frac{c \left(-2 \sqrt{a^2 x^2+1} \left(a^2 x^2+61\right)+9 a x \left(a^2 x^2+3\right) \sinh ^{-1}(a x)^3-9 \sqrt{a^2 x^2+1} \left(a^2 x^2+7\right) \sinh ^{-1}(a x)^2+6 a x \left(a^2 x^2+21\right) \sinh ^{-1}(a x)\right)}{27 a}","\frac{2}{9} a^2 c x^3 \sinh ^{-1}(a x)-\frac{2 c \left(a^2 x^2+1\right)^{3/2}}{27 a}-\frac{40 c \sqrt{a^2 x^2+1}}{9 a}+\frac{1}{3} c x \left(a^2 x^2+1\right) \sinh ^{-1}(a x)^3-\frac{c \left(a^2 x^2+1\right)^{3/2} \sinh ^{-1}(a x)^2}{3 a}-\frac{2 c \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2}{a}+\frac{2}{3} c x \sinh ^{-1}(a x)^3+\frac{14}{3} c x \sinh ^{-1}(a x)",1,"(c*(-2*Sqrt[1 + a^2*x^2]*(61 + a^2*x^2) + 6*a*x*(21 + a^2*x^2)*ArcSinh[a*x] - 9*Sqrt[1 + a^2*x^2]*(7 + a^2*x^2)*ArcSinh[a*x]^2 + 9*a*x*(3 + a^2*x^2)*ArcSinh[a*x]^3))/(27*a)","A",1
331,1,454,174,0.2270219,"\int \frac{\sinh ^{-1}(a x)^3}{c+a^2 c x^2} \, dx","Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2),x]","-\frac{i \left(192 \sinh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(a x)}\right)+192 i \pi  \sinh ^{-1}(a x) \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)+384 \sinh ^{-1}(a x) \text{Li}_3\left(-i e^{-\sinh ^{-1}(a x)}\right)-384 \sinh ^{-1}(a x) \text{Li}_3\left(-i e^{\sinh ^{-1}(a x)}\right)-48 \left(\pi -2 i \sinh ^{-1}(a x)\right)^2 \text{Li}_2\left(-i e^{-\sinh ^{-1}(a x)}\right)-48 \pi ^2 \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)+192 i \pi  \text{Li}_3\left(-i e^{-\sinh ^{-1}(a x)}\right)-192 i \pi  \text{Li}_3\left(i e^{\sinh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{-\sinh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{\sinh ^{-1}(a x)}\right)-16 \sinh ^{-1}(a x)^4-32 i \pi  \sinh ^{-1}(a x)^3+24 \pi ^2 \sinh ^{-1}(a x)^2+8 i \pi ^3 \sinh ^{-1}(a x)-64 \sinh ^{-1}(a x)^3 \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)+64 \sinh ^{-1}(a x)^3 \log \left(1+i e^{\sinh ^{-1}(a x)}\right)-96 i \pi  \sinh ^{-1}(a x)^2 \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)+96 i \pi  \sinh ^{-1}(a x)^2 \log \left(1-i e^{\sinh ^{-1}(a x)}\right)+48 \pi ^2 \sinh ^{-1}(a x) \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)-48 \pi ^2 \sinh ^{-1}(a x) \log \left(1-i e^{\sinh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)-8 i \pi ^3 \log \left(1+i e^{\sinh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(a x)\right)\right)\right)+7 \pi ^4\right)}{64 a c}","-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(a x)}\right)}{a c}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)}{a c}+\frac{6 i \sinh ^{-1}(a x) \text{Li}_3\left(-i e^{\sinh ^{-1}(a x)}\right)}{a c}-\frac{6 i \sinh ^{-1}(a x) \text{Li}_3\left(i e^{\sinh ^{-1}(a x)}\right)}{a c}-\frac{6 i \text{Li}_4\left(-i e^{\sinh ^{-1}(a x)}\right)}{a c}+\frac{6 i \text{Li}_4\left(i e^{\sinh ^{-1}(a x)}\right)}{a c}+\frac{2 \sinh ^{-1}(a x)^3 \tan ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)}{a c}",1,"((-1/64*I)*(7*Pi^4 + (8*I)*Pi^3*ArcSinh[a*x] + 24*Pi^2*ArcSinh[a*x]^2 - (32*I)*Pi*ArcSinh[a*x]^3 - 16*ArcSinh[a*x]^4 + (8*I)*Pi^3*Log[1 + I/E^ArcSinh[a*x]] + 48*Pi^2*ArcSinh[a*x]*Log[1 + I/E^ArcSinh[a*x]] - (96*I)*Pi*ArcSinh[a*x]^2*Log[1 + I/E^ArcSinh[a*x]] - 64*ArcSinh[a*x]^3*Log[1 + I/E^ArcSinh[a*x]] - 48*Pi^2*ArcSinh[a*x]*Log[1 - I*E^ArcSinh[a*x]] + (96*I)*Pi*ArcSinh[a*x]^2*Log[1 - I*E^ArcSinh[a*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcSinh[a*x]] + 64*ArcSinh[a*x]^3*Log[1 + I*E^ArcSinh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcSinh[a*x])/4]] - 48*(Pi - (2*I)*ArcSinh[a*x])^2*PolyLog[2, (-I)/E^ArcSinh[a*x]] + 192*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcSinh[a*x]] + (192*I)*Pi*ArcSinh[a*x]*PolyLog[2, I*E^ArcSinh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcSinh[a*x]] + 384*ArcSinh[a*x]*PolyLog[3, (-I)/E^ArcSinh[a*x]] - 384*ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcSinh[a*x]] + 384*PolyLog[4, (-I)/E^ArcSinh[a*x]] + 384*PolyLog[4, (-I)*E^ArcSinh[a*x]]))/(a*c)","B",0
332,1,568,294,2.60054,"\int \frac{\sinh ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^2} \, dx","Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2)^2,x]","-\frac{i \left(\frac{64 i a x \sinh ^{-1}(a x)^3}{a^2 x^2+1}+\frac{192 i \sinh ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+192 \sinh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(a x)}\right)+192 i \pi  \sinh ^{-1}(a x) \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)+384 \sinh ^{-1}(a x) \text{Li}_3\left(-i e^{-\sinh ^{-1}(a x)}\right)-384 \sinh ^{-1}(a x) \text{Li}_3\left(-i e^{\sinh ^{-1}(a x)}\right)-48 \left(-4 \sinh ^{-1}(a x)^2-4 i \pi  \sinh ^{-1}(a x)+\pi ^2+8\right) \text{Li}_2\left(-i e^{-\sinh ^{-1}(a x)}\right)+384 \text{Li}_2\left(i e^{-\sinh ^{-1}(a x)}\right)-48 \pi ^2 \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)+192 i \pi  \text{Li}_3\left(-i e^{-\sinh ^{-1}(a x)}\right)-192 i \pi  \text{Li}_3\left(i e^{\sinh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{-\sinh ^{-1}(a x)}\right)+384 \text{Li}_4\left(-i e^{\sinh ^{-1}(a x)}\right)-16 \sinh ^{-1}(a x)^4-32 i \pi  \sinh ^{-1}(a x)^3+24 \pi ^2 \sinh ^{-1}(a x)^2+8 i \pi ^3 \sinh ^{-1}(a x)-64 \sinh ^{-1}(a x)^3 \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)+64 \sinh ^{-1}(a x)^3 \log \left(1+i e^{\sinh ^{-1}(a x)}\right)-96 i \pi  \sinh ^{-1}(a x)^2 \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)+96 i \pi  \sinh ^{-1}(a x)^2 \log \left(1-i e^{\sinh ^{-1}(a x)}\right)-384 \sinh ^{-1}(a x) \log \left(1-i e^{-\sinh ^{-1}(a x)}\right)+48 \pi ^2 \sinh ^{-1}(a x) \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)+384 \sinh ^{-1}(a x) \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)-48 \pi ^2 \sinh ^{-1}(a x) \log \left(1-i e^{\sinh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)-8 i \pi ^3 \log \left(1+i e^{\sinh ^{-1}(a x)}\right)+8 i \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(a x)\right)\right)\right)+7 \pi ^4\right)}{128 a c^2}","\frac{x \sinh ^{-1}(a x)^3}{2 c^2 \left(a^2 x^2+1\right)}+\frac{3 \sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{a^2 x^2+1}}-\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(a x)}\right)}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x)^2 \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)}{2 a c^2}+\frac{3 i \sinh ^{-1}(a x) \text{Li}_3\left(-i e^{\sinh ^{-1}(a x)}\right)}{a c^2}-\frac{3 i \sinh ^{-1}(a x) \text{Li}_3\left(i e^{\sinh ^{-1}(a x)}\right)}{a c^2}+\frac{3 i \text{Li}_2\left(-i e^{\sinh ^{-1}(a x)}\right)}{a c^2}-\frac{3 i \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)}{a c^2}-\frac{3 i \text{Li}_4\left(-i e^{\sinh ^{-1}(a x)}\right)}{a c^2}+\frac{3 i \text{Li}_4\left(i e^{\sinh ^{-1}(a x)}\right)}{a c^2}+\frac{\sinh ^{-1}(a x)^3 \tan ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)}{a c^2}-\frac{6 \sinh ^{-1}(a x) \tan ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)}{a c^2}",1,"((-1/128*I)*(7*Pi^4 + (8*I)*Pi^3*ArcSinh[a*x] + 24*Pi^2*ArcSinh[a*x]^2 + ((192*I)*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2] - (32*I)*Pi*ArcSinh[a*x]^3 + ((64*I)*a*x*ArcSinh[a*x]^3)/(1 + a^2*x^2) - 16*ArcSinh[a*x]^4 - 384*ArcSinh[a*x]*Log[1 - I/E^ArcSinh[a*x]] + (8*I)*Pi^3*Log[1 + I/E^ArcSinh[a*x]] + 384*ArcSinh[a*x]*Log[1 + I/E^ArcSinh[a*x]] + 48*Pi^2*ArcSinh[a*x]*Log[1 + I/E^ArcSinh[a*x]] - (96*I)*Pi*ArcSinh[a*x]^2*Log[1 + I/E^ArcSinh[a*x]] - 64*ArcSinh[a*x]^3*Log[1 + I/E^ArcSinh[a*x]] - 48*Pi^2*ArcSinh[a*x]*Log[1 - I*E^ArcSinh[a*x]] + (96*I)*Pi*ArcSinh[a*x]^2*Log[1 - I*E^ArcSinh[a*x]] - (8*I)*Pi^3*Log[1 + I*E^ArcSinh[a*x]] + 64*ArcSinh[a*x]^3*Log[1 + I*E^ArcSinh[a*x]] + (8*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcSinh[a*x])/4]] - 48*(8 + Pi^2 - (4*I)*Pi*ArcSinh[a*x] - 4*ArcSinh[a*x]^2)*PolyLog[2, (-I)/E^ArcSinh[a*x]] + 384*PolyLog[2, I/E^ArcSinh[a*x]] + 192*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]] - 48*Pi^2*PolyLog[2, I*E^ArcSinh[a*x]] + (192*I)*Pi*ArcSinh[a*x]*PolyLog[2, I*E^ArcSinh[a*x]] + (192*I)*Pi*PolyLog[3, (-I)/E^ArcSinh[a*x]] + 384*ArcSinh[a*x]*PolyLog[3, (-I)/E^ArcSinh[a*x]] - 384*ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a*x]] - (192*I)*Pi*PolyLog[3, I*E^ArcSinh[a*x]] + 384*PolyLog[4, (-I)/E^ArcSinh[a*x]] + 384*PolyLog[4, (-I)*E^ArcSinh[a*x]]))/(a*c^2)","A",0
333,1,654,409,5.4718603,"\int \frac{\sinh ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^3} \, dx","Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2)^3,x]","-\frac{i \left(-\frac{128 i}{\sqrt{a^2 x^2+1}}+\frac{192 i a x \sinh ^{-1}(a x)^3}{a^2 x^2+1}+\frac{128 i a x \sinh ^{-1}(a x)^3}{\left(a^2 x^2+1\right)^2}+\frac{576 i \sinh ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{128 i \sinh ^{-1}(a x)^2}{\left(a^2 x^2+1\right)^{3/2}}-\frac{128 i a x \sinh ^{-1}(a x)}{a^2 x^2+1}+576 \sinh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(a x)}\right)+576 i \pi  \sinh ^{-1}(a x) \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)+1152 \sinh ^{-1}(a x) \text{Li}_3\left(-i e^{-\sinh ^{-1}(a x)}\right)-1152 \sinh ^{-1}(a x) \text{Li}_3\left(-i e^{\sinh ^{-1}(a x)}\right)-16 \left(-36 \sinh ^{-1}(a x)^2-36 i \pi  \sinh ^{-1}(a x)+9 \pi ^2+80\right) \text{Li}_2\left(-i e^{-\sinh ^{-1}(a x)}\right)+1280 \text{Li}_2\left(i e^{-\sinh ^{-1}(a x)}\right)-144 \pi ^2 \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)+576 i \pi  \text{Li}_3\left(-i e^{-\sinh ^{-1}(a x)}\right)-576 i \pi  \text{Li}_3\left(i e^{\sinh ^{-1}(a x)}\right)+1152 \text{Li}_4\left(-i e^{-\sinh ^{-1}(a x)}\right)+1152 \text{Li}_4\left(-i e^{\sinh ^{-1}(a x)}\right)-48 \sinh ^{-1}(a x)^4-96 i \pi  \sinh ^{-1}(a x)^3+72 \pi ^2 \sinh ^{-1}(a x)^2+24 i \pi ^3 \sinh ^{-1}(a x)-192 \sinh ^{-1}(a x)^3 \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)+192 \sinh ^{-1}(a x)^3 \log \left(1+i e^{\sinh ^{-1}(a x)}\right)-288 i \pi  \sinh ^{-1}(a x)^2 \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)+288 i \pi  \sinh ^{-1}(a x)^2 \log \left(1-i e^{\sinh ^{-1}(a x)}\right)-1280 \sinh ^{-1}(a x) \log \left(1-i e^{-\sinh ^{-1}(a x)}\right)+144 \pi ^2 \sinh ^{-1}(a x) \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)+1280 \sinh ^{-1}(a x) \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)-144 \pi ^2 \sinh ^{-1}(a x) \log \left(1-i e^{\sinh ^{-1}(a x)}\right)+24 i \pi ^3 \log \left(1+i e^{-\sinh ^{-1}(a x)}\right)-24 i \pi ^3 \log \left(1+i e^{\sinh ^{-1}(a x)}\right)+24 i \pi ^3 \log \left(\tan \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(a x)\right)\right)\right)+21 \pi ^4\right)}{512 a c^3}","-\frac{1}{4 a c^3 \sqrt{a^2 x^2+1}}+\frac{3 x \sinh ^{-1}(a x)^3}{8 c^3 \left(a^2 x^2+1\right)}+\frac{x \sinh ^{-1}(a x)^3}{4 c^3 \left(a^2 x^2+1\right)^2}+\frac{9 \sinh ^{-1}(a x)^2}{8 a c^3 \sqrt{a^2 x^2+1}}+\frac{\sinh ^{-1}(a x)^2}{4 a c^3 \left(a^2 x^2+1\right)^{3/2}}-\frac{x \sinh ^{-1}(a x)}{4 c^3 \left(a^2 x^2+1\right)}-\frac{9 i \sinh ^{-1}(a x)^2 \text{Li}_2\left(-i e^{\sinh ^{-1}(a x)}\right)}{8 a c^3}+\frac{9 i \sinh ^{-1}(a x)^2 \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)}{8 a c^3}+\frac{9 i \sinh ^{-1}(a x) \text{Li}_3\left(-i e^{\sinh ^{-1}(a x)}\right)}{4 a c^3}-\frac{9 i \sinh ^{-1}(a x) \text{Li}_3\left(i e^{\sinh ^{-1}(a x)}\right)}{4 a c^3}+\frac{5 i \text{Li}_2\left(-i e^{\sinh ^{-1}(a x)}\right)}{2 a c^3}-\frac{5 i \text{Li}_2\left(i e^{\sinh ^{-1}(a x)}\right)}{2 a c^3}-\frac{9 i \text{Li}_4\left(-i e^{\sinh ^{-1}(a x)}\right)}{4 a c^3}+\frac{9 i \text{Li}_4\left(i e^{\sinh ^{-1}(a x)}\right)}{4 a c^3}+\frac{3 \sinh ^{-1}(a x)^3 \tan ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)}{4 a c^3}-\frac{5 \sinh ^{-1}(a x) \tan ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)}{a c^3}",1,"((-1/512*I)*(21*Pi^4 - (128*I)/Sqrt[1 + a^2*x^2] + (24*I)*Pi^3*ArcSinh[a*x] - ((128*I)*a*x*ArcSinh[a*x])/(1 + a^2*x^2) + 72*Pi^2*ArcSinh[a*x]^2 + ((128*I)*ArcSinh[a*x]^2)/(1 + a^2*x^2)^(3/2) + ((576*I)*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2] - (96*I)*Pi*ArcSinh[a*x]^3 + ((128*I)*a*x*ArcSinh[a*x]^3)/(1 + a^2*x^2)^2 + ((192*I)*a*x*ArcSinh[a*x]^3)/(1 + a^2*x^2) - 48*ArcSinh[a*x]^4 - 1280*ArcSinh[a*x]*Log[1 - I/E^ArcSinh[a*x]] + (24*I)*Pi^3*Log[1 + I/E^ArcSinh[a*x]] + 1280*ArcSinh[a*x]*Log[1 + I/E^ArcSinh[a*x]] + 144*Pi^2*ArcSinh[a*x]*Log[1 + I/E^ArcSinh[a*x]] - (288*I)*Pi*ArcSinh[a*x]^2*Log[1 + I/E^ArcSinh[a*x]] - 192*ArcSinh[a*x]^3*Log[1 + I/E^ArcSinh[a*x]] - 144*Pi^2*ArcSinh[a*x]*Log[1 - I*E^ArcSinh[a*x]] + (288*I)*Pi*ArcSinh[a*x]^2*Log[1 - I*E^ArcSinh[a*x]] - (24*I)*Pi^3*Log[1 + I*E^ArcSinh[a*x]] + 192*ArcSinh[a*x]^3*Log[1 + I*E^ArcSinh[a*x]] + (24*I)*Pi^3*Log[Tan[(Pi + (2*I)*ArcSinh[a*x])/4]] - 16*(80 + 9*Pi^2 - (36*I)*Pi*ArcSinh[a*x] - 36*ArcSinh[a*x]^2)*PolyLog[2, (-I)/E^ArcSinh[a*x]] + 1280*PolyLog[2, I/E^ArcSinh[a*x]] + 576*ArcSinh[a*x]^2*PolyLog[2, (-I)*E^ArcSinh[a*x]] - 144*Pi^2*PolyLog[2, I*E^ArcSinh[a*x]] + (576*I)*Pi*ArcSinh[a*x]*PolyLog[2, I*E^ArcSinh[a*x]] + (576*I)*Pi*PolyLog[3, (-I)/E^ArcSinh[a*x]] + 1152*ArcSinh[a*x]*PolyLog[3, (-I)/E^ArcSinh[a*x]] - 1152*ArcSinh[a*x]*PolyLog[3, (-I)*E^ArcSinh[a*x]] - (576*I)*Pi*PolyLog[3, I*E^ArcSinh[a*x]] + 1152*PolyLog[4, (-I)/E^ArcSinh[a*x]] + 1152*PolyLog[4, (-I)*E^ArcSinh[a*x]]))/(a*c^3)","A",0
334,1,177,509,0.6948745,"\int \left(c+a^2 c x^2\right)^{5/2} \sinh ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^3,x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(4320 \sinh ^{-1}(a x)^4+288 \left(45 \sinh \left(2 \sinh ^{-1}(a x)\right)+9 \sinh \left(4 \sinh ^{-1}(a x)\right)+\sinh \left(6 \sinh ^{-1}(a x)\right)\right) \sinh ^{-1}(a x)^3+12 \left(1620 \sinh \left(2 \sinh ^{-1}(a x)\right)+81 \sinh \left(4 \sinh ^{-1}(a x)\right)+4 \sinh \left(6 \sinh ^{-1}(a x)\right)\right) \sinh ^{-1}(a x)-72 \sinh ^{-1}(a x)^2 \left(270 \cosh \left(2 \sinh ^{-1}(a x)\right)+27 \cosh \left(4 \sinh ^{-1}(a x)\right)+2 \cosh \left(6 \sinh ^{-1}(a x)\right)\right)-9720 \cosh \left(2 \sinh ^{-1}(a x)\right)-243 \cosh \left(4 \sinh ^{-1}(a x)\right)-8 \cosh \left(6 \sinh ^{-1}(a x)\right)\right)}{55296 a \sqrt{a^2 x^2+1}}","-\frac{865 a c^2 x^2 \sqrt{a^2 c x^2+c}}{2304 \sqrt{a^2 x^2+1}}-\frac{c^2 \left(a^2 x^2+1\right)^{5/2} \sqrt{a^2 c x^2+c}}{216 a}-\frac{15 a c^2 x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{32 \sqrt{a^2 x^2+1}}+\frac{5}{16} c^2 x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^3+\frac{245}{384} c^2 x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{1}{36} c^2 x \left(a^2 x^2+1\right)^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{65}{576} c^2 x \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{5 c^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^4}{64 a \sqrt{a^2 x^2+1}}-\frac{c^2 \left(a^2 x^2+1\right)^{5/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{12 a}-\frac{5 c^2 \left(a^2 x^2+1\right)^{3/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{32 a}-\frac{115 c^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{768 a \sqrt{a^2 x^2+1}}+\frac{1}{6} x \left(a^2 c x^2+c\right)^{5/2} \sinh ^{-1}(a x)^3+\frac{5}{24} c x \left(a^2 c x^2+c\right)^{3/2} \sinh ^{-1}(a x)^3-\frac{65 a^3 c^2 x^4 \sqrt{a^2 c x^2+c}}{2304 \sqrt{a^2 x^2+1}}",1,"(c^2*Sqrt[c + a^2*c*x^2]*(4320*ArcSinh[a*x]^4 - 9720*Cosh[2*ArcSinh[a*x]] - 243*Cosh[4*ArcSinh[a*x]] - 8*Cosh[6*ArcSinh[a*x]] - 72*ArcSinh[a*x]^2*(270*Cosh[2*ArcSinh[a*x]] + 27*Cosh[4*ArcSinh[a*x]] + 2*Cosh[6*ArcSinh[a*x]]) + 288*ArcSinh[a*x]^3*(45*Sinh[2*ArcSinh[a*x]] + 9*Sinh[4*ArcSinh[a*x]] + Sinh[6*ArcSinh[a*x]]) + 12*ArcSinh[a*x]*(1620*Sinh[2*ArcSinh[a*x]] + 81*Sinh[4*ArcSinh[a*x]] + 4*Sinh[6*ArcSinh[a*x]])))/(55296*a*Sqrt[1 + a^2*x^2])","A",1
335,1,136,348,0.2888594,"\int \left(c+a^2 c x^2\right)^{3/2} \sinh ^{-1}(a x)^3 \, dx","Integrate[(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^3,x]","\frac{c \sqrt{a^2 c x^2+c} \left(96 \sinh ^{-1}(a x)^4+32 \left(8 \sinh \left(2 \sinh ^{-1}(a x)\right)+\sinh \left(4 \sinh ^{-1}(a x)\right)\right) \sinh ^{-1}(a x)^3+12 \left(32 \sinh \left(2 \sinh ^{-1}(a x)\right)+\sinh \left(4 \sinh ^{-1}(a x)\right)\right) \sinh ^{-1}(a x)-24 \sinh ^{-1}(a x)^2 \left(16 \cosh \left(2 \sinh ^{-1}(a x)\right)+\cosh \left(4 \sinh ^{-1}(a x)\right)\right)-3 \left(64 \cosh \left(2 \sinh ^{-1}(a x)\right)+\cosh \left(4 \sinh ^{-1}(a x)\right)\right)\right)}{1024 a \sqrt{a^2 x^2+1}}","-\frac{51 a c x^2 \sqrt{a^2 c x^2+c}}{128 \sqrt{a^2 x^2+1}}-\frac{9 a c x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{16 \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \sinh ^{-1}(a x)^3+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^3+\frac{45}{64} c x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{3}{32} c x \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)+\frac{3 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^4}{32 a \sqrt{a^2 x^2+1}}-\frac{3 c \left(a^2 x^2+1\right)^{3/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{16 a}-\frac{27 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{128 a \sqrt{a^2 x^2+1}}-\frac{3 a^3 c x^4 \sqrt{a^2 c x^2+c}}{128 \sqrt{a^2 x^2+1}}",1,"(c*Sqrt[c + a^2*c*x^2]*(96*ArcSinh[a*x]^4 - 24*ArcSinh[a*x]^2*(16*Cosh[2*ArcSinh[a*x]] + Cosh[4*ArcSinh[a*x]]) - 3*(64*Cosh[2*ArcSinh[a*x]] + Cosh[4*ArcSinh[a*x]]) + 32*ArcSinh[a*x]^3*(8*Sinh[2*ArcSinh[a*x]] + Sinh[4*ArcSinh[a*x]]) + 12*ArcSinh[a*x]*(32*Sinh[2*ArcSinh[a*x]] + Sinh[4*ArcSinh[a*x]])))/(1024*a*Sqrt[1 + a^2*x^2])","A",1
336,1,86,205,0.14965,"\int \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^3 \, dx","Integrate[Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^3,x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(2 \sinh ^{-1}(a x) \left(\sinh ^{-1}(a x)^3+\left(2 \sinh ^{-1}(a x)^2+3\right) \sinh \left(2 \sinh ^{-1}(a x)\right)\right)-3 \left(2 \sinh ^{-1}(a x)^2+1\right) \cosh \left(2 \sinh ^{-1}(a x)\right)\right)}{16 a \sqrt{a^2 x^2+1}}","-\frac{3 a x^2 \sqrt{a^2 c x^2+c}}{8 \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^4}{8 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^3-\frac{3 a x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{4 \sqrt{a^2 x^2+1}}-\frac{3 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^2}{8 a \sqrt{a^2 x^2+1}}+\frac{3}{4} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)",1,"(Sqrt[c*(1 + a^2*x^2)]*(-3*(1 + 2*ArcSinh[a*x]^2)*Cosh[2*ArcSinh[a*x]] + 2*ArcSinh[a*x]*(ArcSinh[a*x]^3 + (3 + 2*ArcSinh[a*x]^2)*Sinh[2*ArcSinh[a*x]])))/(16*a*Sqrt[1 + a^2*x^2])","A",1
337,1,40,40,0.0374744,"\int \frac{\sinh ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcSinh[a*x]^3/Sqrt[c + a^2*c*x^2],x]","\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^4}{4 a \sqrt{a^2 c x^2+c}}","\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^4}{4 a \sqrt{a^2 c x^2+c}}",1,"(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^4)/(4*a*Sqrt[c + a^2*c*x^2])","A",1
338,1,133,218,0.2629398,"\int \frac{\sinh ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2)^(3/2),x]","\frac{6 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(a x)}\right)+3 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{-2 \sinh ^{-1}(a x)}\right)-2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \left(\sinh ^{-1}(a x)+3 \log \left(e^{-2 \sinh ^{-1}(a x)}+1\right)\right)+2 a x \sinh ^{-1}(a x)^3}{2 a c \sqrt{c \left(a^2 x^2+1\right)}}","-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{Li}_2\left(-e^{2 \sinh ^{-1}(a x)}\right)}{a c \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{2 \sinh ^{-1}(a x)}\right)}{2 a c \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^3}{c \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a c \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left(e^{2 \sinh ^{-1}(a x)}+1\right)}{a c \sqrt{a^2 c x^2+c}}",1,"(2*a*x*ArcSinh[a*x]^3 - 2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2*(ArcSinh[a*x] + 3*Log[1 + E^(-2*ArcSinh[a*x])]) + 6*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*PolyLog[2, -E^(-2*ArcSinh[a*x])] + 3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(-2*ArcSinh[a*x])])/(2*a*c*Sqrt[c*(1 + a^2*x^2)])","A",0
339,1,195,363,0.5457265,"\int \frac{\sinh ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2)^(5/2),x]","\frac{\left(a^2 x^2+1\right)^{3/2} \left(3 \log \left(a^2 x^2+1\right)+\frac{4 a x \sinh ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}}+\frac{2 a x \sinh ^{-1}(a x)^3}{\left(a^2 x^2+1\right)^{3/2}}+\frac{3 \sinh ^{-1}(a x)^2}{a^2 x^2+1}-\frac{6 a x \sinh ^{-1}(a x)}{\sqrt{a^2 x^2+1}}+12 \sinh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(a x)}\right)+6 \text{Li}_3\left(-e^{-2 \sinh ^{-1}(a x)}\right)-4 \sinh ^{-1}(a x)^3-12 \sinh ^{-1}(a x)^2 \log \left(e^{-2 \sinh ^{-1}(a x)}+1\right)\right)}{6 a c \left(a^2 c x^2+c\right)^{3/2}}","-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{Li}_2\left(-e^{2 \sinh ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{2 \sinh ^{-1}(a x)}\right)}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \log \left(a^2 x^2+1\right)}{2 a c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \sinh ^{-1}(a x)^3}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a c^2 \sqrt{a^2 c x^2+c}}+\frac{\sinh ^{-1}(a x)^2}{2 a c^2 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}-\frac{x \sinh ^{-1}(a x)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left(e^{2 \sinh ^{-1}(a x)}+1\right)}{a c^2 \sqrt{a^2 c x^2+c}}+\frac{x \sinh ^{-1}(a x)^3}{3 c \left(a^2 c x^2+c\right)^{3/2}}",1,"((1 + a^2*x^2)^(3/2)*((-6*a*x*ArcSinh[a*x])/Sqrt[1 + a^2*x^2] + (3*ArcSinh[a*x]^2)/(1 + a^2*x^2) - 4*ArcSinh[a*x]^3 + (2*a*x*ArcSinh[a*x]^3)/(1 + a^2*x^2)^(3/2) + (4*a*x*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2] - 12*ArcSinh[a*x]^2*Log[1 + E^(-2*ArcSinh[a*x])] + 3*Log[1 + a^2*x^2] + 12*ArcSinh[a*x]*PolyLog[2, -E^(-2*ArcSinh[a*x])] + 6*PolyLog[3, -E^(-2*ArcSinh[a*x])]))/(6*a*c*(c + a^2*c*x^2)^(3/2))","A",0
340,1,297,515,0.6604067,"\int \frac{\sinh ^{-1}(a x)^3}{\left(c+a^2 c x^2\right)^{7/2}} \, dx","Integrate[ArcSinh[a*x]^3/(c + a^2*c*x^2)^(7/2),x]","\frac{96 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(a x)}\right)+48 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{-2 \sinh ^{-1}(a x)}\right)-\frac{3}{\sqrt{a^2 x^2+1}}+30 \sqrt{a^2 x^2+1} \log \left(a^2 x^2+1\right)-32 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3+\frac{16 a x \sinh ^{-1}(a x)^3}{a^2 x^2+1}+\frac{12 a x \sinh ^{-1}(a x)^3}{\left(a^2 x^2+1\right)^2}+\frac{24 \sinh ^{-1}(a x)^2}{\sqrt{a^2 x^2+1}}+\frac{9 \sinh ^{-1}(a x)^2}{\left(a^2 x^2+1\right)^{3/2}}-\frac{6 a x \sinh ^{-1}(a x)}{a^2 x^2+1}-96 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left(e^{-2 \sinh ^{-1}(a x)}+1\right)+32 a x \sinh ^{-1}(a x)^3-60 a x \sinh ^{-1}(a x)}{60 a c^3 \sqrt{a^2 c x^2+c}}","-\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x) \text{Li}_2\left(-e^{2 \sinh ^{-1}(a x)}\right)}{5 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 \sqrt{a^2 x^2+1} \text{Li}_3\left(-e^{2 \sinh ^{-1}(a x)}\right)}{5 a c^3 \sqrt{a^2 c x^2+c}}-\frac{1}{20 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 x^2+1} \log \left(a^2 x^2+1\right)}{2 a c^3 \sqrt{a^2 c x^2+c}}+\frac{8 x \sinh ^{-1}(a x)^3}{15 c^3 \sqrt{a^2 c x^2+c}}+\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{15 a c^3 \sqrt{a^2 c x^2+c}}+\frac{2 \sinh ^{-1}(a x)^2}{5 a c^3 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}+\frac{3 \sinh ^{-1}(a x)^2}{20 a c^3 \left(a^2 x^2+1\right)^{3/2} \sqrt{a^2 c x^2+c}}-\frac{x \sinh ^{-1}(a x)}{c^3 \sqrt{a^2 c x^2+c}}-\frac{x \sinh ^{-1}(a x)}{10 c^3 \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c}}-\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^2 \log \left(e^{2 \sinh ^{-1}(a x)}+1\right)}{5 a c^3 \sqrt{a^2 c x^2+c}}+\frac{4 x \sinh ^{-1}(a x)^3}{15 c^2 \left(a^2 c x^2+c\right)^{3/2}}+\frac{x \sinh ^{-1}(a x)^3}{5 c \left(a^2 c x^2+c\right)^{5/2}}",1,"(-3/Sqrt[1 + a^2*x^2] - 60*a*x*ArcSinh[a*x] - (6*a*x*ArcSinh[a*x])/(1 + a^2*x^2) + (9*ArcSinh[a*x]^2)/(1 + a^2*x^2)^(3/2) + (24*ArcSinh[a*x]^2)/Sqrt[1 + a^2*x^2] + 32*a*x*ArcSinh[a*x]^3 + (12*a*x*ArcSinh[a*x]^3)/(1 + a^2*x^2)^2 + (16*a*x*ArcSinh[a*x]^3)/(1 + a^2*x^2) - 32*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3 - 96*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2*Log[1 + E^(-2*ArcSinh[a*x])] + 30*Sqrt[1 + a^2*x^2]*Log[1 + a^2*x^2] + 96*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]*PolyLog[2, -E^(-2*ArcSinh[a*x])] + 48*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(-2*ArcSinh[a*x])])/(60*a*c^3*Sqrt[c + a^2*c*x^2])","A",0
341,0,0,26,0.4938543,"\int \frac{x^m \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^m*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]","\int \frac{x^m \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx","\text{Int}\left(\frac{x^m \sinh ^{-1}(a x)^3}{\sqrt{a^2 x^2+1}},x\right)",0,"Integrate[(x^m*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2], x]","A",-1
342,1,121,187,0.0782397,"\int \frac{x^4 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^4*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]","\frac{-3 a^4 x^4+45 a^2 x^2+16 a x \sqrt{a^2 x^2+1} \left(2 a^2 x^2-3\right) \sinh ^{-1}(a x)^3+6 a x \sqrt{a^2 x^2+1} \left(2 a^2 x^2-15\right) \sinh ^{-1}(a x)+\left(-24 a^4 x^4+72 a^2 x^2+45\right) \sinh ^{-1}(a x)^2+12 \sinh ^{-1}(a x)^4}{128 a^5}","\frac{3 \sinh ^{-1}(a x)^4}{32 a^5}+\frac{45 \sinh ^{-1}(a x)^2}{128 a^5}+\frac{45 x^2}{128 a^3}+\frac{9 x^2 \sinh ^{-1}(a x)^2}{16 a^3}+\frac{x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{4 a^2}+\frac{3 x^3 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{32 a^2}-\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{8 a^4}-\frac{45 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{64 a^4}-\frac{3 x^4}{128 a}-\frac{3 x^4 \sinh ^{-1}(a x)^2}{16 a}",1,"(45*a^2*x^2 - 3*a^4*x^4 + 6*a*x*Sqrt[1 + a^2*x^2]*(-15 + 2*a^2*x^2)*ArcSinh[a*x] + (45 + 72*a^2*x^2 - 24*a^4*x^4)*ArcSinh[a*x]^2 + 16*a*x*Sqrt[1 + a^2*x^2]*(-3 + 2*a^2*x^2)*ArcSinh[a*x]^3 + 12*ArcSinh[a*x]^4)/(128*a^5)","A",1
343,1,98,153,0.0693907,"\int \frac{x^3 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^3*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]","\frac{-2 a x \left(a^2 x^2-60\right)+9 \left(a^2 x^2-2\right) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3-9 a x \left(a^2 x^2-6\right) \sinh ^{-1}(a x)^2+6 \left(a^2 x^2-20\right) \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{27 a^4}","\frac{40 x}{9 a^3}+\frac{2 x \sinh ^{-1}(a x)^2}{a^3}+\frac{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a^2}+\frac{2 x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^2}-\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{3 a^4}-\frac{40 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{9 a^4}-\frac{2 x^3}{27 a}-\frac{x^3 \sinh ^{-1}(a x)^2}{3 a}",1,"(-2*a*x*(-60 + a^2*x^2) + 6*(-20 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] - 9*a*x*(-6 + a^2*x^2)*ArcSinh[a*x]^2 + 9*(-2 + a^2*x^2)*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(27*a^4)","A",1
344,1,83,105,0.0647369,"\int \frac{x^2 \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^2*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]","-\frac{3 a^2 x^2-4 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3+\left(6 a^2 x^2+3\right) \sinh ^{-1}(a x)^2-6 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+\sinh ^{-1}(a x)^4}{8 a^3}","-\frac{\sinh ^{-1}(a x)^4}{8 a^3}-\frac{3 \sinh ^{-1}(a x)^2}{8 a^3}+\frac{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{2 a^2}+\frac{3 x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{4 a^2}-\frac{3 x^2}{8 a}-\frac{3 x^2 \sinh ^{-1}(a x)^2}{4 a}",1,"-1/8*(3*a^2*x^2 - 6*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + (3 + 6*a^2*x^2)*ArcSinh[a*x]^2 - 4*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3 + ArcSinh[a*x]^4)/a^3","A",1
345,1,58,64,0.0315673,"\int \frac{x \sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x*ArcSinh[a*x]^3)/Sqrt[1 + a^2*x^2],x]","\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3+6 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)-6 a x-3 a x \sinh ^{-1}(a x)^2}{a^2}","\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a^2}+\frac{6 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}{a^2}-\frac{6 x}{a}-\frac{3 x \sinh ^{-1}(a x)^2}{a}",1,"(-6*a*x + 6*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] - 3*a*x*ArcSinh[a*x]^2 + Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/a^2","A",1
346,1,13,13,0.0062274,"\int \frac{\sinh ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^3/Sqrt[1 + a^2*x^2],x]","\frac{\sinh ^{-1}(a x)^4}{4 a}","\frac{\sinh ^{-1}(a x)^4}{4 a}",1,"ArcSinh[a*x]^4/(4*a)","A",1
347,1,146,102,0.136908,"\int \frac{\sinh ^{-1}(a x)^3}{x \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^3/(x*Sqrt[1 + a^2*x^2]),x]","\frac{1}{8} \left(24 \sinh ^{-1}(a x)^2 \text{Li}_2\left(-e^{-\sinh ^{-1}(a x)}\right)+24 \sinh ^{-1}(a x)^2 \text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)+48 \sinh ^{-1}(a x) \text{Li}_3\left(-e^{-\sinh ^{-1}(a x)}\right)-48 \sinh ^{-1}(a x) \text{Li}_3\left(e^{\sinh ^{-1}(a x)}\right)+48 \text{Li}_4\left(-e^{-\sinh ^{-1}(a x)}\right)+48 \text{Li}_4\left(e^{\sinh ^{-1}(a x)}\right)-2 \sinh ^{-1}(a x)^4-8 \sinh ^{-1}(a x)^3 \log \left(e^{-\sinh ^{-1}(a x)}+1\right)+8 \sinh ^{-1}(a x)^3 \log \left(1-e^{\sinh ^{-1}(a x)}\right)+\pi ^4\right)","-3 \sinh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\sinh ^{-1}(a x)}\right)+3 \sinh ^{-1}(a x)^2 \text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)+6 \sinh ^{-1}(a x) \text{Li}_3\left(-e^{\sinh ^{-1}(a x)}\right)-6 \sinh ^{-1}(a x) \text{Li}_3\left(e^{\sinh ^{-1}(a x)}\right)-6 \text{Li}_4\left(-e^{\sinh ^{-1}(a x)}\right)+6 \text{Li}_4\left(e^{\sinh ^{-1}(a x)}\right)-2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)",1,"(Pi^4 - 2*ArcSinh[a*x]^4 - 8*ArcSinh[a*x]^3*Log[1 + E^(-ArcSinh[a*x])] + 8*ArcSinh[a*x]^3*Log[1 - E^ArcSinh[a*x]] + 24*ArcSinh[a*x]^2*PolyLog[2, -E^(-ArcSinh[a*x])] + 24*ArcSinh[a*x]^2*PolyLog[2, E^ArcSinh[a*x]] + 48*ArcSinh[a*x]*PolyLog[3, -E^(-ArcSinh[a*x])] - 48*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] + 48*PolyLog[4, -E^(-ArcSinh[a*x])] + 48*PolyLog[4, E^ArcSinh[a*x]])/8","A",0
348,1,97,88,0.2103178,"\int \frac{\sinh ^{-1}(a x)^3}{x^2 \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^3/(x^2*Sqrt[1 + a^2*x^2]),x]","\frac{1}{8} a \left(-\frac{8 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{a x}+24 \sinh ^{-1}(a x) \text{Li}_2\left(e^{2 \sinh ^{-1}(a x)}\right)-12 \text{Li}_3\left(e^{2 \sinh ^{-1}(a x)}\right)-8 \sinh ^{-1}(a x)^3+24 \sinh ^{-1}(a x)^2 \log \left(1-e^{2 \sinh ^{-1}(a x)}\right)+i \pi ^3\right)","-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{x}+3 a \sinh ^{-1}(a x) \text{Li}_2\left(e^{2 \sinh ^{-1}(a x)}\right)-\frac{3}{2} a \text{Li}_3\left(e^{2 \sinh ^{-1}(a x)}\right)-a \sinh ^{-1}(a x)^3+3 a \sinh ^{-1}(a x)^2 \log \left(1-e^{2 \sinh ^{-1}(a x)}\right)",1,"(a*(I*Pi^3 - 8*ArcSinh[a*x]^3 - (8*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3)/(a*x) + 24*ArcSinh[a*x]^2*Log[1 - E^(2*ArcSinh[a*x])] + 24*ArcSinh[a*x]*PolyLog[2, E^(2*ArcSinh[a*x])] - 12*PolyLog[3, E^(2*ArcSinh[a*x])]))/8","C",1
349,1,304,210,4.6214837,"\int \frac{\sinh ^{-1}(a x)^3}{x^3 \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^3/(x^3*Sqrt[1 + a^2*x^2]),x]","\frac{a \left(-24 a x \sinh ^{-1}(a x)^2 \text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)-48 a x \sinh ^{-1}(a x) \text{Li}_3\left(-e^{-\sinh ^{-1}(a x)}\right)+48 a x \sinh ^{-1}(a x) \text{Li}_3\left(e^{\sinh ^{-1}(a x)}\right)-24 a x \left(\sinh ^{-1}(a x)^2-2\right) \text{Li}_2\left(-e^{-\sinh ^{-1}(a x)}\right)-48 a x \text{Li}_2\left(e^{-\sinh ^{-1}(a x)}\right)-48 a x \text{Li}_4\left(-e^{-\sinh ^{-1}(a x)}\right)-48 a x \text{Li}_4\left(e^{\sinh ^{-1}(a x)}\right)-\pi ^4 a x+2 a x \sinh ^{-1}(a x)^4+8 a x \sinh ^{-1}(a x)^3 \log \left(e^{-\sinh ^{-1}(a x)}+1\right)-8 a x \sinh ^{-1}(a x)^3 \log \left(1-e^{\sinh ^{-1}(a x)}\right)+48 a x \sinh ^{-1}(a x) \log \left(1-e^{-\sinh ^{-1}(a x)}\right)-48 a x \sinh ^{-1}(a x) \log \left(e^{-\sinh ^{-1}(a x)}+1\right)-4 \sinh ^{-1}(a x)^3 \tanh \left(\frac{1}{2} \sinh ^{-1}(a x)\right)+12 a x \sinh ^{-1}(a x)^2 \tanh \left(\frac{1}{2} \sinh ^{-1}(a x)\right)-12 a x \sinh ^{-1}(a x)^2 \coth \left(\frac{1}{2} \sinh ^{-1}(a x)\right)-2 a x \sinh ^{-1}(a x)^3 \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(a x)\right)\right)}{16 x}","\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left(-e^{\sinh ^{-1}(a x)}\right)-\frac{3}{2} a^2 \sinh ^{-1}(a x)^2 \text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)-3 a^2 \sinh ^{-1}(a x) \text{Li}_3\left(-e^{\sinh ^{-1}(a x)}\right)+3 a^2 \sinh ^{-1}(a x) \text{Li}_3\left(e^{\sinh ^{-1}(a x)}\right)-3 a^2 \text{Li}_2\left(-e^{\sinh ^{-1}(a x)}\right)+3 a^2 \text{Li}_2\left(e^{\sinh ^{-1}(a x)}\right)+3 a^2 \text{Li}_4\left(-e^{\sinh ^{-1}(a x)}\right)-3 a^2 \text{Li}_4\left(e^{\sinh ^{-1}(a x)}\right)-\frac{\sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^3}{2 x^2}+a^2 \sinh ^{-1}(a x)^3 \tanh ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)-6 a^2 \sinh ^{-1}(a x) \tanh ^{-1}\left(e^{\sinh ^{-1}(a x)}\right)-\frac{3 a \sinh ^{-1}(a x)^2}{2 x}",1,"(a*(-(a*Pi^4*x) + 2*a*x*ArcSinh[a*x]^4 - 12*a*x*ArcSinh[a*x]^2*Coth[ArcSinh[a*x]/2] - 2*a*x*ArcSinh[a*x]^3*Csch[ArcSinh[a*x]/2]^2 + 48*a*x*ArcSinh[a*x]*Log[1 - E^(-ArcSinh[a*x])] - 48*a*x*ArcSinh[a*x]*Log[1 + E^(-ArcSinh[a*x])] + 8*a*x*ArcSinh[a*x]^3*Log[1 + E^(-ArcSinh[a*x])] - 8*a*x*ArcSinh[a*x]^3*Log[1 - E^ArcSinh[a*x]] - 24*a*x*(-2 + ArcSinh[a*x]^2)*PolyLog[2, -E^(-ArcSinh[a*x])] - 48*a*x*PolyLog[2, E^(-ArcSinh[a*x])] - 24*a*x*ArcSinh[a*x]^2*PolyLog[2, E^ArcSinh[a*x]] - 48*a*x*ArcSinh[a*x]*PolyLog[3, -E^(-ArcSinh[a*x])] + 48*a*x*ArcSinh[a*x]*PolyLog[3, E^ArcSinh[a*x]] - 48*a*x*PolyLog[4, -E^(-ArcSinh[a*x])] - 48*a*x*PolyLog[4, E^ArcSinh[a*x]] + 12*a*x*ArcSinh[a*x]^2*Tanh[ArcSinh[a*x]/2] - 4*ArcSinh[a*x]^3*Tanh[ArcSinh[a*x]/2]))/(16*x)","A",0
350,1,43,67,0.1109046,"\int \frac{\left(c+a^2 c x^2\right)^3}{\sinh ^{-1}(a x)} \, dx","Integrate[(c + a^2*c*x^2)^3/ArcSinh[a*x],x]","\frac{c^3 \left(35 \text{Chi}\left(\sinh ^{-1}(a x)\right)+21 \text{Chi}\left(3 \sinh ^{-1}(a x)\right)+7 \text{Chi}\left(5 \sinh ^{-1}(a x)\right)+\text{Chi}\left(7 \sinh ^{-1}(a x)\right)\right)}{64 a}","\frac{35 c^3 \text{Chi}\left(\sinh ^{-1}(a x)\right)}{64 a}+\frac{21 c^3 \text{Chi}\left(3 \sinh ^{-1}(a x)\right)}{64 a}+\frac{7 c^3 \text{Chi}\left(5 \sinh ^{-1}(a x)\right)}{64 a}+\frac{c^3 \text{Chi}\left(7 \sinh ^{-1}(a x)\right)}{64 a}",1,"(c^3*(35*CoshIntegral[ArcSinh[a*x]] + 21*CoshIntegral[3*ArcSinh[a*x]] + 7*CoshIntegral[5*ArcSinh[a*x]] + CoshIntegral[7*ArcSinh[a*x]]))/(64*a)","A",1
351,1,34,50,0.0752705,"\int \frac{\left(c+a^2 c x^2\right)^2}{\sinh ^{-1}(a x)} \, dx","Integrate[(c + a^2*c*x^2)^2/ArcSinh[a*x],x]","\frac{c^2 \left(10 \text{Chi}\left(\sinh ^{-1}(a x)\right)+5 \text{Chi}\left(3 \sinh ^{-1}(a x)\right)+\text{Chi}\left(5 \sinh ^{-1}(a x)\right)\right)}{16 a}","\frac{5 c^2 \text{Chi}\left(\sinh ^{-1}(a x)\right)}{8 a}+\frac{5 c^2 \text{Chi}\left(3 \sinh ^{-1}(a x)\right)}{16 a}+\frac{c^2 \text{Chi}\left(5 \sinh ^{-1}(a x)\right)}{16 a}",1,"(c^2*(10*CoshIntegral[ArcSinh[a*x]] + 5*CoshIntegral[3*ArcSinh[a*x]] + CoshIntegral[5*ArcSinh[a*x]]))/(16*a)","A",1
352,1,23,29,0.0154254,"\int \frac{c+a^2 c x^2}{\sinh ^{-1}(a x)} \, dx","Integrate[(c + a^2*c*x^2)/ArcSinh[a*x],x]","\frac{c \left(3 \text{Chi}\left(\sinh ^{-1}(a x)\right)+\text{Chi}\left(3 \sinh ^{-1}(a x)\right)\right)}{4 a}","\frac{3 c \text{Chi}\left(\sinh ^{-1}(a x)\right)}{4 a}+\frac{c \text{Chi}\left(3 \sinh ^{-1}(a x)\right)}{4 a}",1,"(c*(3*CoshIntegral[ArcSinh[a*x]] + CoshIntegral[3*ArcSinh[a*x]]))/(4*a)","A",1
353,0,0,22,0.35854,"\int \frac{1}{\left(c+a^2 c x^2\right) \sinh ^{-1}(a x)} \, dx","Integrate[1/((c + a^2*c*x^2)*ArcSinh[a*x]),x]","\int \frac{1}{\left(c+a^2 c x^2\right) \sinh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{\left(a^2 c x^2+c\right) \sinh ^{-1}(a x)},x\right)",0,"Integrate[1/((c + a^2*c*x^2)*ArcSinh[a*x]), x]","A",-1
354,0,0,22,1.6107519,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \sinh ^{-1}(a x)} \, dx","Integrate[1/((c + a^2*c*x^2)^2*ArcSinh[a*x]),x]","\int \frac{1}{\left(c+a^2 c x^2\right)^2 \sinh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{\left(a^2 c x^2+c\right)^2 \sinh ^{-1}(a x)},x\right)",0,"Integrate[1/((c + a^2*c*x^2)^2*ArcSinh[a*x]), x]","A",-1
355,1,152,206,0.3603503,"\int \frac{x^4 \sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^4*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]),x]","\frac{-\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+2 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+2 \log \left(a+b \sinh ^{-1}(c x)\right)}{32 b c^5}","-\frac{\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^5}-\frac{\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^5}+\frac{\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^5}+\frac{\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}-\frac{\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^5}+\frac{\log \left(a+b \sinh ^{-1}(c x)\right)}{16 b c^5}",1,"(-(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])]) - 2*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c*x])] + Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcSinh[c*x])] + 2*Log[a + b*ArcSinh[c*x]] + Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + 2*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])])/(32*b*c^5)","A",1
356,1,135,183,0.2892267,"\int \frac{x^3 \sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^3*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]),x]","\frac{2 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{16 b c^4}","\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^4}+\frac{\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^4}-\frac{\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^4}-\frac{\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^4}-\frac{\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^4}+\frac{\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^4}",1,"(2*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - CoshIntegral[5*(a/b + ArcSinh[c*x])]*Sinh[(5*a)/b] - 2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])])/(16*b*c^4)","A",1
357,1,65,82,0.1751203,"\int \frac{x^2 \sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]),x]","\frac{\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\log \left(a+b \sinh ^{-1}(c x)\right)}{8 b c^3}","\frac{\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b c^3}-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b c^3}-\frac{\log \left(a+b \sinh ^{-1}(c x)\right)}{8 b c^3}",1,"(Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c*x])] - Log[a + b*ArcSinh[c*x]] - Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])])/(8*b*c^3)","A",1
358,1,91,121,0.2014825,"\int \frac{x \sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]),x]","\frac{\sinh \left(\frac{a}{b}\right) \left(-\text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{4 b c^2}","-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b c^2}-\frac{\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b c^2}+\frac{\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b c^2}+\frac{\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b c^2}",1,"(-(CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b]) - CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] + Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(4*b*c^2)","A",1
359,1,63,82,0.1289255,"\int \frac{\sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(a + b*ArcSinh[c*x]),x]","\frac{\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\log \left(a+b \sinh ^{-1}(c x)\right)}{2 b c}","\frac{\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c}-\frac{\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c}+\frac{\log \left(a+b \sinh ^{-1}(c x)\right)}{2 b c}",1,"(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] + Log[a + b*ArcSinh[c*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])])/(2*b*c)","A",1
360,0,0,78,1.9556755,"\int \frac{\sqrt{1+c^2 x^2}}{x \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(x*(a + b*ArcSinh[c*x])),x]","\int \frac{\sqrt{1+c^2 x^2}}{x \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b}+\frac{\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b}",0,"Integrate[Sqrt[1 + c^2*x^2]/(x*(a + b*ArcSinh[c*x])), x]","A",-1
361,0,0,45,1.147571,"\int \frac{\sqrt{1+c^2 x^2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(x^2*(a + b*ArcSinh[c*x])),x]","\int \frac{\sqrt{1+c^2 x^2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)+\frac{c \log \left(a+b \sinh ^{-1}(c x)\right)}{b}",0,"Integrate[Sqrt[1 + c^2*x^2]/(x^2*(a + b*ArcSinh[c*x])), x]","A",-1
362,0,0,30,4.7330545,"\int \frac{\sqrt{1+c^2 x^2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])),x]","\int \frac{\sqrt{1+c^2 x^2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\sqrt{c^2 x^2+1}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])), x]","A",-1
363,0,0,30,0.8275802,"\int \frac{\sqrt{1+c^2 x^2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])),x]","\int \frac{\sqrt{1+c^2 x^2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\sqrt{c^2 x^2+1}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])), x]","A",-1
364,1,179,245,0.6566476,"\int \frac{x^3 \left(1+c^2 x^2\right)^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^3*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x]),x]","\frac{3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-3 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{64 b c^4}","\frac{3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b c^4}+\frac{3 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^4}-\frac{\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^4}-\frac{\sinh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^4}-\frac{3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b c^4}-\frac{3 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^4}+\frac{\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^4}+\frac{\cosh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^4}",1,"(3*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 3*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - CoshIntegral[5*(a/b + ArcSinh[c*x])]*Sinh[(5*a)/b] - CoshIntegral[7*(a/b + ArcSinh[c*x])]*Sinh[(7*a)/b] - 3*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] + Cosh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])])/(64*b*c^4)","A",1
365,1,152,206,0.4764781,"\int \frac{x^2 \left(1+c^2 x^2\right)^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^2*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x]),x]","\frac{-\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+2 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 \log \left(a+b \sinh ^{-1}(c x)\right)}{32 b c^3}","-\frac{\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}+\frac{\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^3}+\frac{\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}+\frac{\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^3}-\frac{\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\log \left(a+b \sinh ^{-1}(c x)\right)}{16 b c^3}",1,"(-(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])]) + 2*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c*x])] + Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcSinh[c*x])] - 2*Log[a + b*ArcSinh[c*x]] + Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 2*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])])/(32*b*c^3)","A",1
366,1,136,183,0.444518,"\int \frac{x \left(1+c^2 x^2\right)^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x]),x]","\frac{-2 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-3 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{16 b c^2}","-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^2}-\frac{3 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^2}-\frac{\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^2}+\frac{\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^2}+\frac{3 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^2}+\frac{\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^2}",1,"(-2*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - 3*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - CoshIntegral[5*(a/b + ArcSinh[c*x])]*Sinh[(5*a)/b] + 2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])])/(16*b*c^2)","A",1
367,1,109,144,0.2527592,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(a + b*ArcSinh[c*x]),x]","\frac{4 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-4 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 \log \left(a+b \sinh ^{-1}(c x)\right)}{8 b c}","\frac{\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c}+\frac{\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b c}-\frac{\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c}-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b c}+\frac{3 \log \left(a+b \sinh ^{-1}(c x)\right)}{8 b c}",1,"(4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] + Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c*x])] + 3*Log[a + b*ArcSinh[c*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])])/(8*b*c)","A",1
368,0,0,139,1.9872245,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(x*(a + b*ArcSinh[c*x])),x]","\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)-\frac{5 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b}-\frac{\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b}+\frac{5 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b}+\frac{\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b}",0,"Integrate[(1 + c^2*x^2)^(3/2)/(x*(a + b*ArcSinh[c*x])), x]","A",-1
369,0,0,106,1.436503,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(x^2*(a + b*ArcSinh[c*x])),x]","\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)+\frac{c \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b}-\frac{c \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b}+\frac{3 c \log \left(a+b \sinh ^{-1}(c x)\right)}{2 b}",0,"Integrate[(1 + c^2*x^2)^(3/2)/(x^2*(a + b*ArcSinh[c*x])), x]","A",-1
370,0,0,30,4.6858767,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])),x]","\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{3/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])), x]","A",-1
371,0,0,30,0.8555354,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(x^4*(a + b*ArcSinh[c*x])),x]","\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{3/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 + c^2*x^2)^(3/2)/(x^4*(a + b*ArcSinh[c*x])), x]","A",-1
372,1,180,245,0.9912864,"\int \frac{x^3 \left(1+c^2 x^2\right)^{5/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^3*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]),x]","\frac{6 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+8 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 \sinh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{9 a}{b}\right) \text{Chi}\left(9 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-6 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-8 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 \cosh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{9 a}{b}\right) \text{Shi}\left(9 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{256 b c^4}","\frac{3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{128 b c^4}+\frac{\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^4}-\frac{3 \sinh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{256 b c^4}-\frac{\sinh \left(\frac{9 a}{b}\right) \text{Chi}\left(\frac{9 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{256 b c^4}-\frac{3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{128 b c^4}-\frac{\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^4}+\frac{3 \cosh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{256 b c^4}+\frac{\cosh \left(\frac{9 a}{b}\right) \text{Shi}\left(\frac{9 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{256 b c^4}",1,"(6*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 8*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - 3*CoshIntegral[7*(a/b + ArcSinh[c*x])]*Sinh[(7*a)/b] - CoshIntegral[9*(a/b + ArcSinh[c*x])]*Sinh[(9*a)/b] - 6*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 8*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 3*Cosh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])] + Cosh[(9*a)/b]*SinhIntegral[9*(a/b + ArcSinh[c*x])])/(256*b*c^4)","A",1
373,1,197,268,0.8825193,"\int \frac{x^2 \left(1+c^2 x^2\right)^{5/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^2*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]),x]","\frac{-4 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+4 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+4 \cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{8 a}{b}\right) \text{Chi}\left(8 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+4 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-4 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-4 \sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{8 a}{b}\right) \text{Shi}\left(8 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-5 \log \left(a+b \sinh ^{-1}(c x)\right)}{128 b c^3}","-\frac{\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}+\frac{\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}+\frac{\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}+\frac{\cosh \left(\frac{8 a}{b}\right) \text{Chi}\left(\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{128 b c^3}+\frac{\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c^3}-\frac{\sinh \left(\frac{8 a}{b}\right) \text{Shi}\left(\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{128 b c^3}-\frac{5 \log \left(a+b \sinh ^{-1}(c x)\right)}{128 b c^3}",1,"(-4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] + 4*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c*x])] + 4*Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcSinh[c*x])] + Cosh[(8*a)/b]*CoshIntegral[8*(a/b + ArcSinh[c*x])] - 5*Log[a + b*ArcSinh[c*x]] + 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 4*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 4*Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])] - Sinh[(8*a)/b]*SinhIntegral[8*(a/b + ArcSinh[c*x])])/(128*b*c^3)","A",1
374,1,180,245,0.804887,"\int \frac{x \left(1+c^2 x^2\right)^{5/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]),x]","\frac{-5 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-9 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-5 \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+9 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{64 b c^2}","-\frac{5 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b c^2}-\frac{9 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^2}-\frac{5 \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^2}-\frac{\sinh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^2}+\frac{5 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b c^2}+\frac{9 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^2}+\frac{5 \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^2}+\frac{\cosh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^2}",1,"(-5*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - 9*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - 5*CoshIntegral[5*(a/b + ArcSinh[c*x])]*Sinh[(5*a)/b] - CoshIntegral[7*(a/b + ArcSinh[c*x])]*Sinh[(7*a)/b] + 5*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 9*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 5*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] + Cosh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])])/(64*b*c^2)","A",1
375,1,153,206,0.5540813,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(a + b*ArcSinh[c*x]),x]","\frac{15 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+6 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-15 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-6 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+10 \log \left(a+b \sinh ^{-1}(c x)\right)}{32 b c}","\frac{15 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c}+\frac{3 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c}+\frac{\cosh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c}-\frac{15 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c}-\frac{3 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c}-\frac{\sinh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b c}+\frac{5 \log \left(a+b \sinh ^{-1}(c x)\right)}{16 b c}",1,"(15*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] + 6*Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c*x])] + Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcSinh[c*x])] + 10*Log[a + b*ArcSinh[c*x]] - 15*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 6*Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])])/(32*b*c)","A",1
376,0,0,195,2.0081011,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(x*(a + b*ArcSinh[c*x])),x]","\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)-\frac{11 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b}-\frac{7 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b}-\frac{\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b}+\frac{11 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b}+\frac{7 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b}+\frac{\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b}",0,"Integrate[(1 + c^2*x^2)^(5/2)/(x*(a + b*ArcSinh[c*x])), x]","A",-1
377,0,0,159,1.3727113,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])),x]","\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)+\frac{c \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b}+\frac{c \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b}-\frac{c \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b}-\frac{c \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b}+\frac{15 c \log \left(a+b \sinh ^{-1}(c x)\right)}{8 b}",0,"Integrate[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])), x]","A",-1
378,0,0,30,4.6773569,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])),x]","\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{5/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])), x]","A",-1
379,0,0,30,0.9333482,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(x^4*(a + b*ArcSinh[c*x])),x]","\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{5/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[(1 + c^2*x^2)^(5/2)/(x^4*(a + b*ArcSinh[c*x])), x]","A",-1
380,1,31,41,0.0755882,"\int \frac{x^4}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","Integrate[x^4/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]","\frac{-4 \text{Chi}\left(2 \sinh ^{-1}(a x)\right)+\text{Chi}\left(4 \sinh ^{-1}(a x)\right)+3 \log \left(\sinh ^{-1}(a x)\right)}{8 a^5}","-\frac{\text{Chi}\left(2 \sinh ^{-1}(a x)\right)}{2 a^5}+\frac{\text{Chi}\left(4 \sinh ^{-1}(a x)\right)}{8 a^5}+\frac{3 \log \left(\sinh ^{-1}(a x)\right)}{8 a^5}",1,"(-4*CoshIntegral[2*ArcSinh[a*x]] + CoshIntegral[4*ArcSinh[a*x]] + 3*Log[ArcSinh[a*x]])/(8*a^5)","A",1
381,1,22,27,0.081385,"\int \frac{x^3}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","Integrate[x^3/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]","\frac{\text{Shi}\left(3 \sinh ^{-1}(a x)\right)-3 \text{Shi}\left(\sinh ^{-1}(a x)\right)}{4 a^4}","\frac{\text{Shi}\left(3 \sinh ^{-1}(a x)\right)}{4 a^4}-\frac{3 \text{Shi}\left(\sinh ^{-1}(a x)\right)}{4 a^4}",1,"(-3*SinhIntegral[ArcSinh[a*x]] + SinhIntegral[3*ArcSinh[a*x]])/(4*a^4)","A",1
382,1,22,27,0.0691697,"\int \frac{x^2}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","Integrate[x^2/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]","\frac{\text{Chi}\left(2 \sinh ^{-1}(a x)\right)-\log \left(\sinh ^{-1}(a x)\right)}{2 a^3}","\frac{\text{Chi}\left(2 \sinh ^{-1}(a x)\right)}{2 a^3}-\frac{\log \left(\sinh ^{-1}(a x)\right)}{2 a^3}",1,"(CoshIntegral[2*ArcSinh[a*x]] - Log[ArcSinh[a*x]])/(2*a^3)","A",1
383,1,22,27,0.0240762,"\int \frac{x^2}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","Integrate[x^2/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]","\frac{\text{Chi}\left(2 \sinh ^{-1}(a x)\right)-\log \left(\sinh ^{-1}(a x)\right)}{2 a^3}","\frac{\text{Chi}\left(2 \sinh ^{-1}(a x)\right)}{2 a^3}-\frac{\log \left(\sinh ^{-1}(a x)\right)}{2 a^3}",1,"(CoshIntegral[2*ArcSinh[a*x]] - Log[ArcSinh[a*x]])/(2*a^3)","A",1
