1,1,168,170,0.0100311,"\int \frac{\sinh ^{-1}(c x)}{d+e x} \, dx","Integrate[ArcSinh[c*x]/(d + e*x),x]","\frac{\text{Li}_2\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)}{e}+\frac{\text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{\sinh ^{-1}(c x) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)}{e}+\frac{\sinh ^{-1}(c x) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)}{e}-\frac{\sinh ^{-1}(c x)^2}{2 e}","\frac{\text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{\text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{\sinh ^{-1}(c x) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)}{e}+\frac{\sinh ^{-1}(c x) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)}{e}-\frac{\sinh ^{-1}(c x)^2}{2 e}",1,"-1/2*ArcSinh[c*x]^2/e + (ArcSinh[c*x]*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/e + (ArcSinh[c*x]*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e + PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])]/e + PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))]/e","A",1
2,1,240,260,0.1363515,"\int \frac{\sinh ^{-1}(c x)^2}{d+e x} \, dx","Integrate[ArcSinh[c*x]^2/(d + e*x),x]","-\frac{-6 \sinh ^{-1}(c x) \text{Li}_2\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)-6 \sinh ^{-1}(c x) \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)+6 \text{Li}_3\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)+6 \text{Li}_3\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)-3 \sinh ^{-1}(c x)^2 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)-3 \sinh ^{-1}(c x)^2 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)+\sinh ^{-1}(c x)^3}{3 e}","\frac{2 \sinh ^{-1}(c x) \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{2 \sinh ^{-1}(c x) \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}-\frac{2 \text{Li}_3\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e}-\frac{2 \text{Li}_3\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{\sinh ^{-1}(c x)^2 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)}{e}+\frac{\sinh ^{-1}(c x)^2 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)}{e}-\frac{\sinh ^{-1}(c x)^3}{3 e}",1,"-1/3*(ArcSinh[c*x]^3 - 3*ArcSinh[c*x]^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])] - 3*ArcSinh[c*x]^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])] - 6*ArcSinh[c*x]*PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] - 6*ArcSinh[c*x]*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))] + 6*PolyLog[3, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] + 6*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/e","A",1
3,1,322,348,0.0431509,"\int \frac{\sinh ^{-1}(c x)^3}{d+e x} \, dx","Integrate[ArcSinh[c*x]^3/(d + e*x),x]","\frac{12 \sinh ^{-1}(c x)^2 \text{Li}_2\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)+12 \sinh ^{-1}(c x)^2 \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)-24 \sinh ^{-1}(c x) \text{Li}_3\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)-24 \sinh ^{-1}(c x) \text{Li}_3\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)+24 \text{Li}_4\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)+24 \text{Li}_4\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)+4 \sinh ^{-1}(c x)^3 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)+4 \sinh ^{-1}(c x)^3 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)-\sinh ^{-1}(c x)^4}{4 e}","\frac{3 \sinh ^{-1}(c x)^2 \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{3 \sinh ^{-1}(c x)^2 \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}-\frac{6 \sinh ^{-1}(c x) \text{Li}_3\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e}-\frac{6 \sinh ^{-1}(c x) \text{Li}_3\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{6 \text{Li}_4\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{6 \text{Li}_4\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{\sinh ^{-1}(c x)^3 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)}{e}+\frac{\sinh ^{-1}(c x)^3 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)}{e}-\frac{\sinh ^{-1}(c x)^4}{4 e}",1,"(-ArcSinh[c*x]^4 + 4*ArcSinh[c*x]^3*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])] + 4*ArcSinh[c*x]^3*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])] + 12*ArcSinh[c*x]^2*PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] + 12*ArcSinh[c*x]^2*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))] - 24*ArcSinh[c*x]*PolyLog[3, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] - 24*ArcSinh[c*x]*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))] + 24*PolyLog[4, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] + 24*PolyLog[4, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/(4*e)","A",1
4,1,166,176,0.1446206,"\int (d+e x)^3 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(a + b*ArcSinh[c*x]),x]","\frac{24 a c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)-b c \sqrt{c^2 x^2+1} \left(c^2 \left(96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right)-e^2 (64 d+9 e x)\right)+3 b \sinh ^{-1}(c x) \left(8 c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+24 c^2 d^2 e-3 e^3\right)}{96 c^4}","\frac{(d+e x)^4 \left(a+b \sinh ^{-1}(c x)\right)}{4 e}-\frac{b \sqrt{c^2 x^2+1} (d+e x)^3}{16 c}-\frac{7 b d \sqrt{c^2 x^2+1} (d+e x)^2}{48 c}-\frac{b \left(8 c^4 d^4-24 c^2 d^2 e^2+3 e^4\right) \sinh ^{-1}(c x)}{32 c^4 e}-\frac{b \sqrt{c^2 x^2+1} \left(e x \left(26 c^2 d^2-9 e^2\right)+4 d \left(19 c^2 d^2-16 e^2\right)\right)}{96 c^3}",1,"(24*a*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - b*c*Sqrt[1 + c^2*x^2]*(-(e^2*(64*d + 9*e*x)) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + 3*b*(24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcSinh[c*x])/(96*c^4)","A",1