384,1,9,9,0.0557749,"\int \frac{x}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","Integrate[x/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]","\frac{\text{Shi}\left(\sinh ^{-1}(a x)\right)}{a^2}","\frac{\text{Shi}\left(\sinh ^{-1}(a x)\right)}{a^2}",1,"SinhIntegral[ArcSinh[a*x]]/a^2","A",1
385,1,9,9,0.0169785,"\int \frac{1}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","Integrate[1/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]","\frac{\log \left(\sinh ^{-1}(a x)\right)}{a}","\frac{\log \left(\sinh ^{-1}(a x)\right)}{a}",1,"Log[ArcSinh[a*x]]/a","A",1
386,0,0,26,1.1174649,"\int \frac{1}{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","Integrate[1/(x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]","\int \frac{1}{x \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)},x\right)",0,"Integrate[1/(x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x]","A",-1
387,0,0,26,0.7481592,"\int \frac{1}{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","Integrate[1/(x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]),x]","\int \frac{1}{x^2 \sqrt{1+a^2 x^2} \sinh ^{-1}(a x)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)},x\right)",0,"Integrate[1/(x^2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]), x]","A",-1
388,1,136,183,0.3381394,"\int \frac{x^5}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[x^5/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","-\frac{10 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-5 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-10 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+5 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{16 b c^6}","-\frac{5 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^6}+\frac{5 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^6}-\frac{\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^6}+\frac{5 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^6}-\frac{5 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^6}+\frac{\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^6}",1,"-1/16*(10*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - 5*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] + CoshIntegral[5*(a/b + ArcSinh[c*x])]*Sinh[(5*a)/b] - 10*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 5*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])])/(b*c^6)","A",1
389,1,109,144,0.2212412,"\int \frac{x^4}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[x^4/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","-\frac{4 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-4 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 \log \left(a+b \sinh ^{-1}(c x)\right)}{8 b c^5}","-\frac{\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c^5}+\frac{\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b c^5}+\frac{\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c^5}-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b c^5}+\frac{3 \log \left(a+b \sinh ^{-1}(c x)\right)}{8 b c^5}",1,"-1/8*(4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] - Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c*x])] - 3*Log[a + b*ArcSinh[c*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])])/(b*c^5)","A",1
390,1,92,121,0.2127599,"\int \frac{x^3}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[x^3/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","\frac{3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{4 b c^4}","\frac{3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b c^4}-\frac{\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b c^4}-\frac{3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b c^4}+\frac{\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b c^4}",1,"(3*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - 3*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(4*b*c^4)","A",1
391,1,65,82,0.180836,"\int \frac{x^2}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[x^2/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","\frac{\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\log \left(a+b \sinh ^{-1}(c x)\right)}{2 b c^3}","\frac{\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c^3}-\frac{\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c^3}-\frac{\log \left(a+b \sinh ^{-1}(c x)\right)}{2 b c^3}",1,"(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] - Log[a + b*ArcSinh[c*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])])/(2*b*c^3)","A",1
392,1,46,54,0.111937,"\int \frac{x}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[x/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{b c^2}","\frac{\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c^2}-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c^2}",1,"-((CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b*c^2))","A",1
393,1,16,16,0.0254649,"\int \frac{1}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","\frac{\log \left(a+b \sinh ^{-1}(c x)\right)}{b c}","\frac{\log \left(a+b \sinh ^{-1}(c x)\right)}{b c}",1,"Log[a + b*ArcSinh[c*x]]/(b*c)","A",1
394,0,0,30,1.7589699,"\int \frac{1}{x \sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{x \sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]","A",-1
395,0,0,30,1.2044133,"\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]","A",-1
396,0,0,30,2.1115142,"\int \frac{x^2}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[x^2/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]","\int \frac{x^2}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^2}{\left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[x^2/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]","A",-1
397,0,0,28,1.0767506,"\int \frac{x}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[x/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]","\int \frac{x}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x}{\left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[x/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]","A",-1
398,0,0,27,0.0596436,"\int \frac{1}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]","A",-1
399,0,0,30,1.4993432,"\int \frac{1}{x \left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{x \left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]","A",-1
400,0,0,30,1.1211313,"\int \frac{1}{x^2 \left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{x^2 \left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]","A",-1
401,0,0,30,0.9150552,"\int \frac{x^m \left(1+c^2 x^2\right)^{5/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^m*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]),x]","\int \frac{x^m \left(1+c^2 x^2\right)^{5/2}}{a+b \sinh ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{5/2} x^m}{a+b \sinh ^{-1}(c x)},x\right)",0,"Integrate[(x^m*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x]), x]","A",-1
402,0,0,30,0.5163795,"\int \frac{x^m \left(1+c^2 x^2\right)^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^m*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x]),x]","\int \frac{x^m \left(1+c^2 x^2\right)^{3/2}}{a+b \sinh ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{3/2} x^m}{a+b \sinh ^{-1}(c x)},x\right)",0,"Integrate[(x^m*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x]), x]","A",-1
403,0,0,30,0.1164667,"\int \frac{x^m \sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]),x]","\int \frac{x^m \sqrt{1+c^2 x^2}}{a+b \sinh ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\sqrt{c^2 x^2+1} x^m}{a+b \sinh ^{-1}(c x)},x\right)",0,"Integrate[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]), x]","A",-1
404,0,0,30,0.3424766,"\int \frac{x^m}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[x^m/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","\int \frac{x^m}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^m}{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[x^m/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]","A",-1
405,0,0,30,0.614799,"\int \frac{x^m}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[x^m/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]","\int \frac{x^m}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{x^m}{\left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[x^m/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]","A",-1
406,1,82,94,0.5001104,"\int \frac{\left(c+a^2 c x^2\right)^3}{\sinh ^{-1}(a x)^2} \, dx","Integrate[(c + a^2*c*x^2)^3/ArcSinh[a*x]^2,x]","\frac{c^3 \left(-64 \left(a^2 x^2+1\right)^{7/2}+35 \sinh ^{-1}(a x) \text{Shi}\left(\sinh ^{-1}(a x)\right)+63 \sinh ^{-1}(a x) \text{Shi}\left(3 \sinh ^{-1}(a x)\right)+35 \sinh ^{-1}(a x) \text{Shi}\left(5 \sinh ^{-1}(a x)\right)+7 \sinh ^{-1}(a x) \text{Shi}\left(7 \sinh ^{-1}(a x)\right)\right)}{64 a \sinh ^{-1}(a x)}","-\frac{c^3 \left(a^2 x^2+1\right)^{7/2}}{a \sinh ^{-1}(a x)}+\frac{35 c^3 \text{Shi}\left(\sinh ^{-1}(a x)\right)}{64 a}+\frac{63 c^3 \text{Shi}\left(3 \sinh ^{-1}(a x)\right)}{64 a}+\frac{35 c^3 \text{Shi}\left(5 \sinh ^{-1}(a x)\right)}{64 a}+\frac{7 c^3 \text{Shi}\left(7 \sinh ^{-1}(a x)\right)}{64 a}",1,"(c^3*(-64*(1 + a^2*x^2)^(7/2) + 35*ArcSinh[a*x]*SinhIntegral[ArcSinh[a*x]] + 63*ArcSinh[a*x]*SinhIntegral[3*ArcSinh[a*x]] + 35*ArcSinh[a*x]*SinhIntegral[5*ArcSinh[a*x]] + 7*ArcSinh[a*x]*SinhIntegral[7*ArcSinh[a*x]]))/(64*a*ArcSinh[a*x])","A",1
407,1,69,77,0.3958219,"\int \frac{\left(c+a^2 c x^2\right)^2}{\sinh ^{-1}(a x)^2} \, dx","Integrate[(c + a^2*c*x^2)^2/ArcSinh[a*x]^2,x]","\frac{c^2 \left(-16 \left(a^2 x^2+1\right)^{5/2}+10 \sinh ^{-1}(a x) \text{Shi}\left(\sinh ^{-1}(a x)\right)+15 \sinh ^{-1}(a x) \text{Shi}\left(3 \sinh ^{-1}(a x)\right)+5 \sinh ^{-1}(a x) \text{Shi}\left(5 \sinh ^{-1}(a x)\right)\right)}{16 a \sinh ^{-1}(a x)}","-\frac{c^2 \left(a^2 x^2+1\right)^{5/2}}{a \sinh ^{-1}(a x)}+\frac{5 c^2 \text{Shi}\left(\sinh ^{-1}(a x)\right)}{8 a}+\frac{15 c^2 \text{Shi}\left(3 \sinh ^{-1}(a x)\right)}{16 a}+\frac{5 c^2 \text{Shi}\left(5 \sinh ^{-1}(a x)\right)}{16 a}",1,"(c^2*(-16*(1 + a^2*x^2)^(5/2) + 10*ArcSinh[a*x]*SinhIntegral[ArcSinh[a*x]] + 15*ArcSinh[a*x]*SinhIntegral[3*ArcSinh[a*x]] + 5*ArcSinh[a*x]*SinhIntegral[5*ArcSinh[a*x]]))/(16*a*ArcSinh[a*x])","A",1
408,1,54,54,0.1733761,"\int \frac{c+a^2 c x^2}{\sinh ^{-1}(a x)^2} \, dx","Integrate[(c + a^2*c*x^2)/ArcSinh[a*x]^2,x]","\frac{c \left(-4 \left(a^2 x^2+1\right)^{3/2}+3 \sinh ^{-1}(a x) \text{Shi}\left(\sinh ^{-1}(a x)\right)+3 \sinh ^{-1}(a x) \text{Shi}\left(3 \sinh ^{-1}(a x)\right)\right)}{4 a \sinh ^{-1}(a x)}","-\frac{c \left(a^2 x^2+1\right)^{3/2}}{a \sinh ^{-1}(a x)}+\frac{3 c \text{Shi}\left(\sinh ^{-1}(a x)\right)}{4 a}+\frac{3 c \text{Shi}\left(3 \sinh ^{-1}(a x)\right)}{4 a}",1,"(c*(-4*(1 + a^2*x^2)^(3/2) + 3*ArcSinh[a*x]*SinhIntegral[ArcSinh[a*x]] + 3*ArcSinh[a*x]*SinhIntegral[3*ArcSinh[a*x]]))/(4*a*ArcSinh[a*x])","A",1
409,0,0,58,1.5316133,"\int \frac{1}{\left(c+a^2 c x^2\right) \sinh ^{-1}(a x)^2} \, dx","Integrate[1/((c + a^2*c*x^2)*ArcSinh[a*x]^2),x]","\int \frac{1}{\left(c+a^2 c x^2\right) \sinh ^{-1}(a x)^2} \, dx","-\frac{a \text{Int}\left(\frac{x}{\left(a^2 x^2+1\right)^{3/2} \sinh ^{-1}(a x)},x\right)}{c}-\frac{1}{a c \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)}",0,"Integrate[1/((c + a^2*c*x^2)*ArcSinh[a*x]^2), x]","A",-1
410,0,0,58,3.7711549,"\int \frac{1}{\left(c+a^2 c x^2\right)^2 \sinh ^{-1}(a x)^2} \, dx","Integrate[1/((c + a^2*c*x^2)^2*ArcSinh[a*x]^2),x]","\int \frac{1}{\left(c+a^2 c x^2\right)^2 \sinh ^{-1}(a x)^2} \, dx","-\frac{3 a \text{Int}\left(\frac{x}{\left(a^2 x^2+1\right)^{5/2} \sinh ^{-1}(a x)},x\right)}{c^2}-\frac{1}{a c^2 \left(a^2 x^2+1\right)^{3/2} \sinh ^{-1}(a x)}",0,"Integrate[1/((c + a^2*c*x^2)^2*ArcSinh[a*x]^2), x]","A",-1
411,1,175,213,0.6739201,"\int \frac{x^3 \sqrt{1+c^2 x^2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^3*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2,x]","-\frac{\frac{16 b c^5 x^5}{a+b \sinh ^{-1}(c x)}+\frac{16 b c^3 x^3}{a+b \sinh ^{-1}(c x)}+2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-5 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{16 b^2 c^4}","-\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2 c^4}-\frac{3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4}+\frac{5 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4}+\frac{\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2 c^4}+\frac{3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4}-\frac{5 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^4}-\frac{x^3 \left(c^2 x^2+1\right)}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/16*((16*b*c^3*x^3)/(a + b*ArcSinh[c*x]) + (16*b*c^5*x^5)/(a + b*ArcSinh[c*x]) + 2*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] + 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 5*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 5*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])])/(b^2*c^4)","A",1
412,1,82,93,0.3218197,"\int \frac{x^2 \sqrt{1+c^2 x^2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2,x]","\frac{-\frac{2 b c^2 x^2 \left(c^2 x^2+1\right)}{a+b \sinh ^{-1}(c x)}-\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{2 b^2 c^3}","-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}+\frac{\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}-\frac{x^2 \left(c^2 x^2+1\right)}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"((-2*b*c^2*x^2*(1 + c^2*x^2))/(a + b*ArcSinh[c*x]) - CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] + Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])])/(2*b^2*c^3)","A",1
413,1,126,149,0.3770148,"\int \frac{x \sqrt{1+c^2 x^2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2,x]","-\frac{\frac{4 b c^3 x^3}{a+b \sinh ^{-1}(c x)}-\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\frac{4 b c x}{a+b \sinh ^{-1}(c x)}}{4 b^2 c^2}","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b^2 c^2}+\frac{3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^2}-\frac{\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b^2 c^2}-\frac{3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^2}-\frac{x \left(c^2 x^2+1\right)}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/4*((4*b*c*x)/(a + b*ArcSinh[c*x]) + (4*b*c^3*x^3)/(a + b*ArcSinh[c*x]) - Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(b^2*c^2)","A",1
414,1,73,85,0.1716456,"\int \frac{\sqrt{1+c^2 x^2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(a + b*ArcSinh[c*x])^2,x]","\frac{-\frac{b c^2 x^2+b}{a+b \sinh ^{-1}(c x)}-\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{b^2 c}","-\frac{\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c}+\frac{\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c}-\frac{c^2 x^2+1}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"(-((b + b*c^2*x^2)/(a + b*ArcSinh[c*x])) - CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] + Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])])/(b^2*c)","A",1
415,0,0,103,10.8652472,"\int \frac{\sqrt{1+c^2 x^2}}{x \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(x*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\sqrt{1+c^2 x^2}}{x \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}+\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2}-\frac{\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2}-\frac{c^2 x^2+1}{b c x \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[Sqrt[1 + c^2*x^2]/(x*(a + b*ArcSinh[c*x])^2), x]","A",-1
416,0,0,56,2.8252408,"\int \frac{\sqrt{1+c^2 x^2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(x^2*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\sqrt{1+c^2 x^2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^3 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}-\frac{c^2 x^2+1}{b c x^2 \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[Sqrt[1 + c^2*x^2]/(x^2*(a + b*ArcSinh[c*x])^2), x]","A",-1
417,0,0,30,19.0877374,"\int \frac{\sqrt{1+c^2 x^2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\sqrt{1+c^2 x^2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{c^2 x^2+1}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[Sqrt[1 + c^2*x^2]/(x^3*(a + b*ArcSinh[c*x])^2), x]","A",-1
418,0,0,30,3.6754846,"\int \frac{\sqrt{1+c^2 x^2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\sqrt{1+c^2 x^2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{c^2 x^2+1}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[Sqrt[1 + c^2*x^2]/(x^4*(a + b*ArcSinh[c*x])^2), x]","A",-1
419,1,399,277,0.8160666,"\int \frac{x^3 \left(1+c^2 x^2\right)^{3/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^3*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]","\frac{-3 \cosh \left(\frac{a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-9 \cosh \left(\frac{3 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 a \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 b \cosh \left(\frac{5 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+7 a \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+7 b \cosh \left(\frac{7 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 a \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 b \sinh \left(\frac{a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+9 a \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+9 b \sinh \left(\frac{3 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-5 a \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-5 b \sinh \left(\frac{5 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-7 a \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-7 b \sinh \left(\frac{7 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-64 b c^7 x^7-128 b c^5 x^5-64 b c^3 x^3}{64 b^2 c^4 \left(a+b \sinh ^{-1}(c x)\right)}","-\frac{3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b^2 c^4}-\frac{9 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}+\frac{5 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}+\frac{7 \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}+\frac{3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b^2 c^4}+\frac{9 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}-\frac{5 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}-\frac{7 \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^4}-\frac{x^3 \left(c^2 x^2+1\right)^2}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"(-64*b*c^3*x^3 - 128*b*c^5*x^5 - 64*b*c^7*x^7 - 3*(a + b*ArcSinh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - 9*(a + b*ArcSinh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] + 5*a*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] + 5*b*ArcSinh[c*x]*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] + 7*a*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] + 7*b*ArcSinh[c*x]*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] + 3*a*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 3*b*ArcSinh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 9*a*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 9*b*ArcSinh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - 5*a*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] - 5*b*ArcSinh[c*x]*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] - 7*a*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])] - 7*b*ArcSinh[c*x]*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])])/(64*b^2*c^4*(a + b*ArcSinh[c*x]))","A",1
420,1,306,219,0.7371507,"\int \frac{x^2 \left(1+c^2 x^2\right)^{3/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^2*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]","-\frac{-\sinh \left(\frac{2 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+4 \sinh \left(\frac{4 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 a \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 b \sinh \left(\frac{6 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+a \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+b \cosh \left(\frac{2 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-4 a \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-4 b \cosh \left(\frac{4 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 a \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 b \cosh \left(\frac{6 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+16 b c^6 x^6+32 b c^4 x^4+16 b c^2 x^2}{16 b^2 c^3 \left(a+b \sinh ^{-1}(c x)\right)}","\frac{\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}-\frac{3 \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}+\frac{\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}+\frac{3 \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{x^2 \left(c^2 x^2+1\right)^2}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/16*(16*b*c^2*x^2 + 32*b*c^4*x^4 + 16*b*c^6*x^6 - (a + b*ArcSinh[c*x])*CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] + 4*(a + b*ArcSinh[c*x])*CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] + 3*a*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + 3*b*ArcSinh[c*x]*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + b*ArcSinh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 4*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 4*b*ArcSinh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])] - 3*b*ArcSinh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])])/(b^2*c^3*(a + b*ArcSinh[c*x]))","A",1
421,1,295,213,0.6090402,"\int \frac{x \left(1+c^2 x^2\right)^{3/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]","-\frac{-2 \cosh \left(\frac{a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-9 \cosh \left(\frac{3 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-5 a \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-5 b \cosh \left(\frac{5 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+2 a \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+2 b \sinh \left(\frac{a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+9 a \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+9 b \sinh \left(\frac{3 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 a \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 b \sinh \left(\frac{5 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+16 b c^5 x^5+32 b c^3 x^3+16 b c x}{16 b^2 c^2 \left(a+b \sinh ^{-1}(c x)\right)}","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2 c^2}+\frac{9 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2}+\frac{5 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2}-\frac{\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2 c^2}-\frac{9 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2}-\frac{5 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^2}-\frac{x \left(c^2 x^2+1\right)^2}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/16*(16*b*c*x + 32*b*c^3*x^3 + 16*b*c^5*x^5 - 2*(a + b*ArcSinh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - 9*(a + b*ArcSinh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 5*a*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 5*b*ArcSinh[c*x]*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] + 2*a*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 2*b*ArcSinh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 9*a*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 9*b*ArcSinh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 5*a*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] + 5*b*ArcSinh[c*x]*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])])/(b^2*c^2*(a + b*ArcSinh[c*x]))","A",1
422,1,122,149,0.4734923,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(a + b*ArcSinh[c*x])^2,x]","\frac{-\frac{2 b \left(c^2 x^2+1\right)^2}{a+b \sinh ^{-1}(c x)}-2 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+2 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{2 b^2 c}","-\frac{\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c}-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c}+\frac{\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c}+\frac{\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c}-\frac{\left(c^2 x^2+1\right)^2}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"((-2*b*(1 + c^2*x^2)^2)/(a + b*ArcSinh[c*x]) - 2*CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] - CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] + 2*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])])/(2*b^2*c)","A",1
423,0,0,175,7.7003961,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(x*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","-\frac{\text{Int}\left(\frac{c^2 x^2+1}{x^2 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}+\frac{9 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b^2}+\frac{3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2}-\frac{9 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b^2}-\frac{3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2}-\frac{\left(c^2 x^2+1\right)^2}{b c x \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[(1 + c^2*x^2)^(3/2)/(x*(a + b*ArcSinh[c*x])^2), x]","A",-1
424,0,0,99,4.3098506,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(x^2*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\frac{2 c \text{Int}\left(\frac{c^2 x^2+1}{x \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b}-\frac{2 \text{Int}\left(\frac{c^2 x^2+1}{x^3 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}-\frac{\left(c^2 x^2+1\right)^2}{b c x^2 \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[(1 + c^2*x^2)^(3/2)/(x^2*(a + b*ArcSinh[c*x])^2), x]","A",-1
425,0,0,30,14.5073988,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{3/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(1 + c^2*x^2)^(3/2)/(x^3*(a + b*ArcSinh[c*x])^2), x]","A",-1
426,0,0,67,2.7774363,"\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(3/2)/(x^4*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\left(1+c^2 x^2\right)^{3/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","-\frac{4 \text{Int}\left(\frac{c^2 x^2+1}{x^5 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}-\frac{\left(c^2 x^2+1\right)^2}{b c x^4 \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[(1 + c^2*x^2)^(3/2)/(x^4*(a + b*ArcSinh[c*x])^2), x]","A",-1
427,1,408,277,1.3761851,"\int \frac{x^3 \left(1+c^2 x^2\right)^{5/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^3*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]","-\frac{6 \cosh \left(\frac{a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+24 \cosh \left(\frac{3 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-21 a \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-21 b \cosh \left(\frac{7 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-9 a \cosh \left(\frac{9 a}{b}\right) \text{Chi}\left(9 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-9 b \cosh \left(\frac{9 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(9 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-6 a \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-6 b \sinh \left(\frac{a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-24 a \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-24 b \sinh \left(\frac{3 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+21 a \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+21 b \sinh \left(\frac{7 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+9 a \sinh \left(\frac{9 a}{b}\right) \text{Shi}\left(9 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+9 b \sinh \left(\frac{9 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(9 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+256 b c^9 x^9+768 b c^7 x^7+768 b c^5 x^5+256 b c^3 x^3}{256 b^2 c^4 \left(a+b \sinh ^{-1}(c x)\right)}","-\frac{3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{128 b^2 c^4}-\frac{3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b^2 c^4}+\frac{21 \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{256 b^2 c^4}+\frac{9 \cosh \left(\frac{9 a}{b}\right) \text{Chi}\left(\frac{9 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{256 b^2 c^4}+\frac{3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{128 b^2 c^4}+\frac{3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 b^2 c^4}-\frac{21 \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{256 b^2 c^4}-\frac{9 \sinh \left(\frac{9 a}{b}\right) \text{Shi}\left(\frac{9 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{256 b^2 c^4}-\frac{x^3 \left(c^2 x^2+1\right)^3}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/256*(256*b*c^3*x^3 + 768*b*c^5*x^5 + 768*b*c^7*x^7 + 256*b*c^9*x^9 + 6*(a + b*ArcSinh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] + 24*(a + b*ArcSinh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 21*a*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] - 21*b*ArcSinh[c*x]*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] - 9*a*Cosh[(9*a)/b]*CoshIntegral[9*(a/b + ArcSinh[c*x])] - 9*b*ArcSinh[c*x]*Cosh[(9*a)/b]*CoshIntegral[9*(a/b + ArcSinh[c*x])] - 6*a*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 6*b*ArcSinh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 24*a*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - 24*b*ArcSinh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 21*a*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])] + 21*b*ArcSinh[c*x]*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])] + 9*a*Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcSinh[c*x])] + 9*b*ArcSinh[c*x]*Sinh[(9*a)/b]*SinhIntegral[9*(a/b + ArcSinh[c*x])])/(b^2*c^4*(a + b*ArcSinh[c*x]))","A",1
428,1,413,281,1.174741,"\int \frac{x^2 \left(1+c^2 x^2\right)^{5/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^2*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]","-\frac{-\sinh \left(\frac{2 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+2 \sinh \left(\frac{4 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 a \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 b \sinh \left(\frac{6 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+a \sinh \left(\frac{8 a}{b}\right) \text{Chi}\left(8 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+b \sinh \left(\frac{8 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(8 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+a \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+b \cosh \left(\frac{2 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 a \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 b \cosh \left(\frac{4 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 a \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 b \cosh \left(\frac{6 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-a \cosh \left(\frac{8 a}{b}\right) \text{Shi}\left(8 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-b \cosh \left(\frac{8 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(8 \left(\frac{a}{b}+\sinh ^{-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}{16 b^2 c^3 \left(a+b \sinh ^{-1}(c x)\right)}","\frac{\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b^2 c^3}-\frac{3 \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\sinh \left(\frac{8 a}{b}\right) \text{Chi}\left(\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}+\frac{\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 b^2 c^3}+\frac{3 \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}+\frac{\cosh \left(\frac{8 a}{b}\right) \text{Shi}\left(\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^3}-\frac{x^2 \left(c^2 x^2+1\right)^3}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/16*(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*ArcSinh[c*x])*CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] + 2*(a + b*ArcSinh[c*x])*CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] + 3*a*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + 3*b*ArcSinh[c*x]*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + a*CoshIntegral[8*(a/b + ArcSinh[c*x])]*Sinh[(8*a)/b] + b*ArcSinh[c*x]*CoshIntegral[8*(a/b + ArcSinh[c*x])]*Sinh[(8*a)/b] + a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + b*ArcSinh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 2*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 2*b*ArcSinh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])] - 3*b*ArcSinh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])] - a*Cosh[(8*a)/b]*SinhIntegral[8*(a/b + ArcSinh[c*x])] - b*ArcSinh[c*x]*Cosh[(8*a)/b]*SinhIntegral[8*(a/b + ArcSinh[c*x])])/(b^2*c^3*(a + b*ArcSinh[c*x]))","A",1
429,1,404,275,1.0170446,"\int \frac{x \left(1+c^2 x^2\right)^{5/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]","-\frac{-5 \cosh \left(\frac{a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-27 \cosh \left(\frac{3 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-25 a \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-25 b \cosh \left(\frac{5 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-7 a \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-7 b \cosh \left(\frac{7 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 a \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+5 b \sinh \left(\frac{a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+27 a \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+27 b \sinh \left(\frac{3 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+25 a \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+25 b \sinh \left(\frac{5 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+7 a \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+7 b \sinh \left(\frac{7 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-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}{64 b^2 c^2 \left(a+b \sinh ^{-1}(c x)\right)}","\frac{5 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b^2 c^2}+\frac{27 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}+\frac{25 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}+\frac{7 \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}-\frac{5 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b^2 c^2}-\frac{27 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}-\frac{25 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}-\frac{7 \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b^2 c^2}-\frac{x \left(c^2 x^2+1\right)^3}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/64*(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*ArcSinh[c*x])*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - 27*(a + b*ArcSinh[c*x])*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 25*a*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 25*b*ArcSinh[c*x]*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 7*a*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] - 7*b*ArcSinh[c*x]*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] + 5*a*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 5*b*ArcSinh[c*x]*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 27*a*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 27*b*ArcSinh[c*x]*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 25*a*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] + 25*b*ArcSinh[c*x]*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] + 7*a*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])] + 7*b*ArcSinh[c*x]*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])])/(b^2*c^2*(a + b*ArcSinh[c*x]))","A",1
430,1,311,216,0.7203142,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(a + b*ArcSinh[c*x])^2,x]","-\frac{15 \sinh \left(\frac{2 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+12 \sinh \left(\frac{4 a}{b}\right) \left(a+b \sinh ^{-1}(c x)\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 a \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 b \sinh \left(\frac{6 a}{b}\right) \sinh ^{-1}(c x) \text{Chi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-15 a \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-15 b \cosh \left(\frac{2 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-12 a \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-12 b \cosh \left(\frac{4 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 a \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-3 b \cosh \left(\frac{6 a}{b}\right) \sinh ^{-1}(c x) \text{Shi}\left(6 \left(\frac{a}{b}+\sinh ^{-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}{16 b^2 c \left(a+b \sinh ^{-1}(c x)\right)}","-\frac{15 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c}-\frac{3 \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c}-\frac{3 \sinh \left(\frac{6 a}{b}\right) \text{Chi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c}+\frac{15 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c}+\frac{3 \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c}+\frac{3 \cosh \left(\frac{6 a}{b}\right) \text{Shi}\left(\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c}-\frac{\left(c^2 x^2+1\right)^3}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/16*(16*b + 48*b*c^2*x^2 + 48*b*c^4*x^4 + 16*b*c^6*x^6 + 15*(a + b*ArcSinh[c*x])*CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] + 12*(a + b*ArcSinh[c*x])*CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] + 3*a*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] + 3*b*ArcSinh[c*x]*CoshIntegral[6*(a/b + ArcSinh[c*x])]*Sinh[(6*a)/b] - 15*a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 15*b*ArcSinh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 12*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 12*b*ArcSinh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])] - 3*b*ArcSinh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcSinh[c*x])])/(b^2*c*(a + b*ArcSinh[c*x]))","A",1
431,0,0,233,9.518219,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(x*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","-\frac{\text{Int}\left(\frac{\left(c^2 x^2+1\right)^2}{x^2 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}+\frac{25 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2}+\frac{25 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2}+\frac{5 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2}-\frac{25 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2}-\frac{25 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2}-\frac{5 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2}-\frac{\left(c^2 x^2+1\right)^3}{b c x \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[(1 + c^2*x^2)^(5/2)/(x*(a + b*ArcSinh[c*x])^2), x]","A",-1
432,0,0,103,3.4223253,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^2 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\frac{4 c \text{Int}\left(\frac{\left(c^2 x^2+1\right)^2}{x \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b}-\frac{2 \text{Int}\left(\frac{\left(c^2 x^2+1\right)^2}{x^3 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}-\frac{\left(c^2 x^2+1\right)^3}{b c x^2 \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[(1 + c^2*x^2)^(5/2)/(x^2*(a + b*ArcSinh[c*x])^2), x]","A",-1
433,0,0,30,14.5738667,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{5/2}}{x^3 \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(1 + c^2*x^2)^(5/2)/(x^3*(a + b*ArcSinh[c*x])^2), x]","A",-1
434,0,0,30,3.083084,"\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(1 + c^2*x^2)^(5/2)/(x^4*(a + b*ArcSinh[c*x])^2),x]","\int \frac{\left(1+c^2 x^2\right)^{5/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{5/2}}{x^4 \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(1 + c^2*x^2)^(5/2)/(x^4*(a + b*ArcSinh[c*x])^2), x]","A",-1
435,1,158,204,0.3406674,"\int \frac{x^5}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^5/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]","\frac{5 \left(2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)\right)}{16 b^2 c^6}-\frac{x^5}{b c \left(a+b \sinh ^{-1}(c x)\right)}","\frac{5 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2 c^6}-\frac{15 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6}+\frac{5 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6}-\frac{5 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2 c^6}+\frac{15 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6}-\frac{5 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^6}-\frac{x^5}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-(x^5/(b*c*(a + b*ArcSinh[c*x]))) + (5*(2*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] + Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])]))/(16*b^2*c^6)","A",1
436,1,117,141,0.2862476,"\int \frac{x^4}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^4/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]","\frac{-\frac{2 b c^4 x^4}{a+b \sinh ^{-1}(c x)}+2 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{2 b^2 c^5}","\frac{\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c^5}-\frac{\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^5}-\frac{\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c^5}+\frac{\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^5}-\frac{x^4}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"((-2*b*c^4*x^4)/(a + b*ArcSinh[c*x]) + 2*CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] - CoshIntegral[4*(a/b + ArcSinh[c*x])]*Sinh[(4*a)/b] - 2*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] + Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c*x])])/(2*b^2*c^5)","A",1
437,1,113,142,0.2696661,"\int \frac{x^3}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^3/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]","\frac{3 \left(-\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)\right)}{4 b^2 c^4}-\frac{x^3}{b c \left(a+b \sinh ^{-1}(c x)\right)}","-\frac{3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b^2 c^4}+\frac{3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^4}+\frac{3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b^2 c^4}-\frac{3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^4}-\frac{x^3}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-(x^3/(b*c*(a + b*ArcSinh[c*x]))) + (3*(-(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]]) + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])]))/(4*b^2*c^4)","A",1
438,1,70,79,0.1410474,"\int \frac{x^2}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]","\frac{-\frac{b c^2 x^2}{a+b \sinh ^{-1}(c x)}-\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{b^2 c^3}","-\frac{\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c^3}+\frac{\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c^3}-\frac{x^2}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"(-((b*c^2*x^2)/(a + b*ArcSinh[c*x])) - CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] + Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])])/(b^2*c^3)","A",1
439,1,60,73,0.1241018,"\int \frac{x}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-\frac{b c x}{a+b \sinh ^{-1}(c x)}}{b^2 c^2}","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2 c^2}-\frac{\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2 c^2}-\frac{x}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"(-((b*c*x)/(a + b*ArcSinh[c*x])) + Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b^2*c^2)","A",1
440,1,18,18,0.0123308,"\int \frac{1}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]","-\frac{1}{b c \left(a+b \sinh ^{-1}(c x)\right)}","-\frac{1}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-(1/(b*c*(a + b*ArcSinh[c*x])))","A",1
441,0,0,47,5.4415961,"\int \frac{1}{x \sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{x \sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","-\frac{\text{Int}\left(\frac{1}{x^2 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}-\frac{1}{b c x \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[1/(x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x]","A",-1
442,0,0,47,1.2487492,"\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{x^2 \sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","-\frac{2 \text{Int}\left(\frac{1}{x^3 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}-\frac{1}{b c x^2 \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[1/(x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x]","A",-1
443,0,0,30,8.0950311,"\int \frac{x^3}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^3/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{x^3}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^3}{\left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^3/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
444,0,0,67,3.6032438,"\int \frac{x^2}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{x^2}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\frac{2 \text{Int}\left(\frac{x}{\left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b c}-\frac{x^2}{b c \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[x^2/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
445,0,0,28,5.3618087,"\int \frac{x}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{x}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x}{\left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
446,0,0,62,2.4379199,"\int \frac{1}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","-\frac{2 c \text{Int}\left(\frac{x}{\left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b}-\frac{1}{b c \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
447,0,0,30,7.8266752,"\int \frac{1}{x \left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{x \left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
448,0,0,30,20.0045139,"\int \frac{1}{x^2 \left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{x^2 \left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
449,0,0,30,14.3650495,"\int \frac{x^3}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^3/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{x^3}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^3}{\left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^3/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
450,0,0,30,5.8738082,"\int \frac{x^2}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^2/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{x^2}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^2}{\left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^2/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
451,0,0,28,11.0574746,"\int \frac{x}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{x}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x}{\left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
452,0,0,62,2.8591024,"\int \frac{1}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","-\frac{4 c \text{Int}\left(\frac{x}{\left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)},x\right)}{b}-\frac{1}{b c \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[1/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
453,0,0,30,12.7792237,"\int \frac{1}{x \left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{x \left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
454,0,0,30,12.0880784,"\int \frac{1}{x^2 \left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(x^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{x^2 \left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(x^2*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
455,0,0,30,1.0253428,"\int \frac{x^m \left(1+c^2 x^2\right)^{5/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^m*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2,x]","\int \frac{x^m \left(1+c^2 x^2\right)^{5/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{5/2} x^m}{\left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(x^m*(1 + c^2*x^2)^(5/2))/(a + b*ArcSinh[c*x])^2, x]","A",-1
456,0,0,30,0.5666701,"\int \frac{x^m \left(1+c^2 x^2\right)^{3/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^m*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2,x]","\int \frac{x^m \left(1+c^2 x^2\right)^{3/2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\left(c^2 x^2+1\right)^{3/2} x^m}{\left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(x^m*(1 + c^2*x^2)^(3/2))/(a + b*ArcSinh[c*x])^2, x]","A",-1
457,0,0,30,0.1259363,"\int \frac{x^m \sqrt{1+c^2 x^2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2,x]","\int \frac{x^m \sqrt{1+c^2 x^2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{c^2 x^2+1} x^m}{\left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(x^m*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])^2, x]","A",-1
458,0,0,49,0.3581655,"\int \frac{x^m}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^m/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]","\int \frac{x^m}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\frac{m \text{Int}\left(\frac{x^{m-1}}{a+b \sinh ^{-1}(c x)},x\right)}{b c}-\frac{x^m}{b c \left(a+b \sinh ^{-1}(c x)\right)}",0,"Integrate[x^m/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2), x]","A",-1
459,0,0,30,0.6671956,"\int \frac{x^m}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^m/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{x^m}{\left(1+c^2 x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^m}{\left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^m/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
460,0,0,30,0.9030181,"\int \frac{x^m}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[x^m/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{x^m}{\left(1+c^2 x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{x^m}{\left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[x^m/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
461,1,13,13,0.0087677,"\int \frac{1}{\sqrt{1+a^2 x^2} \sinh ^{-1}(a x)^3} \, dx","Integrate[1/(Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^3),x]","-\frac{1}{2 a \sinh ^{-1}(a x)^2}","-\frac{1}{2 a \sinh ^{-1}(a x)^2}",1,"-1/2*1/(a*ArcSinh[a*x]^2)","A",1
462,1,232,254,0.9224964,"\int \frac{x^3 \left(d+c^2 d x^2\right)}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^3*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2),x]","\frac{d e^{-\frac{6 a}{b}} \left(-8 e^{\frac{6 a}{b}} \sinh ^3\left(2 \sinh ^{-1}(c x)\right)+\sqrt{6} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-3 \sqrt{2} e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+3 \sqrt{2} e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-\sqrt{6} e^{\frac{12 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{32 b c^4 \sqrt{a+b \sinh ^{-1}(c x)}}","-\frac{3 \sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^4}-\frac{2 d x^3 \left(c^2 x^2+1\right)^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"(d*(Sqrt[6]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcSinh[c*x]))/b] - 3*Sqrt[2]*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c*x]))/b] + 3*Sqrt[2]*E^((8*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c*x]))/b] - Sqrt[6]*E^((12*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (6*(a + b*ArcSinh[c*x]))/b] - 8*E^((6*a)/b)*Sinh[2*ArcSinh[c*x]]^3))/(32*b*c^4*E^((6*a)/b)*Sqrt[a + b*ArcSinh[c*x]])","A",0
463,1,436,335,0.7452978,"\int \frac{x^2 \left(d+c^2 d x^2\right)}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^2*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2),x]","\frac{d e^{-5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)} \left(-e^{\frac{5 a}{b}+2 \sinh ^{-1}(c x)}+2 e^{\frac{5 a}{b}+4 \sinh ^{-1}(c x)}+2 e^{\frac{5 a}{b}+6 \sinh ^{-1}(c x)}-e^{\frac{5 a}{b}+8 \sinh ^{-1}(c x)}-e^{\frac{5 a}{b}+10 \sinh ^{-1}(c x)}-2 e^{\frac{6 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)+\sqrt{5} e^{5 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+\sqrt{3} e^{\frac{2 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-2 e^{\frac{4 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)+\sqrt{3} e^{\frac{8 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+\sqrt{5} e^{5 \left(\frac{2 a}{b}+\sinh ^{-1}(c x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-e^{\frac{5 a}{b}}\right)}{16 b c^3 \sqrt{a+b \sinh ^{-1}(c x)}}","\frac{\sqrt{\pi } d e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^3}-\frac{2 d x^2 \left(c^2 x^2+1\right)^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"(d*(-E^((5*a)/b) - E^((5*a)/b + 2*ArcSinh[c*x]) + 2*E^((5*a)/b + 4*ArcSinh[c*x]) + 2*E^((5*a)/b + 6*ArcSinh[c*x]) - E^((5*a)/b + 8*ArcSinh[c*x]) - E^((5*a)/b + 10*ArcSinh[c*x]) - 2*E^((6*a)/b + 5*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + Sqrt[5]*E^(5*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcSinh[c*x]))/b] + Sqrt[3]*E^((2*a)/b + 5*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] - 2*E^((4*a)/b + 5*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] + Sqrt[3]*E^((8*a)/b + 5*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b] + Sqrt[5]*E^(5*((2*a)/b + ArcSinh[c*x]))*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (5*(a + b*ArcSinh[c*x]))/b]))/(16*b*c^3*E^(5*(a/b + ArcSinh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])","A",0
464,1,227,236,0.4629494,"\int \frac{x \left(d+c^2 d x^2\right)}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x*(d + c^2*d*x^2))/(a + b*ArcSinh[c*x])^(3/2),x]","\frac{d e^{-\frac{4 a}{b}} \left(\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+\sqrt{2} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-e^{\frac{4 a}{b}} \left(\sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+2 \sinh \left(2 \sinh ^{-1}(c x)\right)+\sinh \left(4 \sinh ^{-1}(c x)\right)\right)\right)}{4 b c^2 \sqrt{a+b \sinh ^{-1}(c x)}}","\frac{\sqrt{\pi } d e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2} c^2}-\frac{2 d x \left(c^2 x^2+1\right)^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"(d*(Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c*x]))/b] + Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c*x]))/b] - E^((4*a)/b)*(Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c*x]))/b] + E^((4*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (4*(a + b*ArcSinh[c*x]))/b] + 2*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])))/(4*b*c^2*E^((4*a)/b)*Sqrt[a + b*ArcSinh[c*x]])","A",0
465,1,295,228,0.9262684,"\int \frac{d+c^2 d x^2}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d + c^2*d*x^2)/(a + b*ArcSinh[c*x])^(3/2),x]","\frac{d e^{-3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)} \left(-3 e^{\frac{3 a}{b}+2 \sinh ^{-1}(c x)}-3 e^{\frac{3 a}{b}+4 \sinh ^{-1}(c x)}-e^{\frac{3 a}{b}+6 \sinh ^{-1}(c x)}+3 e^{\frac{4 a}{b}+3 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)+\sqrt{3} e^{3 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+3 e^{\frac{2 a}{b}+3 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)+\sqrt{3} e^{\frac{6 a}{b}+3 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-e^{\frac{3 a}{b}}\right)}{4 b c \sqrt{a+b \sinh ^{-1}(c x)}}","-\frac{3 \sqrt{\pi } d e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2} c}-\frac{2 d \left(c^2 x^2+1\right)^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"(d*(-E^((3*a)/b) - 3*E^((3*a)/b + 2*ArcSinh[c*x]) - 3*E^((3*a)/b + 4*ArcSinh[c*x]) - E^((3*a)/b + 6*ArcSinh[c*x]) + 3*E^((4*a)/b + 3*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + Sqrt[3]*E^(3*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] + 3*E^((2*a)/b + 3*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] + Sqrt[3]*E^((6*a)/b + 3*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b]))/(4*b*c*E^(3*(a/b + ArcSinh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])","A",0
466,0,0,175,4.4348605,"\int \frac{d+c^2 d x^2}{x \left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d + c^2*d*x^2)/(x*(a + b*ArcSinh[c*x])^(3/2)),x]","\int \frac{d+c^2 d x^2}{x \left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","-\frac{2 d \text{Int}\left(\frac{1}{x^2 \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2}}-\frac{2 d \left(c^2 x^2+1\right)^{3/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}",0,"Integrate[(d + c^2*d*x^2)/(x*(a + b*ArcSinh[c*x])^(3/2)), x]","A",-1