5,1,121,124,0.0899685,"\int (d+e x)^2 \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcSinh[c*x]),x]","\frac{6 a c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)-b \sqrt{c^2 x^2+1} \left(c^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)-4 e^2\right)+3 b c \sinh ^{-1}(c x) \left(6 c^2 d^2 x+3 d \left(2 c^2 e x^2+e\right)+2 c^2 e^2 x^3\right)}{18 c^3}","\frac{(d+e x)^3 \left(a+b \sinh ^{-1}(c x)\right)}{3 e}-\frac{b d \left(2 d^2-\frac{3 e^2}{c^2}\right) \sinh ^{-1}(c x)}{6 e}-\frac{b \sqrt{c^2 x^2+1} (d+e x)^2}{9 c}-\frac{b \sqrt{c^2 x^2+1} \left(4 \left(4 c^2 d^2-e^2\right)+5 c^2 d e x\right)}{18 c^3}",1,"(6*a*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) - b*Sqrt[1 + c^2*x^2]*(-4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + 3*b*c*(6*c^2*d^2*x + 2*c^2*e^2*x^3 + 3*d*(e + 2*c^2*e*x^2))*ArcSinh[c*x])/(18*c^3)","A",1
6,1,91,97,0.0410951,"\int (d+e x) \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcSinh[c*x]),x]","a d x+\frac{1}{2} a e x^2-\frac{b d \sqrt{c^2 x^2+1}}{c}-\frac{b e x \sqrt{c^2 x^2+1}}{4 c}+\frac{b e \sinh ^{-1}(c x)}{4 c^2}+b d x \sinh ^{-1}(c x)+\frac{1}{2} b e x^2 \sinh ^{-1}(c x)","\frac{(d+e x)^2 \left(a+b \sinh ^{-1}(c x)\right)}{2 e}-\frac{b \left(2 d^2-\frac{e^2}{c^2}\right) \sinh ^{-1}(c x)}{4 e}-\frac{b \sqrt{c^2 x^2+1} (d+e x)}{4 c}-\frac{3 b d \sqrt{c^2 x^2+1}}{4 c}",1,"a*d*x + (a*e*x^2)/2 - (b*d*Sqrt[1 + c^2*x^2])/c - (b*e*x*Sqrt[1 + c^2*x^2])/(4*c) + (b*e*ArcSinh[c*x])/(4*c^2) + b*d*x*ArcSinh[c*x] + (b*e*x^2*ArcSinh[c*x])/2","A",1
7,1,30,30,0.0086765,"\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
8,1,175,187,0.0491124,"\int \frac{a+b \sinh ^{-1}(c x)}{d+e x} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + e*x),x]","\frac{-\left(a+b \sinh ^{-1}(c x)\right) \left(a-2 b \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)-2 b \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)+b \sinh ^{-1}(c x)\right)+2 b^2 \text{Li}_2\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)+2 b^2 \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{2 b e}","\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)}{e}+\frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)}{e}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 b e}+\frac{b \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{b \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}",1,"(-((a + b*ArcSinh[c*x])*(a + b*ArcSinh[c*x] - 2*b*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])] - 2*b*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])) + 2*b^2*PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] + 2*b^2*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/(2*b*e)","A",1
9,1,79,82,0.1005441,"\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{\frac{a+b \sinh ^{-1}(c x)}{d+e x}+\frac{b c \tanh ^{-1}\left(\frac{e-c^2 d x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d^2+e^2}}\right)}{\sqrt{c^2 d^2+e^2}}}{e}","-\frac{a+b \sinh ^{-1}(c x)}{e (d+e x)}-\frac{b c \tanh ^{-1}\left(\frac{e-c^2 d x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d^2+e^2}}\right)}{e \sqrt{c^2 d^2+e^2}}",1,"-(((a + b*ArcSinh[c*x])/(d + e*x) + (b*c*ArcTanh[(e - c^2*d*x)/(Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^2*x^2])])/Sqrt[c^2*d^2 + e^2])/e)","A",1
10,1,166,128,0.3595923,"\int \frac{a+b \sinh ^{-1}(c x)}{(d+e x)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + e*x)^3,x]","\frac{1}{2} \left(-\frac{a}{e (d+e x)^2}-\frac{b c \sqrt{c^2 x^2+1}}{\left(c^2 d^2+e^2\right) (d+e x)}-\frac{b c^3 d \log \left(\sqrt{c^2 x^2+1} \sqrt{c^2 d^2+e^2}+c^2 (-d) x+e\right)}{e \left(c^2 d^2+e^2\right)^{3/2}}+\frac{b c^3 d \log (d+e x)}{e \left(c^2 d^2+e^2\right)^{3/2}}-\frac{b \sinh ^{-1}(c x)}{e (d+e x)^2}\right)","-\frac{a+b \sinh ^{-1}(c x)}{2 e (d+e x)^2}-\frac{b c \sqrt{c^2 x^2+1}}{2 \left(c^2 d^2+e^2\right) (d+e x)}-\frac{b c^3 d \tanh ^{-1}\left(\frac{e-c^2 d x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d^2+e^2}}\right)}{2 e \left(c^2 d^2+e^2\right)^{3/2}}",1,"(-(a/(e*(d + e*x)^2)) - (b*c*Sqrt[1 + c^2*x^2])/((c^2*d^2 + e^2)*(d + e*x)) - (b*ArcSinh[c*x])/(e*(d + e*x)^2) + (b*c^3*d*Log[d + e*x])/(e*(c^2*d^2 + e^2)^(3/2)) - (b*c^3*d*Log[e - c^2*d*x + Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^2*x^2]])/(e*(c^2*d^2 + e^2)^(3/2)))/2","A",1
11,1,205,183,0.4316886,"\int \frac{a+b \sinh ^{-1}(c x)}{(d+e x)^4} \, dx","Integrate[(a + b*ArcSinh[c*x])/(d + e*x)^4,x]","\frac{1}{6} \left(-\frac{2 a}{e (d+e x)^3}-\frac{b c \sqrt{c^2 x^2+1} \left(c^2 d (4 d+3 e x)+e^2\right)}{\left(c^2 d^2+e^2\right)^2 (d+e x)^2}+\frac{b c^3 \left(e^2-2 c^2 d^2\right) \log \left(\sqrt{c^2 x^2+1} \sqrt{c^2 d^2+e^2}+c^2 (-d) x+e\right)}{e \left(c^2 d^2+e^2\right)^{5/2}}-\frac{b c^3 \left(e^2-2 c^2 d^2\right) \log (d+e x)}{e \left(c^2 d^2+e^2\right)^{5/2}}-\frac{2 b \sinh ^{-1}(c x)}{e (d+e x)^3}\right)","-\frac{a+b \sinh ^{-1}(c x)}{3 e (d+e x)^3}-\frac{b c \sqrt{c^2 x^2+1}}{6 \left(c^2 d^2+e^2\right) (d+e x)^2}-\frac{b c^3 d \sqrt{c^2 x^2+1}}{2 \left(c^2 d^2+e^2\right)^2 (d+e x)}-\frac{b c^3 \left(2 c^2 d^2-e^2\right) \tanh ^{-1}\left(\frac{e-c^2 d x}{\sqrt{c^2 x^2+1} \sqrt{c^2 d^2+e^2}}\right)}{6 e \left(c^2 d^2+e^2\right)^{5/2}}",1,"((-2*a)/(e*(d + e*x)^3) - (b*c*Sqrt[1 + c^2*x^2]*(e^2 + c^2*d*(4*d + 3*e*x)))/((c^2*d^2 + e^2)^2*(d + e*x)^2) - (2*b*ArcSinh[c*x])/(e*(d + e*x)^3) - (b*c^3*(-2*c^2*d^2 + e^2)*Log[d + e*x])/(e*(c^2*d^2 + e^2)^(5/2)) + (b*c^3*(-2*c^2*d^2 + e^2)*Log[e - c^2*d*x + Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^2*x^2]])/(e*(c^2*d^2 + e^2)^(5/2)))/6","A",1