467,1,462,474,1.0105504,"\int \frac{x^3 \left(d+c^2 d x^2\right)^2}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^3*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2),x]","\frac{d^2 e^{-\frac{8 a}{b}} \left(6 e^{\frac{8 a}{b}} \sinh \left(2 \sinh ^{-1}(c x)\right)+2 e^{\frac{8 a}{b}} \sinh \left(4 \sinh ^{-1}(c x)\right)-2 e^{\frac{8 a}{b}} \sinh \left(6 \sinh ^{-1}(c x)\right)-e^{\frac{8 a}{b}} \sinh \left(8 \sinh ^{-1}(c x)\right)+\sqrt{2} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+\sqrt{6} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-3 \sqrt{2} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+3 \sqrt{2} e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+2 e^{\frac{12 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-\sqrt{6} e^{\frac{14 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-\sqrt{2} e^{\frac{16 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{64 b c^4 \sqrt{a+b \sinh ^{-1}(c x)}}","-\frac{\sqrt{\pi } d^2 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{32 b^{3/2} c^4}-\frac{2 d^2 x^3 \left(c^2 x^2+1\right)^{5/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"(d^2*(Sqrt[2]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-8*(a + b*ArcSinh[c*x]))/b] + Sqrt[6]*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcSinh[c*x]))/b] - 2*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c*x]))/b] - 3*Sqrt[2]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c*x]))/b] + 3*Sqrt[2]*E^((10*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c*x]))/b] + 2*E^((12*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (4*(a + b*ArcSinh[c*x]))/b] - Sqrt[6]*E^((14*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (6*(a + b*ArcSinh[c*x]))/b] - Sqrt[2]*E^((16*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (8*(a + b*ArcSinh[c*x]))/b] + 6*E^((8*a)/b)*Sinh[2*ArcSinh[c*x]] + 2*E^((8*a)/b)*Sinh[4*ArcSinh[c*x]] - 2*E^((8*a)/b)*Sinh[6*ArcSinh[c*x]] - E^((8*a)/b)*Sinh[8*ArcSinh[c*x]]))/(64*b*c^4*E^((8*a)/b)*Sqrt[a + b*ArcSinh[c*x]])","A",0
468,1,577,457,1.6372598,"\int \frac{x^2 \left(d+c^2 d x^2\right)^2}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x^2*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2),x]","-\frac{d^2 e^{-7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)} \left(3 e^{\frac{7 a}{b}+2 \sinh ^{-1}(c x)}+e^{\frac{7 a}{b}+4 \sinh ^{-1}(c x)}-5 e^{\frac{7 a}{b}+6 \sinh ^{-1}(c x)}-5 e^{\frac{7 a}{b}+8 \sinh ^{-1}(c x)}+e^{\frac{7 a}{b}+10 \sinh ^{-1}(c x)}+3 e^{\frac{7 a}{b}+12 \sinh ^{-1}(c x)}+e^{\frac{7 a}{b}+14 \sinh ^{-1}(c x)}+5 e^{\frac{8 a}{b}+7 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)-\sqrt{7} e^{7 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-3 \sqrt{5} e^{\frac{2 a}{b}+7 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e^{\frac{4 a}{b}+7 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+5 e^{\frac{6 a}{b}+7 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)-\sqrt{3} e^{\frac{10 a}{b}+7 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-3 \sqrt{5} e^{\frac{12 a}{b}+7 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-\sqrt{7} e^{7 \left(\frac{2 a}{b}+\sinh ^{-1}(c x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+e^{\frac{7 a}{b}}\right)}{64 b c^3 \sqrt{a+b \sinh ^{-1}(c x)}}","\frac{5 \sqrt{\pi } d^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{64 b^{3/2} c^3}-\frac{2 d^2 x^2 \left(c^2 x^2+1\right)^{5/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"-1/64*(d^2*(E^((7*a)/b) + 3*E^((7*a)/b + 2*ArcSinh[c*x]) + E^((7*a)/b + 4*ArcSinh[c*x]) - 5*E^((7*a)/b + 6*ArcSinh[c*x]) - 5*E^((7*a)/b + 8*ArcSinh[c*x]) + E^((7*a)/b + 10*ArcSinh[c*x]) + 3*E^((7*a)/b + 12*ArcSinh[c*x]) + E^((7*a)/b + 14*ArcSinh[c*x]) + 5*E^((8*a)/b + 7*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] - Sqrt[7]*E^(7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-7*(a + b*ArcSinh[c*x]))/b] - 3*Sqrt[5]*E^((2*a)/b + 7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcSinh[c*x]))/b] - Sqrt[3]*E^((4*a)/b + 7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] + 5*E^((6*a)/b + 7*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] - Sqrt[3]*E^((10*a)/b + 7*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b] - 3*Sqrt[5]*E^((12*a)/b + 7*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (5*(a + b*ArcSinh[c*x]))/b] - Sqrt[7]*E^(7*((2*a)/b + ArcSinh[c*x]))*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (7*(a + b*ArcSinh[c*x]))/b]))/(b*c^3*E^(7*(a/b + ArcSinh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])","A",0
469,1,351,358,0.7057504,"\int \frac{x \left(d+c^2 d x^2\right)^2}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(x*(d + c^2*d*x^2)^2)/(a + b*ArcSinh[c*x])^(3/2),x]","-\frac{d^2 e^{-\frac{6 a}{b}} \left(10 e^{\frac{6 a}{b}} \sinh \left(2 \sinh ^{-1}(c x)\right)+8 e^{\frac{6 a}{b}} \sinh \left(4 \sinh ^{-1}(c x)\right)+2 e^{\frac{6 a}{b}} \sinh \left(6 \sinh ^{-1}(c x)\right)-\sqrt{6} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-8 e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-5 \sqrt{2} e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+5 \sqrt{2} e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+8 e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+\sqrt{6} e^{\frac{12 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{32 b c^2 \sqrt{a+b \sinh ^{-1}(c x)}}","\frac{\sqrt{\pi } d^2 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c^2}-\frac{2 d^2 x \left(c^2 x^2+1\right)^{5/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"-1/32*(d^2*(-(Sqrt[6]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-6*(a + b*ArcSinh[c*x]))/b]) - 8*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c*x]))/b] - 5*Sqrt[2]*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c*x]))/b] + 5*Sqrt[2]*E^((8*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c*x]))/b] + 8*E^((10*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (4*(a + b*ArcSinh[c*x]))/b] + Sqrt[6]*E^((12*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (6*(a + b*ArcSinh[c*x]))/b] + 10*E^((6*a)/b)*Sinh[2*ArcSinh[c*x]] + 8*E^((6*a)/b)*Sinh[4*ArcSinh[c*x]] + 2*E^((6*a)/b)*Sinh[6*ArcSinh[c*x]]))/(b*c^2*E^((6*a)/b)*Sqrt[a + b*ArcSinh[c*x]])","A",0
470,1,440,346,2.0851084,"\int \frac{\left(d+c^2 d x^2\right)^2}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d + c^2*d*x^2)^2/(a + b*ArcSinh[c*x])^(3/2),x]","\frac{d^2 e^{-5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)} \left(-5 e^{\frac{5 a}{b}+2 \sinh ^{-1}(c x)}-10 e^{\frac{5 a}{b}+4 \sinh ^{-1}(c x)}-10 e^{\frac{5 a}{b}+6 \sinh ^{-1}(c x)}-5 e^{\frac{5 a}{b}+8 \sinh ^{-1}(c x)}-e^{\frac{5 a}{b}+10 \sinh ^{-1}(c x)}+10 e^{\frac{6 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)+\sqrt{5} e^{5 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+5 \sqrt{3} e^{\frac{2 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+10 e^{\frac{4 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)+5 \sqrt{3} e^{\frac{8 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+\sqrt{5} e^{5 \left(\frac{2 a}{b}+\sinh ^{-1}(c x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-e^{\frac{5 a}{b}}\right)}{16 b c \sqrt{a+b \sinh ^{-1}(c x)}}","-\frac{5 \sqrt{\pi } d^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{16 b^{3/2} c}-\frac{2 d^2 \left(c^2 x^2+1\right)^{5/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"(d^2*(-E^((5*a)/b) - 5*E^((5*a)/b + 2*ArcSinh[c*x]) - 10*E^((5*a)/b + 4*ArcSinh[c*x]) - 10*E^((5*a)/b + 6*ArcSinh[c*x]) - 5*E^((5*a)/b + 8*ArcSinh[c*x]) - E^((5*a)/b + 10*ArcSinh[c*x]) + 10*E^((6*a)/b + 5*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + Sqrt[5]*E^(5*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcSinh[c*x]))/b] + 5*Sqrt[3]*E^((2*a)/b + 5*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] + 10*E^((4*a)/b + 5*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] + 5*Sqrt[3]*E^((8*a)/b + 5*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b] + Sqrt[5]*E^(5*((2*a)/b + ArcSinh[c*x]))*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (5*(a + b*ArcSinh[c*x]))/b]))/(16*b*c*E^(5*(a/b + ArcSinh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])","A",0
471,0,0,375,3.3090436,"\int \frac{\left(d+c^2 d x^2\right)^2}{x \left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d + c^2*d*x^2)^2/(x*(a + b*ArcSinh[c*x])^(3/2)),x]","\int \frac{\left(d+c^2 d x^2\right)^2}{x \left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","-\frac{2 d^2 \text{Int}\left(\frac{1}{x^2 \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2}}+\frac{\sqrt{2 \pi } d^2 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} d^2 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 b^{3/2}}+\frac{\sqrt{2 \pi } d^2 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} d^2 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 b^{3/2}}-\frac{2 d^2 \left(c^2 x^2+1\right)^{5/2}}{b c x \sqrt{a+b \sinh ^{-1}(c x)}}",0,"Integrate[(d + c^2*d*x^2)^2/(x*(a + b*ArcSinh[c*x])^(3/2)), x]","A",-1
472,1,142,319,0.1937775,"\int \left(c+a^2 c x^2\right)^{3/2} \sqrt{\sinh ^{-1}(a x)} \, dx","Integrate[(c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]],x]","\frac{c \sqrt{a^2 c x^2+c} \left(-\sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{3}{2},-4 \sinh ^{-1}(a x)\right)-8 \sqrt{2} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{3}{2},-2 \sinh ^{-1}(a x)\right)+\sqrt{\sinh ^{-1}(a x)} \left(32 \sinh ^{-1}(a x)^{3/2}-8 \sqrt{2} \Gamma \left(\frac{3}{2},2 \sinh ^{-1}(a x)\right)-\Gamma \left(\frac{3}{2},4 \sinh ^{-1}(a x)\right)\right)\right)}{128 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erf}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{256 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{16 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erfi}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{256 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{16 a \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \sqrt{\sinh ^{-1}(a x)}+\frac{c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt{a^2 x^2+1}}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}",1,"(c*Sqrt[c + a^2*c*x^2]*(-(Sqrt[-ArcSinh[a*x]]*Gamma[3/2, -4*ArcSinh[a*x]]) - 8*Sqrt[2]*Sqrt[-ArcSinh[a*x]]*Gamma[3/2, -2*ArcSinh[a*x]] + Sqrt[ArcSinh[a*x]]*(32*ArcSinh[a*x]^(3/2) - 8*Sqrt[2]*Gamma[3/2, 2*ArcSinh[a*x]] - Gamma[3/2, 4*ArcSinh[a*x]])))/(128*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",0
473,1,104,186,0.1010014,"\int \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)} \, dx","Integrate[Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]],x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(16 \sinh ^{-1}(a x)^2-3 \sqrt{2} \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{3}{2},2 \sinh ^{-1}(a x)\right)-3 \sqrt{2} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{3}{2},-2 \sinh ^{-1}(a x)\right)\right)}{48 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{16 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{16 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}",1,"(Sqrt[c*(1 + a^2*x^2)]*(16*ArcSinh[a*x]^2 - 3*Sqrt[2]*Sqrt[-ArcSinh[a*x]]*Gamma[3/2, -2*ArcSinh[a*x]] - 3*Sqrt[2]*Sqrt[ArcSinh[a*x]]*Gamma[3/2, 2*ArcSinh[a*x]]))/(48*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",0
474,1,42,42,0.037704,"\int \frac{\sqrt{\sinh ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[Sqrt[ArcSinh[a*x]]/Sqrt[c + a^2*c*x^2],x]","\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{a^2 c x^2+c}}","\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{a^2 c x^2+c}}",1,"(2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))/(3*a*Sqrt[c + a^2*c*x^2])","A",1
475,0,0,87,0.6159591,"\int \frac{\sqrt{\sinh ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(3/2),x]","\int \frac{\sqrt{\sinh ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\frac{x \sqrt{\sinh ^{-1}(a x)}}{c \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 x^2+1} \text{Int}\left(\frac{x}{\left(a^2 x^2+1\right) \sqrt{\sinh ^{-1}(a x)}},x\right)}{2 c \sqrt{a^2 c x^2+c}}",0,"Integrate[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(3/2), x]","A",-1
476,0,0,179,1.4172978,"\int \frac{\sqrt{\sinh ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","Integrate[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(5/2),x]","\int \frac{\sqrt{\sinh ^{-1}(a x)}}{\left(c+a^2 c x^2\right)^{5/2}} \, dx","-\frac{a \sqrt{a^2 x^2+1} \text{Int}\left(\frac{x}{\left(a^2 x^2+1\right)^2 \sqrt{\sinh ^{-1}(a x)}},x\right)}{6 c^2 \sqrt{a^2 c x^2+c}}-\frac{a \sqrt{a^2 x^2+1} \text{Int}\left(\frac{x}{\left(a^2 x^2+1\right) \sqrt{\sinh ^{-1}(a x)}},x\right)}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 x \sqrt{\sinh ^{-1}(a x)}}{3 c^2 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{\sinh ^{-1}(a x)}}{3 c \left(a^2 c x^2+c\right)^{3/2}}",0,"Integrate[Sqrt[ArcSinh[a*x]]/(c + a^2*c*x^2)^(5/2), x]","A",-1
477,1,186,449,0.3169485,"\int \left(c+a^2 c x^2\right)^{3/2} \sinh ^{-1}(a x)^{3/2} \, dx","Integrate[(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2),x]","\frac{c \sqrt{a^2 c x^2+c} \left(60 \sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)+60 \sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)+384 \sinh ^{-1}(a x)^3+640 \sinh \left(2 \sinh ^{-1}(a x)\right) \sinh ^{-1}(a x)^2-480 \sinh ^{-1}(a x) \cosh \left(2 \sinh ^{-1}(a x)\right)-5 \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{5}{2},4 \sinh ^{-1}(a x)\right)+5 \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{5}{2},-4 \sinh ^{-1}(a x)\right)\right)}{2560 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","\frac{3 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erf}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{2048 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erfi}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{2048 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{3 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}}{20 a \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \sinh ^{-1}(a x)^{3/2}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}-\frac{3 c \left(a^2 x^2+1\right)^{3/2} \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{32 a}-\frac{9 a c x^2 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{32 \sqrt{a^2 x^2+1}}-\frac{27 c \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{256 a \sqrt{a^2 x^2+1}}",1,"(c*Sqrt[c + a^2*c*x^2]*(384*ArcSinh[a*x]^3 - 480*ArcSinh[a*x]*Cosh[2*ArcSinh[a*x]] + 60*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + 60*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + 5*Sqrt[-ArcSinh[a*x]]*Gamma[5/2, -4*ArcSinh[a*x]] - 5*Sqrt[ArcSinh[a*x]]*Gamma[5/2, 4*ArcSinh[a*x]] + 640*ArcSinh[a*x]^2*Sinh[2*ArcSinh[a*x]]))/(2560*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",0
478,1,126,271,0.2722137,"\int \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2} \, dx","Integrate[Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2),x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(15 \sqrt{2 \pi } \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)+15 \sqrt{2 \pi } \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)+8 \sqrt{\sinh ^{-1}(a x)} \left(4 \sinh ^{-1}(a x) \left(4 \sinh ^{-1}(a x)+5 \sinh \left(2 \sinh ^{-1}(a x)\right)\right)-15 \cosh \left(2 \sinh ^{-1}(a x)\right)\right)\right)}{640 a \sqrt{a^2 x^2+1}}","\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}-\frac{3 a x^2 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{8 \sqrt{a^2 x^2+1}}-\frac{3 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{16 a \sqrt{a^2 x^2+1}}",1,"(Sqrt[c*(1 + a^2*x^2)]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + 15*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + 8*Sqrt[ArcSinh[a*x]]*(-15*Cosh[2*ArcSinh[a*x]] + 4*ArcSinh[a*x]*(4*ArcSinh[a*x] + 5*Sinh[2*ArcSinh[a*x]]))))/(640*a*Sqrt[1 + a^2*x^2])","A",1
479,1,42,42,0.0422328,"\int \frac{\sinh ^{-1}(a x)^{3/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcSinh[a*x]^(3/2)/Sqrt[c + a^2*c*x^2],x]","\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{a^2 c x^2+c}}","\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt{a^2 c x^2+c}}",1,"(2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(5/2))/(5*a*Sqrt[c + a^2*c*x^2])","A",1
480,0,0,87,0.6617088,"\int \frac{\sinh ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcSinh[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2),x]","\int \frac{\sinh ^{-1}(a x)^{3/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\frac{x \sinh ^{-1}(a x)^{3/2}}{c \sqrt{a^2 c x^2+c}}-\frac{3 a \sqrt{a^2 x^2+1} \text{Int}\left(\frac{x \sqrt{\sinh ^{-1}(a x)}}{a^2 x^2+1},x\right)}{2 c \sqrt{a^2 c x^2+c}}",0,"Integrate[ArcSinh[a*x]^(3/2)/(c + a^2*c*x^2)^(3/2), x]","A",-1
481,1,201,514,0.3453934,"\int \left(c+a^2 c x^2\right)^{3/2} \sinh ^{-1}(a x)^{5/2} \, dx","Integrate[(c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(5/2),x]","\frac{c \sqrt{a^2 c x^2+c} \left(420 \sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)-420 \sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)+1536 \sinh ^{-1}(a x)^4+3584 \sinh \left(2 \sinh ^{-1}(a x)\right) \sinh ^{-1}(a x)^3+3360 \sinh \left(2 \sinh ^{-1}(a x)\right) \sinh ^{-1}(a x)-4480 \sinh ^{-1}(a x)^2 \cosh \left(2 \sinh ^{-1}(a x)\right)-7 \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{7}{2},4 \sinh ^{-1}(a x)\right)-7 \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{7}{2},-4 \sinh ^{-1}(a x)\right)\right)}{14336 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","\frac{15 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erf}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{16384 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{256 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erfi}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{16384 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{256 a \sqrt{a^2 x^2+1}}+\frac{3 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{7/2}}{28 a \sqrt{a^2 x^2+1}}+\frac{1}{4} x \left(a^2 c x^2+c\right)^{3/2} \sinh ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}-\frac{5 c \left(a^2 x^2+1\right)^{3/2} \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{32 a}-\frac{15 a c x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{32 \sqrt{a^2 x^2+1}}-\frac{45 c \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{256 a \sqrt{a^2 x^2+1}}+\frac{225}{512} c x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}+\frac{15}{256} c x \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}",1,"(c*Sqrt[c + a^2*c*x^2]*(1536*ArcSinh[a*x]^4 - 4480*ArcSinh[a*x]^2*Cosh[2*ArcSinh[a*x]] + 420*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 420*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 7*Sqrt[-ArcSinh[a*x]]*Gamma[7/2, -4*ArcSinh[a*x]] - 7*Sqrt[ArcSinh[a*x]]*Gamma[7/2, 4*ArcSinh[a*x]] + 3360*ArcSinh[a*x]*Sinh[2*ArcSinh[a*x]] + 3584*ArcSinh[a*x]^3*Sinh[2*ArcSinh[a*x]]))/(14336*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",0
482,1,135,298,0.2934963,"\int \sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2} \, dx","Integrate[Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2),x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(105 \sqrt{2 \pi } \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)-105 \sqrt{2 \pi } \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)+8 \sqrt{\sinh ^{-1}(a x)} \left(64 \sinh ^{-1}(a x)^3+7 \left(16 \sinh ^{-1}(a x)^2+15\right) \sinh \left(2 \sinh ^{-1}(a x)\right)-140 \sinh ^{-1}(a x) \cosh \left(2 \sinh ^{-1}(a x)\right)\right)\right)}{3584 a \sqrt{a^2 x^2+1}}","\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{256 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{256 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}-\frac{5 a x^2 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt{a^2 x^2+1}}-\frac{5 \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt{a^2 x^2+1}}+\frac{15}{32} x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}",1,"(Sqrt[c*(1 + a^2*x^2)]*(105*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 105*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + 8*Sqrt[ArcSinh[a*x]]*(64*ArcSinh[a*x]^3 - 140*ArcSinh[a*x]*Cosh[2*ArcSinh[a*x]] + 7*(15 + 16*ArcSinh[a*x]^2)*Sinh[2*ArcSinh[a*x]])))/(3584*a*Sqrt[1 + a^2*x^2])","A",1
483,1,42,42,0.0372269,"\int \frac{\sinh ^{-1}(a x)^{5/2}}{\sqrt{c+a^2 c x^2}} \, dx","Integrate[ArcSinh[a*x]^(5/2)/Sqrt[c + a^2*c*x^2],x]","\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{a^2 c x^2+c}}","\frac{2 \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt{a^2 c x^2+c}}",1,"(2*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(7/2))/(7*a*Sqrt[c + a^2*c*x^2])","A",1
484,0,0,87,0.6452215,"\int \frac{\sinh ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","Integrate[ArcSinh[a*x]^(5/2)/(c + a^2*c*x^2)^(3/2),x]","\int \frac{\sinh ^{-1}(a x)^{5/2}}{\left(c+a^2 c x^2\right)^{3/2}} \, dx","\frac{x \sinh ^{-1}(a x)^{5/2}}{c \sqrt{a^2 c x^2+c}}-\frac{5 a \sqrt{a^2 x^2+1} \text{Int}\left(\frac{x \sinh ^{-1}(a x)^{3/2}}{a^2 x^2+1},x\right)}{2 c \sqrt{a^2 c x^2+c}}",0,"Integrate[ArcSinh[a*x]^(5/2)/(c + a^2*c*x^2)^(3/2), x]","A",-1
485,1,156,309,0.1770127,"\int \left(a^2+x^2\right)^{3/2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)} \, dx","Integrate[(a^2 + x^2)^(3/2)*Sqrt[ArcSinh[x/a]],x]","\frac{a^3 \sqrt{a^2+x^2} \left(-\sqrt{-\sinh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},-4 \sinh ^{-1}\left(\frac{x}{a}\right)\right)-8 \sqrt{2} \sqrt{-\sinh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},-2 \sinh ^{-1}\left(\frac{x}{a}\right)\right)+\sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)} \left(32 \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}-8 \sqrt{2} \Gamma \left(\frac{3}{2},2 \sinh ^{-1}\left(\frac{x}{a}\right)\right)-\Gamma \left(\frac{3}{2},4 \sinh ^{-1}\left(\frac{x}{a}\right)\right)\right)\right)}{128 \sqrt{\frac{x^2}{a^2}+1} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}","\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}+\frac{1}{4} x \left(a^2+x^2\right)^{3/2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}+\frac{\sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{erf}\left(2 \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{256 \sqrt{\frac{x^2}{a^2}+1}}+\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{16 \sqrt{\frac{x^2}{a^2}+1}}-\frac{\sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{erfi}\left(2 \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{256 \sqrt{\frac{x^2}{a^2}+1}}-\frac{\sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{16 \sqrt{\frac{x^2}{a^2}+1}}+\frac{a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{4 \sqrt{\frac{x^2}{a^2}+1}}",1,"(a^3*Sqrt[a^2 + x^2]*(-(Sqrt[-ArcSinh[x/a]]*Gamma[3/2, -4*ArcSinh[x/a]]) - 8*Sqrt[2]*Sqrt[-ArcSinh[x/a]]*Gamma[3/2, -2*ArcSinh[x/a]] + Sqrt[ArcSinh[x/a]]*(32*ArcSinh[x/a]^(3/2) - 8*Sqrt[2]*Gamma[3/2, 2*ArcSinh[x/a]] - Gamma[3/2, 4*ArcSinh[x/a]])))/(128*Sqrt[1 + x^2/a^2]*Sqrt[ArcSinh[x/a]])","A",0
486,1,110,176,0.0886974,"\int \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)} \, dx","Integrate[Sqrt[a^2 + x^2]*Sqrt[ArcSinh[x/a]],x]","\frac{a \sqrt{a^2+x^2} \left(16 \sinh ^{-1}\left(\frac{x}{a}\right)^2-3 \sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},2 \sinh ^{-1}\left(\frac{x}{a}\right)\right)-3 \sqrt{2} \sqrt{-\sinh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{3}{2},-2 \sinh ^{-1}\left(\frac{x}{a}\right)\right)\right)}{48 \sqrt{\frac{x^2}{a^2}+1} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}","\frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{16 \sqrt{\frac{x^2}{a^2}+1}}-\frac{\sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{16 \sqrt{\frac{x^2}{a^2}+1}}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{3 \sqrt{\frac{x^2}{a^2}+1}}+\frac{1}{2} x \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}",1,"(a*Sqrt[a^2 + x^2]*(16*ArcSinh[x/a]^2 - 3*Sqrt[2]*Sqrt[-ArcSinh[x/a]]*Gamma[3/2, -2*ArcSinh[x/a]] - 3*Sqrt[2]*Sqrt[ArcSinh[x/a]]*Gamma[3/2, 2*ArcSinh[x/a]]))/(48*Sqrt[1 + x^2/a^2]*Sqrt[ArcSinh[x/a]])","A",0
487,1,39,39,0.0273216,"\int \frac{\sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{\sqrt{a^2+x^2}} \, dx","Integrate[Sqrt[ArcSinh[x/a]]/Sqrt[a^2 + x^2],x]","\frac{2 a \sqrt{\frac{x^2}{a^2}+1} \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{3 \sqrt{a^2+x^2}}","\frac{2 a \sqrt{\frac{x^2}{a^2}+1} \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{3 \sqrt{a^2+x^2}}",1,"(2*a*Sqrt[1 + x^2/a^2]*ArcSinh[x/a]^(3/2))/(3*Sqrt[a^2 + x^2])","A",1
488,0,0,84,0.3502928,"\int \frac{\sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2+x^2\right)^{3/2}} \, dx","Integrate[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(3/2),x]","\int \frac{\sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2+x^2\right)^{3/2}} \, dx","\frac{x \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{a^2 \sqrt{a^2+x^2}}-\frac{\sqrt{\frac{x^2}{a^2}+1} \text{Int}\left(\frac{x}{\left(\frac{x^2}{a^2}+1\right) \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}},x\right)}{2 a^3 \sqrt{a^2+x^2}}",0,"Integrate[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(3/2), x]","A",-1
489,0,0,173,0.9253347,"\int \frac{\sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2+x^2\right)^{5/2}} \, dx","Integrate[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(5/2),x]","\int \frac{\sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{\left(a^2+x^2\right)^{5/2}} \, dx","-\frac{\sqrt{\frac{x^2}{a^2}+1} \text{Int}\left(\frac{x}{\left(\frac{x^2}{a^2}+1\right)^2 \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}},x\right)}{6 a^5 \sqrt{a^2+x^2}}-\frac{\sqrt{\frac{x^2}{a^2}+1} \text{Int}\left(\frac{x}{\left(\frac{x^2}{a^2}+1\right) \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}},x\right)}{3 a^5 \sqrt{a^2+x^2}}+\frac{x \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{3 a^2 \left(a^2+x^2\right)^{3/2}}+\frac{2 x \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{3 a^4 \sqrt{a^2+x^2}}",0,"Integrate[Sqrt[ArcSinh[x/a]]/(a^2 + x^2)^(5/2), x]","A",-1
490,1,210,433,0.3007197,"\int \left(a^2+x^2\right)^{3/2} \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2} \, dx","Integrate[(a^2 + x^2)^(3/2)*ArcSinh[x/a]^(3/2),x]","\frac{a^3 \sqrt{a^2+x^2} \left(60 \sqrt{2 \pi } \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)+60 \sqrt{2 \pi } \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)+384 \sinh ^{-1}\left(\frac{x}{a}\right)^3+640 \sinh \left(2 \sinh ^{-1}\left(\frac{x}{a}\right)\right) \sinh ^{-1}\left(\frac{x}{a}\right)^2-480 \sinh ^{-1}\left(\frac{x}{a}\right) \cosh \left(2 \sinh ^{-1}\left(\frac{x}{a}\right)\right)-5 \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{5}{2},4 \sinh ^{-1}\left(\frac{x}{a}\right)\right)+5 \sqrt{-\sinh ^{-1}\left(\frac{x}{a}\right)} \Gamma \left(\frac{5}{2},-4 \sinh ^{-1}\left(\frac{x}{a}\right)\right)\right)}{2560 \sqrt{\frac{x^2}{a^2}+1} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}","\frac{3}{8} a^2 x \sqrt{a^2+x^2} \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}-\frac{9 a x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{32 \sqrt{\frac{x^2}{a^2}+1}}+\frac{1}{4} x \left(a^2+x^2\right)^{3/2} \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}-\frac{3 \left(a^2+x^2\right)^{5/2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{32 a \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{erf}\left(2 \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{2048 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\pi } a^3 \sqrt{a^2+x^2} \text{erfi}\left(2 \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{2048 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a^3 \sqrt{a^2+x^2} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 a^3 \sqrt{a^2+x^2} \sinh ^{-1}\left(\frac{x}{a}\right)^{5/2}}{20 \sqrt{\frac{x^2}{a^2}+1}}-\frac{27 a^3 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{256 \sqrt{\frac{x^2}{a^2}+1}}",1,"(a^3*Sqrt[a^2 + x^2]*(384*ArcSinh[x/a]^3 - 480*ArcSinh[x/a]*Cosh[2*ArcSinh[x/a]] + 60*Sqrt[2*Pi]*Sqrt[ArcSinh[x/a]]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + 60*Sqrt[2*Pi]*Sqrt[ArcSinh[x/a]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + 5*Sqrt[-ArcSinh[x/a]]*Gamma[5/2, -4*ArcSinh[x/a]] - 5*Sqrt[ArcSinh[x/a]]*Gamma[5/2, 4*ArcSinh[x/a]] + 640*ArcSinh[x/a]^2*Sinh[2*ArcSinh[x/a]]))/(2560*Sqrt[1 + x^2/a^2]*Sqrt[ArcSinh[x/a]])","A",0
491,1,133,259,0.2467731,"\int \sqrt{a^2+x^2} \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2} \, dx","Integrate[Sqrt[a^2 + x^2]*ArcSinh[x/a]^(3/2),x]","\frac{a \sqrt{a^2+x^2} \left(15 \sqrt{2 \pi } \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)+15 \sqrt{2 \pi } \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)+8 \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)} \left(16 \sinh ^{-1}\left(\frac{x}{a}\right)^2+20 \sinh \left(2 \sinh ^{-1}\left(\frac{x}{a}\right)\right) \sinh ^{-1}\left(\frac{x}{a}\right)-15 \cosh \left(2 \sinh ^{-1}\left(\frac{x}{a}\right)\right)\right)\right)}{640 \sqrt{\frac{x^2}{a^2}+1}}","\frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{3 \sqrt{\frac{\pi }{2}} a \sqrt{a^2+x^2} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}\right)}{64 \sqrt{\frac{x^2}{a^2}+1}}+\frac{a \sqrt{a^2+x^2} \sinh ^{-1}\left(\frac{x}{a}\right)^{5/2}}{5 \sqrt{\frac{x^2}{a^2}+1}}+\frac{1}{2} x \sqrt{a^2+x^2} \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}-\frac{3 x^2 \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{8 a \sqrt{\frac{x^2}{a^2}+1}}-\frac{3 a \sqrt{a^2+x^2} \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{16 \sqrt{\frac{x^2}{a^2}+1}}",1,"(a*Sqrt[a^2 + x^2]*(15*Sqrt[2*Pi]*Erf[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + 15*Sqrt[2*Pi]*Erfi[Sqrt[2]*Sqrt[ArcSinh[x/a]]] + 8*Sqrt[ArcSinh[x/a]]*(16*ArcSinh[x/a]^2 - 15*Cosh[2*ArcSinh[x/a]] + 20*ArcSinh[x/a]*Sinh[2*ArcSinh[x/a]])))/(640*Sqrt[1 + x^2/a^2])","A",1
492,1,39,39,0.0279687,"\int \frac{\sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{\sqrt{a^2+x^2}} \, dx","Integrate[ArcSinh[x/a]^(3/2)/Sqrt[a^2 + x^2],x]","\frac{2 a \sqrt{\frac{x^2}{a^2}+1} \sinh ^{-1}\left(\frac{x}{a}\right)^{5/2}}{5 \sqrt{a^2+x^2}}","\frac{2 a \sqrt{\frac{x^2}{a^2}+1} \sinh ^{-1}\left(\frac{x}{a}\right)^{5/2}}{5 \sqrt{a^2+x^2}}",1,"(2*a*Sqrt[1 + x^2/a^2]*ArcSinh[x/a]^(5/2))/(5*Sqrt[a^2 + x^2])","A",1
493,0,0,84,0.3793499,"\int \frac{\sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{\left(a^2+x^2\right)^{3/2}} \, dx","Integrate[ArcSinh[x/a]^(3/2)/(a^2 + x^2)^(3/2),x]","\int \frac{\sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{\left(a^2+x^2\right)^{3/2}} \, dx","\frac{x \sinh ^{-1}\left(\frac{x}{a}\right)^{3/2}}{a^2 \sqrt{a^2+x^2}}-\frac{3 \sqrt{\frac{x^2}{a^2}+1} \text{Int}\left(\frac{x \sqrt{\sinh ^{-1}\left(\frac{x}{a}\right)}}{\frac{x^2}{a^2}+1},x\right)}{2 a^3 \sqrt{a^2+x^2}}",0,"Integrate[ArcSinh[x/a]^(3/2)/(a^2 + x^2)^(3/2), x]","A",-1
494,1,34,33,0.064004,"\int \frac{x}{\sqrt{1+x^2} \sqrt{\sinh ^{-1}(x)}} \, dx","Integrate[x/(Sqrt[1 + x^2]*Sqrt[ArcSinh[x]]),x]","\frac{1}{2} \left(\frac{\sqrt{-\sinh ^{-1}(x)} \Gamma \left(\frac{1}{2},-\sinh ^{-1}(x)\right)}{\sqrt{\sinh ^{-1}(x)}}+\Gamma \left(\frac{1}{2},\sinh ^{-1}(x)\right)\right)","\frac{1}{2} \sqrt{\pi } \text{erfi}\left(\sqrt{\sinh ^{-1}(x)}\right)-\frac{1}{2} \sqrt{\pi } \text{erf}\left(\sqrt{\sinh ^{-1}(x)}\right)",1,"((Sqrt[-ArcSinh[x]]*Gamma[1/2, -ArcSinh[x]])/Sqrt[ArcSinh[x]] + Gamma[1/2, ArcSinh[x]])/2","A",0
495,1,197,396,0.3111327,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\sqrt{\sinh ^{-1}(a x)}} \, dx","Integrate[(c + a^2*c*x^2)^(5/2)/Sqrt[ArcSinh[a*x]],x]","\frac{c^2 \sqrt{a^2 c x^2+c} \left(240 \sinh ^{-1}(a x)-45 \sqrt{2} \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 \sinh ^{-1}(a x)\right)-18 \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 \sinh ^{-1}(a x)\right)-\sqrt{6} \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},6 \sinh ^{-1}(a x)\right)+\sqrt{6} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-6 \sinh ^{-1}(a x)\right)+18 \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \sinh ^{-1}(a x)\right)+45 \sqrt{2} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \sinh ^{-1}(a x)\right)\right)}{384 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","\frac{3 \sqrt{\pi } c^2 \sqrt{a^2 c x^2+c} \text{erf}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{6}} c^2 \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\pi } c^2 \sqrt{a^2 c x^2+c} \text{erfi}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{6}} c^2 \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right)}{64 a \sqrt{a^2 x^2+1}}+\frac{5 c^2 \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{8 a \sqrt{a^2 x^2+1}}",1,"(c^2*Sqrt[c + a^2*c*x^2]*(240*ArcSinh[a*x] + Sqrt[6]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -6*ArcSinh[a*x]] + 18*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -4*ArcSinh[a*x]] + 45*Sqrt[2]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -2*ArcSinh[a*x]] - 45*Sqrt[2]*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 2*ArcSinh[a*x]] - 18*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 4*ArcSinh[a*x]] - Sqrt[6]*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 6*ArcSinh[a*x]]))/(384*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",0
496,1,141,264,0.1760661,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\sqrt{\sinh ^{-1}(a x)}} \, dx","Integrate[(c + a^2*c*x^2)^(3/2)/Sqrt[ArcSinh[a*x]],x]","\frac{c \sqrt{a^2 c x^2+c} \left(\sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \sinh ^{-1}(a x)\right)+4 \sqrt{2} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \sinh ^{-1}(a x)\right)+\sqrt{\sinh ^{-1}(a x)} \left(24 \sqrt{\sinh ^{-1}(a x)}-4 \sqrt{2} \Gamma \left(\frac{1}{2},2 \sinh ^{-1}(a x)\right)-\Gamma \left(\frac{1}{2},4 \sinh ^{-1}(a x)\right)\right)\right)}{32 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erf}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{32 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{4 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erfi}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{32 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{4 a \sqrt{a^2 x^2+1}}+\frac{3 c \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{4 a \sqrt{a^2 x^2+1}}",1,"(c*Sqrt[c + a^2*c*x^2]*(Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -4*ArcSinh[a*x]] + 4*Sqrt[2]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -2*ArcSinh[a*x]] + Sqrt[ArcSinh[a*x]]*(24*Sqrt[ArcSinh[a*x]] - 4*Sqrt[2]*Gamma[1/2, 2*ArcSinh[a*x]] - Gamma[1/2, 4*ArcSinh[a*x]])))/(32*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",0
497,1,101,156,0.0982521,"\int \frac{\sqrt{c+a^2 c x^2}}{\sqrt{\sinh ^{-1}(a x)}} \, dx","Integrate[Sqrt[c + a^2*c*x^2]/Sqrt[ArcSinh[a*x]],x]","\frac{\sqrt{c \left(a^2 x^2+1\right)} \left(8 \sinh ^{-1}(a x)-\sqrt{2} \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 \sinh ^{-1}(a x)\right)+\sqrt{2} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \sinh ^{-1}(a x)\right)\right)}{8 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{4 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{4 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}{a \sqrt{a^2 x^2+1}}",1,"(Sqrt[c*(1 + a^2*x^2)]*(8*ArcSinh[a*x] + Sqrt[2]*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -2*ArcSinh[a*x]] - Sqrt[2]*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 2*ArcSinh[a*x]]))/(8*a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",0
498,1,40,40,0.0387354,"\int \frac{1}{\sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}} \, dx","Integrate[1/(Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]]),x]","\frac{2 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{a \sqrt{a^2 c x^2+c}}","\frac{2 \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}{a \sqrt{a^2 c x^2+c}}",1,"(2*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])/(a*Sqrt[c + a^2*c*x^2])","A",1
499,0,0,26,0.7380171,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\sinh ^{-1}(a x)}} \, dx","Integrate[1/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]]),x]","\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \sqrt{\sinh ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{\left(a^2 c x^2+c\right)^{3/2} \sqrt{\sinh ^{-1}(a x)}},x\right)",0,"Integrate[1/((c + a^2*c*x^2)^(3/2)*Sqrt[ArcSinh[a*x]]), x]","A",-1
500,0,0,26,1.6644892,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\sinh ^{-1}(a x)}} \, dx","Integrate[1/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcSinh[a*x]]),x]","\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \sqrt{\sinh ^{-1}(a x)}} \, dx","\text{Int}\left(\frac{1}{\left(a^2 c x^2+c\right)^{5/2} \sqrt{\sinh ^{-1}(a x)}},x\right)",0,"Integrate[1/((c + a^2*c*x^2)^(5/2)*Sqrt[ArcSinh[a*x]]), x]","A",-1
501,1,399,391,1.0655642,"\int \frac{\left(c+a^2 c x^2\right)^{5/2}}{\sinh ^{-1}(a x)^{3/2}} \, dx","Integrate[(c + a^2*c*x^2)^(5/2)/ArcSinh[a*x]^(3/2),x]","\frac{c^2 \sqrt{a^2 c x^2+c} e^{-6 \sinh ^{-1}(a x)} \left(-64 a^2 x^2 e^{6 \sinh ^{-1}(a x)}-16 \sqrt{2 \pi } e^{6 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)+16 \sqrt{2 \pi } e^{6 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)-6 e^{2 \sinh ^{-1}(a x)}+e^{4 \sinh ^{-1}(a x)}-52 e^{6 \sinh ^{-1}(a x)}+e^{8 \sinh ^{-1}(a x)}-6 e^{10 \sinh ^{-1}(a x)}-e^{12 \sinh ^{-1}(a x)}+\sqrt{6} e^{6 \sinh ^{-1}(a x)} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-6 \sinh ^{-1}(a x)\right)+12 e^{6 \sinh ^{-1}(a x)} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \sinh ^{-1}(a x)\right)-\sqrt{2} e^{6 \sinh ^{-1}(a x)} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-2 \sinh ^{-1}(a x)\right)-\sqrt{2} e^{6 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},2 \sinh ^{-1}(a x)\right)+12 e^{6 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 \sinh ^{-1}(a x)\right)+\sqrt{6} e^{6 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},6 \sinh ^{-1}(a x)\right)-1\right)}{32 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","-\frac{3 \sqrt{\pi } c^2 \sqrt{a^2 c x^2+c} \text{erf}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{8 a \sqrt{a^2 x^2+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{16 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{3 \pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right)}{16 a \sqrt{a^2 x^2+1}}+\frac{3 \sqrt{\pi } c^2 \sqrt{a^2 c x^2+c} \text{erfi}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{8 a \sqrt{a^2 x^2+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{16 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{3 \pi }{2}} c^2 \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{6} \sqrt{\sinh ^{-1}(a x)}\right)}{16 a \sqrt{a^2 x^2+1}}-\frac{2 \sqrt{a^2 x^2+1} \left(a^2 c x^2+c\right)^{5/2}}{a \sqrt{\sinh ^{-1}(a x)}}",1,"(c^2*Sqrt[c + a^2*c*x^2]*(-1 - 6*E^(2*ArcSinh[a*x]) + E^(4*ArcSinh[a*x]) - 52*E^(6*ArcSinh[a*x]) + E^(8*ArcSinh[a*x]) - 6*E^(10*ArcSinh[a*x]) - E^(12*ArcSinh[a*x]) - 64*a^2*E^(6*ArcSinh[a*x])*x^2 - 16*E^(6*ArcSinh[a*x])*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + 16*E^(6*ArcSinh[a*x])*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] + Sqrt[6]*E^(6*ArcSinh[a*x])*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -6*ArcSinh[a*x]] + 12*E^(6*ArcSinh[a*x])*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -4*ArcSinh[a*x]] - Sqrt[2]*E^(6*ArcSinh[a*x])*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -2*ArcSinh[a*x]] - Sqrt[2]*E^(6*ArcSinh[a*x])*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 2*ArcSinh[a*x]] + 12*E^(6*ArcSinh[a*x])*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 4*ArcSinh[a*x]] + Sqrt[6]*E^(6*ArcSinh[a*x])*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 6*ArcSinh[a*x]]))/(32*a*E^(6*ArcSinh[a*x])*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",0
502,1,225,256,0.3734245,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\sinh ^{-1}(a x)^{3/2}} \, dx","Integrate[(c + a^2*c*x^2)^(3/2)/ArcSinh[a*x]^(3/2),x]","-\frac{c \sqrt{a^2 c x^2+c} e^{-4 \sinh ^{-1}(a x)} \left(16 a^2 x^2 e^{4 \sinh ^{-1}(a x)}+4 \sqrt{2 \pi } e^{4 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)-4 \sqrt{2 \pi } e^{4 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)+14 e^{4 \sinh ^{-1}(a x)}+e^{8 \sinh ^{-1}(a x)}-2 e^{4 \sinh ^{-1}(a x)} \sqrt{-\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},-4 \sinh ^{-1}(a x)\right)-2 e^{4 \sinh ^{-1}(a x)} \sqrt{\sinh ^{-1}(a x)} \Gamma \left(\frac{1}{2},4 \sinh ^{-1}(a x)\right)+1\right)}{8 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","-\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erf}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{4 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erfi}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{4 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} c \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{a \sqrt{a^2 x^2+1}}-\frac{2 \sqrt{a^2 x^2+1} \left(a^2 c x^2+c\right)^{3/2}}{a \sqrt{\sinh ^{-1}(a x)}}",1,"-1/8*(c*Sqrt[c + a^2*c*x^2]*(1 + 14*E^(4*ArcSinh[a*x]) + E^(8*ArcSinh[a*x]) + 16*a^2*E^(4*ArcSinh[a*x])*x^2 + 4*E^(4*ArcSinh[a*x])*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 4*E^(4*ArcSinh[a*x])*Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - 2*E^(4*ArcSinh[a*x])*Sqrt[-ArcSinh[a*x]]*Gamma[1/2, -4*ArcSinh[a*x]] - 2*E^(4*ArcSinh[a*x])*Sqrt[ArcSinh[a*x]]*Gamma[1/2, 4*ArcSinh[a*x]]))/(a*E^(4*ArcSinh[a*x])*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",0
503,1,115,152,0.1877051,"\int \frac{\sqrt{c+a^2 c x^2}}{\sinh ^{-1}(a x)^{3/2}} \, dx","Integrate[Sqrt[c + a^2*c*x^2]/ArcSinh[a*x]^(3/2),x]","-\frac{\sqrt{a^2 c x^2+c} \left(4 a^2 x^2+\sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)-\sqrt{2 \pi } \sqrt{\sinh ^{-1}(a x)} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)+4\right)}{2 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{a \sqrt{a^2 x^2+1}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{a \sqrt{a^2 x^2+1}}-\frac{2 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}{a \sqrt{\sinh ^{-1}(a x)}}",1,"-1/2*(Sqrt[c + a^2*c*x^2]*(4 + 4*a^2*x^2 + Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcSinh[a*x]]] - Sqrt[2*Pi]*Sqrt[ArcSinh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcSinh[a*x]]]))/(a*Sqrt[1 + a^2*x^2]*Sqrt[ArcSinh[a*x]])","A",1
504,1,40,40,0.0358741,"\int \frac{1}{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}} \, dx","Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2)),x]","-\frac{2 \sqrt{a^2 x^2+1}}{a \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}","-\frac{2 \sqrt{a^2 x^2+1}}{a \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)}}",1,"(-2*Sqrt[1 + a^2*x^2])/(a*Sqrt[c + a^2*c*x^2]*Sqrt[ArcSinh[a*x]])","A",1
505,0,0,98,0.7412621,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \sinh ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2)),x]","\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \sinh ^{-1}(a x)^{3/2}} \, dx","-\frac{4 a \sqrt{a^2 x^2+1} \text{Int}\left(\frac{x}{\left(a^2 x^2+1\right)^2 \sqrt{\sinh ^{-1}(a x)}},x\right)}{c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{a \left(a^2 c x^2+c\right)^{3/2} \sqrt{\sinh ^{-1}(a x)}}",0,"Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(3/2)), x]","A",-1
506,0,0,98,1.5444406,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \sinh ^{-1}(a x)^{3/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^(3/2)),x]","\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \sinh ^{-1}(a x)^{3/2}} \, dx","-\frac{8 a \sqrt{a^2 x^2+1} \text{Int}\left(\frac{x}{\left(a^2 x^2+1\right)^3 \sqrt{\sinh ^{-1}(a x)}},x\right)}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{a \left(a^2 c x^2+c\right)^{5/2} \sqrt{\sinh ^{-1}(a x)}}",0,"Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^(3/2)), x]","A",-1
507,1,262,296,0.4021235,"\int \frac{\left(c+a^2 c x^2\right)^{3/2}}{\sinh ^{-1}(a x)^{5/2}} \, dx","Integrate[(c + a^2*c*x^2)^(3/2)/ArcSinh[a*x]^(5/2),x]","-\frac{c \sqrt{a^2 c x^2+c} e^{-4 \sinh ^{-1}(a x)} \left(16 a^2 x^2 e^{4 \sinh ^{-1}(a x)}+64 a x \sqrt{a^2 x^2+1} e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)+14 e^{4 \sinh ^{-1}(a x)}+e^{8 \sinh ^{-1}(a x)}+8 e^{8 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)-8 \sinh ^{-1}(a x)+16 e^{4 \sinh ^{-1}(a x)} \left(-\sinh ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-4 \sinh ^{-1}(a x)\right)+16 \sqrt{2} e^{4 \sinh ^{-1}(a x)} \left(-\sinh ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 \sinh ^{-1}(a x)\right)+16 \sqrt{2} e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \Gamma \left(\frac{1}{2},2 \sinh ^{-1}(a x)\right)+16 e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \Gamma \left(\frac{1}{2},4 \sinh ^{-1}(a x)\right)+1\right)}{24 a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}","\frac{2 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erf}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{\pi } c \sqrt{a^2 c x^2+c} \text{erfi}\left(2 \sqrt{\sinh ^{-1}(a x)}\right)}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{2 \pi } c \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{3 a \sqrt{a^2 x^2+1}}-\frac{2 \sqrt{a^2 x^2+1} \left(a^2 c x^2+c\right)^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac{16 c x \left(a^2 x^2+1\right) \sqrt{a^2 c x^2+c}}{3 \sqrt{\sinh ^{-1}(a x)}}",1,"-1/24*(c*Sqrt[c + a^2*c*x^2]*(1 + 14*E^(4*ArcSinh[a*x]) + E^(8*ArcSinh[a*x]) + 16*a^2*E^(4*ArcSinh[a*x])*x^2 - 8*ArcSinh[a*x] + 8*E^(8*ArcSinh[a*x])*ArcSinh[a*x] + 64*a*E^(4*ArcSinh[a*x])*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + 16*E^(4*ArcSinh[a*x])*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -4*ArcSinh[a*x]] + 16*Sqrt[2]*E^(4*ArcSinh[a*x])*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -2*ArcSinh[a*x]] + 16*Sqrt[2]*E^(4*ArcSinh[a*x])*ArcSinh[a*x]^(3/2)*Gamma[1/2, 2*ArcSinh[a*x]] + 16*E^(4*ArcSinh[a*x])*ArcSinh[a*x]^(3/2)*Gamma[1/2, 4*ArcSinh[a*x]]))/(a*E^(4*ArcSinh[a*x])*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))","A",0
508,1,122,182,0.1547245,"\int \frac{\sqrt{c+a^2 c x^2}}{\sinh ^{-1}(a x)^{5/2}} \, dx","Integrate[Sqrt[c + a^2*c*x^2]/ArcSinh[a*x]^(5/2),x]","-\frac{2 \sqrt{a^2 c x^2+c} \left(a^2 x^2+4 a x \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)+\sqrt{2} \left(-\sinh ^{-1}(a x)\right)^{3/2} \Gamma \left(\frac{1}{2},-2 \sinh ^{-1}(a x)\right)+\sqrt{2} \sinh ^{-1}(a x)^{3/2} \Gamma \left(\frac{1}{2},2 \sinh ^{-1}(a x)\right)+1\right)}{3 a \sqrt{a^2 x^2+1} \sinh ^{-1}(a x)^{3/2}}","\frac{2 \sqrt{2 \pi } \sqrt{a^2 c x^2+c} \text{erf}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{3 a \sqrt{a^2 x^2+1}}+\frac{2 \sqrt{2 \pi } \sqrt{a^2 c x^2+c} \text{erfi}\left(\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right)}{3 a \sqrt{a^2 x^2+1}}-\frac{8 x \sqrt{a^2 c x^2+c}}{3 \sqrt{\sinh ^{-1}(a x)}}-\frac{2 \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}{3 a \sinh ^{-1}(a x)^{3/2}}",1,"(-2*Sqrt[c + a^2*c*x^2]*(1 + a^2*x^2 + 4*a*x*Sqrt[1 + a^2*x^2]*ArcSinh[a*x] + Sqrt[2]*(-ArcSinh[a*x])^(3/2)*Gamma[1/2, -2*ArcSinh[a*x]] + Sqrt[2]*ArcSinh[a*x]^(3/2)*Gamma[1/2, 2*ArcSinh[a*x]]))/(3*a*Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^(3/2))","A",0
509,1,42,42,0.0331156,"\int \frac{1}{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}} \, dx","Integrate[1/(Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(5/2)),x]","-\frac{2 \sqrt{a^2 x^2+1}}{3 a \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}","-\frac{2 \sqrt{a^2 x^2+1}}{3 a \sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}",1,"(-2*Sqrt[1 + a^2*x^2])/(3*a*Sqrt[c + a^2*c*x^2]*ArcSinh[a*x]^(3/2))","A",1
510,0,0,102,0.7291823,"\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \sinh ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(5/2)),x]","\int \frac{1}{\left(c+a^2 c x^2\right)^{3/2} \sinh ^{-1}(a x)^{5/2}} \, dx","-\frac{4 a \sqrt{a^2 x^2+1} \text{Int}\left(\frac{x}{\left(a^2 x^2+1\right)^2 \sinh ^{-1}(a x)^{3/2}},x\right)}{3 c \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{3 a \left(a^2 c x^2+c\right)^{3/2} \sinh ^{-1}(a x)^{3/2}}",0,"Integrate[1/((c + a^2*c*x^2)^(3/2)*ArcSinh[a*x]^(5/2)), x]","A",-1
511,0,0,102,1.5515999,"\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \sinh ^{-1}(a x)^{5/2}} \, dx","Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^(5/2)),x]","\int \frac{1}{\left(c+a^2 c x^2\right)^{5/2} \sinh ^{-1}(a x)^{5/2}} \, dx","-\frac{8 a \sqrt{a^2 x^2+1} \text{Int}\left(\frac{x}{\left(a^2 x^2+1\right)^3 \sinh ^{-1}(a x)^{3/2}},x\right)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{3 a \left(a^2 c x^2+c\right)^{5/2} \sinh ^{-1}(a x)^{3/2}}",0,"Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcSinh[a*x]^(5/2)), x]","A",-1
512,1,170,235,0.9936562,"\int x^2 \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^n \, dx","Integrate[x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n,x]","\frac{d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(4^{-n} e^{-\frac{4 a}{b}} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-e^{\frac{8 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)-\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b n+b}\right)}{64 c^3 \sqrt{d \left(c^2 x^2+1\right)}}","-\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^{n+1}}{8 b c^3 (n+1) \sqrt{c^2 x^2+1}}+\frac{2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}-\frac{2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}",1,"(d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*((-8*(a + b*ArcSinh[c*x]))/(b + b*n) + ((a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b] - E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(4^n*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^n)))/(64*c^3*Sqrt[d*(1 + c^2*x^2)])","A",1
513,1,229,355,0.9498989,"\int x \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^n \, dx","Integrate[x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n,x]","\frac{d e^{-\frac{3 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \left(3^{-n} e^{\frac{6 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{2 n} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+3^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+3 e^{\frac{2 a}{b}} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right)\right)+3 e^{\frac{4 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^{-n} \Gamma \left(n+1,\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{24 c^2 \sqrt{d \left(c^2 x^2+1\right)}}","\frac{3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 c^2 \sqrt{c^2 x^2+1}}+\frac{e^{-\frac{a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 c^2 \sqrt{c^2 x^2+1}}+\frac{e^{a/b} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 c^2 \sqrt{c^2 x^2+1}}+\frac{3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 c^2 \sqrt{c^2 x^2+1}}",1,"(d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*((3*E^((4*a)/b)*Gamma[1 + n, a/b + ArcSinh[c*x]])/(a/b + ArcSinh[c*x])^n + (Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b]/3^n + 3*E^((2*a)/b)*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)] + (E^((6*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b])/(3^n*(-((a + b*ArcSinh[c*x])^2/b^2))^n))/(-((a + b*ArcSinh[c*x])/b))^n))/(24*c^2*E^((3*a)/b)*Sqrt[d*(1 + c^2*x^2)])","A",1
514,1,160,235,0.6048149,"\int \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^n \, dx","Integrate[Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n,x]","\frac{d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{4 a+4 b \sinh ^{-1}(c x)}{b n+b}-2^{-n} e^{\frac{2 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+2^{-n} e^{-\frac{2 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{8 c \sqrt{d \left(c^2 x^2+1\right)}}","\frac{\sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^{n+1}}{2 b c (n+1) \sqrt{c^2 x^2+1}}+\frac{2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}-\frac{2^{-n-3} e^{\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}",1,"(d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*((4*a + 4*b*ArcSinh[c*x])/(b + b*n) + Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b]/(2^n*E^((2*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n) - (E^((2*a)/b)*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(2^n*(a/b + ArcSinh[c*x])^n)))/(8*c*Sqrt[d*(1 + c^2*x^2)])","A",1
515,0,0,199,0.2197979,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x,x]","\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x} \, dx","d \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^n}{x \sqrt{c^2 d x^2+d}},x\right)+\frac{d e^{-\frac{a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{2 \sqrt{c^2 d x^2+d}}+\frac{d e^{a/b} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{2 \sqrt{c^2 d x^2+d}}",0,"Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x, x]","A",-1
516,0,0,84,0.2209967,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x^2} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x^2,x]","\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x^2} \, dx","d \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^n}{x^2 \sqrt{c^2 d x^2+d}},x\right)+\frac{c d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^{n+1}}{b (n+1) \sqrt{c^2 d x^2+d}}",0,"Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^n)/x^2, x]","A",-1
517,1,429,616,3.1912225,"\int x^2 \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^n \, dx","Integrate[x^2*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n,x]","-\frac{d^2 2^{-2 n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \sinh ^{-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 \sinh ^{-1}(c x)\right) \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n+b (n+1) e^{\frac{6 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)+b \left(-2^n\right) (n+1) \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-b 3^{n+1} (n+1) e^{\frac{2 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+b 2^n 3^{n+1} (n+1) e^{\frac{4 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-b 2^n 3^{n+1} (n+1) e^{\frac{8 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+b 3^{n+1} (n+1) e^{\frac{10 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{b c^3 (n+1) \sqrt{c^2 d x^2+d}}","-\frac{d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^{n+1}}{16 b c^3 (n+1) \sqrt{c^2 x^2+1}}+\frac{d 2^{-n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}+\frac{d 2^{-2 n-7} e^{-\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}-\frac{d 2^{-n-7} e^{-\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}+\frac{d 2^{-n-7} e^{\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}-\frac{d 2^{-2 n-7} e^{\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}-\frac{d 2^{-n-7} 3^{-n-1} e^{\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}",1,"-((2^(-7 - 2*n)*3^(-1 - n)*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(-(2^n*b*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b]) - 3^(1 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b] + 2^n*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b] - 2^n*3^(1 + n)*b*E^((8*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((10*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b] + 2^n*E^((6*a)/b)*(2^(3 + n)*3^(1 + n)*(a + b*ArcSinh[c*x])*(-((a + b*ArcSinh[c*x])^2/b^2))^n + b*E^((6*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b])))/(b*c^3*E^((6*a)/b)*(1 + n)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])^2/b^2))^n))","A",1
518,1,390,542,1.7912163,"\int x \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^n \, dx","Integrate[x*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n,x]","\frac{d^2 15^{-n-1} e^{-\frac{5 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{-2 n} \left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \left(3^n \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+5^{n+1} e^{\frac{2 a}{b}} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+2\ 3^n 5^{n+1} e^{\frac{4 a}{b}} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right)+5^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{2 n} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+3^n e^{\frac{10 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{2 n} \Gamma \left(n+1,\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)+2\ 15^{n+1} e^{\frac{6 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{32 c^2 \sqrt{c^2 d x^2+d}}","\frac{d 5^{-n-1} e^{-\frac{5 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{c^2 x^2+1}}+\frac{d 3^{-n} e^{-\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{c^2 x^2+1}}+\frac{d e^{-\frac{a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{16 c^2 \sqrt{c^2 x^2+1}}+\frac{d e^{a/b} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{16 c^2 \sqrt{c^2 x^2+1}}+\frac{d 3^{-n} e^{\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{c^2 x^2+1}}+\frac{d 5^{-n-1} e^{\frac{5 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 c^2 \sqrt{c^2 x^2+1}}",1,"(15^(-1 - n)*d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(2*15^(1 + n)*E^((6*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, a/b + ArcSinh[c*x]] + 3*(a/b + ArcSinh[c*x])^n*(3^n*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b] + 5^(1 + n)*E^((2*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b] + 2*3^n*5^(1 + n)*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)] + 5^(1 + n)*E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b] + 3^n*E^((10*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b])))/(32*c^2*E^((5*a)/b)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n))","A",1