12,1,354,368,0.5574372,"\int (d+e x)^3 \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)^3*(a + b*ArcSinh[c*x])^2,x]","\frac{c \left(72 a^2 c^3 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)-6 a b \sqrt{c^2 x^2+1} \left(c^2 \left(96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right)-e^2 (64 d+9 e x)\right)+b^2 c x \left(c^2 \left(576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right)-3 e^2 (128 d+9 e x)\right)\right)-6 b \sinh ^{-1}(c x) \left(b c \sqrt{c^2 x^2+1} \left(c^2 \left(96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right)-e^2 (64 d+9 e x)\right)-3 a \left(8 c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+24 c^2 d^2 e-3 e^3\right)\right)+9 b^2 \sinh ^{-1}(c x)^2 \left(8 c^4 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+24 c^2 d^2 e-3 e^3\right)}{288 c^4}","-\frac{3 e^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{32 c^4}-\frac{2 b d^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c}-\frac{3 b d^2 e x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 c}+\frac{3 d^2 e \left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^2}-\frac{2 b d e^2 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 c}-\frac{b e^3 x^3 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{8 c}+\frac{4 b d e^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^3}+\frac{3 b e^3 x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{16 c^3}-\frac{d^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{4 e}+\frac{(d+e x)^4 \left(a+b \sinh ^{-1}(c x)\right)^2}{4 e}-\frac{4 b^2 d e^2 x}{3 c^2}-\frac{3 b^2 e^3 x^2}{32 c^2}+2 b^2 d^3 x+\frac{3}{4} b^2 d^2 e x^2+\frac{2}{9} b^2 d e^2 x^3+\frac{1}{32} b^2 e^3 x^4",1,"(c*(72*a^2*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - 6*a*b*Sqrt[1 + c^2*x^2]*(-(e^2*(64*d + 9*e*x)) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) + b^2*c*x*(-3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216*d^2*e*x + 64*d*e^2*x^2 + 9*e^3*x^3))) - 6*b*(-3*a*(24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)) + b*c*Sqrt[1 + c^2*x^2]*(-(e^2*(64*d + 9*e*x)) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)))*ArcSinh[c*x] + 9*b^2*(24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcSinh[c*x]^2)/(288*c^4)","A",1
13,1,248,239,0.3824694,"\int (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{18 a^2 c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)-6 a b \sqrt{c^2 x^2+1} \left(c^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)-4 e^2\right)-6 b \sinh ^{-1}(c x) \left(b \sqrt{c^2 x^2+1} \left(c^2 \left(18 d^2+9 d e x+2 e^2 x^2\right)-4 e^2\right)-3 a \left(2 c^3 x \left(3 d^2+3 d e x+e^2 x^2\right)+3 c d e\right)\right)+b^2 c x \left(c^2 \left(108 d^2+27 d e x+4 e^2 x^2\right)-24 e^2\right)+9 b^2 c \sinh ^{-1}(c x)^2 \left(6 c^2 d^2 x+3 d \left(2 c^2 e x^2+e\right)+2 c^2 e^2 x^3\right)}{54 c^3}","-\frac{2 b d^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c}-\frac{b d e x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c}+\frac{d e \left(a+b \sinh ^{-1}(c x)\right)^2}{2 c^2}-\frac{2 b e^2 x^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c}+\frac{4 b e^2 \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{9 c^3}-\frac{d^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 e}+\frac{(d+e x)^3 \left(a+b \sinh ^{-1}(c x)\right)^2}{3 e}-\frac{4 b^2 e^2 x}{9 c^2}+2 b^2 d^2 x+\frac{1}{2} b^2 d e x^2+\frac{2}{27} b^2 e^2 x^3",1,"(18*a^2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) - 6*a*b*Sqrt[1 + c^2*x^2]*(-4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + b^2*c*x*(-24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) - 6*b*(-3*a*(3*c*d*e + 2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)) + b*Sqrt[1 + c^2*x^2]*(-4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)))*ArcSinh[c*x] + 9*b^2*c*(6*c^2*d^2*x + 2*c^2*e^2*x^3 + 3*d*(e + 2*c^2*e*x^2))*ArcSinh[c*x]^2)/(54*c^3)","A",1
14,1,142,140,0.3420754,"\int (d+e x) \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)*(a + b*ArcSinh[c*x])^2,x]","\frac{c \left(2 a^2 c x (2 d+e x)-2 a b \sqrt{c^2 x^2+1} (4 d+e x)+b^2 c x (8 d+e x)\right)+2 b \sinh ^{-1}(c x) \left(a \left(4 c^2 d x+2 c^2 e x^2+e\right)-b c \sqrt{c^2 x^2+1} (4 d+e x)\right)+b^2 \sinh ^{-1}(c x)^2 \left(4 c^2 d x+2 c^2 e x^2+e\right)}{4 c^2}","-\frac{2 b d \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{c}-\frac{b e x \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{2 c}+\frac{e \left(a+b \sinh ^{-1}(c x)\right)^2}{4 c^2}-\frac{d^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 e}+\frac{(d+e x)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 e}+2 b^2 d x+\frac{1}{4} b^2 e x^2",1,"(c*(2*a^2*c*x*(2*d + e*x) + b^2*c*x*(8*d + e*x) - 2*a*b*(4*d + e*x)*Sqrt[1 + c^2*x^2]) + 2*b*(-(b*c*(4*d + e*x)*Sqrt[1 + c^2*x^2]) + a*(e + 4*c^2*d*x + 2*c^2*e*x^2))*ArcSinh[c*x] + b^2*(e + 4*c^2*d*x + 2*c^2*e*x^2)*ArcSinh[c*x]^2)/(4*c^2)","A",1
15,1,74,46,0.0617464,"\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
16,1,273,291,0.2240888,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d+e x} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + e*x),x]","\frac{6 b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)+6 b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)+3 \left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)+3 \left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)-\frac{\left(a+b \sinh ^{-1}(c x)\right)^3}{b}-6 b^2 \text{Li}_3\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)-6 b^2 \text{Li}_3\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{3 e}","\frac{2 b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{2 b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)}{e}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)}{e}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^3}{3 b e}-\frac{2 b^2 \text{Li}_3\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e}-\frac{2 b^2 \text{Li}_3\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e}",1,"(-((a + b*ArcSinh[c*x])^3/b) + 3*(a + b*ArcSinh[c*x])^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])] + 3*(a + b*ArcSinh[c*x])^2*Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])] + 6*b*(a + b*ArcSinh[c*x])*PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] + 6*b*(a + b*ArcSinh[c*x])*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))] - 6*b^2*PolyLog[3, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] - 6*b^2*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))])/(3*e)","A",1