519,1,287,420,1.4307347,"\int \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^n \, dx","Integrate[(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n,x]","\frac{d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(4^{-n} e^{-\frac{4 a}{b}} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{-n} \left(\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-e^{\frac{8 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)-\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b n+b}+8 \left(\frac{4 a+4 b \sinh ^{-1}(c x)}{b n+b}-2^{-n} e^{\frac{2 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+2^{-n} e^{-\frac{2 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)\right)}{64 c \sqrt{c^2 d x^2+d}}","\frac{3 d \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^{n+1}}{8 b c (n+1) \sqrt{c^2 x^2+1}}+\frac{d 2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}+\frac{d 2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}-\frac{d 2^{-n-3} e^{\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}-\frac{d 2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}",1,"(d^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*((-8*(a + b*ArcSinh[c*x]))/(b + b*n) + 8*((4*a + 4*b*ArcSinh[c*x])/(b + b*n) + Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b]/(2^n*E^((2*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n) - (E^((2*a)/b)*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b])/(2^n*(a/b + ArcSinh[c*x])^n)) + ((a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b] - E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b])/(4^n*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^n)))/(64*c*Sqrt[d + c^2*d*x^2])","A",1
520,0,0,390,0.2462043,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n)/x,x]","\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x} \, dx","d^2 \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^n}{x \sqrt{c^2 d x^2+d}},x\right)+\frac{d^2 3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{c^2 d x^2+d}}+\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 \sqrt{c^2 d x^2+d}}+\frac{5 d^2 e^{a/b} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 \sqrt{c^2 d x^2+d}}+\frac{d^2 3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{c^2 d x^2+d}}",0,"Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n)/x, x]","A",-1
521,0,0,273,0.6418805,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n)/x^2,x]","\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x^2} \, dx","d^2 \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^n}{x^2 \sqrt{c^2 d x^2+d}},x\right)+\frac{3 c d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^{n+1}}{2 b (n+1) \sqrt{c^2 d x^2+d}}+\frac{c d^2 2^{-n-3} e^{-\frac{2 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{\sqrt{c^2 d x^2+d}}-\frac{c d^2 2^{-n-3} e^{\frac{2 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{\sqrt{c^2 d x^2+d}}",0,"Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^n)/x^2, x]","A",-1
522,1,667,816,6.9436363,"\int x^2 \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^n \, dx","Integrate[x^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n,x]","-\frac{d^3 2^{-3 n-11} 3^{-n-1} e^{-\frac{8 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \sinh ^{-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 \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n+5 b 2^{3 n+4} 3^{n+1} \sinh ^{-1}(c x) \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n+b \left(-3^{n+1}\right) 4^{n+2} (n+1) e^{\frac{2 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+b 2^{n+3} 3^{n+1} (n+1) e^{\frac{4 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+b 4^{n+2} e^{\frac{6 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+b 4^{n+2} n e^{\frac{6 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+b 3^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+b 3^{n+1} n e^{\frac{8 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)+b \left(-3^{n+1}\right) (n+1) \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-b 4^{n+2} (n+1) e^{\frac{2 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-b 2^{n+3} 3^{n+1} (n+1) e^{\frac{4 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+b 3^{n+1} 4^{n+2} (n+1) e^{\frac{6 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{b c^3 (n+1) \sqrt{c^2 d x^2+d}}","\frac{2^{-3 n-11} d^2 e^{-\frac{8 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n}}{c^3 \sqrt{c^2 x^2+1}}+\frac{2^{-n-7} 3^{-n-1} d^2 e^{-\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n}}{c^3 \sqrt{c^2 x^2+1}}+\frac{2^{-2 (n+4)} d^2 e^{-\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n}}{c^3 \sqrt{c^2 x^2+1}}-\frac{2^{-n-7} d^2 e^{-\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right) \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n}}{c^3 \sqrt{c^2 x^2+1}}-\frac{5 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^{n+1}}{128 b c^3 (n+1) \sqrt{c^2 x^2+1}}+\frac{2^{-n-7} d^2 e^{\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}-\frac{2^{-2 (n+4)} d^2 e^{\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}-\frac{2^{-n-7} 3^{-n-1} d^2 e^{\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}-\frac{2^{-3 n-11} d^2 e^{\frac{8 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{8 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c^3 \sqrt{c^2 x^2+1}}",1,"-((2^(-11 - 3*n)*3^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(-(3^(1 + n)*b*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-8*(a + b*ArcSinh[c*x]))/b]) - 4^(2 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b] - 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*4^(2 + n)*b*E^((6*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b] + E^((8*a)/b)*(5*2^(4 + 3*n)*3^(1 + n)*a*(-((a + b*ArcSinh[c*x])^2/b^2))^n + 5*2^(4 + 3*n)*3^(1 + n)*b*ArcSinh[c*x]*(-((a + b*ArcSinh[c*x])^2/b^2))^n - 3^(1 + n)*4^(2 + n)*b*E^((2*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b] + 2^(3 + n)*3^(1 + n)*b*E^((4*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b] + 4^(2 + n)*b*E^((6*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b] + 4^(2 + n)*b*E^((6*a)/b)*n*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b] + 3^(1 + n)*b*E^((8*a)/b)*n*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (8*(a + b*ArcSinh[c*x]))/b])))/(b*c^3*E^((8*a)/b)*(1 + n)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])^2/b^2))^n))","A",1
523,1,685,745,2.9559844,"\int x \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^n \, dx","Integrate[x*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n,x]","\frac{d^3 105^{-n-1} e^{-\frac{7 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{-2 n} \left(5^{n+2} 21^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{2 n} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{a}{b}+\sinh ^{-1}(c x)\right)+15^{n+1} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+e^{\frac{2 a}{b}} \left(5\ 21^{n+1} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+9\ 35^{n+1} e^{\frac{2 a}{b}} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+5^{n+2} 21^{n+1} e^{\frac{4 a}{b}} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right)+8\ 35^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{2 n} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+8\ 21^{n+1} e^{\frac{10 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{2 n} \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+35^{n+1} e^{\frac{8 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{3 n} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-3^{n+2} 7^{n+1} e^{\frac{10 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{3 n} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+15^{n+1} e^{\frac{12 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{3 n} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)\right)}{128 c^2 \sqrt{c^2 d x^2+d}}","\frac{d^2 7^{-n-1} e^{-\frac{7 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 5^{-n} e^{-\frac{5 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 3^{1-n} e^{-\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{5 d^2 e^{-\frac{a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{5 d^2 e^{a/b} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 3^{1-n} e^{\frac{3 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 5^{-n} e^{\frac{5 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c^2 x^2+1}}+\frac{d^2 7^{-n-1} e^{\frac{7 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{128 c^2 \sqrt{c^2 x^2+1}}",1,"(105^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(5^(2 + n)*21^(1 + n)*E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, a/b + ArcSinh[c*x]] + 15^(1 + n)*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-7*(a + b*ArcSinh[c*x]))/b] + E^((2*a)/b)*(5*21^(1 + n)*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-5*(a + b*ArcSinh[c*x]))/b] + 9*35^(1 + n)*E^((2*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, (-3*(a + b*ArcSinh[c*x]))/b] + 5^(2 + n)*21^(1 + n)*E^((4*a)/b)*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n)*Gamma[1 + n, -((a + b*ArcSinh[c*x])/b)] + 35^(1 + n)*E^((8*a)/b)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh[c*x])/b))^(3*n)*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b] + 8*35^(1 + n)*E^((8*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (3*(a + b*ArcSinh[c*x]))/b] - 3^(2 + n)*7^(1 + n)*E^((10*a)/b)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh[c*x])/b))^(3*n)*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b] + 8*21^(1 + n)*E^((10*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(2*n)*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (5*(a + b*ArcSinh[c*x]))/b] + 15^(1 + n)*E^((12*a)/b)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh[c*x])/b))^(3*n)*Gamma[1 + n, (7*(a + b*ArcSinh[c*x]))/b])))/(128*c^2*E^((7*a)/b)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])/b))^n*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n))","A",1
524,1,529,632,5.3676261,"\int \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^n \, dx","Integrate[(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n,x]","\frac{d^3 2^{-2 n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \sinh ^{-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{a}{b}+\sinh ^{-1}(c x)\right)^n \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-e^{\frac{6 a}{b}} \left(5 b 2^n 3^{n+2} (n+1) e^{\frac{2 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+2^n \left(b (n+1) e^{\frac{6 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{2 n} \Gamma \left(n+1,\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-5\ 2^{n+3} 3^{n+1} \left(a+b \sinh ^{-1}(c x)\right) \left(-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}\right)^{2 n}\right)+b 3^{n+2} (n+1) e^{\frac{4 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{2 n} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^n \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)+b 2^n (n+1) \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^{2 n} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+b 3^{n+2} (n+1) e^{\frac{2 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)^{2 n} \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^n \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{b c (n+1) \sqrt{c^2 d x^2+d}}","\frac{5 d^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^{n+1}}{16 b c (n+1) \sqrt{c^2 x^2+1}}+\frac{d^2 2^{-n-7} 3^{-n-1} e^{-\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}+\frac{3 d^2 2^{-2 n-7} e^{-\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}+\frac{15 d^2 2^{-n-7} e^{-\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}-\frac{15 d^2 2^{-n-7} e^{\frac{2 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}-\frac{3 d^2 2^{-2 n-7} e^{\frac{4 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}-\frac{d^2 2^{-n-7} 3^{-n-1} e^{\frac{6 a}{b}} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{6 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{c \sqrt{c^2 x^2+1}}",1,"(2^(-7 - 2*n)*3^(-1 - n)*d^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^n*(2^n*b*(1 + n)*(a/b + ArcSinh[c*x])^(2*n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (-6*(a + b*ArcSinh[c*x]))/b] + 3^(2 + n)*b*E^((2*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^(2*n)*(-((a + b*ArcSinh[c*x])/b))^n*Gamma[1 + n, (-4*(a + b*ArcSinh[c*x]))/b] + 5*2^n*3^(2 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c*x]))/b] - E^((6*a)/b)*(5*2^n*3^(2 + n)*b*E^((2*a)/b)*(1 + n)*(-((a + b*ArcSinh[c*x])/b))^n*(-((a + b*ArcSinh[c*x])^2/b^2))^n*Gamma[1 + n, (2*(a + b*ArcSinh[c*x]))/b] + 3^(2 + n)*b*E^((4*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh[c*x])/b))^(2*n)*Gamma[1 + n, (4*(a + b*ArcSinh[c*x]))/b] + 2^n*(-5*2^(3 + n)*3^(1 + n)*(a + b*ArcSinh[c*x])*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n) + b*E^((6*a)/b)*(1 + n)*(a/b + ArcSinh[c*x])^n*(-((a + b*ArcSinh[c*x])/b))^(2*n)*Gamma[1 + n, (6*(a + b*ArcSinh[c*x]))/b]))))/(b*c*E^((6*a)/b)*(1 + n)*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])^2/b^2))^(2*n))","A",1
525,0,0,756,0.2712706,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x,x]","\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x} \, dx","d^3 \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^n}{x \sqrt{c^2 d x^2+d}},x\right)+\frac{d^3 5^{-n-1} e^{-\frac{5 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 \sqrt{c^2 d x^2+d}}-\frac{5 d^3 3^{-n-1} e^{-\frac{3 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 \sqrt{c^2 d x^2+d}}+\frac{d^3 3^{-n} e^{-\frac{3 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{c^2 d x^2+d}}+\frac{11 d^3 e^{-\frac{a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{16 \sqrt{c^2 d x^2+d}}+\frac{11 d^3 e^{a/b} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{16 \sqrt{c^2 d x^2+d}}-\frac{5 d^3 3^{-n-1} e^{\frac{3 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 \sqrt{c^2 d x^2+d}}+\frac{d^3 3^{-n} e^{\frac{3 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{8 \sqrt{c^2 d x^2+d}}+\frac{d^3 5^{-n-1} e^{\frac{5 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{32 \sqrt{c^2 d x^2+d}}",0,"Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x, x]","A",-1
526,0,0,455,0.6509762,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x^2} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x^2,x]","\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^n}{x^2} \, dx","d^3 \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^n}{x^2 \sqrt{c^2 d x^2+d}},x\right)+\frac{15 c d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^{n+1}}{8 b (n+1) \sqrt{c^2 d x^2+d}}+\frac{c d^3 2^{-2 (n+3)} e^{-\frac{4 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{\sqrt{c^2 d x^2+d}}+\frac{c d^3 2^{-n-2} e^{-\frac{2 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{\sqrt{c^2 d x^2+d}}-\frac{c d^3 2^{-n-2} e^{\frac{2 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{\sqrt{c^2 d x^2+d}}-\frac{c d^3 2^{-2 (n+3)} e^{\frac{4 a}{b}} \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^n \left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{4 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{\sqrt{c^2 d x^2+d}}",0,"Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^n)/x^2, x]","A",-1
527,0,0,26,0.5147649,"\int \frac{x^m \sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^m*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2],x]","\int \frac{x^m \sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx","\text{Int}\left(\frac{x^m \sinh ^{-1}(a x)^n}{\sqrt{a^2 x^2+1}},x\right)",0,"Integrate[(x^m*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2], x]","A",-1
528,1,100,113,0.196064,"\int \frac{x^3 \sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^3*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2],x]","\frac{3^{-n-1} \left(-\sinh ^{-1}(a x)\right)^{-n} \left(\left(-\sinh ^{-1}(a x)\right)^n \left(\Gamma \left(n+1,3 \sinh ^{-1}(a x)\right)-3^{n+2} \Gamma \left(n+1,\sinh ^{-1}(a x)\right)\right)+\sinh ^{-1}(a x)^n \Gamma \left(n+1,-3 \sinh ^{-1}(a x)\right)-3^{n+2} \sinh ^{-1}(a x)^n \Gamma \left(n+1,-\sinh ^{-1}(a x)\right)\right)}{8 a^4}","\frac{3^{-n-1} \sinh ^{-1}(a x)^n \left(-\sinh ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-3 \sinh ^{-1}(a x)\right)}{8 a^4}-\frac{3 \sinh ^{-1}(a x)^n \left(-\sinh ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-\sinh ^{-1}(a x)\right)}{8 a^4}-\frac{3 \Gamma \left(n+1,\sinh ^{-1}(a x)\right)}{8 a^4}+\frac{3^{-n-1} \Gamma \left(n+1,3 \sinh ^{-1}(a x)\right)}{8 a^4}",1,"(3^(-1 - n)*(ArcSinh[a*x]^n*Gamma[1 + n, -3*ArcSinh[a*x]] - 3^(2 + n)*ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]] + (-ArcSinh[a*x])^n*(-(3^(2 + n)*Gamma[1 + n, ArcSinh[a*x]]) + Gamma[1 + n, 3*ArcSinh[a*x]])))/(8*a^4*(-ArcSinh[a*x])^n)","A",1
529,1,86,80,0.2093734,"\int \frac{x^2 \sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x^2*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2],x]","\frac{2^{-n-3} \left(-\sinh ^{-1}(a x)\right)^{-n} \left((n+1) \sinh ^{-1}(a x)^n \Gamma \left(n+1,-2 \sinh ^{-1}(a x)\right)-\left(-\sinh ^{-1}(a x)\right)^n \left(2^{n+2} \sinh ^{-1}(a x)^{n+1}+(n+1) \Gamma \left(n+1,2 \sinh ^{-1}(a x)\right)\right)\right)}{a^3 (n+1)}","-\frac{\sinh ^{-1}(a x)^{n+1}}{2 a^3 (n+1)}+\frac{2^{-n-3} \sinh ^{-1}(a x)^n \left(-\sinh ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-2 \sinh ^{-1}(a x)\right)}{a^3}-\frac{2^{-n-3} \Gamma \left(n+1,2 \sinh ^{-1}(a x)\right)}{a^3}",1,"(2^(-3 - n)*((1 + n)*ArcSinh[a*x]^n*Gamma[1 + n, -2*ArcSinh[a*x]] - (-ArcSinh[a*x])^n*(2^(2 + n)*ArcSinh[a*x]^(1 + n) + (1 + n)*Gamma[1 + n, 2*ArcSinh[a*x]])))/(a^3*(1 + n)*(-ArcSinh[a*x])^n)","A",1
530,1,43,49,0.0750444,"\int \frac{x \sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx","Integrate[(x*ArcSinh[a*x]^n)/Sqrt[1 + a^2*x^2],x]","\frac{\sinh ^{-1}(a x)^n \left(-\sinh ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-\sinh ^{-1}(a x)\right)+\Gamma \left(n+1,\sinh ^{-1}(a x)\right)}{2 a^2}","\frac{\sinh ^{-1}(a x)^n \left(-\sinh ^{-1}(a x)\right)^{-n} \Gamma \left(n+1,-\sinh ^{-1}(a x)\right)}{2 a^2}+\frac{\Gamma \left(n+1,\sinh ^{-1}(a x)\right)}{2 a^2}",1,"((ArcSinh[a*x]^n*Gamma[1 + n, -ArcSinh[a*x]])/(-ArcSinh[a*x])^n + Gamma[1 + n, ArcSinh[a*x]])/(2*a^2)","A",1
531,1,17,17,0.0092528,"\int \frac{\sinh ^{-1}(a x)^n}{\sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^n/Sqrt[1 + a^2*x^2],x]","\frac{\sinh ^{-1}(a x)^{n+1}}{a (n+1)}","\frac{\sinh ^{-1}(a x)^{n+1}}{a (n+1)}",1,"ArcSinh[a*x]^(1 + n)/(a*(1 + n))","A",1
532,0,0,26,5.9339819,"\int \frac{\sinh ^{-1}(a x)^n}{x \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^n/(x*Sqrt[1 + a^2*x^2]),x]","\int \frac{\sinh ^{-1}(a x)^n}{x \sqrt{1+a^2 x^2}} \, dx","\text{Int}\left(\frac{\sinh ^{-1}(a x)^n}{x \sqrt{a^2 x^2+1}},x\right)",0,"Integrate[ArcSinh[a*x]^n/(x*Sqrt[1 + a^2*x^2]), x]","A",-1
533,0,0,26,1.9306409,"\int \frac{\sinh ^{-1}(a x)^n}{x^2 \sqrt{1+a^2 x^2}} \, dx","Integrate[ArcSinh[a*x]^n/(x^2*Sqrt[1 + a^2*x^2]),x]","\int \frac{\sinh ^{-1}(a x)^n}{x^2 \sqrt{1+a^2 x^2}} \, dx","\text{Int}\left(\frac{\sinh ^{-1}(a x)^n}{x^2 \sqrt{a^2 x^2+1}},x\right)",0,"Integrate[ArcSinh[a*x]^n/(x^2*Sqrt[1 + a^2*x^2]), x]","A",-1
534,1,361,416,1.4814991,"\int (d+i c d x)^{5/2} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]),x]","\frac{720 a d^{5/2} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+48 a d^2 \sqrt{c^2 x^2+1} \left(-6 c^3 x^3+16 i c^2 x^2+9 c x+16 i\right) \sqrt{d+i c d x} \sqrt{f-i c f x}-64 i b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(-3 \sinh ^{-1}(c x) \left(3 \sqrt{c^2 x^2+1}+\cosh \left(3 \sinh ^{-1}(c x)\right)\right)+9 c x+\sinh \left(3 \sinh ^{-1}(c x)\right)\right)+144 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \sinh ^{-1}(c x)^2+2 \sinh \left(2 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-\cosh \left(2 \sinh ^{-1}(c x)\right)\right)+9 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(8 \sinh ^{-1}(c x)^2-4 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+\cosh \left(4 \sinh ^{-1}(c x)\right)\right)}{1152 c \sqrt{c^2 x^2+1}}","-\frac{1}{4} c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c \sqrt{c^2 x^2+1}}+\frac{2 i d^2 \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{3 c}+\frac{3}{8} d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)-\frac{3 b c d^2 x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{c^2 x^2+1}}-\frac{2 i b d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}}{3 \sqrt{c^2 x^2+1}}-\frac{2 i b c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 \sqrt{c^2 x^2+1}}+\frac{b c^3 d^2 x^4 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{c^2 x^2+1}}",1,"(48*a*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2]*(16*I + 9*c*x + (16*I)*c^2*x^2 - 6*c^3*x^3) + 720*a*d^(5/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 144*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(2*ArcSinh[c*x]^2 - Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]) - (64*I)*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(9*c*x - 3*ArcSinh[c*x]*(3*Sqrt[1 + c^2*x^2] + Cosh[3*ArcSinh[c*x]]) + Sinh[3*ArcSinh[c*x]]) + 9*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(1152*c*Sqrt[1 + c^2*x^2])","A",1
535,1,273,304,1.5759492,"\int (d+i c d x)^{3/2} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)^(3/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]),x]","\frac{12 a d \left(2 i c^2 x^2+3 c x+2 i\right) \sqrt{d+i c d x} \sqrt{f-i c f x}+36 a d^{3/2} \sqrt{f} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+\frac{9 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \sinh ^{-1}(c x)^2+2 \sinh \left(2 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-\cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}-\frac{2 i b d \sqrt{d+i c d x} \sqrt{f-i c f x} \left(-3 \sinh ^{-1}(c x) \left(3 \sqrt{c^2 x^2+1}+\cosh \left(3 \sinh ^{-1}(c x)\right)\right)+9 c x+\sinh \left(3 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}}{72 c}","\frac{d \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c \sqrt{c^2 x^2+1}}+\frac{i d \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{3 c}+\frac{1}{2} d x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)-\frac{b c d x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{4 \sqrt{c^2 x^2+1}}-\frac{i b d x \sqrt{d+i c d x} \sqrt{f-i c f x}}{3 \sqrt{c^2 x^2+1}}-\frac{i b c^2 d x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 \sqrt{c^2 x^2+1}}",1,"(12*a*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(2*I + 3*c*x + (2*I)*c^2*x^2) + 36*a*d^(3/2)*Sqrt[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + (9*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(2*ArcSinh[c*x]^2 - Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2] - ((2*I)*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(9*c*x - 3*ArcSinh[c*x]*(3*Sqrt[1 + c^2*x^2] + Cosh[3*ArcSinh[c*x]]) + Sinh[3*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2])/(72*c)","A",1
536,1,233,147,0.6024807,"\int \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]),x]","\frac{1}{2} a x \sqrt{i d (c x-i)} \sqrt{-i f (c x+i)}+\frac{a \sqrt{d} \sqrt{f} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i d (c x-i)} \sqrt{-i f (c x+i)}\right)}{2 c}-\frac{b \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(2 \sinh ^{-1}(c x)\right)-2 \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)\right)}{8 c \sqrt{c^2 x^2+1} \sqrt{-((c d x-i d) (c f x+i f))}}","\frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c \sqrt{c^2 x^2+1}}+\frac{1}{2} x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)-\frac{b c x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{4 \sqrt{c^2 x^2+1}}",1,"(a*x*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)])/2 + (a*Sqrt[d]*Sqrt[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(2*c) - (b*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])))/(8*c*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2])","A",1
537,1,227,158,0.6100146,"\int \frac{\sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+i c d x}} \, dx","Integrate[(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/Sqrt[d + I*c*d*x],x]","\frac{2 i \sqrt{d+i c d x} \sqrt{f-i c f x} \left(b c x-a \sqrt{c^2 x^2+1}\right)+2 a \sqrt{d} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)-2 i b \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)+b \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2}{2 c d \sqrt{c^2 x^2+1}}","\frac{f \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{i f \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i b f x \sqrt{c^2 x^2+1}}{\sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((2*I)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(b*c*x - a*Sqrt[1 + c^2*x^2]) - (2*I)*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 + 2*a*Sqrt[d]*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(2*c*d*Sqrt[1 + c^2*x^2])","A",1
538,1,283,181,1.5158577,"\int \frac{\sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{(d+i c d x)^{3/2}} \, dx","Integrate[(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(3/2),x]","-\frac{-\frac{4 a \sqrt{d+i c d x} \sqrt{f-i c f x}}{c x-i}+2 a \sqrt{d} \sqrt{f} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+\frac{b \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\log \left(c^2 x^2+1\right)+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\sinh ^{-1}(c x) \left(-4 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{\sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}}{2 c d^2}","-\frac{f^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 i f^2 (1-i c x) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 b f^2 \left(c^2 x^2+1\right)^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"-1/2*((-4*a*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(-I + c*x) + 2*a*Sqrt[d]*Sqrt[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + (b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(ArcSinh[c*x]*((-4*I)*Cosh[ArcSinh[c*x]/2] - 4*Sinh[ArcSinh[c*x]/2]) + ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + 2*((4*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + Log[1 + c^2*x^2])*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(c*d^2)","A",1
539,1,141,187,0.4986308,"\int \frac{\sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{(d+i c d x)^{5/2}} \, dx","Integrate[(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(5/2),x]","\frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left(-(c x+i) \left(a \sqrt{c^2 x^2+1}+b c x-i b\right)-b (c x+i) \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+b (c x-i)^2 \log (d+i c d x)\right)}{3 c d^3 (c x-i)^2 \sqrt{c^2 x^2+1}}","\frac{i f^3 (1-i c x)^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i b f^3 \left(c^2 x^2+1\right)^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b f^3 \left(c^2 x^2+1\right)^{5/2} \log (-c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-((I + c*x)*((-I)*b + b*c*x + a*Sqrt[1 + c^2*x^2])) - b*(I + c*x)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + b*(-I + c*x)^2*Log[d + I*c*d*x]))/(3*c*d^3*(-I + c*x)^2*Sqrt[1 + c^2*x^2])","A",1
540,1,683,459,1.7638576,"\int (d+i c d x)^{5/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{3600 a d^{5/2} f^{3/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+3840 i a c^2 d^2 f x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+6000 a c d^2 f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+1920 i a d^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+1920 i a c^4 d^2 f x^4 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+2400 a c^3 d^2 f x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+60 b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(5 \left(4 i \sqrt{c^2 x^2+1}+8 \sinh \left(2 \sinh ^{-1}(c x)\right)+\sinh \left(4 \sinh ^{-1}(c x)\right)\right)+10 i \cosh \left(3 \sinh ^{-1}(c x)\right)+2 i \cosh \left(5 \sinh ^{-1}(c x)\right)\right)-1200 i b c d^2 f x \sqrt{d+i c d x} \sqrt{f-i c f x}+1800 b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2-200 i b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)-24 i b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(5 \sinh ^{-1}(c x)\right)-1200 b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-75 b d^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)}{9600 c \sqrt{c^2 x^2+1}}","\frac{3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 \left(c^2 x^2+1\right)}+\frac{3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c \left(c^2 x^2+1\right)^{3/2}}+\frac{i d \left(c^2 x^2+1\right) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 c}+\frac{1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{5 b c d x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left(c^2 x^2+1\right)^{3/2}}-\frac{i b d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left(c^2 x^2+1\right)^{3/2}}-\frac{2 i b c^2 d x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left(c^2 x^2+1\right)^{3/2}}-\frac{i b c^4 d x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left(c^2 x^2+1\right)^{3/2}}-\frac{b c^3 d x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left(c^2 x^2+1\right)^{3/2}}",1,"((-1200*I)*b*c*d^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (1920*I)*a*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 6000*a*c*d^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (3840*I)*a*c^2*d^2*f*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2400*a*c^3*d^2*f*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (1920*I)*a*c^4*d^2*f*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 1800*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 - 1200*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 75*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 3600*a*d^(5/2)*f^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - (200*I)*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 60*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((10*I)*Cosh[3*ArcSinh[c*x]] + (2*I)*Cosh[5*ArcSinh[c*x]] + 5*((4*I)*Sqrt[1 + c^2*x^2] + 8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])) - (24*I)*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[5*ArcSinh[c*x]])/(9600*c*Sqrt[1 + c^2*x^2])","A",1
541,1,352,247,0.9031286,"\int (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{48 a d^{3/2} f^{3/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+80 a c d f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+32 a c^3 d f x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+24 b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2+4 b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(8 \sinh \left(2 \sinh ^{-1}(c x)\right)+\sinh \left(4 \sinh ^{-1}(c x)\right)\right)-16 b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)}{128 c \sqrt{c^2 x^2+1}}","\frac{3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 \left(c^2 x^2+1\right)}+\frac{3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c \left(c^2 x^2+1\right)^{3/2}}+\frac{1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{5 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left(c^2 x^2+1\right)^{3/2}}-\frac{b c^3 x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left(c^2 x^2+1\right)^{3/2}}",1,"(80*a*c*d*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 32*a*c^3*d*f*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 24*b*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 - 16*b*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - b*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 48*a*d^(3/2)*f^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 4*b*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]]))/(128*c*Sqrt[1 + c^2*x^2])","A",1
542,1,273,304,1.5890175,"\int \sqrt{d+i c d x} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{12 a f \left(-2 i c^2 x^2+3 c x-2 i\right) \sqrt{d+i c d x} \sqrt{f-i c f x}+36 a \sqrt{d} f^{3/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+\frac{9 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \sinh ^{-1}(c x)^2+2 \sinh \left(2 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-\cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}+\frac{2 i b f \sqrt{d+i c d x} \sqrt{f-i c f x} \left(-3 \sinh ^{-1}(c x) \left(3 \sqrt{c^2 x^2+1}+\cosh \left(3 \sinh ^{-1}(c x)\right)\right)+9 c x+\sinh \left(3 \sinh ^{-1}(c x)\right)\right)}{\sqrt{c^2 x^2+1}}}{72 c}","\frac{f \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c \sqrt{c^2 x^2+1}}-\frac{i f \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{3 c}+\frac{1}{2} f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)-\frac{b c f x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{4 \sqrt{c^2 x^2+1}}+\frac{i b f x \sqrt{d+i c d x} \sqrt{f-i c f x}}{3 \sqrt{c^2 x^2+1}}+\frac{i b c^2 f x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 \sqrt{c^2 x^2+1}}",1,"(12*a*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-2*I + 3*c*x - (2*I)*c^2*x^2) + 36*a*Sqrt[d]*f^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + (9*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(2*ArcSinh[c*x]^2 - Cosh[2*ArcSinh[c*x]] + 2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2] + ((2*I)*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(9*c*x - 3*ArcSinh[c*x]*(3*Sqrt[1 + c^2*x^2] + Cosh[3*ArcSinh[c*x]]) + Sinh[3*ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2])/(72*c)","A",1
543,1,344,266,1.0826747,"\int \frac{(f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+i c d x}} \, dx","Integrate[((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/Sqrt[d + I*c*d*x],x]","\frac{12 a \sqrt{d} f^{3/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)-16 i a f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-4 a c f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-4 b f (c x+4 i) \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)+16 i b c f x \sqrt{d+i c d x} \sqrt{f-i c f x}+6 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2+b f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)}{8 c d \sqrt{c^2 x^2+1}}","\frac{3 f^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{f^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i f^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c f^2 x^2 \sqrt{c^2 x^2+1}}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i b f^2 x \sqrt{c^2 x^2+1}}{\sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((16*I)*b*c*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (16*I)*a*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 4*a*c*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 4*b*f*(4*I + c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 6*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 + b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + 12*a*Sqrt[d]*f^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(8*c*d*Sqrt[1 + c^2*x^2])","A",1
544,1,514,284,2.3832594,"\int \frac{(f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{(d+i c d x)^{3/2}} \, dx","Integrate[((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(3/2),x]","\frac{-\frac{6 a f^{3/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)}{d^{3/2}}+\frac{2 a f (5+i c x) \sqrt{d+i c d x} \sqrt{f-i c f x}}{d^2 (c x-i)}-\frac{b f \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\log \left(c^2 x^2+1\right)+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\sinh ^{-1}(c x) \left(-4 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{d^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{2 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \left(\sinh ^{-1}(c x) \left(-\left(\sqrt{c^2 x^2+1}-2\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \left(\sqrt{c^2 x^2+1}+2\right) \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(-i \log \left(c^2 x^2+1\right)+c x-4 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)-\left(\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{d^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}}{2 c}","-\frac{3 f^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{i f^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i f^3 (1-i c x) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{i b f^3 x \left(c^2 x^2+1\right)^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b f^3 \left(c^2 x^2+1\right)^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"((2*a*f*(5 + I*c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(d^2*(-I + c*x)) - (6*a*f^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/d^(3/2) - (b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(ArcSinh[c*x]*((-4*I)*Cosh[ArcSinh[c*x]/2] - 4*Sinh[ArcSinh[c*x]/2]) + ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + 2*((4*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + Log[1 + c^2*x^2])*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(d^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (2*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (c*x - 4*ArcTan[Coth[ArcSinh[c*x]/2]] - I*Log[1 + c^2*x^2])*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + ArcSinh[c*x]*(I*(2 + Sqrt[1 + c^2*x^2])*Cosh[ArcSinh[c*x]/2] - (-2 + Sqrt[1 + c^2*x^2])*Sinh[ArcSinh[c*x]/2])))/(d^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(2*c)","A",1
545,1,706,364,5.8409401,"\int \frac{(f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{(d+i c d x)^{5/2}} \, dx","Integrate[((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(5/2),x]","\frac{\frac{12 a f^{3/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)}{d^{5/2}}-\frac{16 a f (2 c x-i) \sqrt{d+i c d x} \sqrt{f-i c f x}}{d^3 (c x-i)^2}-\frac{b f \sqrt{d+i c d x} \sqrt{f-i c f x} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(2 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(14 \log \left(c^2 x^2+1\right)+\sqrt{c^2 x^2+1} \left(7 \log \left(c^2 x^2+1\right)+\sinh ^{-1}(c x) \left(3 \sinh ^{-1}(c x)-14 i\right)+28 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+6 \sinh ^{-1}(c x)^2-4 i \sinh ^{-1}(c x)+56 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4\right)+\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(7 i \log \left(c^2 x^2+1\right)+\left(-14+3 i \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-28 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(84 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-i \left(21 \log \left(c^2 x^2+1\right)+9 \sinh ^{-1}(c x)^2-6 i \sinh ^{-1}(c x)+8\right)\right)\right)}{d^3 (c x+i) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}+\frac{2 i b f \sqrt{d+i c d x} \sqrt{f-i c f x} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(2 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\frac{1}{2} i \left(\left(\sqrt{c^2 x^2+1}+2\right) \log \left(c^2 x^2+1\right)+4\right)+\left(\sqrt{c^2 x^2+1}+2\right) \sinh ^{-1}(c x)+2 \left(\sqrt{c^2 x^2+1}+2\right) \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)-i \cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(-\frac{1}{2} i \log \left(c^2 x^2+1\right)+\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\frac{3}{2} \log \left(c^2 x^2+1\right)+3 i \sinh ^{-1}(c x)-6 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4\right)\right)}{d^3 (c x+i) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}}{12 c}","-\frac{2 i f^4 (1-i c x) \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i f^4 (1-i c x)^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{f^4 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 i b f^4 \left(c^2 x^2+1\right)^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{8 b f^4 \left(c^2 x^2+1\right)^{5/2} \log (-c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b f^4 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"((-16*a*f*(-I + 2*c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(d^3*(-I + c*x)^2) + (12*a*f^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/d^(5/2) - (b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSinh[c*x])/2]*((-14 + (3*I)*ArcSinh[c*x])*ArcSinh[c*x] - 28*ArcTan[Tanh[ArcSinh[c*x]/2]] + (7*I)*Log[1 + c^2*x^2]) + Cosh[ArcSinh[c*x]/2]*(84*ArcTan[Tanh[ArcSinh[c*x]/2]] - I*(8 - (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 + 21*Log[1 + c^2*x^2])) + 2*(4 - (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 + (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*Log[1 + c^2*x^2] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(-14*I + 3*ArcSinh[c*x]) + (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 7*Log[1 + c^2*x^2]))*Sinh[ArcSinh[c*x]/2]))/(d^3*(I + c*x)*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) + ((2*I)*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*((-I)*Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] - (I/2)*Log[1 + c^2*x^2]) + Cosh[ArcSinh[c*x]/2]*(4 + (3*I)*ArcSinh[c*x] - (6*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*Log[1 + c^2*x^2])/2) + 2*((2 + Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + 2*(2 + Sqrt[1 + c^2*x^2])*ArcTan[Coth[ArcSinh[c*x]/2]] + (I/2)*(4 + (2 + Sqrt[1 + c^2*x^2])*Log[1 + c^2*x^2]))*Sinh[ArcSinh[c*x]/2]))/(d^3*(I + c*x)*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4))/(12*c)","A",0
546,1,481,344,1.1751789,"\int (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{720 a d^{5/2} f^{5/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+1584 a c d^2 f^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+384 a c^5 d^2 f^2 x^5 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+1248 a c^3 d^2 f^2 x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+360 b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2+12 b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(45 \sinh \left(2 \sinh ^{-1}(c x)\right)+9 \sinh \left(4 \sinh ^{-1}(c x)\right)+\sinh \left(6 \sinh ^{-1}(c x)\right)\right)-270 b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-27 b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)-2 b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(6 \sinh ^{-1}(c x)\right)}{2304 c \sqrt{c^2 x^2+1}}","\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{24 \left(c^2 x^2+1\right)}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{16 \left(c^2 x^2+1\right)^2}+\frac{5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{32 b c \left(c^2 x^2+1\right)^{5/2}}+\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{25 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{96 \left(c^2 x^2+1\right)^{5/2}}-\frac{b \sqrt{c^2 x^2+1} (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{36 c}-\frac{5 b c^3 x^4 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{96 \left(c^2 x^2+1\right)^{5/2}}",1,"(1584*a*c*d^2*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 1248*a*c^3*d^2*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 384*a*c^5*d^2*f^2*x^5*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 360*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 - 270*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 27*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] - 2*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[6*ArcSinh[c*x]] + 720*a*d^(5/2)*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 12*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(45*Sinh[2*ArcSinh[c*x]] + 9*Sinh[4*ArcSinh[c*x]] + Sinh[6*ArcSinh[c*x]]))/(2304*c*Sqrt[1 + c^2*x^2])","A",1
547,1,683,459,1.6670384,"\int (d+i c d x)^{3/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{3600 a d^{3/2} f^{5/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)-3840 i a c^2 d f^2 x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+6000 a c d f^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-1920 i a d f^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-1920 i a c^4 d f^2 x^4 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+2400 a c^3 d f^2 x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+60 b d f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(5 \left(-4 i \sqrt{c^2 x^2+1}+8 \sinh \left(2 \sinh ^{-1}(c x)\right)+\sinh \left(4 \sinh ^{-1}(c x)\right)\right)-10 i \cosh \left(3 \sinh ^{-1}(c x)\right)-2 i \cosh \left(5 \sinh ^{-1}(c x)\right)\right)+1200 i b c d f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}+1800 b d f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2+200 i b d f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)+24 i b d f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(5 \sinh ^{-1}(c x)\right)-1200 b d f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-75 b d f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)}{9600 c \sqrt{c^2 x^2+1}}","\frac{3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 \left(c^2 x^2+1\right)}+\frac{3 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c \left(c^2 x^2+1\right)^{3/2}}-\frac{i f \left(c^2 x^2+1\right) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 c}+\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)-\frac{5 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left(c^2 x^2+1\right)^{3/2}}+\frac{i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left(c^2 x^2+1\right)^{3/2}}+\frac{2 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left(c^2 x^2+1\right)^{3/2}}+\frac{i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left(c^2 x^2+1\right)^{3/2}}-\frac{b c^3 f x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left(c^2 x^2+1\right)^{3/2}}",1,"((1200*I)*b*c*d*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (1920*I)*a*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 6000*a*c*d*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (3840*I)*a*c^2*d*f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2400*a*c^3*d*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (1920*I)*a*c^4*d*f^2*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 1800*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 - 1200*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 75*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 3600*a*d^(3/2)*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + (200*I)*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 60*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((-10*I)*Cosh[3*ArcSinh[c*x]] - (2*I)*Cosh[5*ArcSinh[c*x]] + 5*((-4*I)*Sqrt[1 + c^2*x^2] + 8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])) + (24*I)*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[5*ArcSinh[c*x]])/(9600*c*Sqrt[1 + c^2*x^2])","A",1
548,1,565,416,1.233179,"\int \sqrt{d+i c d x} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{720 a \sqrt{d} f^{5/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)-768 i a c^2 f^2 x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+432 a c f^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-768 i a f^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-288 a c^3 f^2 x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+12 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(-48 i \sqrt{c^2 x^2+1}+24 \sinh \left(2 \sinh ^{-1}(c x)\right)-3 \sinh \left(4 \sinh ^{-1}(c x)\right)-16 i \cosh \left(3 \sinh ^{-1}(c x)\right)\right)+576 i b c f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}+360 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2+64 i b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)-144 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)+9 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)}{1152 c \sqrt{c^2 x^2+1}}","-\frac{1}{4} c^2 f^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 b c \sqrt{c^2 x^2+1}}-\frac{2 i f^2 \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{3 c}+\frac{3}{8} f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)-\frac{3 b c f^2 x^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{c^2 x^2+1}}+\frac{2 i b f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}}{3 \sqrt{c^2 x^2+1}}+\frac{2 i b c^2 f^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 \sqrt{c^2 x^2+1}}+\frac{b c^3 f^2 x^4 \sqrt{d+i c d x} \sqrt{f-i c f x}}{16 \sqrt{c^2 x^2+1}}",1,"((576*I)*b*c*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (768*I)*a*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 432*a*c*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (768*I)*a*c^2*f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 288*a*c^3*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 360*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 - 144*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + 9*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 720*a*Sqrt[d]*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + (64*I)*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 12*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((-48*I)*Sqrt[1 + c^2*x^2] - (16*I)*Cosh[3*ArcSinh[c*x]] + 24*Sinh[2*ArcSinh[c*x]] - 3*Sinh[4*ArcSinh[c*x]]))/(1152*c*Sqrt[1 + c^2*x^2])","A",1
549,1,465,381,1.7137358,"\int \frac{(f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+i c d x}} \, dx","Integrate[((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/Sqrt[d + I*c*d*x],x]","\frac{180 a \sqrt{d} f^{5/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+24 i a c^2 f^2 x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-108 a c f^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-264 i a f^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-8 i b c^3 f^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}-6 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(9 (2 c x+5 i) \sqrt{c^2 x^2+1}-i \cosh \left(3 \sinh ^{-1}(c x)\right)\right)+264 i b c f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}+90 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2+27 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)}{72 c d \sqrt{c^2 x^2+1}}","\frac{5 f^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i c f^3 x^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 f^3 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{11 i f^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b c f^3 x^2 \sqrt{c^2 x^2+1}}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{11 i b f^3 x \sqrt{c^2 x^2+1}}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{i b c^2 f^3 x^3 \sqrt{c^2 x^2+1}}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((264*I)*b*c*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (8*I)*b*c^3*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (264*I)*a*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 108*a*c*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (24*I)*a*c^2*f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 90*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 + 27*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 6*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(9*(5*I + 2*c*x)*Sqrt[1 + c^2*x^2] - I*Cosh[3*ArcSinh[c*x]]) + 180*a*Sqrt[d]*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(72*c*d*Sqrt[1 + c^2*x^2])","A",1
550,1,779,518,3.9669181,"\int \frac{(f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{(d+i c d x)^{3/2}} \, dx","Integrate[((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(3/2),x]","\frac{\frac{4 a f^2 \left(c^2 x^2+7 i c x+24\right) \sqrt{d+i c d x} \sqrt{f-i c f x}}{d^2 (c x-i)}-\frac{60 a f^{5/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)}{d^{3/2}}+\frac{b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \sinh ^{-1}(c x) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(-8 \sqrt{c^2 x^2+1}+i \sinh \left(2 \sinh ^{-1}(c x)\right)+8\right)+\left(\sinh \left(2 \sinh ^{-1}(c x)\right)+8 i \left(\sqrt{c^2 x^2+1}+1\right)\right) \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\cosh \left(2 \sinh ^{-1}(c x)\right)+8 \left(\log \left(c^2 x^2+1\right)+2 i c x+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)-10 \sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{d^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}-\frac{4 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\log \left(c^2 x^2+1\right)+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\sinh ^{-1}(c x) \left(-4 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{d^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{16 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(\sinh ^{-1}(c x) \left(-\left(\sqrt{c^2 x^2+1}-2\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \left(\sqrt{c^2 x^2+1}+2\right) \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(-i \log \left(c^2 x^2+1\right)+c x-4 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)-\left(\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{d^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}}{8 c}","\frac{5 i f^4 (1-i c x) \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{15 i f^4 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 i f^4 (1-i c x)^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{15 f^4 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \left(a+b \sinh ^{-1}(c x)\right)}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b c f^4 x^2 \left(c^2 x^2+1\right)^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{5 b f^4 (1-i c x)^2 \left(c^2 x^2+1\right)^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{3 i b f^4 x \left(c^2 x^2+1\right)^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b f^4 \left(c^2 x^2+1\right)^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{15 b f^4 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"((4*a*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(24 + (7*I)*c*x + c^2*x^2))/(d^2*(-I + c*x)) - (60*a*f^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/d^(3/2) - (4*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(ArcSinh[c*x]*((-4*I)*Cosh[ArcSinh[c*x]/2] - 4*Sinh[ArcSinh[c*x]/2]) + ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + 2*((4*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + Log[1 + c^2*x^2])*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(d^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (16*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (c*x - 4*ArcTan[Coth[ArcSinh[c*x]/2]] - I*Log[1 + c^2*x^2])*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + ArcSinh[c*x]*(I*(2 + Sqrt[1 + c^2*x^2])*Cosh[ArcSinh[c*x]/2] - (-2 + Sqrt[1 + c^2*x^2])*Sinh[ArcSinh[c*x]/2])))/(d^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-10*ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) - (Cosh[2*ArcSinh[c*x]] + 8*((2*I)*c*x + (4*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + Log[1 + c^2*x^2]))*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + 2*ArcSinh[c*x]*(Sinh[ArcSinh[c*x]/2]*(8 - 8*Sqrt[1 + c^2*x^2] + I*Sinh[2*ArcSinh[c*x]]) + Cosh[ArcSinh[c*x]/2]*((8*I)*(1 + Sqrt[1 + c^2*x^2]) + Sinh[2*ArcSinh[c*x]]))))/(d^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(8*c)","A",1
551,1,1005,472,7.4285707,"\int \frac{(f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{(d+i c d x)^{5/2}} \, dx","Integrate[((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(d + I*c*d*x)^(5/2),x]","\frac{\frac{60 a \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i c x d+d} \sqrt{f-i c f x}\right) f^{5/2}}{d^{5/2}}-\frac{4 i a \sqrt{i c x d+d} \sqrt{f-i c f x} \left(3 c^2 x^2-34 i c x-23\right) f^2}{d^3 (c x-i)^2}-\frac{2 b \sqrt{i c x d+d} \sqrt{f-i c f x} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\left(3 i \sinh ^{-1}(c x)-14\right) \sinh ^{-1}(c x)-28 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+7 i \log \left(c^2 x^2+1\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(84 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-i \left(9 \sinh ^{-1}(c x)^2-6 i \sinh ^{-1}(c x)+21 \log \left(c^2 x^2+1\right)+8\right)\right)+2 \left(6 \sinh ^{-1}(c x)^2-4 i \sinh ^{-1}(c x)+56 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+14 \log \left(c^2 x^2+1\right)+\sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x) \left(3 \sinh ^{-1}(c x)-14 i\right)+28 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+7 \log \left(c^2 x^2+1\right)\right)+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) f^2}{d^3 (c x+i) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}+\frac{2 i b \sqrt{i c x d+d} \sqrt{f-i c f x} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(-i \cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-\frac{1}{2} i \log \left(c^2 x^2+1\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(3 i \sinh ^{-1}(c x)-6 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\frac{3}{2} \log \left(c^2 x^2+1\right)+4\right)+2 \left(\left(\sqrt{c^2 x^2+1}+2\right) \sinh ^{-1}(c x)+2 \left(\sqrt{c^2 x^2+1}+2\right) \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\frac{1}{2} i \left(\left(\sqrt{c^2 x^2+1}+2\right) \log \left(c^2 x^2+1\right)+4\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) f^2}{d^3 (c x+i) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}+\frac{b \sqrt{i c x d+d} \sqrt{f-i c f x} \left(i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(18 \sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3+2 \left(-6 c^2 x^2+6 i c x+52 (c x-i) \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+13 (i c x+1) \log \left(c^2 x^2+1\right)+4\right) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\sinh ^{-1}(c x) \left(-24 i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-35 i \cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right)+3 i \cosh \left(\frac{5}{2} \sinh ^{-1}(c x)\right)-24 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+35 \sinh \left(\frac{3}{2} \sinh ^{-1}(c x)\right)+3 \sinh \left(\frac{5}{2} \sinh ^{-1}(c x)\right)\right)\right) f^2}{d^3 (c x+i) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}}{12 c}","-\frac{5 i f^5 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{10 i f^5 (1-i c x)^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i f^5 (1-i c x)^4 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{5 f^5 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b f^5 x \left(c^2 x^2+1\right)^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{8 i b f^5 \left(c^2 x^2+1\right)^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{28 b f^5 \left(c^2 x^2+1\right)^{5/2} \log (-c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{5 b f^5 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(((-4*I)*a*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-23 - (34*I)*c*x + 3*c^2*x^2))/(d^3*(-I + c*x)^2) + (60*a*f^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/d^(5/2) - (2*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSinh[c*x])/2]*((-14 + (3*I)*ArcSinh[c*x])*ArcSinh[c*x] - 28*ArcTan[Tanh[ArcSinh[c*x]/2]] + (7*I)*Log[1 + c^2*x^2]) + Cosh[ArcSinh[c*x]/2]*(84*ArcTan[Tanh[ArcSinh[c*x]/2]] - I*(8 - (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 + 21*Log[1 + c^2*x^2])) + 2*(4 - (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 + (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*Log[1 + c^2*x^2] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(-14*I + 3*ArcSinh[c*x]) + (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 7*Log[1 + c^2*x^2]))*Sinh[ArcSinh[c*x]/2]))/(d^3*(I + c*x)*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) + ((2*I)*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*((-I)*Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] - (I/2)*Log[1 + c^2*x^2]) + Cosh[ArcSinh[c*x]/2]*(4 + (3*I)*ArcSinh[c*x] - (6*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*Log[1 + c^2*x^2])/2) + 2*((2 + Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + 2*(2 + Sqrt[1 + c^2*x^2])*ArcTan[Coth[ArcSinh[c*x]/2]] + (I/2)*(4 + (2 + Sqrt[1 + c^2*x^2])*Log[1 + c^2*x^2]))*Sinh[ArcSinh[c*x]/2]))/(d^3*(I + c*x)*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) + (b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])*(2*(4 + (6*I)*c*x - 6*c^2*x^2 + 52*(-I + c*x)*ArcTan[Coth[ArcSinh[c*x]/2]] + 13*(1 + I*c*x)*Log[1 + c^2*x^2])*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + 18*ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + ArcSinh[c*x]*((-24*I)*Cosh[ArcSinh[c*x]/2] - (35*I)*Cosh[(3*ArcSinh[c*x])/2] + (3*I)*Cosh[(5*ArcSinh[c*x])/2] - 24*Sinh[ArcSinh[c*x]/2] + 35*Sinh[(3*ArcSinh[c*x])/2] + 3*Sinh[(5*ArcSinh[c*x])/2])))/(d^3*(I + c*x)*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4))/(12*c)","B",0
552,1,465,381,1.677589,"\int \frac{(d+i c d x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{f-i c f x}} \, dx","Integrate[((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x]))/Sqrt[f - I*c*f*x],x]","\frac{180 a d^{5/2} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)-24 i a c^2 d^2 x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-108 a c d^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+264 i a d^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+8 i b c^3 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}-6 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(9 \sqrt{c^2 x^2+1} (2 c x-5 i)+i \cosh \left(3 \sinh ^{-1}(c x)\right)\right)-264 i b c d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}+90 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2+27 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)}{72 c f \sqrt{c^2 x^2+1}}","\frac{5 d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{i c d^3 x^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 d^3 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{11 i d^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b c d^3 x^2 \sqrt{c^2 x^2+1}}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{11 i b d^3 x \sqrt{c^2 x^2+1}}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i b c^2 d^3 x^3 \sqrt{c^2 x^2+1}}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((-264*I)*b*c*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (8*I)*b*c^3*d^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (264*I)*a*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 108*a*c*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (24*I)*a*c^2*d^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 90*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 + 27*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 6*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(9*(-5*I + 2*c*x)*Sqrt[1 + c^2*x^2] + I*Cosh[3*ArcSinh[c*x]]) + 180*a*d^(5/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(72*c*f*Sqrt[1 + c^2*x^2])","A",1
553,1,344,266,1.0254835,"\int \frac{(d+i c d x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{f-i c f x}} \, dx","Integrate[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x]))/Sqrt[f - I*c*f*x],x]","\frac{12 a d^{3/2} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+16 i a d \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-4 a c d x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-4 b d (c x-4 i) \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)-16 i b c d x \sqrt{d+i c d x} \sqrt{f-i c f x}+6 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2+b d \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)}{8 c f \sqrt{c^2 x^2+1}}","\frac{3 d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{4 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{d^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c d^2 x^2 \sqrt{c^2 x^2+1}}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b d^2 x \sqrt{c^2 x^2+1}}{\sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((-16*I)*b*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (16*I)*a*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 4*a*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 4*b*d*(-4*I + c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 6*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 + b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + 12*a*d^(3/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(8*c*f*Sqrt[1 + c^2*x^2])","A",1
554,1,227,158,0.5707426,"\int \frac{\sqrt{d+i c d x} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{f-i c f x}} \, dx","Integrate[(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x]))/Sqrt[f - I*c*f*x],x]","\frac{-2 i \sqrt{d+i c d x} \sqrt{f-i c f x} \left(b c x-a \sqrt{c^2 x^2+1}\right)+2 a \sqrt{d} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+2 i b \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)+b \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2}{2 c f \sqrt{c^2 x^2+1}}","\frac{d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{i b d x \sqrt{c^2 x^2+1}}{\sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((-2*I)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(b*c*x - a*Sqrt[1 + c^2*x^2]) + (2*I)*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 + 2*a*Sqrt[d]*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(2*c*f*Sqrt[1 + c^2*x^2])","A",1
555,1,113,59,0.4503797,"\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{d+i c d x} \sqrt{f-i c f x}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]),x]","\frac{a \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)}{c \sqrt{d} \sqrt{f}}+\frac{b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2}{2 c \sqrt{d+i c d x} \sqrt{f-i c f x}}","\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"(b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2)/(2*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (a*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(c*Sqrt[d]*Sqrt[f])","A",1
556,1,113,111,0.3795648,"\int \frac{a+b \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} \sqrt{f-i c f x}} \, dx","Integrate[(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*Sqrt[f - I*c*f*x]),x]","\frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left(a \sqrt{c^2 x^2+1}+b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+b (-c x+i) \log (d+i c d x)\right)}{c d^2 f (c x-i) \sqrt{c^2 x^2+1}}","\frac{f (c x+i) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b f \left(c^2 x^2+1\right)^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a*Sqrt[1 + c^2*x^2] + b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + b*(I - c*x)*Log[d + I*c*d*x]))/(c*d^2*f*(-I + c*x)*Sqrt[1 + c^2*x^2])","A",1
557,1,143,295,0.5046878,"\int \frac{a+b \sinh ^{-1}(c x)}{(d+i c d x)^{5/2} \sqrt{f-i c f x}} \, dx","Integrate[(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]),x]","\frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left((c x-2 i) \left(a \sqrt{c^2 x^2+1}+b c x-i b\right)+b (c x-2 i) \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-b (c x-i)^2 \log (d+i c d x)\right)}{3 c d^3 f (c x-i)^2 \sqrt{c^2 x^2+1}}","\frac{f^2 x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i f^2 (1-i c x) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b f^2 \left(c^2 x^2+1\right)^{5/2}}{3 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b f^2 \left(c^2 x^2+1\right)^{5/2} \log \left(c^2 x^2+1\right)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i b f^2 \left(c^2 x^2+1\right)^{5/2} \tan ^{-1}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((-2*I + c*x)*((-I)*b + b*c*x + a*Sqrt[1 + c^2*x^2]) + b*(-2*I + c*x)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - b*(-I + c*x)^2*Log[d + I*c*d*x]))/(3*c*d^3*f*(-I + c*x)^2*Sqrt[1 + c^2*x^2])","A",1
558,1,781,517,4.0194157,"\int \frac{(d+i c d x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{(f-i c f x)^{3/2}} \, dx","Integrate[((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(3/2),x]","\frac{\frac{4 a d^2 \left(c^2 x^2-7 i c x+24\right) \sqrt{d+i c d x} \sqrt{f-i c f x}}{f^2 (c x+i)}-\frac{60 a d^{5/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)}{f^{3/2}}+\frac{4 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(4 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i \log \left(c^2 x^2+1\right)\right)-\left(\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+4 \sinh ^{-1}(c x) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{f^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \sinh ^{-1}(c x) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(-8 \sqrt{c^2 x^2+1}-i \sinh \left(2 \sinh ^{-1}(c x)\right)+8\right)+\left(\sinh \left(2 \sinh ^{-1}(c x)\right)-8 i \left(\sqrt{c^2 x^2+1}+1\right)\right) \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(8 i \log \left(c^2 x^2+1\right)+16 c x+i \cosh \left(2 \sinh ^{-1}(c x)\right)+32 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)-10 \sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{f^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{16 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(\sinh ^{-1}(c x) \left(-\left(\sqrt{c^2 x^2+1}-2\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \left(\sqrt{c^2 x^2+1}+2\right) \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(i \log \left(c^2 x^2+1\right)+c x-4 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)-\left(\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{f^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}}{8 c}","-\frac{5 i d^4 (1+i c x) \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{15 i d^4 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 i d^4 (1+i c x)^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{15 d^4 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) \left(a+b \sinh ^{-1}(c x)\right)}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{b c d^4 x^2 \left(c^2 x^2+1\right)^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{5 b d^4 (1+i c x)^2 \left(c^2 x^2+1\right)^{3/2}}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{3 i b d^4 x \left(c^2 x^2+1\right)^{3/2}}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b d^4 \left(c^2 x^2+1\right)^{3/2} \log (c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{15 b d^4 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)^2}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"((4*a*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(24 - (7*I)*c*x + c^2*x^2))/(f^2*(I + c*x)) - (60*a*d^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/f^(3/2) + (4*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + 4*ArcSinh[c*x]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + 2*(4*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(f^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + (16*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + (c*x - 4*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + ArcSinh[c*x]*((-I)*(2 + Sqrt[1 + c^2*x^2])*Cosh[ArcSinh[c*x]/2] - (-2 + Sqrt[1 + c^2*x^2])*Sinh[ArcSinh[c*x]/2])))/(f^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + (b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-10*ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]) + (16*c*x + 32*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*Cosh[2*ArcSinh[c*x]] + (8*I)*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + 2*ArcSinh[c*x]*(Sinh[ArcSinh[c*x]/2]*(8 - 8*Sqrt[1 + c^2*x^2] - I*Sinh[2*ArcSinh[c*x]]) + Cosh[ArcSinh[c*x]/2]*((-8*I)*(1 + Sqrt[1 + c^2*x^2]) + Sinh[2*ArcSinh[c*x]]))))/(f^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(8*c)","A",1
559,1,515,283,2.4307881,"\int \frac{(d+i c d x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{(f-i c f x)^{3/2}} \, dx","Integrate[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(3/2),x]","\frac{-\frac{6 a d^{3/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)}{f^{3/2}}+\frac{2 a d (5-i c x) \sqrt{d+i c d x} \sqrt{f-i c f x}}{f^2 (c x+i)}+\frac{b d \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(4 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i \log \left(c^2 x^2+1\right)\right)-\left(\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+4 \sinh ^{-1}(c x) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{f^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{2 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \left(\sinh ^{-1}(c x) \left(-\left(\sqrt{c^2 x^2+1}-2\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \left(\sqrt{c^2 x^2+1}+2\right) \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(i \log \left(c^2 x^2+1\right)+c x-4 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)-\left(\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{f^2 \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}}{2 c}","-\frac{3 d^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{i d^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 i d^3 (1+i c x) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{i b d^3 x \left(c^2 x^2+1\right)^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b d^3 \left(c^2 x^2+1\right)^{3/2} \log (c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"((2*a*d*(5 - I*c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(f^2*(I + c*x)) - (6*a*d^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/f^(3/2) + (b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + 4*ArcSinh[c*x]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + 2*(4*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(f^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + (2*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + (c*x - 4*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + ArcSinh[c*x]*((-I)*(2 + Sqrt[1 + c^2*x^2])*Cosh[ArcSinh[c*x]/2] - (-2 + Sqrt[1 + c^2*x^2])*Sinh[ArcSinh[c*x]/2])))/(f^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(2*c)","A",1
560,1,285,180,1.2753374,"\int \frac{\sqrt{d+i c d x} \left(a+b \sinh ^{-1}(c x)\right)}{(f-i c f x)^{3/2}} \, dx","Integrate[(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(3/2),x]","\frac{\frac{4 a \sqrt{d+i c d x} \sqrt{f-i c f x}}{c x+i}-2 a \sqrt{d} \sqrt{f} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+\frac{b \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(4 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i \log \left(c^2 x^2+1\right)\right)-\left(\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+4 \sinh ^{-1}(c x) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{\sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}}{2 c f^2}","-\frac{d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 i d^2 (1+i c x) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 b d^2 \left(c^2 x^2+1\right)^{3/2} \log (c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"((4*a*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(I + c*x) - 2*a*Sqrt[d]*Sqrt[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + (b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + 4*ArcSinh[c*x]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + 2*(4*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(2*c*f^2)","A",1
561,1,94,112,0.4136074,"\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{d+i c d x} (f-i c f x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(3/2)),x]","\frac{\sqrt{f-i c f x} \left(i a c x+a-i b \sqrt{c^2 x^2+1} \log (d (-1+i c x))+(b+i b c x) \sinh ^{-1}(c x)\right)}{c f^2 (c x+i) \sqrt{d+i c d x}}","-\frac{d (-c x+i) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b d \left(c^2 x^2+1\right)^{3/2} \log (c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"(Sqrt[f - I*c*f*x]*(a + I*a*c*x + (b + I*b*c*x)*ArcSinh[c*x] - I*b*Sqrt[1 + c^2*x^2]*Log[d*(-1 + I*c*x)]))/(c*f^2*(I + c*x)*Sqrt[d + I*c*d*x])","A",1
562,1,118,103,0.4530133,"\int \frac{a+b \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)),x]","\frac{i \sqrt{f-i c f x} \left(2 a c x-b \sqrt{c^2 x^2+1} \log (d (-1+i c x))-b \sqrt{c^2 x^2+1} \log (d+i c d x)+2 b c x \sinh ^{-1}(c x)\right)}{2 c d f^2 (c x+i) \sqrt{d+i c d x}}","\frac{x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b \left(c^2 x^2+1\right)^{3/2} \log \left(c^2 x^2+1\right)}{2 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"((I/2)*Sqrt[f - I*c*f*x]*(2*a*c*x + 2*b*c*x*ArcSinh[c*x] - b*Sqrt[1 + c^2*x^2]*Log[d*(-1 + I*c*x)] - b*Sqrt[1 + c^2*x^2]*Log[d + I*c*d*x]))/(c*d*f^2*(I + c*x)*Sqrt[d + I*c*d*x])","A",1
563,1,201,282,0.6363719,"\int \frac{a+b \sinh ^{-1}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)),x]","\frac{\sqrt{f-i c f x} \left(8 i a c^2 x^2+8 a c x+4 i a-5 i b c x \sqrt{c^2 x^2+1} \log (d+i c d x)+3 b (-1-i c x) \sqrt{c^2 x^2+1} \log (d (-1+i c x))-5 b \sqrt{c^2 x^2+1} \log (d+i c d x)+2 b \sqrt{c^2 x^2+1}+4 b \left(2 i c^2 x^2+2 c x+i\right) \sinh ^{-1}(c x)\right)}{12 d^2 f^2 \left(c^3 x^2+c\right) \sqrt{d+i c d x}}","\frac{2 f x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{f (c x+i) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b f \left(c^2 x^2+1\right)^{5/2}}{6 c (-c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b f \left(c^2 x^2+1\right)^{5/2} \log \left(c^2 x^2+1\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i b f \left(c^2 x^2+1\right)^{5/2} \tan ^{-1}(c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[f - I*c*f*x]*((4*I)*a + 8*a*c*x + (8*I)*a*c^2*x^2 + 2*b*Sqrt[1 + c^2*x^2] + 4*b*(I + 2*c*x + (2*I)*c^2*x^2)*ArcSinh[c*x] + 3*b*(-1 - I*c*x)*Sqrt[1 + c^2*x^2]*Log[d*(-1 + I*c*x)] - 5*b*Sqrt[1 + c^2*x^2]*Log[d + I*c*d*x] - (5*I)*b*c*x*Sqrt[1 + c^2*x^2]*Log[d + I*c*d*x]))/(12*d^2*f^2*Sqrt[d + I*c*d*x]*(c + c^3*x^2))","A",1
564,1,1331,470,8.7527436,"\int \frac{(d+i c d x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{(f-i c f x)^{5/2}} \, dx","Integrate[((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(5/2),x]","\frac{5 a \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i d (c x-i)} \sqrt{-i f (c x+i)}\right) d^{5/2}}{c f^{5/2}}-\frac{i b \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(-\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i \log \left(\sqrt{c^2 x^2+1}\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(3 \sinh ^{-1}(c x)-6 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+3 i \log \left(\sqrt{c^2 x^2+1}\right)+4 i\right)+2 \left(\sqrt{c^2 x^2+1} \left(i \sinh ^{-1}(c x)+2 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\log \left(\sqrt{c^2 x^2+1}\right)\right)+2 \left(i \sinh ^{-1}(c x)+2 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\log \left(\sqrt{c^2 x^2+1}\right)+1\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) d^2}{6 c f^3 (i c x+1) \sqrt{-((c d x-i d) (c x f+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}+\frac{b \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\left(14 i-3 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+28 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-14 \log \left(\sqrt{c^2 x^2+1}\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(9 \sinh ^{-1}(c x)^2+6 i \sinh ^{-1}(c x)-84 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+42 \log \left(\sqrt{c^2 x^2+1}\right)+8\right)-2 i \left(6 \sinh ^{-1}(c x)^2+4 i \sinh ^{-1}(c x)-56 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+28 \log \left(\sqrt{c^2 x^2+1}\right)+\sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x) \left(3 \sinh ^{-1}(c x)+14 i\right)-28 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+14 \log \left(\sqrt{c^2 x^2+1}\right)\right)+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) d^2}{6 c f^3 (i c x+1) \sqrt{-((c d x-i d) (c x f+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}-\frac{i b \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(-\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(9 \sinh ^{-1}(c x)^2-35 i \sinh ^{-1}(c x)+52 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+26 \log \left(\sqrt{c^2 x^2+1}\right)+9\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(27 \sinh ^{-1}(c x)^2+24 i \sinh ^{-1}(c x)+156 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+78 \log \left(\sqrt{c^2 x^2+1}\right)+20\right)-i \left(3 \left(\sinh ^{-1}(c x)-i\right) \cosh \left(\frac{5}{2} \sinh ^{-1}(c x)\right)+2 \left(18 \sinh ^{-1}(c x)^2+7 i \sinh ^{-1}(c x)+104 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+3 i \left(\sinh ^{-1}(c x)+i\right) \cosh \left(2 \sinh ^{-1}(c x)\right)+52 \log \left(\sqrt{c^2 x^2+1}\right)+\sqrt{c^2 x^2+1} \left(9 \sinh ^{-1}(c x)^2+38 i \sinh ^{-1}(c x)+52 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+26 \log \left(\sqrt{c^2 x^2+1}\right)+6\right)+13\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right) d^2}{12 c f^3 (c x-i) \sqrt{-((c d x-i d) (c x f+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}+\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left(-\frac{28 a d^2}{3 f^3 (c x+i)}+\frac{8 i a d^2}{3 f^3 (c x+i)^2}+\frac{i a d^2}{f^3}\right)}{c}","\frac{5 i d^5 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{10 i d^5 (1+i c x)^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i d^5 (1+i c x)^4 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{5 d^5 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i b d^5 x \left(c^2 x^2+1\right)^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{8 i b d^5 \left(c^2 x^2+1\right)^{5/2}}{3 c (c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{28 b d^5 \left(c^2 x^2+1\right)^{5/2} \log (c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{5 b d^5 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((I*a*d^2)/f^3 + (((8*I)/3)*a*d^2)/(f^3*(I + c*x)^2) - (28*a*d^2)/(3*f^3*(I + c*x))))/c + (5*a*d^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(c*f^(5/2)) - ((I/6)*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(-(Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]])) + Cosh[ArcSinh[c*x]/2]*(4*I + 3*ArcSinh[c*x] - 6*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*I)*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x^2]*(I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[Sqrt[1 + c^2*x^2]]) + 2*(1 + I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*f^3*(1 + I*c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) + (b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSinh[c*x])/2]*((14*I - 3*ArcSinh[c*x])*ArcSinh[c*x] + (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] - 14*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(8 + (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 - (84*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 42*Log[Sqrt[1 + c^2*x^2]]) - (2*I)*(4 + (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 - (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 28*Log[Sqrt[1 + c^2*x^2]] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(14*I + 3*ArcSinh[c*x]) - (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(6*c*f^3*(1 + I*c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) - ((I/12)*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(-(Cosh[(3*ArcSinh[c*x])/2]*(9 - (35*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 + (52*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 26*Log[Sqrt[1 + c^2*x^2]])) + Cosh[ArcSinh[c*x]/2]*(20 + (24*I)*ArcSinh[c*x] + 27*ArcSinh[c*x]^2 + (156*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 78*Log[Sqrt[1 + c^2*x^2]]) - I*(3*(-I + ArcSinh[c*x])*Cosh[(5*ArcSinh[c*x])/2] + 2*(13 + (7*I)*ArcSinh[c*x] + 18*ArcSinh[c*x]^2 + (104*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*I)*(I + ArcSinh[c*x])*Cosh[2*ArcSinh[c*x]] + 52*Log[Sqrt[1 + c^2*x^2]] + Sqrt[1 + c^2*x^2]*(6 + (38*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 + (52*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 26*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2])))/(c*f^3*(-I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4)","B",0
565,1,706,362,5.6406344,"\int \frac{(d+i c d x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{(f-i c f x)^{5/2}} \, dx","Integrate[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(5/2),x]","\frac{\frac{12 a d^{3/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)}{f^{5/2}}-\frac{16 a d (2 c x+i) \sqrt{d+i c d x} \sqrt{f-i c f x}}{f^3 (c x+i)^2}+\frac{b d \sqrt{d+i c d x} \sqrt{f-i c f x} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(-2 i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(14 \log \left(c^2 x^2+1\right)+\sqrt{c^2 x^2+1} \left(7 \log \left(c^2 x^2+1\right)+\sinh ^{-1}(c x) \left(3 \sinh ^{-1}(c x)+14 i\right)-28 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+6 \sinh ^{-1}(c x)^2+4 i \sinh ^{-1}(c x)-56 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4\right)+\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(-7 \log \left(c^2 x^2+1\right)+\left(-3 \sinh ^{-1}(c x)+14 i\right) \sinh ^{-1}(c x)+28 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(21 \log \left(c^2 x^2+1\right)+9 \sinh ^{-1}(c x)^2+6 i \sinh ^{-1}(c x)-84 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+8\right)\right)}{f^3 (1+i c x) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}-\frac{2 i b d \sqrt{d+i c d x} \sqrt{f-i c f x} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(2 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\log \left(c^2 x^2+1\right)+\frac{1}{2} \sqrt{c^2 x^2+1} \left(\log \left(c^2 x^2+1\right)+2 i \sinh ^{-1}(c x)+4 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+2 i \sinh ^{-1}(c x)+4 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+2\right)-\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\frac{1}{2} i \log \left(c^2 x^2+1\right)+\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\frac{3}{2} i \log \left(c^2 x^2+1\right)+3 \sinh ^{-1}(c x)-6 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4 i\right)\right)}{f^3 (1+i c x) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}}{12 c}","\frac{2 i d^4 (1+i c x) \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i d^4 (1+i c x)^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{d^4 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 i b d^4 \left(c^2 x^2+1\right)^{5/2}}{3 c (c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{8 b d^4 \left(c^2 x^2+1\right)^{5/2} \log (c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b d^4 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)^2}{2 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"((-16*a*d*(I + 2*c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(f^3*(I + c*x)^2) + (12*a*d^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/f^(5/2) - ((2*I)*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(-(Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] + (I/2)*Log[1 + c^2*x^2])) + Cosh[ArcSinh[c*x]/2]*(4*I + 3*ArcSinh[c*x] - 6*ArcTan[Coth[ArcSinh[c*x]/2]] + ((3*I)/2)*Log[1 + c^2*x^2]) + 2*(2 + (2*I)*ArcSinh[c*x] + (4*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[1 + c^2*x^2] + (Sqrt[1 + c^2*x^2]*((2*I)*ArcSinh[c*x] + (4*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[1 + c^2*x^2]))/2)*Sinh[ArcSinh[c*x]/2]))/(f^3*(1 + I*c*x)*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) + (b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSinh[c*x])/2]*((14*I - 3*ArcSinh[c*x])*ArcSinh[c*x] + (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] - 7*Log[1 + c^2*x^2]) + Cosh[ArcSinh[c*x]/2]*(8 + (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 - (84*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 21*Log[1 + c^2*x^2]) - (2*I)*(4 + (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 - (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*Log[1 + c^2*x^2] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(14*I + 3*ArcSinh[c*x]) - (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 7*Log[1 + c^2*x^2]))*Sinh[ArcSinh[c*x]/2]))/(f^3*(1 + I*c*x)*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4))/(12*c)","A",0
566,1,131,185,0.4304842,"\int \frac{\sqrt{d+i c d x} \left(a+b \sinh ^{-1}(c x)\right)}{(f-i c f x)^{5/2}} \, dx","Integrate[(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x]))/(f - I*c*f*x)^(5/2),x]","-\frac{i d \sqrt{f-i c f x} \left((c x-i) \left(a c x-i a+b \sqrt{c^2 x^2+1}\right)-b (c x+i) \sqrt{c^2 x^2+1} \log (d (-1+i c x))+b (c x-i)^2 \sinh ^{-1}(c x)\right)}{3 c f^3 (c x+i)^2 \sqrt{d+i c d x}}","-\frac{i d^3 (1+i c x)^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i b d^3 \left(c^2 x^2+1\right)^{5/2}}{3 c (c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b d^3 \left(c^2 x^2+1\right)^{5/2} \log (c x+i)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"((-1/3*I)*d*Sqrt[f - I*c*f*x]*((-I + c*x)*((-I)*a + a*c*x + b*Sqrt[1 + c^2*x^2]) + b*(-I + c*x)^2*ArcSinh[c*x] - b*(I + c*x)*Sqrt[1 + c^2*x^2]*Log[d*(-1 + I*c*x)]))/(c*f^3*(I + c*x)^2*Sqrt[d + I*c*d*x])","A",1
567,1,139,294,0.4654795,"\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{d+i c d x} (f-i c f x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(5/2)),x]","\frac{\sqrt{f-i c f x} \left((c x+2 i) \left(i a c x+a+i b \sqrt{c^2 x^2+1}\right)+b (1-i c x) \sqrt{c^2 x^2+1} \log (d (-1+i c x))+i b \left(c^2 x^2+i c x+2\right) \sinh ^{-1}(c x)\right)}{3 c f^3 (c x+i)^2 \sqrt{d+i c d x}}","\frac{d^2 x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i d^2 (1+i c x) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b d^2 \left(c^2 x^2+1\right)^{5/2}}{3 c (c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b d^2 \left(c^2 x^2+1\right)^{5/2} \log \left(c^2 x^2+1\right)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b d^2 \left(c^2 x^2+1\right)^{5/2} \tan ^{-1}(c x)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[f - I*c*f*x]*((2*I + c*x)*(a + I*a*c*x + I*b*Sqrt[1 + c^2*x^2]) + I*b*(2 + I*c*x + c^2*x^2)*ArcSinh[c*x] + b*(1 - I*c*x)*Sqrt[1 + c^2*x^2]*Log[d*(-1 + I*c*x)]))/(3*c*f^3*(I + c*x)^2*Sqrt[d + I*c*d*x])","A",1
568,1,202,282,0.6042149,"\int \frac{a+b \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)),x]","\frac{\sqrt{f-i c f x} \left(8 i a c^2 x^2-8 a c x+4 i a-3 i b c x \sqrt{c^2 x^2+1} \log (d+i c d x)+5 b (1-i c x) \sqrt{c^2 x^2+1} \log (d (-1+i c x))+3 b \sqrt{c^2 x^2+1} \log (d+i c d x)-2 b \sqrt{c^2 x^2+1}+4 i b \left(2 c^2 x^2+2 i c x+1\right) \sinh ^{-1}(c x)\right)}{12 c d f^3 (c x+i)^2 \sqrt{d+i c d x}}","\frac{2 d x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{d (-c x+i) \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b d \left(c^2 x^2+1\right)^{5/2}}{6 c (c x+i) (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b d \left(c^2 x^2+1\right)^{5/2} \log \left(c^2 x^2+1\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b d \left(c^2 x^2+1\right)^{5/2} \tan ^{-1}(c x)}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[f - I*c*f*x]*((4*I)*a - 8*a*c*x + (8*I)*a*c^2*x^2 - 2*b*Sqrt[1 + c^2*x^2] + (4*I)*b*(1 + (2*I)*c*x + 2*c^2*x^2)*ArcSinh[c*x] + 5*b*(1 - I*c*x)*Sqrt[1 + c^2*x^2]*Log[d*(-1 + I*c*x)] + 3*b*Sqrt[1 + c^2*x^2]*Log[d + I*c*d*x] - (3*I)*b*c*x*Sqrt[1 + c^2*x^2]*Log[d + I*c*d*x]))/(12*c*d*f^3*(I + c*x)^2*Sqrt[d + I*c*d*x])","A",1
569,1,193,203,0.5723919,"\int \frac{a+b \sinh ^{-1}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)),x]","\frac{i \sqrt{f-i c f x} \left(4 a c^3 x^3+6 a c x-2 b c^2 x^2 \sqrt{c^2 x^2+1} \log (d+i c d x)-2 b \left(c^2 x^2+1\right)^{3/2} \log (d (-1+i c x))-2 b \sqrt{c^2 x^2+1} \log (d+i c d x)+b \sqrt{c^2 x^2+1}+2 b c x \left(2 c^2 x^2+3\right) \sinh ^{-1}(c x)\right)}{6 c d^2 f^3 (c x-i) (c x+i)^2 \sqrt{d+i c d x}}","\frac{2 x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b \left(c^2 x^2+1\right)^{3/2}}{6 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b \left(c^2 x^2+1\right)^{5/2} \log \left(c^2 x^2+1\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"((I/6)*Sqrt[f - I*c*f*x]*(6*a*c*x + 4*a*c^3*x^3 + b*Sqrt[1 + c^2*x^2] + 2*b*c*x*(3 + 2*c^2*x^2)*ArcSinh[c*x] - 2*b*(1 + c^2*x^2)^(3/2)*Log[d*(-1 + I*c*x)] - 2*b*Sqrt[1 + c^2*x^2]*Log[d + I*c*d*x] - 2*b*c^2*x^2*Sqrt[1 + c^2*x^2]*Log[d + I*c*d*x]))/(c*d^2*f^3*(-I + c*x)*(I + c*x)^2*Sqrt[d + I*c*d*x])","A",1
570,1,890,680,2.4102089,"\int (d+i c d x)^{5/2} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2,x]","\frac{-1728 a^2 c^3 d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^3+4608 i a^2 c^2 d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^2-6912 i a b c d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} x+2592 a^2 c d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x+1440 b^2 d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-1728 a b d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)+256 i b^2 d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(3 \sinh ^{-1}(c x)\right)+108 a b d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)+4320 a^2 d^{5/2} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i c x d+d} \sqrt{f-i c f x}\right)+864 b^2 d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)-768 i a b d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)-27 b^2 d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(4 \sinh ^{-1}(c x)\right)+12 b d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(192 i \cosh \left(3 \sinh ^{-1}(c x)\right) a+288 \sinh \left(2 \sinh ^{-1}(c x)\right) a-36 \sinh \left(4 \sinh ^{-1}(c x)\right) a+576 i \sqrt{c^2 x^2+1} a-576 i b c x-144 b \cosh \left(2 \sinh ^{-1}(c x)\right)+9 b \cosh \left(4 \sinh ^{-1}(c x)\right)-64 i b \sinh \left(3 \sinh ^{-1}(c x)\right)\right)+72 b d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(60 a+16 i b \cosh \left(3 \sinh ^{-1}(c x)\right)+24 b \sinh \left(2 \sinh ^{-1}(c x)\right)-3 b \sinh \left(4 \sinh ^{-1}(c x)\right)+48 i b \sqrt{c^2 x^2+1}\right)+4608 i a^2 d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}+6912 i b^2 d^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}}{6912 c \sqrt{c^2 x^2+1}}","-\frac{1}{4} c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{3 b c d^2 x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{8 \sqrt{c^2 x^2+1}}-\frac{4 i b d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 x^2+1}}+\frac{5 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^3}{24 b c \sqrt{c^2 x^2+1}}+\frac{2 i d^2 \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c}-\frac{4 i b c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{9 \sqrt{c^2 x^2+1}}+\frac{b c^3 d^2 x^4 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{8 \sqrt{c^2 x^2+1}}+\frac{3}{8} d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{1}{32} b^2 c^2 d^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}+\frac{4 i b^2 d^2 \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x}}{27 c}-\frac{15 b^2 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)}{64 c \sqrt{c^2 x^2+1}}+\frac{15}{64} b^2 d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}+\frac{8 i b^2 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 c}",1,"((-6912*I)*a*b*c*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (4608*I)*a^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (6912*I)*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2592*a^2*c*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (4608*I)*a^2*c^2*d^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 1728*a^2*c^3*d^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 1440*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 1728*a*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + (256*I)*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] + 108*a*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 4320*a^2*d^(5/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 864*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] - (768*I)*a*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] - 27*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c*x]] + 12*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((-576*I)*b*c*x + (576*I)*a*Sqrt[1 + c^2*x^2] - 144*b*Cosh[2*ArcSinh[c*x]] + (192*I)*a*Cosh[3*ArcSinh[c*x]] + 9*b*Cosh[4*ArcSinh[c*x]] + 288*a*Sinh[2*ArcSinh[c*x]] - (64*I)*b*Sinh[3*ArcSinh[c*x]] - 36*a*Sinh[4*ArcSinh[c*x]]) + 72*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a + (48*I)*b*Sqrt[1 + c^2*x^2] + (16*I)*b*Cosh[3*ArcSinh[c*x]] + 24*b*Sinh[2*ArcSinh[c*x]] - 3*b*Sinh[4*ArcSinh[c*x]]))/(6912*c*Sqrt[1 + c^2*x^2])","A",1
571,1,705,508,1.8434598,"\int (d+i c d x)^{3/2} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + I*c*d*x)^(3/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2,x]","\frac{108 a^2 d^{3/2} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+72 i a^2 c^2 d x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+72 i a^2 d \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+108 a^2 c d x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+18 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(6 a+3 i b \sqrt{c^2 x^2+1}+3 b \sinh \left(2 \sinh ^{-1}(c x)\right)+i b \cosh \left(3 \sinh ^{-1}(c x)\right)\right)+6 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(-9 b \cosh \left(2 \sinh ^{-1}(c x)\right)+2 \left(9 i a \sqrt{c^2 x^2+1}+9 a \sinh \left(2 \sinh ^{-1}(c x)\right)+3 i a \cosh \left(3 \sinh ^{-1}(c x)\right)-9 i b c x-i b \sinh \left(3 \sinh ^{-1}(c x)\right)\right)\right)-108 i a b c d x \sqrt{d+i c d x} \sqrt{f-i c f x}-12 i a b d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)-54 a b d \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)+108 i b^2 d \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+36 b^2 d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3+27 b^2 d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)+4 i b^2 d \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(3 \sinh ^{-1}(c x)\right)}{216 c \sqrt{c^2 x^2+1}}","-\frac{b c d x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{c^2 x^2+1}}-\frac{2 i b d x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 x^2+1}}+\frac{d \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b c \sqrt{c^2 x^2+1}}+\frac{i d \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c}-\frac{2 i b c^2 d x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{9 \sqrt{c^2 x^2+1}}+\frac{1}{2} d x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2 i b^2 d \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x}}{27 c}-\frac{b^2 d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt{c^2 x^2+1}}+\frac{1}{4} b^2 d x \sqrt{d+i c d x} \sqrt{f-i c f x}+\frac{4 i b^2 d \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 c}",1,"((-108*I)*a*b*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (72*I)*a^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (108*I)*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 108*a^2*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (72*I)*a^2*c^2*d*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 36*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 54*a*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + (4*I)*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] + 108*a^2*d^(3/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 27*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 18*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(6*a + (3*I)*b*Sqrt[1 + c^2*x^2] + I*b*Cosh[3*ArcSinh[c*x]] + 3*b*Sinh[2*ArcSinh[c*x]]) - (12*I)*a*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 6*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(-9*b*Cosh[2*ArcSinh[c*x]] + 2*((-9*I)*b*c*x + (9*I)*a*Sqrt[1 + c^2*x^2] + (3*I)*a*Cosh[3*ArcSinh[c*x]] + 9*a*Sinh[2*ArcSinh[c*x]] - I*b*Sinh[3*ArcSinh[c*x]])))/(216*c*Sqrt[1 + c^2*x^2])","A",1
572,1,352,244,1.0212038,"\int \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2,x]","\frac{12 a^2 c x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+12 a^2 \sqrt{d} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+6 b \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(2 a+b \sinh \left(2 \sinh ^{-1}(c x)\right)\right)-6 b \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(b \cosh \left(2 \sinh ^{-1}(c x)\right)-2 a \sinh \left(2 \sinh ^{-1}(c x)\right)\right)-6 a b \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)+4 b^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3+3 b^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)}{24 c \sqrt{c^2 x^2+1}}","\frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b c \sqrt{c^2 x^2+1}}-\frac{b c x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{c^2 x^2+1}}+\frac{1}{2} x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{b^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt{c^2 x^2+1}}+\frac{1}{4} b^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}",1,"(12*a^2*c*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 4*b^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 6*a*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + 12*a^2*Sqrt[d]*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 3*b^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] - 6*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(b*Cosh[2*ArcSinh[c*x]] - 2*a*Sinh[2*ArcSinh[c*x]]) + 6*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(2*a + b*Sinh[2*ArcSinh[c*x]]))/(24*c*Sqrt[1 + c^2*x^2])","A",1
573,1,315,259,1.1700326,"\int \frac{\sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+i c d x}} \, dx","Integrate[(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/Sqrt[d + I*c*d*x],x]","\frac{-3 i \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a^2 \sqrt{c^2 x^2+1}-2 a b c x+2 b^2 \sqrt{c^2 x^2+1}\right)+3 a^2 \sqrt{d} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+3 b \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(a-i b \sqrt{c^2 x^2+1}\right)+6 i b \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(b c x-a \sqrt{c^2 x^2+1}\right)+b^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3}{3 c d \sqrt{c^2 x^2+1}}","\frac{2 i a b f x \sqrt{c^2 x^2+1}}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{f \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{i f \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b^2 f \left(c^2 x^2+1\right)}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i b^2 f x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{\sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((-3*I)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-2*a*b*c*x + a^2*Sqrt[1 + c^2*x^2] + 2*b^2*Sqrt[1 + c^2*x^2]) + (6*I)*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(b*c*x - a*Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + 3*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a - I*b*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 + b^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 + 3*a^2*Sqrt[d]*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(3*c*d*Sqrt[1 + c^2*x^2])","A",1
574,1,594,544,4.2438474,"\int \frac{\sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2}} \, dx","Integrate[(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(3/2),x]","\frac{\frac{6 a^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{c x-i}-3 a^2 \sqrt{d} \sqrt{f} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)-\frac{3 a b \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\log \left(c^2 x^2+1\right)+4 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\sinh ^{-1}(c x) \left(-4 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-4 i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{\sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{b^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(24 \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-\left(\sinh ^{-1}(c x)^3 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)-(6-6 i) \sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+6 \sinh ^{-1}(c x) \left(\pi -4 i \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)\right) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+12 i \pi  \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+2 \log \left(e^{\sinh ^{-1}(c x)}+1\right)-\log \left(\sin \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)-2 \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}}{3 c d^2}","-\frac{f^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 f^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 i f^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 f^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b f^2 \left(c^2 x^2+1\right)^{3/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b f^2 \left(c^2 x^2+1\right)^{3/2} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b^2 f^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 b^2 f^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 b^2 f^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"((6*a^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(-I + c*x) - 3*a^2*Sqrt[d]*Sqrt[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - (3*a*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(ArcSinh[c*x]*((-4*I)*Cosh[ArcSinh[c*x]/2] - 4*Sinh[ArcSinh[c*x]/2]) + ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + 2*((4*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + Log[1 + c^2*x^2])*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (b^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((-6 + 6*I)*ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - Sinh[ArcSinh[c*x]/2]) - ArcSinh[c*x]^3*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + (12*I)*Pi*(Log[1 - I/E^ArcSinh[c*x]] + 2*Log[1 + E^ArcSinh[c*x]] - 2*Log[Cosh[ArcSinh[c*x]/2]] - Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + 24*PolyLog[2, I/E^ArcSinh[c*x]]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + 6*ArcSinh[c*x]*(Pi - (4*I)*Log[1 - I/E^ArcSinh[c*x]])*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(3*c*d^2)","A",0
575,1,783,518,8.179378,"\int \frac{\sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{5/2}} \, dx","Integrate[(Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(5/2),x]","\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left(-\frac{a^2}{3 d^3 (c x-i)}-\frac{2 i a^2}{3 d^3 (c x-i)^2}\right)}{c}+\frac{i a b \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(2 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\sqrt{c^2 x^2+1} \left(i \log \left(\sqrt{c^2 x^2+1}\right)+\sinh ^{-1}(c x)+2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+2 \left(i \log \left(\sqrt{c^2 x^2+1}\right)+\sinh ^{-1}(c x)+2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i\right)\right)-i \cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(-i \log \left(\sqrt{c^2 x^2+1}\right)+\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(3 \log \left(\sqrt{c^2 x^2+1}\right)+3 i \sinh ^{-1}(c x)-6 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4\right)\right)}{3 c d^3 (c x+i) \sqrt{-((c d x-i d) (c f x+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}+\frac{i b^2 (c x+i) \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(4 \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-(1-i) \sinh ^{-1}(c x)^2-\frac{2 \left(\sinh ^{-1}(c x)-2 i\right) \sinh ^{-1}(c x)}{c x-i}+2 i \left(\pi +2 i \sinh ^{-1}(c x)\right) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-\frac{4 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}+\frac{2 \left(\sinh ^{-1}(c x)^2+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}-i \pi  \left(\sinh ^{-1}(c x)-4 \log \left(e^{\sinh ^{-1}(c x)}+1\right)+2 \log \left(\sin \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)+4 \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{3 c d^3 \sqrt{c^2 x^2+1} \sqrt{-((c d x-i d) (c f x+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^2}","-\frac{f^3 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b f^3 \left(c^2 x^2+1\right)^{5/2} \log \left(1+i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i f^3 \left(c^2 x^2+1\right)^{5/2} \cot \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b f^3 \left(c^2 x^2+1\right)^{5/2} \csc ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f^3 \left(c^2 x^2+1\right)^{5/2} \cot \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \csc ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b^2 f^3 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{4 i b^2 f^3 \left(c^2 x^2+1\right)^{5/2} \cot \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((((-2*I)/3)*a^2)/(d^3*(-I + c*x)^2) - a^2/(3*d^3*(-I + c*x))))/c + ((I/3)*a*b*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*((-I)*Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] - I*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(4 + (3*I)*ArcSinh[c*x] - (6*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 3*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x^2]*(ArcSinh[c*x] + 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]]) + 2*(I + ArcSinh[c*x] + 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) + ((I/3)*b^2*(I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*((-1 + I)*ArcSinh[c*x]^2 - (2*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/(-I + c*x) + (2*I)*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - I*Pi*(ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] + 4*Log[Cosh[ArcSinh[c*x]/2]] + 2*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) + 4*PolyLog[2, I/E^ArcSinh[c*x]] - (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(c*d^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^2)","A",0
576,1,1084,774,3.1785753,"\int (d+i c d x)^{5/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{57600 i a^2 c^4 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^4+72000 a^2 c^3 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^3+115200 i a^2 c^2 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^2-72000 i a b c d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} x+180000 a^2 c d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x+36000 b^2 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-72000 a b d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)+4000 i b^2 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(3 \sinh ^{-1}(c x)\right)-4500 a b d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)+288 i b^2 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(5 \sinh ^{-1}(c x)\right)+108000 a^2 d^{5/2} f^{3/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i c x d+d} \sqrt{f-i c f x}\right)+36000 b^2 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)-12000 i a b d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)+1125 b^2 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(4 \sinh ^{-1}(c x)\right)+1800 b d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(60 a+10 i b \cosh \left(3 \sinh ^{-1}(c x)\right)+2 i b \cosh \left(5 \sinh ^{-1}(c x)\right)+40 b \sinh \left(2 \sinh ^{-1}(c x)\right)+5 b \sinh \left(4 \sinh ^{-1}(c x)\right)+20 i b \sqrt{c^2 x^2+1}\right)-1440 i a b d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(5 \sinh ^{-1}(c x)\right)+60 b d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(600 i \cosh \left(3 \sinh ^{-1}(c x)\right) a+120 i \cosh \left(5 \sinh ^{-1}(c x)\right) a+2400 \sinh \left(2 \sinh ^{-1}(c x)\right) a+300 \sinh \left(4 \sinh ^{-1}(c x)\right) a+1200 i \sqrt{c^2 x^2+1} a-1200 i b c x-1200 b \cosh \left(2 \sinh ^{-1}(c x)\right)-75 b \cosh \left(4 \sinh ^{-1}(c x)\right)-200 i b \sinh \left(3 \sinh ^{-1}(c x)\right)-24 i b \sinh \left(5 \sinh ^{-1}(c x)\right)\right)+57600 i a^2 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}+72000 i b^2 d^2 f \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}}{288000 c \sqrt{c^2 x^2+1}}","-\frac{3 b c d x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 \left(c^2 x^2+1\right)^{3/2}}+\frac{3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{8 \left(c^2 x^2+1\right)}-\frac{2 i b d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 \left(c^2 x^2+1\right)^{3/2}}+\frac{d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{8 b c \left(c^2 x^2+1\right)^{3/2}}+\frac{i d \left(c^2 x^2+1\right) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{5 c}-\frac{b d \sqrt{c^2 x^2+1} (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 c}-\frac{4 i b c^2 d x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{15 \left(c^2 x^2+1\right)^{3/2}}-\frac{2 i b c^4 d x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{25 \left(c^2 x^2+1\right)^{3/2}}+\frac{1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{15 b^2 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left(c^2 x^2+1\right)}+\frac{2 i b^2 d \left(c^2 x^2+1\right) (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{125 c}+\frac{16 i b^2 d (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{75 c \left(c^2 x^2+1\right)}-\frac{9 b^2 d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left(c^2 x^2+1\right)^{3/2}}+\frac{1}{32} b^2 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}+\frac{8 i b^2 d (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{225 c}",1,"((-72000*I)*a*b*c*d^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (57600*I)*a^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (72000*I)*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 180000*a^2*c*d^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (115200*I)*a^2*c^2*d^2*f*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 72000*a^2*c^3*d^2*f*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (57600*I)*a^2*c^4*d^2*f*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 36000*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 72000*a*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + (4000*I)*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] - 4500*a*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + (288*I)*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[5*ArcSinh[c*x]] + 108000*a^2*d^(5/2)*f^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 36000*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] - (12000*I)*a*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 1125*b^2*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c*x]] + 1800*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a + (20*I)*b*Sqrt[1 + c^2*x^2] + (10*I)*b*Cosh[3*ArcSinh[c*x]] + (2*I)*b*Cosh[5*ArcSinh[c*x]] + 40*b*Sinh[2*ArcSinh[c*x]] + 5*b*Sinh[4*ArcSinh[c*x]]) - (1440*I)*a*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[5*ArcSinh[c*x]] + 60*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((-1200*I)*b*c*x + (1200*I)*a*Sqrt[1 + c^2*x^2] - 1200*b*Cosh[2*ArcSinh[c*x]] + (600*I)*a*Cosh[3*ArcSinh[c*x]] - 75*b*Cosh[4*ArcSinh[c*x]] + (120*I)*a*Cosh[5*ArcSinh[c*x]] + 2400*a*Sinh[2*ArcSinh[c*x]] - (200*I)*b*Sinh[3*ArcSinh[c*x]] + 300*a*Sinh[4*ArcSinh[c*x]] - (24*I)*b*Sinh[5*ArcSinh[c*x]]))/(288000*c*Sqrt[1 + c^2*x^2])","A",1