17,1,191,263,0.2179956,"\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{\frac{2 b c \left(\left(a+b \sinh ^{-1}(c x)\right) \left(\log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)-\log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)\right)+b \text{Li}_2\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)-b \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)\right)}{\sqrt{c^2 d^2+e^2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{d+e x}}{e}","\frac{2 b c \left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)}{e \sqrt{c^2 d^2+e^2}}-\frac{2 b c \left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)}{e \sqrt{c^2 d^2+e^2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{e (d+e x)}+\frac{2 b^2 c \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e \sqrt{c^2 d^2+e^2}}-\frac{2 b^2 c \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e \sqrt{c^2 d^2+e^2}}",1,"(-((a + b*ArcSinh[c*x])^2/(d + e*x)) + (2*b*c*((a + b*ArcSinh[c*x])*(Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])] - Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])]) + b*PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] - b*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))]))/Sqrt[c^2*d^2 + e^2])/e","A",1
18,1,270,349,0.7075221,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{(d+e x)^3} \, dx","Integrate[(a + b*ArcSinh[c*x])^2/(d + e*x)^3,x]","\frac{-\frac{2 b c e \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{\left(c^2 d^2+e^2\right) (d+e x)}+\frac{2 b c^3 d \left(\left(a+b \sinh ^{-1}(c x)\right) \left(\log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)-\log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)\right)+b \text{Li}_2\left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}-c d}\right)-b \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)\right)}{\left(c^2 d^2+e^2\right)^{3/2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{(d+e x)^2}+\frac{2 b^2 c^2 \log (d+e x)}{c^2 d^2+e^2}}{2 e}","-\frac{b c \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)}{\left(c^2 d^2+e^2\right) (d+e x)}+\frac{b c^3 d \left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}+1\right)}{e \left(c^2 d^2+e^2\right)^{3/2}}-\frac{b c^3 d \left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{e e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 d^2+e^2}+c d}+1\right)}{e \left(c^2 d^2+e^2\right)^{3/2}}-\frac{\left(a+b \sinh ^{-1}(c x)\right)^2}{2 e (d+e x)^2}+\frac{b^2 c^2 \log (d+e x)}{e \left(c^2 d^2+e^2\right)}+\frac{b^2 c^3 d \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d-\sqrt{c^2 d^2+e^2}}\right)}{e \left(c^2 d^2+e^2\right)^{3/2}}-\frac{b^2 c^3 d \text{Li}_2\left(-\frac{e e^{\sinh ^{-1}(c x)}}{c d+\sqrt{c^2 d^2+e^2}}\right)}{e \left(c^2 d^2+e^2\right)^{3/2}}",1,"((-2*b*c*e*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/((c^2*d^2 + e^2)*(d + e*x)) - (a + b*ArcSinh[c*x])^2/(d + e*x)^2 + (2*b^2*c^2*Log[d + e*x])/(c^2*d^2 + e^2) + (2*b*c^3*d*((a + b*ArcSinh[c*x])*(Log[1 + (e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])] - Log[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])]) + b*PolyLog[2, (e*E^ArcSinh[c*x])/(-(c*d) + Sqrt[c^2*d^2 + e^2])] - b*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2]))]))/(c^2*d^2 + e^2)^(3/2))/(2*e)","A",1
19,1,305,394,0.6944813,"\int \frac{(d+e x)^3}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(d + e*x)^3/(a + b*ArcSinh[c*x]),x]","\frac{e^3 \left(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)\right)}{8 b c^4}+\frac{3 d e^2 \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 c^3}-\frac{3 d^2 e \left(\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)-\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)\right)}{2 b c^2}+\frac{d^3 \left(\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)\right)}{b c}","\frac{e^3 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)}{4 b c^4}-\frac{e^3 \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right)}{8 b c^4}-\frac{e^3 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)}{4 b c^4}+\frac{e^3 \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 a}{b}+4 \sinh ^{-1}(c x)\right)}{8 b c^4}-\frac{3 d e^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{4 b c^3}+\frac{3 d e^2 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right)}{4 b c^3}+\frac{3 d e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{4 b c^3}-\frac{3 d e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right)}{4 b c^3}-\frac{3 d^2 e \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)}{2 b c^2}+\frac{3 d^2 e \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)}{2 b c^2}+\frac{d^3 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{b c}-\frac{d^3 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{b c}",1,"(d^3*(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]]))/(b*c) + (3*d*e^2*(-(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*c^3) + (e^3*(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])]))/(8*b*c^4) - (3*d^2*e*(CoshIntegral[(2*a)/b + 2*ArcSinh[c*x]]*Sinh[(2*a)/b] - Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[c*x]]))/(2*b*c^2)","A",1