577,1,524,396,1.7377346,"\int (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{96 a^2 d^{3/2} f^{3/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+160 a^2 c d f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+64 a^2 c^3 d f x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+8 b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(12 a+8 b \sinh \left(2 \sinh ^{-1}(c x)\right)+b \sinh \left(4 \sinh ^{-1}(c x)\right)\right)-64 a b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-4 a b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)-4 b d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(-4 a \left(8 \sinh \left(2 \sinh ^{-1}(c x)\right)+\sinh \left(4 \sinh ^{-1}(c x)\right)\right)+16 b \cosh \left(2 \sinh ^{-1}(c x)\right)+b \cosh \left(4 \sinh ^{-1}(c x)\right)\right)+32 b^2 d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3+32 b^2 d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)+b^2 d f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(4 \sinh ^{-1}(c x)\right)}{256 c \sqrt{c^2 x^2+1}}","\frac{(d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{8 b c \left(c^2 x^2+1\right)^{3/2}}+\frac{3 x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{8 \left(c^2 x^2+1\right)}-\frac{b \sqrt{c^2 x^2+1} (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 c}-\frac{3 b c x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 \left(c^2 x^2+1\right)^{3/2}}+\frac{1}{4} x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{15 b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left(c^2 x^2+1\right)}-\frac{9 b^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left(c^2 x^2+1\right)^{3/2}}+\frac{1}{32} b^2 x (d+i c d x)^{3/2} (f-i c f x)^{3/2}",1,"(160*a^2*c*d*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 64*a^2*c^3*d*f*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 32*b^2*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 64*a*b*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 4*a*b*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 96*a^2*d^(3/2)*f^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 32*b^2*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + b^2*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c*x]] + 8*b*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(12*a + 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]) - 4*b*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(16*b*Cosh[2*ArcSinh[c*x]] + b*Cosh[4*ArcSinh[c*x]] - 4*a*(8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])))/(256*c*Sqrt[1 + c^2*x^2])","A",1
578,1,705,508,1.831304,"\int \sqrt{d+i c d x} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{108 a^2 \sqrt{d} f^{3/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)-72 i a^2 c^2 f x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-72 i a^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+108 a^2 c f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+18 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(6 a-3 i b \sqrt{c^2 x^2+1}+3 b \sinh \left(2 \sinh ^{-1}(c x)\right)-i b \cosh \left(3 \sinh ^{-1}(c x)\right)\right)+6 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(-9 b \cosh \left(2 \sinh ^{-1}(c x)\right)+2 \left(-9 i a \sqrt{c^2 x^2+1}+9 a \sinh \left(2 \sinh ^{-1}(c x)\right)-3 i a \cosh \left(3 \sinh ^{-1}(c x)\right)+9 i b c x+i b \sinh \left(3 \sinh ^{-1}(c x)\right)\right)\right)+108 i a b c f x \sqrt{d+i c d x} \sqrt{f-i c f x}+12 i a b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)-54 a b f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-108 i b^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+36 b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3+27 b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)-4 i b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(3 \sinh ^{-1}(c x)\right)}{216 c \sqrt{c^2 x^2+1}}","-\frac{b c f x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{c^2 x^2+1}}+\frac{2 i b f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 x^2+1}}+\frac{f \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b c \sqrt{c^2 x^2+1}}-\frac{i f \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c}+\frac{2 i b c^2 f x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{9 \sqrt{c^2 x^2+1}}+\frac{1}{2} f x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{2 i b^2 f \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x}}{27 c}-\frac{b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)}{4 c \sqrt{c^2 x^2+1}}+\frac{1}{4} b^2 f x \sqrt{d+i c d x} \sqrt{f-i c f x}-\frac{4 i b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 c}",1,"((108*I)*a*b*c*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (72*I)*a^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (108*I)*b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 108*a^2*c*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (72*I)*a^2*c^2*f*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 36*b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 54*a*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - (4*I)*b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] + 108*a^2*Sqrt[d]*f^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 27*b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 18*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(6*a - (3*I)*b*Sqrt[1 + c^2*x^2] - I*b*Cosh[3*ArcSinh[c*x]] + 3*b*Sinh[2*ArcSinh[c*x]]) + (12*I)*a*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 6*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(-9*b*Cosh[2*ArcSinh[c*x]] + 2*((9*I)*b*c*x - (9*I)*a*Sqrt[1 + c^2*x^2] - (3*I)*a*Cosh[3*ArcSinh[c*x]] + 9*a*Sinh[2*ArcSinh[c*x]] + I*b*Sinh[3*ArcSinh[c*x]])))/(216*c*Sqrt[1 + c^2*x^2])","A",1
579,1,532,436,2.2633357,"\int \frac{(f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+i c d x}} \, dx","Integrate[((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[d + I*c*d*x],x]","\frac{12 a^2 \sqrt{d} f^{3/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)-16 i a^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-4 a^2 c f x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+2 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(6 a-8 i b \sqrt{c^2 x^2+1}-b \sinh \left(2 \sinh ^{-1}(c x)\right)\right)+2 b f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(b \cosh \left(2 \sinh ^{-1}(c x)\right)+4 i \left(4 b c x+a (-4+i c x) \sqrt{c^2 x^2+1}\right)\right)+32 i a b c f x \sqrt{d+i c d x} \sqrt{f-i c f x}+2 a b f \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-32 i b^2 f \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+4 b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-b^2 f \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)}{8 c d \sqrt{c^2 x^2+1}}","\frac{f^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i f^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{f^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c f^2 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{4 i b f^2 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{4 i b^2 f^2 \left(c^2 x^2+1\right)}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{b^2 f^2 x \left(c^2 x^2+1\right)}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b^2 f^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((32*I)*a*b*c*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (16*I)*a^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (32*I)*b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 4*a^2*c*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 4*b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 + 2*a*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + 2*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((4*I)*(4*b*c*x + a*(-4 + I*c*x)*Sqrt[1 + c^2*x^2]) + b*Cosh[2*ArcSinh[c*x]]) + 12*a^2*Sqrt[d]*f^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - b^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 2*b*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(6*a - (8*I)*b*Sqrt[1 + c^2*x^2] - b*Sinh[2*ArcSinh[c*x]]))/(8*c*d*Sqrt[1 + c^2*x^2])","A",1
580,1,1546,752,9.1030101,"\int \frac{(f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2}} \, dx","Integrate[((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(3/2),x]","-\frac{3 f^{3/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i d (c x-i)} \sqrt{-i f (c x+i)}\right) a^2}{c d^{3/2}}+\frac{2 i b f \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(i \sinh ^{-1}(c x)^2+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 \sinh ^{-1}(c x)-c x+4 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+2 i \log \left(\sqrt{c^2 x^2+1}\right)\right)+i \left(i \sinh ^{-1}(c x)^2+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-2 \sinh ^{-1}(c x)-c x+4 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+2 i \log \left(\sqrt{c^2 x^2+1}\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) a}{c d^2 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}-\frac{b f \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 i\right)+8 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4 \log \left(\sqrt{c^2 x^2+1}\right)\right)+i \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+4 i\right)+8 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4 \log \left(\sqrt{c^2 x^2+1}\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) a}{c d^2 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left(\frac{i f a^2}{d^2}+\frac{4 f a^2}{d^2 (c x-i)}\right)}{c}-\frac{b^2 f \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\sinh ^{-1}(c x)^3+(6-6 i) \sinh ^{-1}(c x)^2+6 i \pi  \sinh ^{-1}(c x)+12 \left(2 \sinh ^{-1}(c x)-i \pi \right) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-24 i \pi  \log \left(1+e^{\sinh ^{-1}(c x)}\right)+24 i \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+12 i \pi  \log \left(\sin \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)\right)-24 \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\left(i \sinh ^{-1}(c x)^3-(6-6 i) \sinh ^{-1}(c x)^2-6 \pi  \sinh ^{-1}(c x)+12 \left(2 i \sinh ^{-1}(c x)+\pi \right) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+24 \pi  \log \left(1+e^{\sinh ^{-1}(c x)}\right)-24 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-12 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}{3 c d^2 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{i b^2 f \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(2 i \sinh ^{-1}(c x)^3+(6+6 i) \sinh ^{-1}(c x)^2-6 c x \sinh ^{-1}(c x)+24 i \log \left(1-i e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)-6 \pi  \sinh ^{-1}(c x)+3 \sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x)^2+2\right)+12 \pi  \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+24 \pi  \log \left(1+e^{\sinh ^{-1}(c x)}\right)-24 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-12 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)\right)+i \left(2 i \sinh ^{-1}(c x)^3-(6-6 i) \sinh ^{-1}(c x)^2-6 c x \sinh ^{-1}(c x)+24 i \log \left(1-i e^{-\sinh ^{-1}(c x)}\right) \sinh ^{-1}(c x)-6 \pi  \sinh ^{-1}(c x)+3 \sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x)^2+2\right)+12 \pi  \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+24 \pi  \log \left(1+e^{\sinh ^{-1}(c x)}\right)-24 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-12 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+24 \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{3 c d^2 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}","-\frac{2 i a b f^3 x \left(c^2 x^2+1\right)^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{f^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{i f^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 f^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 f^3 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i f^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b f^3 \left(c^2 x^2+1\right)^{3/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{16 i b f^3 \left(c^2 x^2+1\right)^{3/2} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b^2 f^3 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 b^2 f^3 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b^2 f^3 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 i b^2 f^3 \left(c^2 x^2+1\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 i b^2 f^3 x \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((I*a^2*f)/d^2 + (4*a^2*f)/(d^2*(-I + c*x))))/c - (3*a^2*f^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(c*d^(3/2)) + ((2*I)*a*b*f*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(-(c*x) + 2*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + I*ArcSinh[c*x]^2 + 4*ArcTan[Coth[ArcSinh[c*x]/2]] + (2*I)*Log[Sqrt[1 + c^2*x^2]]) + I*(-(c*x) - 2*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + I*ArcSinh[c*x]^2 + 4*ArcTan[Coth[ArcSinh[c*x]/2]] + (2*I)*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) - (a*b*f*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(ArcSinh[c*x]*(-4*I + ArcSinh[c*x]) + (8*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]]) + I*(ArcSinh[c*x]*(4*I + ArcSinh[c*x]) + (8*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) - (b^2*f*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*((6*I)*Pi*ArcSinh[c*x] + (6 - 6*I)*ArcSinh[c*x]^2 + ArcSinh[c*x]^3 + 12*((-I)*Pi + 2*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - (24*I)*Pi*Log[1 + E^ArcSinh[c*x]] + (24*I)*Pi*Log[Cosh[ArcSinh[c*x]/2]] + (12*I)*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) - 24*PolyLog[2, I/E^ArcSinh[c*x]]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + (-6*Pi*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 + 12*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - 12*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])*Sinh[ArcSinh[c*x]/2]))/(3*c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + ((I/3)*b^2*f*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(-6*Pi*ArcSinh[c*x] - 6*c*x*ArcSinh[c*x] + (6 + 6*I)*ArcSinh[c*x]^2 + (2*I)*ArcSinh[c*x]^3 + 3*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + 12*Pi*Log[1 - I/E^ArcSinh[c*x]] + (24*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - 12*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) + I*(-6*Pi*ArcSinh[c*x] - 6*c*x*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + (2*I)*ArcSinh[c*x]^3 + 3*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + 12*Pi*Log[1 - I/E^ArcSinh[c*x]] + (24*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - 12*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])*Sinh[ArcSinh[c*x]/2] + 24*PolyLog[2, I/E^ArcSinh[c*x]]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]))","B",0
581,1,1609,580,10.2425033,"\int \frac{(f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{5/2}} \, dx","Integrate[((f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(5/2),x]","\frac{f^{3/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i d (c x-i)} \sqrt{-i f (c x+i)}\right) a^2}{c d^{5/2}}+\frac{i b f \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(-i \cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-i \log \left(\sqrt{c^2 x^2+1}\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(3 i \sinh ^{-1}(c x)-6 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+3 \log \left(\sqrt{c^2 x^2+1}\right)+4\right)+2 \left(\sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x)+2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i \log \left(\sqrt{c^2 x^2+1}\right)\right)+2 \left(\sinh ^{-1}(c x)+2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i \log \left(\sqrt{c^2 x^2+1}\right)+i\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) a}{3 c d^3 (c x+i) \sqrt{-((c d x-i d) (c x f+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}-\frac{b f \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\left(3 i \sinh ^{-1}(c x)-14\right) \sinh ^{-1}(c x)-28 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+14 i \log \left(\sqrt{c^2 x^2+1}\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(84 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-i \left(9 \sinh ^{-1}(c x)^2-6 i \sinh ^{-1}(c x)+42 \log \left(\sqrt{c^2 x^2+1}\right)+8\right)\right)+2 \left(6 \sinh ^{-1}(c x)^2-4 i \sinh ^{-1}(c x)+56 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+28 \log \left(\sqrt{c^2 x^2+1}\right)+\sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x) \left(3 \sinh ^{-1}(c x)-14 i\right)+28 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+14 \log \left(\sqrt{c^2 x^2+1}\right)\right)+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) a}{6 c d^3 (c x+i) \sqrt{-((c d x-i d) (c x f+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}+\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left(-\frac{8 f a^2}{3 d^3 (c x-i)}-\frac{4 i f a^2}{3 d^3 (c x-i)^2}\right)}{c}+\frac{i b^2 f (c x+i) \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(-\frac{4 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}-(1-i) \sinh ^{-1}(c x)^2-\frac{2 \left(\sinh ^{-1}(c x)-2 i\right) \sinh ^{-1}(c x)}{c x-i}+2 i \left(2 i \sinh ^{-1}(c x)+\pi \right) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-i \pi  \left(\sinh ^{-1}(c x)-4 \log \left(1+e^{\sinh ^{-1}(c x)}\right)+4 \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+2 \log \left(\sin \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)\right)+4 \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+\frac{2 \left(\sinh ^{-1}(c x)^2+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}\right)}{3 c d^3 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^2}+\frac{b^2 f (c x+i) \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(-i \sinh ^{-1}(c x)^3-\frac{4 i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}-(7+7 i) \sinh ^{-1}(c x)^2+\frac{2 \left(\sinh ^{-1}(c x)-2 i\right) \sinh ^{-1}(c x)}{i c x+1}+7 \pi  \sinh ^{-1}(c x)-14 \left(2 i \sinh ^{-1}(c x)+\pi \right) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-28 \pi  \log \left(1+e^{\sinh ^{-1}(c x)}\right)+28 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+14 \pi  \log \left(\sin \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)+28 i \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+\frac{2 \left(7 \sinh ^{-1}(c x)^2+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}\right)}{3 c d^3 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^2}","\frac{f^4 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{8 f^4 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{32 b f^4 \left(c^2 x^2+1\right)^{5/2} \log \left(1+i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{8 i f^4 \left(c^2 x^2+1\right)^{5/2} \cot \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b f^4 \left(c^2 x^2+1\right)^{5/2} \csc ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i f^4 \left(c^2 x^2+1\right)^{5/2} \cot \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \csc ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{32 b^2 f^4 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{8 i b^2 f^4 \left(c^2 x^2+1\right)^{5/2} \cot \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((((-4*I)/3)*a^2*f)/(d^3*(-I + c*x)^2) - (8*a^2*f)/(3*d^3*(-I + c*x))))/c + (a^2*f^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(c*d^(5/2)) + ((I/3)*a*b*f*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*((-I)*Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] - I*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(4 + (3*I)*ArcSinh[c*x] - (6*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 3*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x^2]*(ArcSinh[c*x] + 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]]) + 2*(I + ArcSinh[c*x] + 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) - (a*b*f*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSinh[c*x])/2]*((-14 + (3*I)*ArcSinh[c*x])*ArcSinh[c*x] - 28*ArcTan[Tanh[ArcSinh[c*x]/2]] + (14*I)*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(84*ArcTan[Tanh[ArcSinh[c*x]/2]] - I*(8 - (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 + 42*Log[Sqrt[1 + c^2*x^2]])) + 2*(4 - (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 + (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 28*Log[Sqrt[1 + c^2*x^2]] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(-14*I + 3*ArcSinh[c*x]) + (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(6*c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) + ((I/3)*b^2*f*(I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*((-1 + I)*ArcSinh[c*x]^2 - (2*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/(-I + c*x) + (2*I)*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - I*Pi*(ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] + 4*Log[Cosh[ArcSinh[c*x]/2]] + 2*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) + 4*PolyLog[2, I/E^ArcSinh[c*x]] - (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(c*d^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^2) + (b^2*f*(I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(7*Pi*ArcSinh[c*x] - (7 + 7*I)*ArcSinh[c*x]^2 - I*ArcSinh[c*x]^3 + (2*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/(1 + I*c*x) - 14*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - 28*Pi*Log[1 + E^ArcSinh[c*x]] + 28*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 14*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + (28*I)*PolyLog[2, I/E^ArcSinh[c*x]] - ((4*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + 7*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(3*c*d^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^2)","B",0
582,1,735,548,2.3432948,"\int (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{4320 a^2 d^{5/2} f^{5/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+9504 a^2 c d^2 f^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+2304 a^2 c^5 d^2 f^2 x^5 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+7488 a^2 c^3 d^2 f^2 x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+72 b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(60 a+45 b \sinh \left(2 \sinh ^{-1}(c x)\right)+9 b \sinh \left(4 \sinh ^{-1}(c x)\right)+b \sinh \left(6 \sinh ^{-1}(c x)\right)\right)-3240 a b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-324 a b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)-24 a b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(6 \sinh ^{-1}(c x)\right)-12 b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(-540 a \sinh \left(2 \sinh ^{-1}(c x)\right)-108 a \sinh \left(4 \sinh ^{-1}(c x)\right)-12 a \sinh \left(6 \sinh ^{-1}(c x)\right)+270 b \cosh \left(2 \sinh ^{-1}(c x)\right)+27 b \cosh \left(4 \sinh ^{-1}(c x)\right)+2 b \cosh \left(6 \sinh ^{-1}(c x)\right)\right)+1440 b^2 d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3+1620 b^2 d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)+81 b^2 d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(4 \sinh ^{-1}(c x)\right)+4 b^2 d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(6 \sinh ^{-1}(c x)\right)}{13824 c \sqrt{c^2 x^2+1}}","\frac{5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{48 b c \left(c^2 x^2+1\right)^{5/2}}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{24 \left(c^2 x^2+1\right)}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{16 \left(c^2 x^2+1\right)^2}-\frac{b \sqrt{c^2 x^2+1} (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{18 c}-\frac{5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{48 c \sqrt{c^2 x^2+1}}-\frac{5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{16 \left(c^2 x^2+1\right)^{5/2}}+\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left(c^2 x^2+1\right)}+\frac{245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left(c^2 x^2+1\right)^2}-\frac{115 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sinh ^{-1}(c x)}{1152 c \left(c^2 x^2+1\right)^{5/2}}+\frac{1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}",1,"(9504*a^2*c*d^2*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 7488*a^2*c^3*d^2*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2304*a^2*c^5*d^2*f^2*x^5*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 1440*b^2*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 3240*a*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 324*a*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] - 24*a*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[6*ArcSinh[c*x]] + 4320*a^2*d^(5/2)*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 1620*b^2*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 81*b^2*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c*x]] + 4*b^2*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[6*ArcSinh[c*x]] - 12*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(270*b*Cosh[2*ArcSinh[c*x]] + 27*b*Cosh[4*ArcSinh[c*x]] + 2*b*Cosh[6*ArcSinh[c*x]] - 540*a*Sinh[2*ArcSinh[c*x]] - 108*a*Sinh[4*ArcSinh[c*x]] - 12*a*Sinh[6*ArcSinh[c*x]]) + 72*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a + 45*b*Sinh[2*ArcSinh[c*x]] + 9*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]]))/(13824*c*Sqrt[1 + c^2*x^2])","A",1
583,1,1084,774,3.1884863,"\int (d+i c d x)^{3/2} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{-57600 i a^2 c^4 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^4+72000 a^2 c^3 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^3-115200 i a^2 c^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^2+72000 i a b c d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} x+180000 a^2 c d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x+36000 b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-72000 a b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-4000 i b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(3 \sinh ^{-1}(c x)\right)-4500 a b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)-288 i b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(5 \sinh ^{-1}(c x)\right)+108000 a^2 d^{3/2} f^{5/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i c x d+d} \sqrt{f-i c f x}\right)+36000 b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)+12000 i a b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)+1125 b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(4 \sinh ^{-1}(c x)\right)+1800 b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(60 a-10 i b \cosh \left(3 \sinh ^{-1}(c x)\right)-2 i b \cosh \left(5 \sinh ^{-1}(c x)\right)+40 b \sinh \left(2 \sinh ^{-1}(c x)\right)+5 b \sinh \left(4 \sinh ^{-1}(c x)\right)-20 i b \sqrt{c^2 x^2+1}\right)+1440 i a b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(5 \sinh ^{-1}(c x)\right)+60 b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(-600 i \cosh \left(3 \sinh ^{-1}(c x)\right) a-120 i \cosh \left(5 \sinh ^{-1}(c x)\right) a+2400 \sinh \left(2 \sinh ^{-1}(c x)\right) a+300 \sinh \left(4 \sinh ^{-1}(c x)\right) a-1200 i \sqrt{c^2 x^2+1} a+1200 i b c x-1200 b \cosh \left(2 \sinh ^{-1}(c x)\right)-75 b \cosh \left(4 \sinh ^{-1}(c x)\right)+200 i b \sinh \left(3 \sinh ^{-1}(c x)\right)+24 i b \sinh \left(5 \sinh ^{-1}(c x)\right)\right)-57600 i a^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}-72000 i b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}}{288000 c \sqrt{c^2 x^2+1}}","-\frac{3 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 \left(c^2 x^2+1\right)^{3/2}}+\frac{3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{8 \left(c^2 x^2+1\right)}+\frac{2 i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{5 \left(c^2 x^2+1\right)^{3/2}}+\frac{f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{8 b c \left(c^2 x^2+1\right)^{3/2}}-\frac{i f \left(c^2 x^2+1\right) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{5 c}-\frac{b f \sqrt{c^2 x^2+1} (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{8 c}+\frac{4 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{15 \left(c^2 x^2+1\right)^{3/2}}+\frac{2 i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{25 \left(c^2 x^2+1\right)^{3/2}}+\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{15 b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left(c^2 x^2+1\right)}-\frac{2 i b^2 f \left(c^2 x^2+1\right) (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{125 c}-\frac{16 i b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{75 c \left(c^2 x^2+1\right)}-\frac{9 b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left(c^2 x^2+1\right)^{3/2}}+\frac{1}{32} b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}-\frac{8 i b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{225 c}",1,"((72000*I)*a*b*c*d*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (57600*I)*a^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (72000*I)*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 180000*a^2*c*d*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (115200*I)*a^2*c^2*d*f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 72000*a^2*c^3*d*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (57600*I)*a^2*c^4*d*f^2*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 36000*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 72000*a*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - (4000*I)*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] - 4500*a*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] - (288*I)*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[5*ArcSinh[c*x]] + 108000*a^2*d^(3/2)*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 36000*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + (12000*I)*a*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 1125*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c*x]] + 1800*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a - (20*I)*b*Sqrt[1 + c^2*x^2] - (10*I)*b*Cosh[3*ArcSinh[c*x]] - (2*I)*b*Cosh[5*ArcSinh[c*x]] + 40*b*Sinh[2*ArcSinh[c*x]] + 5*b*Sinh[4*ArcSinh[c*x]]) + (1440*I)*a*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[5*ArcSinh[c*x]] + 60*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((1200*I)*b*c*x - (1200*I)*a*Sqrt[1 + c^2*x^2] - 1200*b*Cosh[2*ArcSinh[c*x]] - (600*I)*a*Cosh[3*ArcSinh[c*x]] - 75*b*Cosh[4*ArcSinh[c*x]] - (120*I)*a*Cosh[5*ArcSinh[c*x]] + 2400*a*Sinh[2*ArcSinh[c*x]] + (200*I)*b*Sinh[3*ArcSinh[c*x]] + 300*a*Sinh[4*ArcSinh[c*x]] + (24*I)*b*Sinh[5*ArcSinh[c*x]]))/(288000*c*Sqrt[1 + c^2*x^2])","A",1
584,1,890,680,2.4283181,"\int \sqrt{d+i c d x} (f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2,x]","\frac{-1728 a^2 c^3 f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^3-4608 i a^2 c^2 f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^2+6912 i a b c f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} x+2592 a^2 c f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x+1440 b^2 f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-1728 a b f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-256 i b^2 f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(3 \sinh ^{-1}(c x)\right)+108 a b f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left(4 \sinh ^{-1}(c x)\right)+4320 a^2 \sqrt{d} f^{5/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i c x d+d} \sqrt{f-i c f x}\right)+864 b^2 f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)+768 i a b f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)-27 b^2 f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left(4 \sinh ^{-1}(c x)\right)+12 b f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(-192 i \cosh \left(3 \sinh ^{-1}(c x)\right) a+288 \sinh \left(2 \sinh ^{-1}(c x)\right) a-36 \sinh \left(4 \sinh ^{-1}(c x)\right) a-576 i \sqrt{c^2 x^2+1} a+576 i b c x-144 b \cosh \left(2 \sinh ^{-1}(c x)\right)+9 b \cosh \left(4 \sinh ^{-1}(c x)\right)+64 i b \sinh \left(3 \sinh ^{-1}(c x)\right)\right)+72 b f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(60 a-16 i b \cosh \left(3 \sinh ^{-1}(c x)\right)+24 b \sinh \left(2 \sinh ^{-1}(c x)\right)-3 b \sinh \left(4 \sinh ^{-1}(c x)\right)-48 i b \sqrt{c^2 x^2+1}\right)-4608 i a^2 f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}-6912 i b^2 f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}}{6912 c \sqrt{c^2 x^2+1}}","-\frac{1}{4} c^2 f^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{3 b c f^2 x^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{8 \sqrt{c^2 x^2+1}}+\frac{4 i b f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{c^2 x^2+1}}+\frac{5 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^3}{24 b c \sqrt{c^2 x^2+1}}-\frac{2 i f^2 \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c}+\frac{4 i b c^2 f^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{9 \sqrt{c^2 x^2+1}}+\frac{b c^3 f^2 x^4 \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)}{8 \sqrt{c^2 x^2+1}}+\frac{3}{8} f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a+b \sinh ^{-1}(c x)\right)^2-\frac{1}{32} b^2 c^2 f^2 x^3 \sqrt{d+i c d x} \sqrt{f-i c f x}-\frac{4 i b^2 f^2 \left(c^2 x^2+1\right) \sqrt{d+i c d x} \sqrt{f-i c f x}}{27 c}-\frac{15 b^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)}{64 c \sqrt{c^2 x^2+1}}+\frac{15}{64} b^2 f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}-\frac{8 i b^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{9 c}",1,"((6912*I)*a*b*c*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (4608*I)*a^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (6912*I)*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2592*a^2*c*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (4608*I)*a^2*c^2*f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 1728*a^2*c^3*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 1440*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 1728*a*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - (256*I)*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] + 108*a*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 4320*a^2*Sqrt[d]*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 864*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + (768*I)*a*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] - 27*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c*x]] + 12*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((576*I)*b*c*x - (576*I)*a*Sqrt[1 + c^2*x^2] - 144*b*Cosh[2*ArcSinh[c*x]] - (192*I)*a*Cosh[3*ArcSinh[c*x]] + 9*b*Cosh[4*ArcSinh[c*x]] + 288*a*Sinh[2*ArcSinh[c*x]] + (64*I)*b*Sinh[3*ArcSinh[c*x]] - 36*a*Sinh[4*ArcSinh[c*x]]) + 72*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a - (48*I)*b*Sqrt[1 + c^2*x^2] - (16*I)*b*Cosh[3*ArcSinh[c*x]] + 24*b*Sinh[2*ArcSinh[c*x]] - 3*b*Sinh[4*ArcSinh[c*x]]))/(6912*c*Sqrt[1 + c^2*x^2])","A",1
585,1,723,615,3.6188592,"\int \frac{(f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+i c d x}} \, dx","Integrate[((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[d + I*c*d*x],x]","\frac{540 a^2 \sqrt{d} f^{5/2} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)-792 i a^2 f^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+72 i a^2 c^2 f^2 x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-324 a^2 c f^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+18 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(30 a-45 i b \sqrt{c^2 x^2+1}-9 b \sinh \left(2 \sinh ^{-1}(c x)\right)+i b \cosh \left(3 \sinh ^{-1}(c x)\right)\right)+6 b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(27 b \cosh \left(2 \sinh ^{-1}(c x)\right)+2 i \left(27 a \sqrt{c^2 x^2+1} (-5+2 i c x)+3 a \cosh \left(3 \sinh ^{-1}(c x)\right)-4 b c x \left(c^2 x^2-33\right)\right)\right)+1620 i a b c f^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}-12 i a b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)+162 a b f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)-1620 i b^2 f^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+180 b^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-81 b^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)+4 i b^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(3 \sinh ^{-1}(c x)\right)}{216 c d \sqrt{c^2 x^2+1}}","\frac{5 f^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i c f^3 x^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 f^3 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{11 i f^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b c f^3 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{22 i b f^3 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b c^2 f^3 x^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i b^2 f^3 \left(c^2 x^2+1\right)^2}{27 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 b^2 f^3 x \left(c^2 x^2+1\right)}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{68 i b^2 f^3 \left(c^2 x^2+1\right)}{9 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b^2 f^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((1620*I)*a*b*c*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (792*I)*a^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (1620*I)*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 324*a^2*c*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (72*I)*a^2*c^2*f^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 180*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 + 162*a*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + (4*I)*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] + 6*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(27*b*Cosh[2*ArcSinh[c*x]] + (2*I)*(-4*b*c*x*(-33 + c^2*x^2) + 27*a*(-5 + (2*I)*c*x)*Sqrt[1 + c^2*x^2] + 3*a*Cosh[3*ArcSinh[c*x]])) + 540*a^2*Sqrt[d]*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - 81*b^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 18*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(30*a - (45*I)*b*Sqrt[1 + c^2*x^2] + I*b*Cosh[3*ArcSinh[c*x]] - 9*b*Sinh[2*ArcSinh[c*x]]) - (12*I)*a*b*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]])/(216*c*d*Sqrt[1 + c^2*x^2])","A",1
586,1,2492,972,11.8313554,"\int \frac{(f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2}} \, dx","Integrate[((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(3/2),x]","\text{Result too large to show}","-\frac{5 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3 f^4}{2 b c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{b^2 x \left(c^2 x^2+1\right)^2 f^4}{4 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i b^2 \left(c^2 x^2+1\right)^2 f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 f^4}{2 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2 f^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2 f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i a b x \left(c^2 x^2+1\right)^{3/2} f^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b^2 x \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) f^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) f^4}{4 c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{b c x^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) f^4}{2 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{32 i b \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \log \left(1+e^{2 \sinh ^{-1}(c x)}\right) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{16 b^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) f^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((4*I)*a^2*f^2)/d^2 + (a^2*c*f^2*x)/(2*d^2) + (8*a^2*f^2)/(d^2*(-I + c*x))))/c - (15*a^2*f^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(2*c*d^(3/2)) + ((4*I)*a*b*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(-(c*x) + 2*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + I*ArcSinh[c*x]^2 + 4*ArcTan[Coth[ArcSinh[c*x]/2]] + (2*I)*Log[Sqrt[1 + c^2*x^2]]) + I*(-(c*x) - 2*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + I*ArcSinh[c*x]^2 + 4*ArcTan[Coth[ArcSinh[c*x]/2]] + (2*I)*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) - (a*b*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(ArcSinh[c*x]*(-4*I + ArcSinh[c*x]) + (8*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]]) + I*(ArcSinh[c*x]*(4*I + ArcSinh[c*x]) + (8*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) - (b^2*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*((6*I)*Pi*ArcSinh[c*x] + (6 - 6*I)*ArcSinh[c*x]^2 + ArcSinh[c*x]^3 + 12*((-I)*Pi + 2*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - (24*I)*Pi*Log[1 + E^ArcSinh[c*x]] + (24*I)*Pi*Log[Cosh[ArcSinh[c*x]/2]] + (12*I)*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) - 24*PolyLog[2, I/E^ArcSinh[c*x]]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + (-6*Pi*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 + 12*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - 12*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])*Sinh[ArcSinh[c*x]/2]))/(3*c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (((2*I)/3)*b^2*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(-6*Pi*ArcSinh[c*x] - 6*c*x*ArcSinh[c*x] + (6 + 6*I)*ArcSinh[c*x]^2 + (2*I)*ArcSinh[c*x]^3 + 3*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + 12*Pi*Log[1 - I/E^ArcSinh[c*x]] + (24*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - 12*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) + I*(-6*Pi*ArcSinh[c*x] - 6*c*x*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + (2*I)*ArcSinh[c*x]^3 + 3*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + 12*Pi*Log[1 - I/E^ArcSinh[c*x]] + (24*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - 12*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])*Sinh[ArcSinh[c*x]/2] + 24*PolyLog[2, I/E^ArcSinh[c*x]]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (b^2*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(96*PolyLog[2, I/E^ArcSinh[c*x]]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + Sinh[ArcSinh[c*x]/2]*(24*Pi*ArcSinh[c*x] + 48*c*x*ArcSinh[c*x] + (24 - 24*I)*ArcSinh[c*x]^2 - (10*I)*ArcSinh[c*x]^3 + (3*I)*Sqrt[1 + c^2*x^2]*(c*x + (8*I)*(2 + ArcSinh[c*x]^2)) - (3*I)*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] - 48*Pi*Log[1 - I/E^ArcSinh[c*x]] - (96*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - 96*Pi*Log[1 + E^ArcSinh[c*x]] + 96*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 48*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + (3*I)*ArcSinh[c*x]^2*Sinh[2*ArcSinh[c*x]]) + Cosh[ArcSinh[c*x]/2]*(3*Sqrt[1 + c^2*x^2]*(c*x + (8*I)*(2 + ArcSinh[c*x]^2)) - 3*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] - I*(24*Pi*ArcSinh[c*x] + 48*c*x*ArcSinh[c*x] - (24 + 24*I)*ArcSinh[c*x]^2 - (10*I)*ArcSinh[c*x]^3 - 48*Pi*Log[1 - I/E^ArcSinh[c*x]] - (96*I)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - 96*Pi*Log[1 + E^ArcSinh[c*x]] + 96*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 48*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + (3*I)*ArcSinh[c*x]^2*Sinh[2*ArcSinh[c*x]]))))/(12*c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])) + (a*b*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(-(Sinh[ArcSinh[c*x]/2]*((-16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + Cosh[2*ArcSinh[c*x]] + 2*((8*I)*c*x + (8*I)*ArcSinh[c*x] + 5*ArcSinh[c*x]^2 + (16*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 8*Log[Sqrt[1 + c^2*x^2]] - ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))) + Cosh[ArcSinh[c*x]/2]*(16*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + I*(Cosh[2*ArcSinh[c*x]] + 2*((8*I)*c*x - (8*I)*ArcSinh[c*x] + 5*ArcSinh[c*x]^2 + (16*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 8*Log[Sqrt[1 + c^2*x^2]] - ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]])))))/(4*c*d^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]))","B",0
587,1,2622,790,13.5247177,"\int \frac{(f-i c f x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{5/2}} \, dx","Integrate[((f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(d + I*c*d*x)^(5/2),x]","\text{Result too large to show}","\frac{2 i a b f^5 x \left(c^2 x^2+1\right)^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{5 f^5 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i f^5 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{28 f^5 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{112 b f^5 \left(c^2 x^2+1\right)^{5/2} \log \left(1+i e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{28 i f^5 \left(c^2 x^2+1\right)^{5/2} \cot \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{8 b f^5 \left(c^2 x^2+1\right)^{5/2} \csc ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 i f^5 \left(c^2 x^2+1\right)^{5/2} \cot \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \csc ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{112 b^2 f^5 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i b^2 f^5 \left(c^2 x^2+1\right)^3}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i b^2 f^5 x \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{16 i b^2 f^5 \left(c^2 x^2+1\right)^{5/2} \cot \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((-I)*a^2*f^2)/d^3 - (((8*I)/3)*a^2*f^2)/(d^3*(-I + c*x)^2) - (28*a^2*f^2)/(3*d^3*(-I + c*x))))/c + (5*a^2*f^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(c*d^(5/2)) + ((I/3)*a*b*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*((-I)*Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] - I*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(4 + (3*I)*ArcSinh[c*x] - (6*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 3*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x^2]*(ArcSinh[c*x] + 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]]) + 2*(I + ArcSinh[c*x] + 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) - (a*b*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSinh[c*x])/2]*((-14 + (3*I)*ArcSinh[c*x])*ArcSinh[c*x] - 28*ArcTan[Tanh[ArcSinh[c*x]/2]] + (14*I)*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(84*ArcTan[Tanh[ArcSinh[c*x]/2]] - I*(8 - (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 + 42*Log[Sqrt[1 + c^2*x^2]])) + 2*(4 - (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 + (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 28*Log[Sqrt[1 + c^2*x^2]] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(-14*I + 3*ArcSinh[c*x]) + (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(3*c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) + ((I/3)*b^2*f^2*(I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*((-1 + I)*ArcSinh[c*x]^2 - (2*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/(-I + c*x) + (2*I)*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - I*Pi*(ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] + 4*Log[Cosh[ArcSinh[c*x]/2]] + 2*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) + 4*PolyLog[2, I/E^ArcSinh[c*x]] - (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(c*d^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^2) - ((I/3)*b^2*f^2*(I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(((6*I)*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ((13 - 13*I)*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (3*ArcSinh[c*x]^3)/Sqrt[1 + c^2*x^2] + (2*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/((-I + c*x)*Sqrt[1 + c^2*x^2]) - (3*I)*(2 + ArcSinh[c*x]^2) + ((13*I)*(-2*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] + Pi*(ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] + 4*Log[Cosh[ArcSinh[c*x]/2]] + 2*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) + (4*I)*PolyLog[2, I/E^ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2] + (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3) - (2*(4 + 13*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]))))/(c*d^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^2) + (2*b^2*f^2*(I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(7*Pi*ArcSinh[c*x] - (7 + 7*I)*ArcSinh[c*x]^2 - I*ArcSinh[c*x]^3 + (2*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/(1 + I*c*x) - 14*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - 28*Pi*Log[1 + E^ArcSinh[c*x]] + 28*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 14*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + (28*I)*PolyLog[2, I/E^ArcSinh[c*x]] - ((4*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + 7*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(3*c*d^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^2) + ((I/6)*a*b*f^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])*(-3*Cosh[(5*ArcSinh[c*x])/2] + (3*I)*ArcSinh[c*x]*Cosh[(5*ArcSinh[c*x])/2] - Cosh[(3*ArcSinh[c*x])/2]*(9 + (35*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 - (52*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 26*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(20 - (24*I)*ArcSinh[c*x] + 27*ArcSinh[c*x]^2 - (156*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 78*Log[Sqrt[1 + c^2*x^2]]) + (20*I)*Sinh[ArcSinh[c*x]/2] - 24*ArcSinh[c*x]*Sinh[ArcSinh[c*x]/2] + (27*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2] + 156*ArcTan[Coth[ArcSinh[c*x]/2]]*Sinh[ArcSinh[c*x]/2] + (78*I)*Log[Sqrt[1 + c^2*x^2]]*Sinh[ArcSinh[c*x]/2] + (9*I)*Sinh[(3*ArcSinh[c*x])/2] + 35*ArcSinh[c*x]*Sinh[(3*ArcSinh[c*x])/2] + (9*I)*ArcSinh[c*x]^2*Sinh[(3*ArcSinh[c*x])/2] + 52*ArcTan[Coth[ArcSinh[c*x]/2]]*Sinh[(3*ArcSinh[c*x])/2] + (26*I)*Log[Sqrt[1 + c^2*x^2]]*Sinh[(3*ArcSinh[c*x])/2] - (3*I)*Sinh[(5*ArcSinh[c*x])/2] + 3*ArcSinh[c*x]*Sinh[(5*ArcSinh[c*x])/2]))/(c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4)","B",0
588,1,723,615,3.4839694,"\int \frac{(d+i c d x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{f-i c f x}} \, dx","Integrate[((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x],x]","\frac{540 a^2 d^{5/2} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+792 i a^2 d^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-72 i a^2 c^2 d^2 x^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-324 a^2 c d^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+18 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(30 a+45 i b \sqrt{c^2 x^2+1}-9 b \sinh \left(2 \sinh ^{-1}(c x)\right)-i b \cosh \left(3 \sinh ^{-1}(c x)\right)\right)+6 b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(27 b \cosh \left(2 \sinh ^{-1}(c x)\right)+2 i \left(27 a \sqrt{c^2 x^2+1} (5+2 i c x)-3 a \cosh \left(3 \sinh ^{-1}(c x)\right)+4 b c x \left(c^2 x^2-33\right)\right)\right)-1620 i a b c d^2 x \sqrt{d+i c d x} \sqrt{f-i c f x}+12 i a b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(3 \sinh ^{-1}(c x)\right)+162 a b d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)+1620 i b^2 d^2 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+180 b^2 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-81 b^2 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)-4 i b^2 d^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(3 \sinh ^{-1}(c x)\right)}{216 c f \sqrt{c^2 x^2+1}}","\frac{5 d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{i c d^3 x^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 d^3 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{11 i d^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b c d^3 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{22 i b d^3 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i b c^2 d^3 x^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b^2 d^3 \left(c^2 x^2+1\right)^2}{27 c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{3 b^2 d^3 x \left(c^2 x^2+1\right)}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{68 i b^2 d^3 \left(c^2 x^2+1\right)}{9 c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{3 b^2 d^3 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((-1620*I)*a*b*c*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (792*I)*a^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (1620*I)*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 324*a^2*c*d^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (72*I)*a^2*c^2*d^2*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 180*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 + 162*a*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - (4*I)*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] + 6*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(27*b*Cosh[2*ArcSinh[c*x]] + (2*I)*(4*b*c*x*(-33 + c^2*x^2) + 27*a*(5 + (2*I)*c*x)*Sqrt[1 + c^2*x^2] - 3*a*Cosh[3*ArcSinh[c*x]])) + 540*a^2*d^(5/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - 81*b^2*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 18*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(30*a + (45*I)*b*Sqrt[1 + c^2*x^2] - I*b*Cosh[3*ArcSinh[c*x]] - 9*b*Sinh[2*ArcSinh[c*x]]) + (12*I)*a*b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]])/(216*c*f*Sqrt[1 + c^2*x^2])","A",1
589,1,529,436,2.2100389,"\int \frac{(d+i c d x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{f-i c f x}} \, dx","Integrate[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x],x]","\frac{12 a^2 d^{3/2} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+16 i a^2 d \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}-4 a^2 c d x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+2 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(6 a+8 i b \sqrt{c^2 x^2+1}-b \sinh \left(2 \sinh ^{-1}(c x)\right)\right)+2 b d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(-4 a (c x-4 i) \sqrt{c^2 x^2+1}-16 i b c x+b \cosh \left(2 \sinh ^{-1}(c x)\right)\right)-32 i a b c d x \sqrt{d+i c d x} \sqrt{f-i c f x}+2 a b d \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left(2 \sinh ^{-1}(c x)\right)+32 i b^2 d \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+4 b^2 d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-b^2 d \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left(2 \sinh ^{-1}(c x)\right)}{8 c f \sqrt{c^2 x^2+1}}","\frac{d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{2 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{d^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b c d^2 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{4 i b d^2 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{4 i b^2 d^2 \left(c^2 x^2+1\right)}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{b^2 d^2 x \left(c^2 x^2+1\right)}{4 \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b^2 d^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{4 c \sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((-32*I)*a*b*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (16*I)*a^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (32*I)*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 4*a^2*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 4*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 + 2*a*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + 2*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((-16*I)*b*c*x - 4*a*(-4*I + c*x)*Sqrt[1 + c^2*x^2] + b*Cosh[2*ArcSinh[c*x]]) + 12*a^2*d^(3/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 2*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(6*a + (8*I)*b*Sqrt[1 + c^2*x^2] - b*Sinh[2*ArcSinh[c*x]]))/(8*c*f*Sqrt[1 + c^2*x^2])","A",1
590,1,315,259,1.1496451,"\int \frac{\sqrt{d+i c d x} \left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{f-i c f x}} \, dx","Integrate[(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x],x]","\frac{3 i \sqrt{d+i c d x} \sqrt{f-i c f x} \left(a^2 \sqrt{c^2 x^2+1}-2 a b c x+2 b^2 \sqrt{c^2 x^2+1}\right)+3 a^2 \sqrt{d} \sqrt{f} \sqrt{c^2 x^2+1} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+3 b \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left(a+i b \sqrt{c^2 x^2+1}\right)-6 i b \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left(b c x-a \sqrt{c^2 x^2+1}\right)+b^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3}{3 c f \sqrt{c^2 x^2+1}}","-\frac{2 i a b d x \sqrt{c^2 x^2+1}}{\sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{i d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{2 i b^2 d \left(c^2 x^2+1\right)}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}-\frac{2 i b^2 d x \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)}{\sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"((3*I)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-2*a*b*c*x + a^2*Sqrt[1 + c^2*x^2] + 2*b^2*Sqrt[1 + c^2*x^2]) - (6*I)*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(b*c*x - a*Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + 3*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(a + I*b*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 + b^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 + 3*a^2*Sqrt[d]*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(3*c*f*Sqrt[1 + c^2*x^2])","A",1
591,1,168,59,0.854043,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+i c d x} \sqrt{f-i c f x}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]),x]","\frac{a^2 \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)}{c \sqrt{d} \sqrt{f}}+\frac{a b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2}{c \sqrt{d+i c d x} \sqrt{f-i c f x}}+\frac{b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^3}{3 c \sqrt{d+i c d x} \sqrt{f-i c f x}}","\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c \sqrt{d+i c d x} \sqrt{f-i c f x}}",1,"(a*b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2)/(c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^3)/(3*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (a^2*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]])/(c*Sqrt[d]*Sqrt[f])","B",1
592,1,508,464,2.2631126,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} \sqrt{f-i c f x}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(3/2)*Sqrt[f - I*c*f*x]),x]","\frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left(\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(a^2 c x+i a^2-a b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)-4 i a b \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+2 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+4 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(e^{\sinh ^{-1}(c x)}+1\right)-2 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(\sin \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)-4 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(2 a-4 i b \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+\pi  b\right)+i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(2 a+4 i b \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-\pi  b\right)\right)+4 b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+(-1+i) b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{c d^2 f (c x-i) (c x+i) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}","\frac{f \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{i f \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{f x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 b f \left(c^2 x^2+1\right)^{3/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 i b f \left(c^2 x^2+1\right)^{3/2} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 b^2 f \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 b^2 f \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 f \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((-1 + I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - Sinh[ArcSinh[c*x]/2]) + (I*a^2 + a^2*c*x - (4*I)*a*b*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] + (2*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[1 - I/E^ArcSinh[c*x]] + (4*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[1 + E^ArcSinh[c*x]] - a*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] - (4*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[Cosh[ArcSinh[c*x]/2]] - (2*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + 4*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]) + b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*(I*Cosh[ArcSinh[c*x]/2]*(2*a - b*Pi + (4*I)*b*Log[1 - I/E^ArcSinh[c*x]]) + (2*a + b*Pi - (4*I)*b*Log[1 - I/E^ArcSinh[c*x]])*Sinh[ArcSinh[c*x]/2])))/(c*d^2*f*(-I + c*x)*(I + c*x)*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2]))","A",0
593,1,524,942,7.1371824,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{5/2} \sqrt{f-i c f x}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]),x]","\frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left(\frac{a^2 (c x-2 i)}{(c x-i)^2}-\frac{a b \left(2 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\frac{1}{2} i \left(\left(\sqrt{c^2 x^2+1}+2\right) \log \left(c^2 x^2+1\right)-2\right)+\left(\sqrt{c^2 x^2+1}-1\right) \sinh ^{-1}(c x)+2 \left(\sqrt{c^2 x^2+1}+2\right) \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)-i \cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(-\frac{1}{2} i \log \left(c^2 x^2+1\right)+\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\frac{3}{2} \log \left(c^2 x^2+1\right)-3 i \sinh ^{-1}(c x)-6 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-2\right)\right)}{\sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}-\frac{b^2 \left(-4 \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+(1-i) \sinh ^{-1}(c x)^2-\frac{\left(\sinh ^{-1}(c x)-2 i\right) \sinh ^{-1}(c x)}{c x-i}+2 \left(2 \sinh ^{-1}(c x)-i \pi \right) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-\frac{2 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}-\frac{2 \left(\sinh ^{-1}(c x)^2-2\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}+i \pi  \left(\sinh ^{-1}(c x)-4 \log \left(e^{\sinh ^{-1}(c x)}+1\right)+2 \log \left(\sin \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)+4 \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{c^2 x^2+1}}\right)}{3 c d^3 f}","-\frac{c^2 f^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2 x^3}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{b c f^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) x^2}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{2 b^2 f^2 \left(c^2 x^2+1\right)^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{2 f^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{f^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i b f^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i b^2 f^2 \left(c^2 x^2+1\right)^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{f^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i f^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{b^2 f^2 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{b f^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{4 i b f^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{2 b f^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \log \left(1+e^{2 \sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{2 b^2 f^2 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b^2 f^2 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{b^2 f^2 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((a^2*(-2*I + c*x))/(-I + c*x)^2 - (a*b*((-I)*Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] - (I/2)*Log[1 + c^2*x^2]) + Cosh[ArcSinh[c*x]/2]*(-2 - (3*I)*ArcSinh[c*x] - (6*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*Log[1 + c^2*x^2])/2) + 2*((-1 + Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + 2*(2 + Sqrt[1 + c^2*x^2])*ArcTan[Coth[ArcSinh[c*x]/2]] + (I/2)*(-2 + (2 + Sqrt[1 + c^2*x^2])*Log[1 + c^2*x^2]))*Sinh[ArcSinh[c*x]/2]))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3) - (b^2*((1 - I)*ArcSinh[c*x]^2 - (ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/(-I + c*x) + 2*((-I)*Pi + 2*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] + I*Pi*(ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] + 4*Log[Cosh[ArcSinh[c*x]/2]] + 2*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]]) - 4*PolyLog[2, I/E^ArcSinh[c*x]] - (2*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 - (2*(-2 + ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/Sqrt[1 + c^2*x^2]))/(3*c*d^3*f)","A",0
594,1,2143,972,14.7034852,"\int \frac{(d+i c d x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{(f-i c f x)^{3/2}} \, dx","Integrate[((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(3/2),x]","\text{Result too large to show}","-\frac{5 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3 d^4}{2 b c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{b^2 x \left(c^2 x^2+1\right)^2 d^4}{4 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i b^2 \left(c^2 x^2+1\right)^2 d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 d^4}{2 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{4 i \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2 d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2 d^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{8 i \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2 d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i a b x \left(c^2 x^2+1\right)^{3/2} d^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i b^2 x \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) d^4}{(i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x) d^4}{4 c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{b c x^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) d^4}{2 (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{32 i b \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \log \left(1+e^{2 \sinh ^{-1}(c x)}\right) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}+\frac{16 b^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{16 b^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right) d^4}{c (i c x d+d)^{3/2} (f-i c f x)^{3/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((-4*I)*a^2*d^2)/f^2 + (a^2*c*d^2*x)/(2*f^2) + (8*a^2*d^2)/(f^2*(I + c*x))))/c - (15*a^2*d^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(2*c*f^(3/2)) - ((4*I)*a*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(-(c*x) + 2*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - I*ArcSinh[c*x]^2 + 4*ArcTan[Coth[ArcSinh[c*x]/2]] - (2*I)*Log[Sqrt[1 + c^2*x^2]]) - ((-I)*c*x - (2*I)*ArcSinh[c*x] + I*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]^2 + (4*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 2*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) - (a*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(8*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*(ArcSinh[c*x]*(4*I + ArcSinh[c*x]) + 4*Log[Sqrt[1 + c^2*x^2]])) + (ArcSinh[c*x]*(-4*I + ArcSinh[c*x]) - (8*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])) - (b^2*d^2*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(-18*Pi*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 - 12*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] + 12*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - (24*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - ((12*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(3*c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) - (((2*I)/3)*b^2*d^2*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(((-6*I)*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ((6 + 6*I)*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (2*ArcSinh[c*x]^3)/Sqrt[1 + c^2*x^2] + (3*I)*(2 + ArcSinh[c*x]^2) + ((6*I)*(2*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + Pi*(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) + (4*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2] - (12*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]))))/(c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + (b^2*d^2*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*((-96*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ((48 - 48*I)*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] - ((20*I)*ArcSinh[c*x]^3)/Sqrt[1 + c^2*x^2] + 48*(2 + ArcSinh[c*x]^2) + (6*I)*c*x*(1 + 2*ArcSinh[c*x]^2) - ((6*I)*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] + (48*(2*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + Pi*(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) + (4*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2] + ((96*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]))))/(24*c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + (a*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Sinh[ArcSinh[c*x]/2]*(-16*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + I*Cosh[2*ArcSinh[c*x]] + 2*(8*c*x + 8*ArcSinh[c*x] + (5*I)*ArcSinh[c*x]^2 + 16*ArcTan[Tanh[ArcSinh[c*x]/2]] + (8*I)*Log[Sqrt[1 + c^2*x^2]] - I*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]])) - Cosh[ArcSinh[c*x]/2]*((16*I)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + Cosh[2*ArcSinh[c*x]] - 2*((8*I)*c*x - (8*I)*ArcSinh[c*x] - 5*ArcSinh[c*x]^2 + (16*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] - 8*Log[Sqrt[1 + c^2*x^2]] + ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]))))/(4*c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]))","B",0
595,1,1346,752,11.1943082,"\int \frac{(d+i c d x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{(f-i c f x)^{3/2}} \, dx","Integrate[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(3/2),x]","-\frac{3 d^{3/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i d (c x-i)} \sqrt{-i f (c x+i)}\right) a^2}{c f^{3/2}}-\frac{2 i b d \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(-i \sinh ^{-1}(c x)^2+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+2 \sinh ^{-1}(c x)-c x+4 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-2 i \log \left(\sqrt{c^2 x^2+1}\right)\right)-\left(\sinh ^{-1}(c x)^2+i \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-2 i \sinh ^{-1}(c x)-i c x+4 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+2 \log \left(\sqrt{c^2 x^2+1}\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) a}{c f^2 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}-\frac{b d \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(8 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i \left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+4 i\right)+4 \log \left(\sqrt{c^2 x^2+1}\right)\right)\right)+\left(\sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)-4 i\right)-8 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4 \log \left(\sqrt{c^2 x^2+1}\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) a}{c f^2 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left(\frac{4 a^2 d}{f^2 (c x+i)}-\frac{i a^2 d}{f^2}\right)}{c}-\frac{b^2 d (c x-i) \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(i \sinh ^{-1}(c x)^3-\frac{12 i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}-(6-6 i) \sinh ^{-1}(c x)^2-18 \pi  \sinh ^{-1}(c x)-12 \left(\pi -2 i \sinh ^{-1}(c x)\right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+24 \pi  \log \left(1+e^{\sinh ^{-1}(c x)}\right)+12 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)-24 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-24 i \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)\right)}{3 c f^2 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^2}-\frac{i b^2 d (c x-i) \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\frac{2 \sinh ^{-1}(c x)^3}{\sqrt{c^2 x^2+1}}-\frac{12 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}+\frac{(6+6 i) \sinh ^{-1}(c x)^2}{\sqrt{c^2 x^2+1}}-\frac{6 i c x \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+3 i \left(\sinh ^{-1}(c x)^2+2\right)+\frac{6 i \left(2 \left(\pi -2 i \sinh ^{-1}(c x)\right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+\pi  \left(3 \sinh ^{-1}(c x)-4 \log \left(1+e^{\sinh ^{-1}(c x)}\right)-2 \log \left(-\cos \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)+4 \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+4 i \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)\right)}{\sqrt{c^2 x^2+1}}\right)}{3 c f^2 \sqrt{-((c d x-i d) (c x f+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^2}","\frac{2 i a b d^3 x \left(c^2 x^2+1\right)^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{d^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{i d^3 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 d^3 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 d^3 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 i d^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b d^3 \left(c^2 x^2+1\right)^{3/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{16 i b d^3 \left(c^2 x^2+1\right)^{3/2} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 b^2 d^3 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{8 b^2 d^3 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b^2 d^3 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 i b^2 d^3 \left(c^2 x^2+1\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 i b^2 d^3 x \left(c^2 x^2+1\right)^{3/2} \sinh ^{-1}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((-I)*a^2*d)/f^2 + (4*a^2*d)/(f^2*(I + c*x))))/c - (3*a^2*d^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(c*f^(3/2)) - ((2*I)*a*b*d*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(-(c*x) + 2*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - I*ArcSinh[c*x]^2 + 4*ArcTan[Coth[ArcSinh[c*x]/2]] - (2*I)*Log[Sqrt[1 + c^2*x^2]]) - ((-I)*c*x - (2*I)*ArcSinh[c*x] + I*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]^2 + (4*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 2*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) - (a*b*d*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(8*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*(ArcSinh[c*x]*(4*I + ArcSinh[c*x]) + 4*Log[Sqrt[1 + c^2*x^2]])) + (ArcSinh[c*x]*(-4*I + ArcSinh[c*x]) - (8*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])) - (b^2*d*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(-18*Pi*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 - 12*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] + 12*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - (24*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - ((12*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(3*c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) - ((I/3)*b^2*d*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(((-6*I)*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ((6 + 6*I)*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (2*ArcSinh[c*x]^3)/Sqrt[1 + c^2*x^2] + (3*I)*(2 + ArcSinh[c*x]^2) + ((6*I)*(2*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + Pi*(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) + (4*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2] - (12*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]))))/(c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2)","A",0
596,1,530,544,5.5313693,"\int \frac{\sqrt{d+i c d x} \left(a+b \sinh ^{-1}(c x)\right)^2}{(f-i c f x)^{3/2}} \, dx","Integrate[(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(3/2),x]","\frac{\frac{6 a^2 \sqrt{d+i c d x} \sqrt{f-i c f x}}{c x+i}-3 a^2 \sqrt{d} \sqrt{f} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right)+\frac{3 a b \sqrt{d+i c d x} \sqrt{f-i c f x} \left(2 \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(4 \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i \log \left(c^2 x^2+1\right)\right)-\left(\sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+4 \sinh ^{-1}(c x) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{\sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}-\frac{b^2 (c x-i) \sqrt{d+i c d x} \sqrt{f-i c f x} \left(-24 i \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+i \sinh ^{-1}(c x)^3-(6-6 i) \sinh ^{-1}(c x)^2-18 \pi  \sinh ^{-1}(c x)-12 \left(\pi -2 i \sinh ^{-1}(c x)\right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+24 \pi  \log \left(e^{\sinh ^{-1}(c x)}+1\right)-\frac{12 i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}+12 \pi  \log \left(-\cos \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)-24 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{\sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^2}}{3 c f^2}","-\frac{d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 i d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 d^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b d^2 \left(c^2 x^2+1\right)^{3/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{8 i b d^2 \left(c^2 x^2+1\right)^{3/2} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 b^2 d^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{4 b^2 d^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 b^2 d^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"((6*a^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(I + c*x) - 3*a^2*Sqrt[d]*Sqrt[f]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - (b^2*(-I + c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-18*Pi*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 - 12*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[c*x]] + 12*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 24*Pi*Log[Cosh[ArcSinh[c*x]/2]] - (24*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - ((12*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + (3*a*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + 4*ArcSinh[c*x]*((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + 2*(4*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(3*c*f^2)","A",0
597,1,511,464,2.2769227,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+i c d x} (f-i c f x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(3/2)),x]","\frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left(\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(a^2 c x-i a^2-a b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)+4 i a b \sqrt{c^2 x^2+1} \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-2 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+4 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(e^{\sinh ^{-1}(c x)}+1\right)+2 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(-\cos \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)-4 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+b \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \left(\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(2 a+4 i b \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-3 \pi  b\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(2 a-4 i b \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+3 \pi  b\right)\right)+4 b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+(-1-i) b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2 \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{c d f^2 (c x-i) (c x+i) \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)}","\frac{d \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{i d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{d x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 b d \left(c^2 x^2+1\right)^{3/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{4 i b d \left(c^2 x^2+1\right)^{3/2} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{2 b^2 d \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 b^2 d \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 d \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((-1 - I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2*(Cosh[ArcSinh[c*x]/2] - Sinh[ArcSinh[c*x]/2]) + ((-I)*a^2 + a^2*c*x + (4*I)*a*b*Sqrt[1 + c^2*x^2]*ArcTan[Tanh[ArcSinh[c*x]/2]] - (2*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[1 + I/E^ArcSinh[c*x]] + (4*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[1 + E^ArcSinh[c*x]] - a*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + (2*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - (4*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[Cosh[ArcSinh[c*x]/2]])*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]) + 4*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]) + b*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*((-I)*Cosh[ArcSinh[c*x]/2]*(2*a + 3*b*Pi - (4*I)*b*Log[1 + I/E^ArcSinh[c*x]]) + (2*a - 3*b*Pi + (4*I)*b*Log[1 + I/E^ArcSinh[c*x]])*Sinh[ArcSinh[c*x]/2])))/(c*d*f^2*(-I + c*x)*(I + c*x)*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]))","A",0
598,1,488,224,1.3616975,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)),x]","\frac{a^2 c x-a b \sqrt{c^2 x^2+1} \log \left(c^2 x^2+1\right)+2 a b c x \sinh ^{-1}(c x)+2 b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+2 b^2 \sqrt{c^2 x^2+1} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2-2 i \pi  b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-2 b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-2 b^2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+4 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(e^{\sinh ^{-1}(c x)}+1\right)-i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(\sin \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)+i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(-\cos \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)-4 i \pi  b^2 \sqrt{c^2 x^2+1} \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+b^2 c x \sinh ^{-1}(c x)^2}{c d f \sqrt{d+i c d x} \sqrt{f-i c f x}}","\frac{\left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac{x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{2 b \left(c^2 x^2+1\right)^{3/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac{b^2 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}",1,"(a^2*c*x + 2*a*b*c*x*ArcSinh[c*x] - (2*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + b^2*c*x*ArcSinh[c*x]^2 - b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 + I*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[1 - I/E^ArcSinh[c*x]] - 2*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - I*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[1 + I/E^ArcSinh[c*x]] - 2*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + (4*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[1 + E^ArcSinh[c*x]] - a*b*Sqrt[1 + c^2*x^2]*Log[1 + c^2*x^2] + I*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - (4*I)*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[Cosh[ArcSinh[c*x]/2]] - I*b^2*Pi*Sqrt[1 + c^2*x^2]*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + 2*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSinh[c*x]] + 2*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(c*d*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])","B",0
599,1,754,743,8.3405509,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{5/2} (f-i c f x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(3/2)),x]","\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left(\frac{5 a^2}{12 d^3 f^2 (c x-i)}+\frac{a^2}{4 d^3 f^2 (c x+i)}-\frac{i a^2}{6 d^3 f^2 (c x-i)^2}\right)}{c}+\frac{i a b \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \left(\sqrt{c^2 x^2+1} \left(-4 \log \left(\sqrt{c^2 x^2+1}\right)+2 c x \left(\tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-2 i \log \left(\sqrt{c^2 x^2+1}\right)\right)-2 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+1\right)+4 c x \sinh ^{-1}(c x)+2 i \sinh ^{-1}(c x) \cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{3 c d^2 f (c x-i) \sqrt{-((c d x-i d) (c f x+i f))} \sqrt{-d f \left(c^2 x^2+1\right)}}+\frac{i b^2 \sqrt{c^2 x^2+1} \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \left(6 i \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+10 i \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-(1+4 i) \sinh ^{-1}(c x)^2+\frac{\left(2+i \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)}{c x-i}+7 \pi  \sinh ^{-1}(c x)-5 \left(\pi +2 i \sinh ^{-1}(c x)\right) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+3 \left(\pi -2 i \sinh ^{-1}(c x)\right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-16 \pi  \log \left(e^{\sinh ^{-1}(c x)}+1\right)+\frac{3 i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}+\frac{2 i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}+\frac{\left(5 \sinh ^{-1}(c x)^2-4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}+5 \pi  \log \left(\sin \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)-3 \pi  \log \left(-\cos \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)+16 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{6 c d^2 f \sqrt{-((c d x-i d) (c f x+i f))} \sqrt{-d f \left(c^2 x^2+1\right)}}","\frac{2 f \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 f x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b f \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i b f x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i f \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{f x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{4 b f \left(c^2 x^2+1\right)^{5/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i b f \left(c^2 x^2+1\right)^{5/2} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b^2 f \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b^2 f \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 b^2 f \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i b^2 f \left(c^2 x^2+1\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b^2 f x \left(c^2 x^2+1\right)^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((-1/6*I)*a^2)/(d^3*f^2*(-I + c*x)^2) + (5*a^2)/(12*d^3*f^2*(-I + c*x)) + a^2/(4*d^3*f^2*(I + c*x))))/c + ((I/3)*a*b*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*(4*c*x*ArcSinh[c*x] + (2*I)*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] + Sqrt[1 + c^2*x^2]*(1 - (2*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 2*c*x*(ArcTan[Tanh[ArcSinh[c*x]/2]] - (2*I)*Log[Sqrt[1 + c^2*x^2]]) - 4*Log[Sqrt[1 + c^2*x^2]])))/(c*d^2*f*(-I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[-(d*f*(1 + c^2*x^2))]) + ((I/6)*b^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[1 + c^2*x^2]*(7*Pi*ArcSinh[c*x] + ((2 + I*ArcSinh[c*x])*ArcSinh[c*x])/(-I + c*x) - (1 + 4*I)*ArcSinh[c*x]^2 - 5*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] + 3*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] - 16*Pi*Log[1 + E^ArcSinh[c*x]] - 3*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 16*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 5*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + (6*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] + (10*I)*PolyLog[2, I/E^ArcSinh[c*x]] + ((3*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]) + ((2*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^3 + ((-4 + 5*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/((-I)*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])))/(c*d^2*f*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[-(d*f*(1 + c^2*x^2))])","A",0
600,1,2552,794,13.9076847,"\int \frac{(d+i c d x)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{(f-i c f x)^{5/2}} \, dx","Integrate[((d + I*c*d*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(5/2),x]","\text{Result too large to show}","-\frac{2 i a b d^5 x \left(c^2 x^2+1\right)^{5/2}}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{5 d^5 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i d^5 \left(c^2 x^2+1\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{28 d^5 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{112 b d^5 \left(c^2 x^2+1\right)^{5/2} \log \left(1+i e^{-\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{8 b d^5 \left(c^2 x^2+1\right)^{5/2} \sec ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{28 i d^5 \left(c^2 x^2+1\right)^{5/2} \tan \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{4 i d^5 \left(c^2 x^2+1\right)^{5/2} \sec ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \tan \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{112 b^2 d^5 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i b^2 d^5 \left(c^2 x^2+1\right)^3}{c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i b^2 d^5 x \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{16 i b^2 d^5 \left(c^2 x^2+1\right)^{5/2} \tan \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((I*a^2*d^2)/f^3 + (((8*I)/3)*a^2*d^2)/(f^3*(I + c*x)^2) - (28*a^2*d^2)/(3*f^3*(I + c*x))))/c + (5*a^2*d^(5/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(c*f^(5/2)) - ((I/3)*a*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(-(Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]])) + Cosh[ArcSinh[c*x]/2]*(4*I + 3*ArcSinh[c*x] - 6*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*I)*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x^2]*(I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[Sqrt[1 + c^2*x^2]]) + 2*(1 + I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*f^3*(1 + I*c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) + (a*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSinh[c*x])/2]*((14*I - 3*ArcSinh[c*x])*ArcSinh[c*x] + (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] - 14*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(8 + (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 - (84*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 42*Log[Sqrt[1 + c^2*x^2]]) - (2*I)*(4 + (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 - (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 28*Log[Sqrt[1 + c^2*x^2]] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(14*I + 3*ArcSinh[c*x]) - (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(3*c*f^3*(1 + I*c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) - ((I/3)*b^2*d^2*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*((-1 - I)*ArcSinh[c*x]^2 - (2*ArcSinh[c*x]*(2*I + ArcSinh[c*x]))/(I + c*x) - (2*I)*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] - I*Pi*(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) + 4*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(c*f^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + ((I/3)*b^2*d^2*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(((-6*I)*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ((13 + 13*I)*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (3*ArcSinh[c*x]^3)/Sqrt[1 + c^2*x^2] + (2*ArcSinh[c*x]*(2*I + ArcSinh[c*x]))/((I + c*x)*Sqrt[1 + c^2*x^2]) + (3*I)*(2 + ArcSinh[c*x]^2) + ((13*I)*(2*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + Pi*(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) + (4*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]]))/Sqrt[1 + c^2*x^2] + (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^3) - (2*(4 + 13*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]))))/(c*f^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + (2*b^2*d^2*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(-21*Pi*ArcSinh[c*x] - (7 - 7*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 + ((2*I)*ArcSinh[c*x]*(2*I + ArcSinh[c*x]))/(I + c*x) - 14*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + 28*Pi*Log[1 + E^ArcSinh[c*x]] + 14*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 28*Pi*Log[Cosh[ArcSinh[c*x]/2]] - (28*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - ((2*I)*(4 + 7*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]) + (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])^3))/(3*c*f^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) - ((I/6)*a*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(-(Cosh[(3*ArcSinh[c*x])/2]*(9 - (35*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 + (52*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 26*Log[Sqrt[1 + c^2*x^2]])) + Cosh[ArcSinh[c*x]/2]*(20 + (24*I)*ArcSinh[c*x] + 27*ArcSinh[c*x]^2 + (156*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 78*Log[Sqrt[1 + c^2*x^2]]) - I*(3*(-I + ArcSinh[c*x])*Cosh[(5*ArcSinh[c*x])/2] + 2*(13 + (7*I)*ArcSinh[c*x] + 18*ArcSinh[c*x]^2 + (104*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*I)*(I + ArcSinh[c*x])*Cosh[2*ArcSinh[c*x]] + 52*Log[Sqrt[1 + c^2*x^2]] + Sqrt[1 + c^2*x^2]*(6 + (38*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 + (52*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 26*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2])))/(c*f^3*(-I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4)","B",0
601,1,1617,584,10.2167267,"\int \frac{(d+i c d x)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{(f-i c f x)^{5/2}} \, dx","Integrate[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(5/2),x]","\frac{d^{3/2} \log \left(c d f x+\sqrt{d} \sqrt{f} \sqrt{i d (c x-i)} \sqrt{-i f (c x+i)}\right) a^2}{c f^{5/2}}-\frac{i b d \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(-\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i \log \left(\sqrt{c^2 x^2+1}\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(3 \sinh ^{-1}(c x)-6 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+3 i \log \left(\sqrt{c^2 x^2+1}\right)+4 i\right)+2 \left(\sqrt{c^2 x^2+1} \left(i \sinh ^{-1}(c x)+2 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\log \left(\sqrt{c^2 x^2+1}\right)\right)+2 \left(i \sinh ^{-1}(c x)+2 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+\log \left(\sqrt{c^2 x^2+1}\right)+1\right)\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) a}{3 c f^3 (i c x+1) \sqrt{-((c d x-i d) (c x f+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}+\frac{b d \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\left(14 i-3 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+28 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-14 \log \left(\sqrt{c^2 x^2+1}\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(9 \sinh ^{-1}(c x)^2+6 i \sinh ^{-1}(c x)-84 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+42 \log \left(\sqrt{c^2 x^2+1}\right)+8\right)-2 i \left(6 \sinh ^{-1}(c x)^2+4 i \sinh ^{-1}(c x)-56 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+28 \log \left(\sqrt{c^2 x^2+1}\right)+\sqrt{c^2 x^2+1} \left(\sinh ^{-1}(c x) \left(3 \sinh ^{-1}(c x)+14 i\right)-28 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+14 \log \left(\sqrt{c^2 x^2+1}\right)\right)+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) a}{6 c f^3 (i c x+1) \sqrt{-((c d x-i d) (c x f+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}+\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left(\frac{4 i a^2 d}{3 f^3 (c x+i)^2}-\frac{8 a^2 d}{3 f^3 (c x+i)}\right)}{c}-\frac{i b^2 d (c x-i) \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(-\frac{4 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}-(1+i) \sinh ^{-1}(c x)^2-\frac{2 \left(\sinh ^{-1}(c x)+2 i\right) \sinh ^{-1}(c x)}{c x+i}-2 i \left(\pi -2 i \sinh ^{-1}(c x)\right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-i \pi  \left(3 \sinh ^{-1}(c x)-4 \log \left(1+e^{\sinh ^{-1}(c x)}\right)-2 \log \left(-\cos \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)+4 \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+4 \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+\frac{2 \left(\sinh ^{-1}(c x)^2+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}\right)}{3 c f^3 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^2}+\frac{b^2 d (c x-i) \sqrt{i (c d x-i d)} \sqrt{-i (c x f+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(i \sinh ^{-1}(c x)^3+\frac{4 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(i \cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+\sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}-(7-7 i) \sinh ^{-1}(c x)^2+\frac{2 i \left(\sinh ^{-1}(c x)+2 i\right) \sinh ^{-1}(c x)}{c x+i}-21 \pi  \sinh ^{-1}(c x)-14 \left(\pi -2 i \sinh ^{-1}(c x)\right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+28 \pi  \log \left(1+e^{\sinh ^{-1}(c x)}\right)+14 \pi  \log \left(-\cos \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)-28 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-28 i \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-\frac{2 i \left(7 \sinh ^{-1}(c x)^2+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}\right)}{3 c f^3 \sqrt{-((c d x-i d) (c x f+i f))} \sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^2}","\frac{d^4 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{8 d^4 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{32 b d^4 \left(c^2 x^2+1\right)^{5/2} \log \left(1+i e^{-\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b d^4 \left(c^2 x^2+1\right)^{5/2} \sec ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{8 i d^4 \left(c^2 x^2+1\right)^{5/2} \tan \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i d^4 \left(c^2 x^2+1\right)^{5/2} \sec ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \tan \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{32 b^2 d^4 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{8 i b^2 d^4 \left(c^2 x^2+1\right)^{5/2} \tan \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((((4*I)/3)*a^2*d)/(f^3*(I + c*x)^2) - (8*a^2*d)/(3*f^3*(I + c*x))))/c + (a^2*d^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(c*f^(5/2)) - ((I/3)*a*b*d*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(-(Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]])) + Cosh[ArcSinh[c*x]/2]*(4*I + 3*ArcSinh[c*x] - 6*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*I)*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x^2]*(I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[Sqrt[1 + c^2*x^2]]) + 2*(1 + I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*f^3*(1 + I*c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) + (a*b*d*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(Cosh[(3*ArcSinh[c*x])/2]*((14*I - 3*ArcSinh[c*x])*ArcSinh[c*x] + (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] - 14*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[c*x]/2]*(8 + (6*I)*ArcSinh[c*x] + 9*ArcSinh[c*x]^2 - (84*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 42*Log[Sqrt[1 + c^2*x^2]]) - (2*I)*(4 + (4*I)*ArcSinh[c*x] + 6*ArcSinh[c*x]^2 - (56*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 28*Log[Sqrt[1 + c^2*x^2]] + Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(14*I + 3*ArcSinh[c*x]) - (28*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 14*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(6*c*f^3*(1 + I*c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) - ((I/3)*b^2*d*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*((-1 - I)*ArcSinh[c*x]^2 - (2*ArcSinh[c*x]*(2*I + ArcSinh[c*x]))/(I + c*x) - (2*I)*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] - I*Pi*(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) + 4*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(c*f^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2) + (b^2*d*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(-21*Pi*ArcSinh[c*x] - (7 - 7*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 + ((2*I)*ArcSinh[c*x]*(2*I + ArcSinh[c*x]))/(I + c*x) - 14*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + 28*Pi*Log[1 + E^ArcSinh[c*x]] + 14*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 28*Pi*Log[Cosh[ArcSinh[c*x]/2]] - (28*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - ((2*I)*(4 + 7*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]) + (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])^3))/(3*c*f^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2)","B",0
602,1,788,522,8.2367996,"\int \frac{\sqrt{d+i c d x} \left(a+b \sinh ^{-1}(c x)\right)^2}{(f-i c f x)^{5/2}} \, dx","Integrate[(Sqrt[d + I*c*d*x]*(a + b*ArcSinh[c*x])^2)/(f - I*c*f*x)^(5/2),x]","\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left(\frac{2 i a^2}{3 f^3 (c x+i)^2}-\frac{a^2}{3 f^3 (c x+i)}\right)}{c}-\frac{i a b \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right) \left(2 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\sqrt{c^2 x^2+1} \left(\log \left(\sqrt{c^2 x^2+1}\right)+i \sinh ^{-1}(c x)+2 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+2 \left(\log \left(\sqrt{c^2 x^2+1}\right)+i \sinh ^{-1}(c x)+2 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+1\right)\right)-\cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(i \log \left(\sqrt{c^2 x^2+1}\right)+\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(3 i \log \left(\sqrt{c^2 x^2+1}\right)+3 \sinh ^{-1}(c x)-6 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+4 i\right)\right)}{3 c f^3 (1+i c x) \sqrt{-((c d x-i d) (c f x+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^4}-\frac{i b^2 (c x-i) \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \sqrt{-d f \left(c^2 x^2+1\right)} \left(4 \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-(1+i) \sinh ^{-1}(c x)^2-\frac{2 \left(\sinh ^{-1}(c x)+2 i\right) \sinh ^{-1}(c x)}{c x+i}-2 i \left(\pi -2 i \sinh ^{-1}(c x)\right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-\frac{4 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}+\frac{2 \left(\sinh ^{-1}(c x)^2+4\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}-i \pi  \left(3 \sinh ^{-1}(c x)-4 \log \left(e^{\sinh ^{-1}(c x)}+1\right)-2 \log \left(-\cos \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)+4 \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{3 c f^3 \sqrt{c^2 x^2+1} \sqrt{-((c d x-i d) (c f x+i f))} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^2}","\frac{d^3 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 b d^3 \left(c^2 x^2+1\right)^{5/2} \log \left(1+i e^{-\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b d^3 \left(c^2 x^2+1\right)^{5/2} \sec ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i d^3 \left(c^2 x^2+1\right)^{5/2} \tan \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i d^3 \left(c^2 x^2+1\right)^{5/2} \sec ^2\left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \tan \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{4 b^2 d^3 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{4 i b^2 d^3 \left(c^2 x^2+1\right)^{5/2} \tan \left(\frac{\pi }{4}+\frac{1}{2} i \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((((2*I)/3)*a^2)/(f^3*(I + c*x)^2) - a^2/(3*f^3*(I + c*x))))/c - ((I/3)*a*b*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])*(-(Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]])) + Cosh[ArcSinh[c*x]/2]*(4*I + 3*ArcSinh[c*x] - 6*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*I)*Log[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x^2]*(I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[Sqrt[1 + c^2*x^2]]) + 2*(1 + I*ArcSinh[c*x] + (2*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x]/2]))/(c*f^3*(1 + I*c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^4) - ((I/3)*b^2*(-I + c*x)*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*((-1 - I)*ArcSinh[c*x]^2 - (2*ArcSinh[c*x]*(2*I + ArcSinh[c*x]))/(I + c*x) - (2*I)*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] - I*Pi*(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) + 4*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (4*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^3 + (2*(4 + ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(c*f^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^2)","A",0
603,1,528,942,7.0531278,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+i c d x} (f-i c f x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(Sqrt[d + I*c*d*x]*(f - I*c*f*x)^(5/2)),x]","\frac{\sqrt{d+i c d x} \sqrt{f-i c f x} \left(\frac{a^2 (c x+2 i)}{(c x+i)^2}-\frac{a b \left(2 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(-\frac{1}{2} i \left(\sqrt{c^2 x^2+1}+2\right) \log \left(c^2 x^2+1\right)+\left(\sqrt{c^2 x^2+1}-1\right) \sinh ^{-1}(c x)+2 \left(\sqrt{c^2 x^2+1}+2\right) \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+i\right)+i \cosh \left(\frac{3}{2} \sinh ^{-1}(c x)\right) \left(\frac{1}{2} i \log \left(c^2 x^2+1\right)+\sinh ^{-1}(c x)-2 \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)+\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \left(\frac{3}{2} \log \left(c^2 x^2+1\right)+3 i \sinh ^{-1}(c x)+6 i \tan ^{-1}\left(\coth \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-2\right)\right)}{\sqrt{c^2 x^2+1} \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}-\frac{b^2 \left(-4 \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)+(1+i) \sinh ^{-1}(c x)^2-\frac{\left(\sinh ^{-1}(c x)+2 i\right) \sinh ^{-1}(c x)}{c x+i}+2 \left(2 \sinh ^{-1}(c x)+i \pi \right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-\frac{2 \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}-\frac{2 \left(\sinh ^{-1}(c x)^2-2\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}+i \pi  \left(3 \sinh ^{-1}(c x)-4 \log \left(e^{\sinh ^{-1}(c x)}+1\right)-2 \log \left(-\cos \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)+4 \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)}{\sqrt{c^2 x^2+1}}\right)}{3 c d f^3}","-\frac{c^2 d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2 x^3}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{b c d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) x^2}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{2 b^2 d^2 \left(c^2 x^2+1\right)^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{2 d^2 \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2 x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i b d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) x}{3 (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i b^2 d^2 \left(c^2 x^2+1\right)^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{d^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{2 i d^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{b^2 d^2 \left(c^2 x^2+1\right)^{5/2} \sinh ^{-1}(c x)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{b d^2 \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{4 i b d^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{2 b d^2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \log \left(1+e^{2 \sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}+\frac{2 b^2 d^2 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{2 b^2 d^2 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}-\frac{b^2 d^2 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 c (i c x d+d)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((a^2*(2*I + c*x))/(I + c*x)^2 - (a*b*(I*Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] + (I/2)*Log[1 + c^2*x^2]) + Cosh[ArcSinh[c*x]/2]*(-2 + (3*I)*ArcSinh[c*x] + (6*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + (3*Log[1 + c^2*x^2])/2) + 2*(I + (-1 + Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + 2*(2 + Sqrt[1 + c^2*x^2])*ArcTan[Coth[ArcSinh[c*x]/2]] - (I/2)*(2 + Sqrt[1 + c^2*x^2])*Log[1 + c^2*x^2])*Sinh[ArcSinh[c*x]/2]))/(Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^3) - (b^2*((1 + I)*ArcSinh[c*x]^2 - (ArcSinh[c*x]*(2*I + ArcSinh[c*x]))/(I + c*x) + 2*(I*Pi + 2*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + I*Pi*(3*ArcSinh[c*x] - 4*Log[1 + E^ArcSinh[c*x]] - 2*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 4*Log[Cosh[ArcSinh[c*x]/2]]) - 4*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (2*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^3 - (2*(-2 + ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/Sqrt[1 + c^2*x^2]))/(3*c*d*f^3)","A",0
604,1,757,743,8.3310652,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{3/2} (f-i c f x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)),x]","\frac{\sqrt{i d (c x-i)} \sqrt{-i f (c x+i)} \left(\frac{a^2}{4 d^2 f^3 (c x-i)}+\frac{5 a^2}{12 d^2 f^3 (c x+i)}+\frac{i a^2}{6 d^2 f^3 (c x+i)^2}\right)}{c}-\frac{i a b \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \left(\sqrt{c^2 x^2+1} \left(-4 \log \left(\sqrt{c^2 x^2+1}\right)+2 c x \left(\tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+2 i \log \left(\sqrt{c^2 x^2+1}\right)\right)+2 i \tan ^{-1}\left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)+1\right)+4 c x \sinh ^{-1}(c x)-2 i \sinh ^{-1}(c x) \cosh \left(2 \sinh ^{-1}(c x)\right)\right)}{3 c d f^2 (c x+i) \sqrt{-((c d x-i d) (c f x+i f))} \sqrt{-d f \left(c^2 x^2+1\right)}}-\frac{i b^2 \sqrt{c^2 x^2+1} \sqrt{i (c d x-i d)} \sqrt{-i (c f x+i f)} \left(-10 i \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-6 i \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)-(1-4 i) \sinh ^{-1}(c x)^2+\frac{\left(2-i \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)}{c x+i}-9 \pi  \sinh ^{-1}(c x)+3 \left(\pi +2 i \sinh ^{-1}(c x)\right) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)-5 \left(\pi -2 i \sinh ^{-1}(c x)\right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)+16 \pi  \log \left(e^{\sinh ^{-1}(c x)}+1\right)-\frac{3 i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)+i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}-\frac{2 i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2}{\left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)^3}+\frac{i \left(4-5 \sinh ^{-1}(c x)^2\right) \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}{\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-i \sinh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)}-3 \pi  \log \left(\sin \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)+5 \pi  \log \left(-\cos \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)-16 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)}{6 c d f^2 \sqrt{-((c d x-i d) (c f x+i f))} \sqrt{-d f \left(c^2 x^2+1\right)}}","\frac{2 d \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 d x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b d \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b d x \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{i d \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{d x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{4 b d \left(c^2 x^2+1\right)^{5/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 i b d \left(c^2 x^2+1\right)^{5/2} \tan ^{-1}\left(e^{\sinh ^{-1}(c x)}\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b^2 d \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-i e^{\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b^2 d \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(i e^{\sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 b^2 d \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{i b^2 d \left(c^2 x^2+1\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b^2 d x \left(c^2 x^2+1\right)^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(a^2/(4*d^2*f^3*(-I + c*x)) + ((I/6)*a^2)/(d^2*f^3*(I + c*x)^2) + (5*a^2)/(12*d^2*f^3*(I + c*x))))/c - ((I/3)*a*b*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*(4*c*x*ArcSinh[c*x] - (2*I)*ArcSinh[c*x]*Cosh[2*ArcSinh[c*x]] + Sqrt[1 + c^2*x^2]*(1 + (2*I)*ArcTan[Tanh[ArcSinh[c*x]/2]] + 2*c*x*(ArcTan[Tanh[ArcSinh[c*x]/2]] + (2*I)*Log[Sqrt[1 + c^2*x^2]]) - 4*Log[Sqrt[1 + c^2*x^2]])))/(c*d*f^2*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[-(d*f*(1 + c^2*x^2))]) - ((I/6)*b^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[1 + c^2*x^2]*(-9*Pi*ArcSinh[c*x] + ((2 - I*ArcSinh[c*x])*ArcSinh[c*x])/(I + c*x) - (1 - 4*I)*ArcSinh[c*x]^2 + 3*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - 5*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + 16*Pi*Log[1 + E^ArcSinh[c*x]] + 5*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] - 16*Pi*Log[Cosh[ArcSinh[c*x]/2]] - 3*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] - (10*I)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (6*I)*PolyLog[2, I/E^ArcSinh[c*x]] - ((2*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^3 + (I*(4 - 5*ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2]) - ((3*I)*ArcSinh[c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])))/(c*d*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[-(d*f*(1 + c^2*x^2))])","A",0