20,1,188,245,0.4125094,"\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^2-e^2\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-4 c^2 d^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-4 c d e \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+e^2 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+4 c d e \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-e^2 \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^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{4 b c^3}+\frac{e^2 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right)}{4 b c^3}+\frac{e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{4 b c^3}-\frac{e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 a}{b}+3 \sinh ^{-1}(c x)\right)}{4 b c^3}-\frac{d e \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)}{b c^2}+\frac{d e \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)}{b c^2}+\frac{d^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{b c}-\frac{d^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{b c}",1,"((4*c^2*d^2 - e^2)*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] + e^2*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c*x])] - 4*c*d*e*CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] - 4*c^2*d^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + e^2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 4*c*d*e*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - e^2*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(4*b*c^3)","A",1
21,1,98,116,0.1663395,"\int \frac{d+e x}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(d + e*x)/(a + b*ArcSinh[c*x]),x]","\frac{2 c d \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-e \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-2 c d \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+e \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{2 b c^2}","-\frac{e \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)}{2 b c^2}+\frac{e \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c x)\right)}{2 b c^2}+\frac{d \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{b c}-\frac{d \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)}{b c}",1,"(2*c*d*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c*x]] - e*CoshIntegral[2*(a/b + ArcSinh[c*x])]*Sinh[(2*a)/b] - 2*c*d*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + e*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])])/(2*b*c^2)","A",1
22,1,45,54,0.0218577,"\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
23,0,0,21,0.2052421,"\int \frac{1}{(d+e x) \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x)*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{(d+e x) \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{(d+e x) \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[1/((d + e*x)*(a + b*ArcSinh[c*x])), x]","A",-1
24,0,0,21,0.3902847,"\int \frac{1}{(d+e x)^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[1/((d + e*x)^2*(a + b*ArcSinh[c*x])),x]","\int \frac{1}{(d+e x)^2 \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{(d+e x)^2 \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
25,1,288,359,1.6098558,"\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^2-e^2\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-4 c^2 d^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+\frac{4 b c^2 d^2 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac{8 b c^2 d e x \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac{4 b c^2 e^2 x^2 \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}-8 c d e \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+3 e^2 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+8 c d e \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)+e^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-3 e^2 \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^2 \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^2 \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^2 \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^2 \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{2 d e \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}-\frac{2 d e \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}-\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 \sqrt{c^2 x^2+1}}{b c \left(a+b \sinh ^{-1}(c x)\right)}-\frac{e^2 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^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (8*b*c^2*d*e*x*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (4*b*c^2*e^2*x^2*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) - 8*c*d*e*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] + (4*c^2*d^2 - e^2)*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] + 3*e^2*CoshIntegral[3*(a/b + ArcSinh[c*x])]*Sinh[(3*a)/b] - 4*c^2*d^2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + e^2*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + 8*c*d*e*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])] - 3*e^2*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c*x])])/(b^2*c^3)","A",1
26,1,150,180,0.7486623,"\int \frac{d+e x}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x)/(a + b*ArcSinh[c*x])^2,x]","-\frac{\frac{b c d \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+\frac{b c e x \sqrt{c^2 x^2+1}}{a+b \sinh ^{-1}(c x)}+c d \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)-e \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)-c d \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c x)\right)+e \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c x)\right)\right)}{b^2 c^2}","\frac{e \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}-\frac{e \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c x)\right)}{b}\right)}{b^2 c^2}-\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 \sqrt{c^2 x^2+1}}{b c \left(a+b \sinh ^{-1}(c x)\right)}",1,"-(((b*c*d*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (b*c*e*x*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) - e*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c*x])] + c*d*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - c*d*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c*x]] + e*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c*x])])/(b^2*c^2))","A",1