605,1,642,386,8.1573767,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)),x]","\frac{4 a^2 c x \left(2 c^2 x^2+3\right)+2 a b \left(\sqrt{c^2 x^2+1} \left(2-3 \log \left(c^2 x^2+1\right)\right)-\log \left(c^2 x^2+1\right) \cosh \left(3 \sinh ^{-1}(c x)\right)+2 \sinh ^{-1}(c x) \left(3 c x+\sinh \left(3 \sinh ^{-1}(c x)\right)\right)\right)-b^2 \left(-16 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(-i e^{-\sinh ^{-1}(c x)}\right)-16 \left(c^2 x^2+1\right)^{3/2} \text{Li}_2\left(i e^{-\sinh ^{-1}(c x)}\right)+2 \sqrt{c^2 x^2+1} \left(\left(6 \sinh ^{-1}(c x)-3 i \pi \right) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right)+i \left(-3 i \sinh ^{-1}(c x)^2+6 \pi  \sinh ^{-1}(c x)+2 i \sinh ^{-1}(c x)+3 \left(\pi -2 i \sinh ^{-1}(c x)\right) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right)-12 \pi  \log \left(e^{\sinh ^{-1}(c x)}+1\right)+3 \pi  \log \left(\sin \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)-3 \pi  \log \left(-\cos \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)+12 \pi  \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)\right)\right)+c x-6 c x \sinh ^{-1}(c x)^2-2 \sinh \left(3 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)^2+\sinh \left(3 \sinh ^{-1}(c x)\right)+2 \sinh ^{-1}(c x)^2 \cosh \left(3 \sinh ^{-1}(c x)\right)+4 i \pi  \sinh ^{-1}(c x) \cosh \left(3 \sinh ^{-1}(c x)\right)+4 \sinh ^{-1}(c x) \log \left(1-i e^{-\sinh ^{-1}(c x)}\right) \cosh \left(3 \sinh ^{-1}(c x)\right)+4 \sinh ^{-1}(c x) \log \left(1+i e^{-\sinh ^{-1}(c x)}\right) \cosh \left(3 \sinh ^{-1}(c x)\right)-2 i \pi  \log \left(1-i e^{-\sinh ^{-1}(c x)}\right) \cosh \left(3 \sinh ^{-1}(c x)\right)+2 i \pi  \log \left(1+i e^{-\sinh ^{-1}(c x)}\right) \cosh \left(3 \sinh ^{-1}(c x)\right)-8 i \pi  \log \left(e^{\sinh ^{-1}(c x)}+1\right) \cosh \left(3 \sinh ^{-1}(c x)\right)+8 i \pi  \cosh \left(3 \sinh ^{-1}(c x)\right) \log \left(\cosh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)\right)-2 i \pi  \cosh \left(3 \sinh ^{-1}(c x)\right) \log \left(-\cos \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)+2 i \pi  \cosh \left(3 \sinh ^{-1}(c x)\right) \log \left(\sin \left(\frac{1}{4} \left(\pi +2 i \sinh ^{-1}(c x)\right)\right)\right)\right)}{12 d^2 f^2 \left(c^3 x^2+c\right) \sqrt{d+i c d x} \sqrt{f-i c f x}}","\frac{2 \left(c^2 x^2+1\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{2 x \left(c^2 x^2+1\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{b \left(c^2 x^2+1\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac{x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{4 b \left(c^2 x^2+1\right)^{5/2} \log \left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{2 b^2 \left(c^2 x^2+1\right)^{5/2} \text{Li}_2\left(-e^{2 \sinh ^{-1}(c x)}\right)}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac{b^2 x \left(c^2 x^2+1\right)^2}{3 (d+i c d x)^{5/2} (f-i c f x)^{5/2}}",1,"(4*a^2*c*x*(3 + 2*c^2*x^2) - b^2*(c*x - 6*c*x*ArcSinh[c*x]^2 + (4*I)*Pi*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]] + 2*ArcSinh[c*x]^2*Cosh[3*ArcSinh[c*x]] - (2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[1 - I/E^ArcSinh[c*x]] + 4*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]]*Log[1 - I/E^ArcSinh[c*x]] + (2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[1 + I/E^ArcSinh[c*x]] + 4*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]]*Log[1 + I/E^ArcSinh[c*x]] - (8*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[1 + E^ArcSinh[c*x]] - (2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + (8*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[Cosh[ArcSinh[c*x]/2]] + (2*I)*Pi*Cosh[3*ArcSinh[c*x]]*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]] + 2*Sqrt[1 + c^2*x^2]*(((-3*I)*Pi + 6*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] + I*((2*I)*ArcSinh[c*x] + 6*Pi*ArcSinh[c*x] - (3*I)*ArcSinh[c*x]^2 + 3*(Pi - (2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] - 12*Pi*Log[1 + E^ArcSinh[c*x]] - 3*Pi*Log[-Cos[(Pi + (2*I)*ArcSinh[c*x])/4]] + 12*Pi*Log[Cosh[ArcSinh[c*x]/2]] + 3*Pi*Log[Sin[(Pi + (2*I)*ArcSinh[c*x])/4]])) - 16*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - 16*(1 + c^2*x^2)^(3/2)*PolyLog[2, I/E^ArcSinh[c*x]] + Sinh[3*ArcSinh[c*x]] - 2*ArcSinh[c*x]^2*Sinh[3*ArcSinh[c*x]]) + 2*a*b*(Sqrt[1 + c^2*x^2]*(2 - 3*Log[1 + c^2*x^2]) - Cosh[3*ArcSinh[c*x]]*Log[1 + c^2*x^2] + 2*ArcSinh[c*x]*(3*c*x + Sinh[3*ArcSinh[c*x]])))/(12*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(c + c^3*x^2))","A",0
606,1,260,312,0.3551715,"\int \left(d+e x^2\right)^4 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^4*(a + b*ArcSinh[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^2 x^2+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 \sinh ^{-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 \sinh ^{-1}(c x)\right)+\frac{4}{3} d^3 e x^3 \left(a+b \sinh ^{-1}(c x)\right)+\frac{6}{5} d^2 e^2 x^5 \left(a+b \sinh ^{-1}(c x)\right)+\frac{4}{7} d e^3 x^7 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{9} e^4 x^9 \left(a+b \sinh ^{-1}(c x)\right)-\frac{4 b e^3 \left(c^2 x^2+1\right)^{7/2} \left(9 c^2 d-7 e\right)}{441 c^9}-\frac{b e^4 \left(c^2 x^2+1\right)^{9/2}}{81 c^9}-\frac{2 b e^2 \left(c^2 x^2+1\right)^{5/2} \left(63 c^4 d^2-90 c^2 d e+35 e^2\right)}{525 c^9}-\frac{4 b e \left(c^2 x^2+1\right)^{3/2} \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}-\frac{b \sqrt{c^2 x^2+1} \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}",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^2*x^2]*(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)*ArcSinh[c*x])/99225","A",1
607,1,187,221,0.2819139,"\int \left(d+e x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^3*(a + b*ArcSinh[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^2 x^2+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}+b \sinh ^{-1}(c x) \left(d^3 x+d^2 e x^3+\frac{3}{5} d e^2 x^5+\frac{e^3 x^7}{7}\right)","d^3 x \left(a+b \sinh ^{-1}(c x)\right)+d^2 e x^3 \left(a+b \sinh ^{-1}(c x)\right)+\frac{3}{5} d e^2 x^5 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{7} e^3 x^7 \left(a+b \sinh ^{-1}(c x)\right)-\frac{3 b e^2 \left(c^2 x^2+1\right)^{5/2} \left(7 c^2 d-5 e\right)}{175 c^7}-\frac{b e^3 \left(c^2 x^2+1\right)^{7/2}}{49 c^7}-\frac{b e \left(c^2 x^2+1\right)^{3/2} \left(35 c^4 d^2-42 c^2 d e+15 e^2\right)}{105 c^7}-\frac{b \sqrt{c^2 x^2+1} \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}",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^2*x^2]*(-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*(d^3*x + d^2*e*x^3 + (3*d*e^2*x^5)/5 + (e^3*x^7)/7)*ArcSinh[c*x]","A",1
608,1,125,147,0.1681133,"\int \left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcSinh[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^2 x^2+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 \sinh ^{-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 \sinh ^{-1}(c x)\right)+\frac{2}{3} d e x^3 \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 \left(a+b \sinh ^{-1}(c x)\right)-\frac{2 b e \left(c^2 x^2+1\right)^{3/2} \left(5 c^2 d-3 e\right)}{45 c^5}-\frac{b e^2 \left(c^2 x^2+1\right)^{5/2}}{25 c^5}-\frac{b \sqrt{c^2 x^2+1} \left(15 c^4 d^2-10 c^2 d e+3 e^2\right)}{15 c^5}",1,"(15*a*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - (b*Sqrt[1 + c^2*x^2]*(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)*ArcSinh[c*x])/225","A",1
609,1,71,81,0.0687395,"\int \left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)*(a + b*ArcSinh[c*x]),x]","\frac{1}{9} \left(3 a x \left(3 d+e x^2\right)-\frac{b \sqrt{c^2 x^2+1} \left(c^2 \left(9 d+e x^2\right)-2 e\right)}{c^3}+3 b x \sinh ^{-1}(c x) \left(3 d+e x^2\right)\right)","d x \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \sinh ^{-1}(c x)\right)-\frac{b \sqrt{c^2 x^2+1} \left(3 c^2 d-e\right)}{3 c^3}-\frac{b e \left(c^2 x^2+1\right)^{3/2}}{9 c^3}",1,"(3*a*x*(3*d + e*x^2) - (b*Sqrt[1 + c^2*x^2]*(-2*e + c^2*(9*d + e*x^2)))/c^3 + 3*b*x*(3*d + e*x^2)*ArcSinh[c*x])/9","A",1
610,1,30,30,0.0084408,"\int \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[a + b*ArcSinh[c*x],x]","a x-\frac{b \sqrt{c^2 x^2+1}}{c}+b x \sinh ^{-1}(c x)","a x-\frac{b \sqrt{c^2 x^2+1}}{c}+b x \sinh ^{-1}(c x)",1,"a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]","A",1
611,1,434,485,0.4392234,"\int \frac{a+b \sinh ^{-1}(c x)}{d+e x^2} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + e*x^2),x]","\frac{2 a \sqrt{-d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)-b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-c \sqrt{-d}}\right)-b \sqrt{d} \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)+b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)-b \sqrt{d} \sinh ^{-1}(c x) \log \left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right)+b \sqrt{d} \sinh ^{-1}(c x) \log \left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-c \sqrt{-d}}+1\right)+b \sqrt{d} \sinh ^{-1}(c x) \log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right)-b \sqrt{d} \sinh ^{-1}(c x) \log \left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right)}{2 \sqrt{-d^2} \sqrt{e}}","\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}",1,"(2*a*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] + b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] + b*Sqrt[d]*ArcSinh[c*x]*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] - b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] + b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] - b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] - b*Sqrt[d]*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))] + b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d^2]*Sqrt[e])","A",0
612,1,622,707,1.7222842,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[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^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{e-c^2 d}}\right)+\sinh ^{-1}(c x) \left(-\sinh ^{-1}(c x)+2 \left(\log \left(1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{-\sqrt{e-c^2 d}+i c \sqrt{d}}\right)+\log \left(1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+i c \sqrt{d}}\right)\right)\right)\right)-i \left(2 \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-i c \sqrt{d}}\right)+2 \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{i \sqrt{d} c+\sqrt{e-c^2 d}}\right)+\sinh ^{-1}(c x) \left(-\sinh ^{-1}(c x)+2 \left(\log \left(1+\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-i c \sqrt{d}}\right)+\log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+i c \sqrt{d}}\right)\right)\right)\right)-2 \sqrt{d} \left(\frac{c \tan ^{-1}\left(\frac{\sqrt{e}-i c^2 \sqrt{d} x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right)}{\sqrt{c^2 d-e}}-\frac{\sinh ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}\right)+2 i \sqrt{d} \left(\frac{c \tanh ^{-1}\left(\frac{-c^2 \sqrt{d} x+i \sqrt{e}}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right)}{\sqrt{c^2 d-e}}+\frac{\sinh ^{-1}(c x)}{\sqrt{d}+i \sqrt{e} x}\right)\right)}{4 d^{3/2} \sqrt{e}}\right)","-\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left(\sqrt{-d}-\sqrt{e} x\right)}+\frac{a+b \sinh ^{-1}(c x)}{4 d \sqrt{e} \left(\sqrt{-d}+\sqrt{e} x\right)}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{b \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{b \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{b c \tan ^{-1}\left(\frac{\sqrt{e}-c^2 \sqrt{-d} x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right)}{4 d \sqrt{e} \sqrt{c^2 d-e}}-\frac{b c \tan ^{-1}\left(\frac{c^2 \sqrt{-d} x+\sqrt{e}}{\sqrt{c^2 x^2+1} \sqrt{c^2 d-e}}\right)}{4 d \sqrt{e} \sqrt{c^2 d-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]*(-(ArcSinh[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) + (c*ArcTan[(Sqrt[e] - I*c^2*Sqrt[d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/Sqrt[c^2*d - e]) + (2*I)*Sqrt[d]*(ArcSinh[c*x]/(Sqrt[d] + I*Sqrt[e]*x) + (c*ArcTanh[(I*Sqrt[e] - c^2*Sqrt[d]*x)/(Sqrt[c^2*d - e]*Sqrt[1 + c^2*x^2])])/Sqrt[c^2*d - e]) + I*(ArcSinh[c*x]*(-ArcSinh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] - Sqrt[-(c^2*d) + e])] + Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e])] + 2*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]))]) - I*(ArcSinh[c*x]*(-ArcSinh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e])] + Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e])])) + 2*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) + e]))] + 2*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e])])))/(4*d^(3/2)*Sqrt[e]))/2","C",0
613,1,443,559,0.6080396,"\int \left(d+e x^2\right)^3 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)^3*(a + b*ArcSinh[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^2 x^2+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 \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+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 \sinh ^{-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{2 b d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c}-\frac{2 b d^2 e x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 c}-\frac{6 b d e^2 x^4 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{25 c}-\frac{2 b e^3 x^6 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{49 c}+\frac{32 b e^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{245 c^7}-\frac{16 b d e^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{25 c^5}-\frac{16 b e^3 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{245 c^5}+\frac{4 b d^2 e \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^3}+\frac{8 b d e^2 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{25 c^3}+\frac{12 b e^3 x^4 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{245 c^3}+d^3 x \left(a+b \sinh ^{-1}(c x)\right)^2+d^2 e x^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{3}{5} d e^2 x^5 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{7} e^3 x^7 \left(a+b \sinh ^{-1}(c x)\right)^2-\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^2*x^2]*(-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^2*x^2]*(-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)))*ArcSinh[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)*ArcSinh[c*x]^2)/(385875*c^7)","A",1
614,1,289,329,0.4285968,"\int \left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcSinh[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^2 x^2+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 \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+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 \sinh ^{-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{2 b d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c}-\frac{4 b d e x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c}-\frac{2 b e^2 x^4 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{25 c}-\frac{16 b e^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{75 c^5}+\frac{8 b d e \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c^3}+\frac{8 b e^2 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{75 c^3}+d^2 x \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{2}{3} d e x^3 \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{5} e^2 x^5 \left(a+b \sinh ^{-1}(c x)\right)^2+\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^2*x^2]*(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^2*x^2]*(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)))*ArcSinh[c*x] + 225*b^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcSinh[c*x]^2)/(3375*c^5)","A",1
615,1,164,153,0.2236053,"\int \left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x^2)*(a + b*ArcSinh[c*x])^2,x]","\frac{9 a^2 c^3 x \left(3 d+e x^2\right)-6 a b \sqrt{c^2 x^2+1} \left(c^2 \left(9 d+e x^2\right)-2 e\right)-6 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+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 \sinh ^{-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{2 b d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c}-\frac{2 b e x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c}+\frac{4 b e \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c^3}+d x \left(a+b \sinh ^{-1}(c x)\right)^2+\frac{1}{3} e x^3 \left(a+b \sinh ^{-1}(c x)\right)^2-\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^2*x^2]*(-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^2*x^2]*(-2*e + c^2*(9*d + e*x^2)))*ArcSinh[c*x] + 9*b^2*c^3*x*(3*d + e*x^2)*ArcSinh[c*x]^2)/(27*c^3)","A",1
616,1,74,46,0.061254,"\int \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcSinh[c*x])^2,x]","x \left(a^2+2 b^2\right)-\frac{2 a b \sqrt{c^2 x^2+1}}{c}+\frac{2 b \sinh ^{-1}(c x) \left(a c x-b \sqrt{c^2 x^2+1}\right)}{c}+b^2 x \sinh ^{-1}(c x)^2","-\frac{2 b \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c}+x \left(a+b \sinh ^{-1}(c x)\right)^2+2 b^2 x",1,"(a^2 + 2*b^2)*x - (2*a*b*Sqrt[1 + c^2*x^2])/c + (2*b*(a*c*x - b*Sqrt[1 + c^2*x^2])*ArcSinh[c*x])/c + b^2*x*ArcSinh[c*x]^2","A",1
617,1,985,739,0.693379,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d+e x^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + e*x^2),x]","\frac{2 \sqrt{-d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a^2-2 b \sqrt{d} \sinh ^{-1}(c x) \log \left(\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right) a+2 b \sqrt{d} \sinh ^{-1}(c x) \log \left(\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{\sqrt{e-c^2 d}-c \sqrt{-d}}+1\right) a+2 b \sqrt{d} \sinh ^{-1}(c x) \log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right) a-2 b \sqrt{d} \sinh ^{-1}(c x) \log \left(\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{e-c^2 d}}+1\right) a-2 b \sqrt{d} \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right) a+2 b \sqrt{d} \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right) a-b^2 \sqrt{d} \sinh ^{-1}(c x)^2 \log \left(\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right)+b^2 \sqrt{d} \sinh ^{-1}(c x)^2 \log \left(\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{\sqrt{e-c^2 d}-c \sqrt{-d}}+1\right)+b^2 \sqrt{d} \sinh ^{-1}(c x)^2 \log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)-b^2 \sqrt{d} \sinh ^{-1}(c x)^2 \log \left(\frac{e^{\sinh ^{-1}(c x)} \sqrt{e}}{\sqrt{-d} c+\sqrt{e-c^2 d}}+1\right)+2 b \sqrt{d} \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)-2 b \sqrt{d} \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-c \sqrt{-d}}\right)-2 b^2 \sqrt{d} \sinh ^{-1}(c x) \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)+2 b^2 \sqrt{d} \sinh ^{-1}(c x) \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)-2 b^2 \sqrt{d} \text{Li}_3\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)+2 b^2 \sqrt{d} \text{Li}_3\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}-c \sqrt{-d}}\right)+2 b^2 \sqrt{d} \text{Li}_3\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)-2 b^2 \sqrt{d} \text{Li}_3\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d^2} \sqrt{e}}","-\frac{b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{\sqrt{-d} \sqrt{e}}+\frac{b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{\sqrt{-d} \sqrt{e}}-\frac{b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)}{\sqrt{-d} \sqrt{e}}+\frac{b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)}{\sqrt{-d} \sqrt{e}}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}+1\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(1-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{e-c^2 d}+c \sqrt{-d}}+1\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{c \sqrt{-d}-\sqrt{e-c^2 d}}\right)}{\sqrt{-d} \sqrt{e}}+\frac{b^2 \text{Li}_3\left(-\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)}{\sqrt{-d} \sqrt{e}}-\frac{b^2 \text{Li}_3\left(\frac{\sqrt{e} e^{\sinh ^{-1}(c x)}}{\sqrt{-d} c+\sqrt{e-c^2 d}}\right)}{\sqrt{-d} \sqrt{e}}",1,"(2*a^2*Sqrt[-d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] - 2*a*b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] - b^2*Sqrt[d]*ArcSinh[c*x]^2*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] + 2*a*b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] + b^2*Sqrt[d]*ArcSinh[c*x]^2*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] + 2*a*b*Sqrt[d]*ArcSinh[c*x]*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] + b^2*Sqrt[d]*ArcSinh[c*x]^2*Log[1 - (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] - 2*a*b*Sqrt[d]*ArcSinh[c*x]*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] - b^2*Sqrt[d]*ArcSinh[c*x]^2*Log[1 + (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] + 2*b*Sqrt[d]*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] - 2*b*Sqrt[d]*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] - 2*a*b*Sqrt[d]*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))] - 2*b^2*Sqrt[d]*ArcSinh[c*x]*PolyLog[2, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))] + 2*a*b*Sqrt[d]*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] + 2*b^2*Sqrt[d]*ArcSinh[c*x]*PolyLog[2, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])] - 2*b^2*Sqrt[d]*PolyLog[3, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) + e])] + 2*b^2*Sqrt[d]*PolyLog[3, (Sqrt[e]*E^ArcSinh[c*x])/(-(c*Sqrt[-d]) + Sqrt[-(c^2*d) + e])] + 2*b^2*Sqrt[d]*PolyLog[3, -((Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e]))] - 2*b^2*Sqrt[d]*PolyLog[3, (Sqrt[e]*E^ArcSinh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) + e])])/(2*Sqrt[-d^2]*Sqrt[e])","A",0
618,1,444,670,1.0261954,"\int \frac{\left(d+e x^2\right)^3}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)^3/(a + b*ArcSinh[c*x]),x]","\frac{-64 c^6 d^3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+48 c^4 d^2 e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-48 c^4 d^2 e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+12 c^2 d e^2 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-24 c^2 d e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+36 c^2 d e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-12 c^2 d e^2 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 e \cosh \left(\frac{3 a}{b}\right) \left(16 c^4 d^2-12 c^2 d e+3 e^2\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+\cosh \left(\frac{a}{b}\right) \left(64 c^6 d^3-48 c^4 d^2 e+24 c^2 d e^2-5 e^3\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-5 e^3 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+e^3 \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 e^3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-9 e^3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+5 e^3 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-e^3 \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(7 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{64 b c^7}","-\frac{5 e^3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b c^7}+\frac{9 e^3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^7}-\frac{5 e^3 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^7}+\frac{e^3 \cosh \left(\frac{7 a}{b}\right) \text{Chi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^7}+\frac{5 e^3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{64 b c^7}-\frac{9 e^3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^7}+\frac{5 e^3 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^7}-\frac{e^3 \sinh \left(\frac{7 a}{b}\right) \text{Shi}\left(\frac{7 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{64 b c^7}+\frac{3 d e^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^5}-\frac{9 d e^2 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{3 d e^2 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}-\frac{3 d e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^5}+\frac{9 d e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}-\frac{3 d e^2 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}-\frac{3 d^2 e \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b c^3}+\frac{3 d^2 e \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b c^3}+\frac{3 d^2 e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b c^3}-\frac{3 d^2 e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b c^3}+\frac{d^3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c}-\frac{d^3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c}",1,"((64*c^6*d^3 - 48*c^4*d^2*e + 24*c^2*d*e^2 - 5*e^3)*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] + 3*e*(16*c^4*d^2 - 12*c^2*d*e + 3*e^2)*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] + 12*c^2*d*e^2*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 5*e^3*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] + e^3*Cosh[(7*a)/b]*CoshIntegral[7*(a/b + ArcSinh[c*x])] - 64*c^6*d^3*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 48*c^4*d^2*e*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 24*c^2*d*e^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 5*e^3*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 48*c^4*d^2*e*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 36*c^2*d*e^2*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - 9*e^3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - 12*c^2*d*e^2*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] + 5*e^3*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])] - e^3*Sinh[(7*a)/b]*SinhIntegral[7*(a/b + ArcSinh[c*x])])/(64*b*c^7)","A",1
619,1,253,388,0.5539596,"\int \frac{\left(d+e x^2\right)^2}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)^2/(a + b*ArcSinh[c*x]),x]","\frac{-16 c^4 d^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+e \cosh \left(\frac{3 a}{b}\right) \left(8 c^2 d-3 e\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+8 c^2 d e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-8 c^2 d e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-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) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+e^2 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-e^2 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{16 b c^5}","\frac{e^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^5}-\frac{3 e^2 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}+\frac{e^2 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}-\frac{e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b c^5}+\frac{3 e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}-\frac{e^2 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b c^5}-\frac{d e \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{2 b c^3}+\frac{d e \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c^3}+\frac{d e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{2 b c^3}-\frac{d e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b c^3}+\frac{d^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c}-\frac{d^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c}",1,"(2*(8*c^4*d^2 - 4*c^2*d*e + e^2)*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] + (8*c^2*d - 3*e)*e*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] + e^2*Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c*x])] - 16*c^4*d^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 8*c^2*d*e*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 2*e^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 8*c^2*d*e*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 3*e^2*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - e^2*Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])])/(16*b*c^5)","A",1
620,1,126,180,0.2466807,"\int \frac{d+e x^2}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)/(a + b*ArcSinh[c*x]),x]","\frac{\cosh \left(\frac{a}{b}\right) \left(4 c^2 d-e\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-4 c^2 d \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+e \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{4 b c^3}","-\frac{e \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b c^3}+\frac{e \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b c^3}+\frac{e \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b c^3}-\frac{e \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b c^3}+\frac{d \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c}-\frac{d \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c}",1,"((4*c^2*d - e)*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] + e*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 4*c^2*d*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + e*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - e*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(4*b*c^3)","A",1
621,1,45,54,0.0195235,"\int \frac{1}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(a + b*ArcSinh[c*x])^(-1),x]","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{b c}","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c}-\frac{\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b c}",1,"(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b*c)","A",1
622,0,0,23,0.7158088,"\int \frac{1}{\left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{\left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)*(a + b*ArcSinh[c*x])), x]","A",-1
623,0,0,23,3.4817696,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])), x]","A",-1
624,1,356,495,2.0770674,"\int \frac{\left(d+e x^2\right)^2}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x^2)^2/(a + b*ArcSinh[c*x])^2,x]","-\frac{-16 c^4 d^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 e \sinh \left(\frac{3 a}{b}\right) \left(8 c^2 d-3 e\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+8 c^2 d e \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-24 c^2 d e \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-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}+\sinh ^{-1}(c x)\right)+\frac{16 b c^4 d^2 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac{32 b c^4 d e x^2 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac{16 b c^4 e^2 x^4 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+5 e^2 \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 e^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+9 e^2 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-5 e^2 \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{16 b^2 c^5}","-\frac{e^2 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2 c^5}+\frac{9 e^2 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}-\frac{5 e^2 \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}+\frac{e^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{8 b^2 c^5}-\frac{9 e^2 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}+\frac{5 e^2 \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{16 b^2 c^5}+\frac{d e \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{2 b^2 c^3}-\frac{3 d e \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}-\frac{d e \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{2 b^2 c^3}+\frac{3 d e \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{2 b^2 c^3}-\frac{d^2 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2 c}+\frac{d^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d^2 \sqrt{c^2 x^2+1}}{b c \left(a+b \sinh ^{-1}(c x)\right)}-\frac{2 d e x^2 \sqrt{c^2 x^2+1}}{b c \left(a+b \sinh ^{-1}(c x)\right)}-\frac{e^2 x^4 \sqrt{c^2 x^2+1}}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/16*((16*b*c^4*d^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (32*b*c^4*d*e*x^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (16*b*c^4*e^2*x^4*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + 2*(8*c^4*d^2 - 4*c^2*d*e + e^2)*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 3*(8*c^2*d - 3*e)*e*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] + 5*e^2*CoshIntegral[5*(a/b + ArcSinh[c*x])]*Sinh[(5*a)/b] - 16*c^4*d^2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 8*c^2*d*e*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 2*e^2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 24*c^2*d*e*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] + 9*e^2*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])] - 5*e^2*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c*x])])/(b^2*c^5)","A",1
625,1,190,247,0.9132947,"\int \frac{d+e x^2}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x^2)/(a + b*ArcSinh[c*x])^2,x]","-\frac{\sinh \left(\frac{a}{b}\right) \left(4 c^2 d-e\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-4 c^2 d \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+\frac{4 b c^2 d \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac{4 b c^2 e x^2 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+3 e \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+e \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-3 e \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{4 b^2 c^3}","\frac{e \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b^2 c^3}-\frac{3 e \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}-\frac{e \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{4 b^2 c^3}+\frac{3 e \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{4 b^2 c^3}-\frac{d \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2 c}+\frac{d \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{d \sqrt{c^2 x^2+1}}{b c \left(a+b \sinh ^{-1}(c x)\right)}-\frac{e x^2 \sqrt{c^2 x^2+1}}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-1/4*((4*b*c^2*d*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (4*b*c^2*e*x^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (4*c^2*d - e)*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 3*e*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - 4*c^2*d*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + e*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] - 3*e*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(b^2*c^3)","A",1
626,1,71,85,0.1450132,"\int \frac{1}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^(-2),x]","\frac{-\frac{b \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}-\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{b^2 c}","-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2 c}+\frac{\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{b^2 c}-\frac{\sqrt{c^2 x^2+1}}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"(-((b*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x])) - CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]])/(b^2*c)","A",1
627,0,0,23,11.894931,"\int \frac{1}{\left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{\left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x^2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
628,0,0,23,28.6506777,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^2), x]","A",-1
629,1,535,672,6.3954402,"\int \left(d+e x^2\right)^2 \sqrt{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)^2*Sqrt[a + b*ArcSinh[c*x]],x]","-\frac{b e^{-\frac{5 a}{b}} \left(25 \sqrt{3} b e e^{\frac{2 a}{b}} \left(8 c^2 d-3 e\right) \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \sqrt{-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}} \Gamma \left(\frac{3}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-b e e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \sqrt{-\frac{\left(a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)\right)}{b}\right)\right)+450 e^{\frac{6 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right) \left(b e \left(4 c^2 d-e\right) \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \sqrt{-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}}+8 a c^4 d^2 \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}+8 b c^4 d^2 \sinh ^{-1}(c x) \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}\right)+450 e^{\frac{4 a}{b}} \Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right) \left(b e \left(e-4 c^2 d\right) \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \sqrt{-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}}+8 a c^4 d^2 \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}+8 b c^4 d^2 \sinh ^{-1}(c x) \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}\right)+9 \sqrt{5} b e^2 \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \sqrt{-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}} \Gamma \left(\frac{3}{2},-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{7200 c^5 \left(a+b \sinh ^{-1}(c x)\right)^{3/2}}","\frac{\sqrt{\pi } \sqrt{b} e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}+d^2 x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{2}{3} d e x^3 \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{5} e^2 x^5 \sqrt{a+b \sinh ^{-1}(c x)}",1,"-1/7200*(b*(450*E^((6*a)/b)*(8*a*c^4*d^2*Sqrt[a/b + ArcSinh[c*x]] + 8*b*c^4*d^2*ArcSinh[c*x]*Sqrt[a/b + ArcSinh[c*x]] + b*(4*c^2*d - e)*e*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, a/b + ArcSinh[c*x]] + 9*Sqrt[5]*b*e^2*Sqrt[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]*Gamma[3/2, (-5*(a + b*ArcSinh[c*x]))/b] + 25*Sqrt[3]*b*(8*c^2*d - 3*e)*e*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]*Gamma[3/2, (-3*(a + b*ArcSinh[c*x]))/b] + 450*E^((4*a)/b)*(8*a*c^4*d^2*Sqrt[-((a + b*ArcSinh[c*x])/b)] + 8*b*c^4*d^2*ArcSinh[c*x]*Sqrt[-((a + b*ArcSinh[c*x])/b)] + b*e*(-4*c^2*d + e)*Sqrt[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, -((a + b*ArcSinh[c*x])/b)] - b*e*E^((8*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]*(25*Sqrt[3]*(8*c^2*d - 3*e)*Gamma[3/2, (3*(a + b*ArcSinh[c*x]))/b] + 9*Sqrt[5]*e*E^((2*a)/b)*Gamma[3/2, (5*(a + b*ArcSinh[c*x]))/b])))/(c^5*E^((5*a)/b)*(a + b*ArcSinh[c*x])^(3/2))","A",0
630,1,319,322,2.8984946,"\int \left(d+e x^2\right) \sqrt{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(d + e*x^2)*Sqrt[a + b*ArcSinh[c*x]],x]","\frac{e e^{-\frac{3 a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left(9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{3}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{3}{2},\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{72 c^3 \sqrt{-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}}}+\frac{d e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}}\right)}{2 c}","-\frac{\sqrt{\pi } \sqrt{b} e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}+d x \sqrt{a+b \sinh ^{-1}(c x)}+\frac{1}{3} e x^3 \sqrt{a+b \sinh ^{-1}(c x)}",1,"(d*Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]])/Sqrt[a/b + ArcSinh[c*x]]) + Gamma[3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[-((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b)) + (e*Sqrt[a + b*ArcSinh[c*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, a/b + ArcSinh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, (-3*(a + b*ArcSinh[c*x]))/b] - 9*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, -((a + b*ArcSinh[c*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcSinh[c*x]))/b]))/(72*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])","A",0
631,1,101,102,0.1986084,"\int \sqrt{a+b \sinh ^{-1}(c x)} \, dx","Integrate[Sqrt[a + b*ArcSinh[c*x]],x]","\frac{e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}}\right)}{2 c}","\frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}-\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{4 c}+x \sqrt{a+b \sinh ^{-1}(c x)}",1,"(Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]])/Sqrt[a/b + ArcSinh[c*x]]) + Gamma[3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[-((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b))","A",0
632,0,0,25,7.5516091,"\int \frac{\sqrt{a+b \sinh ^{-1}(c x)}}{d+e x^2} \, dx","Integrate[Sqrt[a + b*ArcSinh[c*x]]/(d + e*x^2),x]","\int \frac{\sqrt{a+b \sinh ^{-1}(c x)}}{d+e x^2} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{d+e x^2},x\right)",0,"Integrate[Sqrt[a + b*ArcSinh[c*x]]/(d + e*x^2), x]","A",-1
633,0,0,25,16.8255331,"\int \frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\left(d+e x^2\right)^2} \, dx","Integrate[Sqrt[a + b*ArcSinh[c*x]]/(d + e*x^2)^2,x]","\int \frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[Sqrt[a + b*ArcSinh[c*x]]/(d + e*x^2)^2, x]","A",-1
634,1,770,427,4.8358435,"\int \left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^{3/2} \, dx","Integrate[(d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2),x]","\frac{a e e^{-\frac{3 a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left(9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{3}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{3}{2},\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{72 c^3 \sqrt{-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{b^2}}}+\frac{\sqrt{b} d \left(4 \sqrt{b} \left(2 c x \sinh ^{-1}(c x)-3 \sqrt{c^2 x^2+1}\right) \sqrt{a+b \sinh ^{-1}(c x)}+\sqrt{\pi } (3 b-2 a) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)+\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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)\right)}{8 c}+\frac{\sqrt{b} e \left(-9 \left(4 \sqrt{b} \left(2 c x \sinh ^{-1}(c x)-3 \sqrt{c^2 x^2+1}\right) \sqrt{a+b \sinh ^{-1}(c x)}+\sqrt{\pi } (3 b-2 a) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)+\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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)\right)+\sqrt{3 \pi } (b-2 a) \left(\sinh \left(\frac{3 a}{b}\right)+\cosh \left(\frac{3 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)+\sqrt{3 \pi } (2 a+b) \left(\cosh \left(\frac{3 a}{b}\right)-\sinh \left(\frac{3 a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)+12 \sqrt{b} \left(2 \sinh ^{-1}(c x) \sinh \left(3 \sinh ^{-1}(c x)\right)-\cosh \left(3 \sinh ^{-1}(c x)\right)\right) \sqrt{a+b \sinh ^{-1}(c x)}\right)}{288 c^3}+\frac{a d e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}}\right)}{2 c}","-\frac{3 \sqrt{\pi } b^{3/2} e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c}-\frac{3 b d \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}-\frac{b e x^2 \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+\frac{b e \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{3 c^3}+d x \left(a+b \sinh ^{-1}(c x)\right)^{3/2}+\frac{1}{3} e x^3 \left(a+b \sinh ^{-1}(c x)\right)^{3/2}",1,"(a*d*Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]])/Sqrt[a/b + ArcSinh[c*x]]) + Gamma[3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[-((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b)) + (a*e*Sqrt[a + b*ArcSinh[c*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, a/b + ArcSinh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, (-3*(a + b*ArcSinh[c*x]))/b] - 9*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[3/2, -((a + b*ArcSinh[c*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcSinh[c*x]))/b]))/(72*c^3*E^((3*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]) + (Sqrt[b]*d*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-3*Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(8*c) + (Sqrt[b]*e*(-9*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-3*Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] - Sinh[(3*a)/b]) + (-2*a + b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-Cosh[3*ArcSinh[c*x]] + 2*ArcSinh[c*x]*Sinh[3*ArcSinh[c*x]])))/(288*c^3)","A",0
635,1,251,135,0.6361355,"\int \left(a+b \sinh ^{-1}(c x)\right)^{3/2} \, dx","Integrate[(a + b*ArcSinh[c*x])^(3/2),x]","\frac{\sqrt{b} \left(4 \sqrt{b} \left(2 c x \sinh ^{-1}(c x)-3 \sqrt{c^2 x^2+1}\right) \sqrt{a+b \sinh ^{-1}(c x)}+\sqrt{\pi } (3 b-2 a) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)+\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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)\right)}{8 c}+\frac{a e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}}\right)}{2 c}","\frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{8 c}-\frac{3 b \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}+x \left(a+b \sinh ^{-1}(c x)\right)^{3/2}",1,"(a*Sqrt[a + b*ArcSinh[c*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c*x]])/Sqrt[a/b + ArcSinh[c*x]]) + Gamma[3/2, -((a + b*ArcSinh[c*x])/b)]/Sqrt[-((a + b*ArcSinh[c*x])/b)]))/(2*c*E^(a/b)) + (Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c*x]]*(-3*Sqrt[1 + c^2*x^2] + 2*c*x*ArcSinh[c*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(8*c)","A",0
636,0,0,25,2.8887528,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}}{d+e x^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^(3/2)/(d + e*x^2),x]","\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}}{d+e x^2} \, dx","\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}}{d+e x^2},x\right)",0,"Integrate[(a + b*ArcSinh[c*x])^(3/2)/(d + e*x^2), x]","A",-1
637,0,0,25,9.4469821,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcSinh[c*x])^(3/2)/(d + e*x^2)^2,x]","\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[(a + b*ArcSinh[c*x])^(3/2)/(d + e*x^2)^2, x]","A",-1
638,1,530,608,1.2332097,"\int \frac{\left(d+e x^2\right)^2}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx","Integrate[(d + e*x^2)^2/Sqrt[a + b*ArcSinh[c*x]],x]","\frac{e^{-\frac{5 a}{b}} \left(240 c^4 d^2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)+40 \sqrt{3} c^2 d e e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-120 c^2 d e e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)-40 \sqrt{3} c^2 d e e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-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}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 \sqrt{5} e^2 \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-15 \sqrt{3} e^2 e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)+30 e^2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)+15 \sqrt{3} e^2 e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-3 \sqrt{5} e^2 e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{480 c^5 \sqrt{a+b \sinh ^{-1}(c x)}}","\frac{\sqrt{\pi } e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + 3*Sqrt[5]*e^2*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcSinh[c*x]))/b] + 40*Sqrt[3]*c^2*d*e*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] - 15*Sqrt[3]*e^2*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] + 240*c^4*d^2*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] - 120*c^2*d*e*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] + 30*e^2*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] - 40*Sqrt[3]*c^2*d*e*E^((8*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b] + 15*Sqrt[3]*e^2*E^((8*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b] - 3*Sqrt[5]*e^2*E^((10*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (5*(a + b*ArcSinh[c*x]))/b])/(480*c^5*E^((5*a)/b)*Sqrt[a + b*ArcSinh[c*x]])","A",0
639,1,218,287,0.6934288,"\int \frac{d+e x^2}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx","Integrate[(d + e*x^2)/Sqrt[a + b*ArcSinh[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}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)+3 e^{\frac{2 a}{b}} \left(4 c^2 d-e\right) \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)+\sqrt{3} e \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-\sqrt{3} e e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{24 c^3 \sqrt{a+b \sinh ^{-1}(c x)}}","-\frac{\sqrt{\pi } e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c}",1,"(-3*(4*c^2*d - e)*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + Sqrt[3]*e*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] + 3*(4*c^2*d - e)*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] - Sqrt[3]*e*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b])/(24*c^3*E^((3*a)/b)*Sqrt[a + b*ArcSinh[c*x]])","A",0
640,1,101,88,0.0999791,"\int \frac{1}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx","Integrate[1/Sqrt[a + b*ArcSinh[c*x]],x]","\frac{e^{-\frac{a}{b}} \left(\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)-e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{2 c \sqrt{a+b \sinh ^{-1}(c x)}}","\frac{\sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{2 \sqrt{b} c}",1,"(-(E^((2*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]]) + Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)])/(2*c*E^(a/b)*Sqrt[a + b*ArcSinh[c*x]])","A",0
641,0,0,25,0.138518,"\int \frac{1}{\left(d+e x^2\right) \sqrt{a+b \sinh ^{-1}(c x)}} \, dx","Integrate[1/((d + e*x^2)*Sqrt[a + b*ArcSinh[c*x]]),x]","\int \frac{1}{\left(d+e x^2\right) \sqrt{a+b \sinh ^{-1}(c x)}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \sqrt{a+b \sinh ^{-1}(c x)}},x\right)",0,"Integrate[1/((d + e*x^2)*Sqrt[a + b*ArcSinh[c*x]]), x]","A",-1
642,0,0,25,0.2665154,"\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b \sinh ^{-1}(c x)}} \, dx","Integrate[1/((d + e*x^2)^2*Sqrt[a + b*ArcSinh[c*x]]),x]","\int \frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b \sinh ^{-1}(c x)}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \sqrt{a+b \sinh ^{-1}(c x)}},x\right)",0,"Integrate[1/((d + e*x^2)^2*Sqrt[a + b*ArcSinh[c*x]]), x]","A",-1
643,1,303,349,1.5186047,"\int \frac{d+e x^2}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(d + e*x^2)/(a + b*ArcSinh[c*x])^(3/2),x]","\frac{e^{-3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)} \left(\left(4 c^2 d-e\right) e^{\frac{4 a}{b}+3 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)+\left(4 c^2 d-e\right) e^{\frac{2 a}{b}+3 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)+e^{\frac{3 a}{b}} \left(\sqrt{3} e e^{3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)-\left(e^{2 \sinh ^{-1}(c x)}+1\right) \left(4 c^2 d e^{2 \sinh ^{-1}(c x)}+e \left(e^{2 \sinh ^{-1}(c x)}-1\right)^2\right)\right)+\sqrt{3} e e^{3 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)\right)}{4 b c^3 \sqrt{a+b \sinh ^{-1}(c x)}}","\frac{\sqrt{\pi } e e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-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 \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 d \sqrt{c^2 x^2+1}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{2 e x^2 \sqrt{c^2 x^2+1}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"((4*c^2*d - e)*E^((4*a)/b + 3*ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + Sqrt[3]*e*E^(3*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b] + (4*c^2*d - e)*E^((2*a)/b + 3*ArcSinh[c*x])*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)] + E^((3*a)/b)*(-((1 + E^(2*ArcSinh[c*x]))*(4*c^2*d*E^(2*ArcSinh[c*x]) + e*(-1 + E^(2*ArcSinh[c*x]))^2)) + Sqrt[3]*e*E^(3*(a/b + ArcSinh[c*x]))*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b]))/(4*b*c^3*E^(3*(a/b + ArcSinh[c*x]))*Sqrt[a + b*ArcSinh[c*x]])","A",0
644,1,137,116,0.2324941,"\int \frac{1}{\left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^(-3/2),x]","\frac{e^{-\frac{a+b \sinh ^{-1}(c x)}{b}} \left(-e^{a/b} \left(e^{2 \sinh ^{-1}(c x)}+1\right)+e^{\frac{2 a}{b}+\sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right)+e^{\sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right)\right)}{b c \sqrt{a+b \sinh ^{-1}(c x)}}","-\frac{\sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right)}{b^{3/2} c}-\frac{2 \sqrt{c^2 x^2+1}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}",1,"(-(E^(a/b)*(1 + E^(2*ArcSinh[c*x]))) + E^((2*a)/b + ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x]]*Gamma[1/2, a/b + ArcSinh[c*x]] + E^ArcSinh[c*x]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)])/(b*c*E^((a + b*ArcSinh[c*x])/b)*Sqrt[a + b*ArcSinh[c*x]])","A",0
645,0,0,25,0.1523002,"\int \frac{1}{\left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[1/((d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2)),x]","\int \frac{1}{\left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right) \left(a+b \sinh ^{-1}(c x)\right)^{3/2}},x\right)",0,"Integrate[1/((d + e*x^2)*(a + b*ArcSinh[c*x])^(3/2)), x]","A",-1
646,0,0,25,0.278429,"\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","Integrate[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^(3/2)),x]","\int \frac{1}{\left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^{3/2}} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^{3/2}},x\right)",0,"Integrate[1/((d + e*x^2)^2*(a + b*ArcSinh[c*x])^(3/2)), x]","A",-1
647,0,0,23,5.2646346,"\int \sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x]),x]","\int \sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right),x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x]), x]","A",-1
648,0,0,23,3.9246556,"\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/Sqrt[d + e*x^2],x]","\int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \sinh ^{-1}(c x)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSinh[c*x])/Sqrt[d + e*x^2], x]","A",-1
649,1,75,70,0.1156947,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + e*x^2)^(3/2),x]","\frac{x \left(2 \left(a+b \sinh ^{-1}(c x)\right)-b c x \sqrt{\frac{e x^2}{d}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right)\right)}{2 d \sqrt{d+e x^2}}","\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{d \sqrt{d+e x^2}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2+1}}{c \sqrt{d+e x^2}}\right)}{d \sqrt{e}}",1,"(x*(-(b*c*x*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^2), -((e*x^2)/d)]) + 2*(a + b*ArcSinh[c*x])))/(2*d*Sqrt[d + e*x^2])","C",0
650,1,139,146,0.3390928,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + e*x^2)^(5/2),x]","\frac{a x \left(3 d+2 e x^2\right)-b c x^2 \left(d+e x^2\right) \sqrt{\frac{e x^2}{d}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right)-\frac{b c d \sqrt{c^2 x^2+1} \left(d+e x^2\right)}{c^2 d-e}+b x \sinh ^{-1}(c x) \left(3 d+2 e x^2\right)}{3 d^2 \left(d+e x^2\right)^{3/2}}","\frac{2 x \left(a+b \sinh ^{-1}(c x)\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2+1}}{c \sqrt{d+e x^2}}\right)}{3 d^2 \sqrt{e}}-\frac{b c \sqrt{c^2 x^2+1}}{3 d \left(c^2 d-e\right) \sqrt{d+e x^2}}",1,"(-((b*c*d*Sqrt[1 + c^2*x^2]*(d + e*x^2))/(c^2*d - e)) + a*x*(3*d + 2*e*x^2) - b*c*x^2*(d + e*x^2)*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^2), -((e*x^2)/d)] + b*x*(3*d + 2*e*x^2)*ArcSinh[c*x])/(3*d^2*(d + e*x^2)^(3/2))","C",0
651,1,191,227,0.4226288,"\int \frac{a+b \sinh ^{-1}(c x)}{\left(d+e x^2\right)^{7/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + e*x^2)^(7/2),x]","\frac{a x \left(15 d^2+20 d e x^2+8 e^2 x^4\right)-4 b c x^2 \sqrt{\frac{e x^2}{d}+1} \left(d+e x^2\right)^2 F_1\left(1;\frac{1}{2},\frac{1}{2};2;-c^2 x^2,-\frac{e x^2}{d}\right)-\frac{b c d \sqrt{c^2 x^2+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)}{\left(e-c^2 d\right)^2}+b x \sinh ^{-1}(c x) \left(15 d^2+20 d e x^2+8 e^2 x^4\right)}{15 d^3 \left(d+e x^2\right)^{5/2}}","\frac{8 x \left(a+b \sinh ^{-1}(c x)\right)}{15 d^3 \sqrt{d+e x^2}}+\frac{4 x \left(a+b \sinh ^{-1}(c x)\right)}{15 d^2 \left(d+e x^2\right)^{3/2}}+\frac{x \left(a+b \sinh ^{-1}(c x)\right)}{5 d \left(d+e x^2\right)^{5/2}}-\frac{8 b \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2+1}}{c \sqrt{d+e x^2}}\right)}{15 d^3 \sqrt{e}}-\frac{2 b c \sqrt{c^2 x^2+1} \left(3 c^2 d-2 e\right)}{15 d^2 \left(c^2 d-e\right)^2 \sqrt{d+e x^2}}-\frac{b c \sqrt{c^2 x^2+1}}{15 d \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) - (b*c*d*Sqrt[1 + c^2*x^2]*(d + e*x^2)*(-(e*(5*d + 4*e*x^2)) + c^2*d*(7*d + 6*e*x^2)))/(-(c^2*d) + e)^2 - 4*b*c*x^2*(d + e*x^2)^2*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^2), -((e*x^2)/d)] + b*x*(15*d^2 + 20*d*e*x^2 + 8*e^2*x^4)*ArcSinh[c*x])/(15*d^3*(d + e*x^2)^(5/2))","C",0
652,0,0,25,14.3999318,"\int \sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2,x]","\int \sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right)^2,x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2, x]","A",-1
653,0,0,25,11.4125763,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/Sqrt[d + e*x^2],x]","\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcSinh[c*x])^2/Sqrt[d + e*x^2], x]","A",-1
654,0,0,25,3.1291953,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + e*x^2)^(3/2),x]","\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcSinh[c*x])^2/(d + e*x^2)^(3/2), x]","A",-1
655,0,0,25,5.9043668,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + e*x^2)^(5/2),x]","\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcSinh[c*x])^2/(d + e*x^2)^(5/2), x]","A",-1
656,0,0,25,1.2085071,"\int \frac{\sqrt{d+e x^2}}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[Sqrt[d + e*x^2]/(a + b*ArcSinh[c*x]),x]","\int \frac{\sqrt{d+e x^2}}{a+b \sinh ^{-1}(c x)} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2}}{a+b \sinh ^{-1}(c x)},x\right)",0,"Integrate[Sqrt[d + e*x^2]/(a + b*ArcSinh[c*x]), x]","A",-1
657,0,0,25,1.0804804,"\int \frac{1}{\sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{\sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])), x]","A",-1
658,0,0,25,1.5329849,"\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])), x]","A",-1
659,0,0,25,3.719435,"\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])), x]","A",-1
660,0,0,25,4.6695365,"\int \frac{\sqrt{d+e x^2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[Sqrt[d + e*x^2]/(a + b*ArcSinh[c*x])^2,x]","\int \frac{\sqrt{d+e x^2}}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2}}{\left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[Sqrt[d + e*x^2]/(a + b*ArcSinh[c*x])^2, x]","A",-1
661,0,0,25,6.4531858,"\int \frac{1}{\sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{\sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\sqrt{d+e x^2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2), x]","A",-1
662,0,0,25,16.7675997,"\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1
663,0,0,25,29.1262374,"\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(d+e x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]","A",-1