27,1,71,85,0.1875393,"\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
28,0,0,21,3.2182068,"\int \frac{1}{(d+e x) \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[1/((d + e*x)*(a + b*ArcSinh[c*x])^2),x]","\int \frac{1}{(d+e x) \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{(d+e x) \left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[1/((d + e*x)*(a + b*ArcSinh[c*x])^2), x]","A",-1
29,0,0,21,5.2094742,"\int \frac{1}{(d+e x)^2 \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}{(d+e x)^2 \left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{1}{(d+e x)^2 \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
30,0,0,75,4.1730002,"\int (d+e x)^m \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","Integrate[(d + e*x)^m*(a + b*ArcSinh[c*x])^2,x]","\int (d+e x)^m \left(a+b \sinh ^{-1}(c x)\right)^2 \, dx","\frac{(d+e x)^{m+1} \left(a+b \sinh ^{-1}(c x)\right)^2}{e (m+1)}-\frac{2 b c \text{Int}\left(\frac{(d+e x)^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 x^2+1}},x\right)}{e (m+1)}",0,"Integrate[(d + e*x)^m*(a + b*ArcSinh[c*x])^2, x]","A",-1
31,0,0,179,0.045826,"\int (d+e x)^m \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^m*(a + b*ArcSinh[c*x]),x]","\int (d+e x)^m \left(a+b \sinh ^{-1}(c x)\right) \, dx","\frac{(d+e x)^{m+1} \left(a+b \sinh ^{-1}(c x)\right)}{e (m+1)}-\frac{b c \sqrt{1-\frac{d+e x}{d-\frac{e}{\sqrt{-c^2}}}} \sqrt{1-\frac{d+e x}{\frac{e}{\sqrt{-c^2}}+d}} (d+e x)^{m+2} F_1\left(m+2;\frac{1}{2},\frac{1}{2};m+3;\frac{d+e x}{d-\frac{e}{\sqrt{-c^2}}},\frac{d+e x}{d+\frac{e}{\sqrt{-c^2}}}\right)}{e^2 (m+1) (m+2) \sqrt{c^2 x^2+1}}",1,"Integrate[(d + e*x)^m*(a + b*ArcSinh[c*x]), x]","F",-1
32,0,0,21,0.3671302,"\int \frac{(d+e x)^m}{a+b \sinh ^{-1}(c x)} \, dx","Integrate[(d + e*x)^m/(a + b*ArcSinh[c*x]),x]","\int \frac{(d+e x)^m}{a+b \sinh ^{-1}(c x)} \, dx","\text{Int}\left(\frac{(d+e x)^m}{a+b \sinh ^{-1}(c x)},x\right)",0,"Integrate[(d + e*x)^m/(a + b*ArcSinh[c*x]), x]","A",-1
33,0,0,21,0.7806823,"\int \frac{(d+e x)^m}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","Integrate[(d + e*x)^m/(a + b*ArcSinh[c*x])^2,x]","\int \frac{(d+e x)^m}{\left(a+b \sinh ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(d+e x)^m}{\left(a+b \sinh ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(d + e*x)^m/(a + b*ArcSinh[c*x])^2, x]","A",-1
34,1,413,640,1.4932439,"\int (f+g x)^3 \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{3600 a c \sqrt{d} f \sqrt{c^2 x^2+1} \left(4 c^2 f^2-3 g^2\right) \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+240 a \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(6 c^4 x \left(10 f^3+20 f^2 g x+15 f g^2 x^2+4 g^3 x^3\right)+c^2 g \left(120 f^2+45 f g x+8 g^2 x^2\right)-16 g^3\right)-675 b c f g^2 \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 b g^3 \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)-3600 b c^3 f^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)-9600 b c^2 f^2 g \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)}{28800 c^4 \sqrt{c^2 x^2+1}}","\frac{1}{2} f^3 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{f^3 \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{f^2 g \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{c^2}+\frac{3 f g^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{8 c^2}+\frac{3}{4} f g^2 x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{g^3 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{5 c^4}-\frac{g^3 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^4}-\frac{3 f g^2 \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 c f^3 x^2 \sqrt{c^2 d x^2+d}}{4 \sqrt{c^2 x^2+1}}-\frac{b f^2 g x \sqrt{c^2 d x^2+d}}{c \sqrt{c^2 x^2+1}}-\frac{b c f^2 g x^3 \sqrt{c^2 d x^2+d}}{3 \sqrt{c^2 x^2+1}}-\frac{3 b f g^2 x^2 \sqrt{c^2 d x^2+d}}{16 c \sqrt{c^2 x^2+1}}-\frac{3 b c f g^2 x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{b c g^3 x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}-\frac{b g^3 x^3 \sqrt{c^2 d x^2+d}}{45 c \sqrt{c^2 x^2+1}}+\frac{2 b g^3 x \sqrt{c^2 d x^2+d}}{15 c^3 \sqrt{c^2 x^2+1}}",1,"(240*a*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(-16*g^3 + c^2*g*(120*f^2 + 45*f*g*x + 8*g^2*x^2) + 6*c^4*x*(10*f^3 + 20*f^2*g*x + 15*f*g^2*x^2 + 4*g^3*x^3)) - 9600*b*c^2*f^2*g*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]) - 128*b*g^3*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]) + 3600*a*c*Sqrt[d]*f*(4*c^2*f^2 - 3*g^2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 3600*b*c^3*f^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]])) - 675*b*c*f*g^2*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]]))/(28800*c^4*Sqrt[1 + c^2*x^2])","A",1
35,1,301,431,0.9583132,"\int (f+g x)^2 \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{48 a c \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(12 c^2 f^2 x+16 f \left(c^2 g x^2+g\right)+3 g^2 x \left(2 c^2 x^2+1\right)\right)+144 a \sqrt{d} \sqrt{c^2 x^2+1} (2 c f-g) (2 c f+g) \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)-144 b c^2 f^2 \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)-9 b g^2 \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)-256 b c f g \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)}{1152 c^3 \sqrt{c^2 x^2+1}}","\frac{1}{2} f^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{f^2 \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{2 f g \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^2}+\frac{g^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{8 c^2}+\frac{1}{4} g^2 x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{g^2 \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 c f^2 x^2 \sqrt{c^2 d x^2+d}}{4 \sqrt{c^2 x^2+1}}-\frac{2 b f g x \sqrt{c^2 d x^2+d}}{3 c \sqrt{c^2 x^2+1}}-\frac{2 b c f g x^3 \sqrt{c^2 d x^2+d}}{9 \sqrt{c^2 x^2+1}}-\frac{b g^2 x^2 \sqrt{c^2 d x^2+d}}{16 c \sqrt{c^2 x^2+1}}-\frac{b c g^2 x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}",1,"(48*a*c*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(12*c^2*f^2*x + 3*g^2*x*(1 + 2*c^2*x^2) + 16*f*(g + c^2*g*x^2)) - 256*b*c*f*g*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]) + 144*a*Sqrt[d]*(2*c*f - g)*(2*c*f + g)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 144*b*c^2*f^2*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])) - 9*b*g^2*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]]))/(1152*c^3*Sqrt[1 + c^2*x^2])","A",1
36,1,208,227,1.2118034,"\int (f+g x) \sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]),x]","\frac{1}{6} a \sqrt{c^2 d x^2+d} \left(\frac{2 g}{c^2}+x (3 f+2 g x)\right)+\frac{a \sqrt{d} f \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)}{2 c}+\frac{b f \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 g \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 c^2 \sqrt{c^2 x^2+1}}","\frac{1}{2} f x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{f \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{g \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 c^2}-\frac{b c f x^2 \sqrt{c^2 d x^2+d}}{4 \sqrt{c^2 x^2+1}}-\frac{b g x \sqrt{c^2 d x^2+d}}{3 c \sqrt{c^2 x^2+1}}-\frac{b c g x^3 \sqrt{c^2 d x^2+d}}{9 \sqrt{c^2 x^2+1}}",1,"(a*Sqrt[d + c^2*d*x^2]*((2*g)/c^2 + x*(3*f + 2*g*x)))/6 - (b*g*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*c^2*Sqrt[1 + c^2*x^2]) + (a*Sqrt[d]*f*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(2*c) + (b*f*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])","A",1
37,1,1353,664,6.367849,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)}{f+g x} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(f + g*x),x]","\frac{2 a \sqrt{c^2 d x^2+d} g+2 a \sqrt{d} \sqrt{c^2 f^2+g^2} \log (f+g x)-2 a c \sqrt{d} f \log \left(c d x+\sqrt{d} \sqrt{c^2 d x^2+d}\right)-2 a \sqrt{d} \sqrt{c^2 f^2+g^2} \log \left(d \left(g-c^2 f x\right)+\sqrt{d} \sqrt{c^2 f^2+g^2} \sqrt{c^2 d x^2+d}\right)+b \sqrt{c^2 d x^2+d} \left(-\frac{c f \sinh ^{-1}(c x)^2}{\sqrt{c^2 x^2+1}}+2 g \sinh ^{-1}(c x)+\frac{2 \left(c^2 f^2+g^2\right) \left(-\frac{i \pi  \tanh ^{-1}\left(\frac{c f \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-g}{\sqrt{c^2 f^2+g^2}}\right)}{\sqrt{c^2 f^2+g^2}}-\frac{2 \cos ^{-1}\left(-\frac{i c f}{g}\right) \tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)+\left(\pi -2 i \sinh ^{-1}(c x)\right) \tanh ^{-1}\left(\frac{(c f-i g) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)+\left(\cos ^{-1}\left(-\frac{i c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)-2 i \tanh ^{-1}\left(\frac{(c f-i g) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)\right) \log \left(\frac{\left(\frac{1}{2}-\frac{i}{2}\right) e^{-\frac{1}{2} \sinh ^{-1}(c x)} \sqrt{-c^2 f^2-g^2}}{\sqrt{-i g} \sqrt{c (f+g x)}}\right)+\left(\cos ^{-1}\left(-\frac{i c f}{g}\right)+2 i \left(\tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)+\tanh ^{-1}\left(\frac{(c f-i g) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)\right)\right) \log \left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) e^{\frac{1}{2} \sinh ^{-1}(c x)} \sqrt{-c^2 f^2-g^2}}{\sqrt{-i g} \sqrt{c (f+g x)}}\right)-\left(\cos ^{-1}\left(-\frac{i c f}{g}\right)+2 i \tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)\right) \log \left(\frac{(i c f+g) \left(-i c f+g+\sqrt{-c^2 f^2-g^2}\right) \left(i \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)+1\right)}{g \left(i c f+g+i \sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}\right)-\left(\cos ^{-1}\left(-\frac{i c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)\right) \log \left(\frac{(i c f+g) \left(i c f-g+\sqrt{-c^2 f^2-g^2}\right) \left(\cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)+i\right)}{g \left(c f-i g+\sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}\right)+i \left(\text{Li}_2\left(\frac{\left(i c f+\sqrt{-c^2 f^2-g^2}\right) \left(i c f+g-i \sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}{g \left(i c f+g+i \sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(c f+i \sqrt{-c^2 f^2-g^2}\right) \left(-c f+i g+\sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}{g \left(i c f+g+i \sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}\right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)}{\sqrt{c^2 x^2+1}}-\frac{2 c g x}{\sqrt{c^2 x^2+1}}\right)}{2 g^2}","-\frac{\sqrt{c^2 d x^2+d} \left(\frac{c^2 f^2}{g^2}+1\right) \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c \sqrt{c^2 x^2+1} (f+g x)}+\frac{\sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c (f+g x)}-\frac{c x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b g \sqrt{c^2 x^2+1}}-\frac{a \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \tanh ^{-1}\left(\frac{g-c^2 f x}{\sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}\right)}{g^2 \sqrt{c^2 x^2+1}}+\frac{a \sqrt{c^2 d x^2+d}}{g}+\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{g^2 \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{g^2 \sqrt{c^2 x^2+1}}+\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \sinh ^{-1}(c x) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)}{g^2 \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2} \sinh ^{-1}(c x) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)}{g^2 \sqrt{c^2 x^2+1}}-\frac{b c x \sqrt{c^2 d x^2+d}}{g \sqrt{c^2 x^2+1}}+\frac{b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g}",1,"(2*a*g*Sqrt[d + c^2*d*x^2] + 2*a*Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Log[f + g*x] - 2*a*c*Sqrt[d]*f*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 2*a*Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]] + b*Sqrt[d + c^2*d*x^2]*((-2*c*g*x)/Sqrt[1 + c^2*x^2] + 2*g*ArcSinh[c*x] - (c*f*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (2*(c^2*f^2 + g^2)*(((-I)*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/Sqrt[c^2*f^2 + g^2]])/Sqrt[c^2*f^2 + g^2] - (2*ArcCos[((-I)*c*f)/g]*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2*I)*ArcSinh[c*x])*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] - (2*I)*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh[c*x]/2)*Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])] + (ArcCos[((-I)*c*f)/g] + (2*I)*(ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]]))*Log[((1/2 + I/2)*E^(ArcSinh[c*x]/2)*Sqrt[-(c^2*f^2) - g^2])/(Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])] - (ArcCos[((-I)*c*f)/g] + (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*((-I)*c*f + g + Sqrt[-(c^2*f^2) - g^2])*(1 + I*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*(I*c*f - g + Sqrt[-(c^2*f^2) - g^2])*(I + Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(c*f - I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] + I*(PolyLog[2, ((I*c*f + Sqrt[-(c^2*f^2) - g^2])*(I*c*f + g - I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) - g^2])*(-(c*f) + I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))]))/Sqrt[-(c^2*f^2) - g^2]))/Sqrt[1 + c^2*x^2]))/(2*g^2)","C",0
38,1,1384,781,9.653372,"\int \frac{\sqrt{d+c^2 d x^2} \left(a+b \sinh ^{-1}(c x)\right)}{(f+g x)^2} \, dx","Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(f + g*x)^2,x]","\frac{-\frac{2 a \sqrt{d} f \log (f+g x) c^2}{\sqrt{c^2 f^2+g^2}}+\frac{2 a \sqrt{d} f \log \left(d \left(g-c^2 f x\right)+\sqrt{d} \sqrt{c^2 f^2+g^2} \sqrt{c^2 d x^2+d}\right) c^2}{\sqrt{c^2 f^2+g^2}}+2 a \sqrt{d} \log \left(c d x+\sqrt{d} \sqrt{c^2 d x^2+d}\right) c+b \sqrt{c^2 d x^2+d} \left(\frac{\sinh ^{-1}(c x)^2}{\sqrt{c^2 x^2+1}}-\frac{2 g \sinh ^{-1}(c x)}{c f+c g x}+\frac{2 i c f \pi  \tanh ^{-1}\left(\frac{c f \tanh \left(\frac{1}{2} \sinh ^{-1}(c x)\right)-g}{\sqrt{c^2 f^2+g^2}}\right)}{\sqrt{c^2 f^2+g^2} \sqrt{c^2 x^2+1}}+\frac{2 \log \left(\frac{g x}{f}+1\right)}{\sqrt{c^2 x^2+1}}+\frac{2 c f \left(2 \cos ^{-1}\left(-\frac{i c f}{g}\right) \tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)+\left(\pi -2 i \sinh ^{-1}(c x)\right) \tanh ^{-1}\left(\frac{(c f-i g) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)+\left(\cos ^{-1}\left(-\frac{i c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)-2 i \tanh ^{-1}\left(\frac{(c f-i g) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)\right) \log \left(\frac{\left(\frac{1}{2}-\frac{i}{2}\right) e^{-\frac{1}{2} \sinh ^{-1}(c x)} \sqrt{-c^2 f^2-g^2}}{\sqrt{-i g} \sqrt{c (f+g x)}}\right)+\left(\cos ^{-1}\left(-\frac{i c f}{g}\right)+2 i \left(\tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)+\tanh ^{-1}\left(\frac{(c f-i g) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)\right)\right) \log \left(\frac{\left(\frac{1}{2}+\frac{i}{2}\right) e^{\frac{1}{2} \sinh ^{-1}(c x)} \sqrt{-c^2 f^2-g^2}}{\sqrt{-i g} \sqrt{c (f+g x)}}\right)-\left(\cos ^{-1}\left(-\frac{i c f}{g}\right)+2 i \tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)\right) \log \left(\frac{(i c f+g) \left(-i c f+g+\sqrt{-c^2 f^2-g^2}\right) \left(i \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)+1\right)}{g \left(i c f+g+i \sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}\right)-\left(\cos ^{-1}\left(-\frac{i c f}{g}\right)-2 i \tanh ^{-1}\left(\frac{(c f+i g) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)}{\sqrt{-c^2 f^2-g^2}}\right)\right) \log \left(\frac{(i c f+g) \left(i c f-g+\sqrt{-c^2 f^2-g^2}\right) \left(\cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)+i\right)}{g \left(c f-i g+\sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}\right)+i \left(\text{Li}_2\left(\frac{\left(i c f+\sqrt{-c^2 f^2-g^2}\right) \left(i c f+g-i \sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}{g \left(i c f+g+i \sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(c f+i \sqrt{-c^2 f^2-g^2}\right) \left(-c f+i g+\sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}{g \left(i c f+g+i \sqrt{-c^2 f^2-g^2} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(c x)+\pi \right)\right)\right)}\right)\right)\right)}{\sqrt{-c^2 f^2-g^2} \sqrt{c^2 x^2+1}}\right) c-\frac{2 a g \sqrt{c^2 d x^2+d}}{f+g x}}{2 g^2}","-\frac{\sqrt{c^2 d x^2+d} \left(g-c^2 f x\right)^2 \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c \sqrt{c^2 x^2+1} \left(c^2 f^2+g^2\right) (f+g x)^2}+\frac{\sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b c (f+g x)^2}+\frac{a c^2 f \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(\frac{g-c^2 f x}{\sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}\right)}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}-\frac{a \sqrt{c^2 d x^2+d}}{g (f+g x)}+\frac{a c^3 f^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g^2 \sqrt{c^2 x^2+1} \left(c^2 f^2+g^2\right)}-\frac{b c^2 f \sqrt{c^2 d x^2+d} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}+\frac{b c^2 f \sqrt{c^2 d x^2+d} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}-\frac{b c^2 f \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}+\frac{b c^2 f \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)}{g^2 \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2}}+\frac{b c \sqrt{c^2 d x^2+d} \log (f+g x)}{g^2 \sqrt{c^2 x^2+1}}-\frac{b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2}{2 g^2 \sqrt{c^2 x^2+1} \left(c^2 f^2+g^2\right)}",1,"((-2*a*g*Sqrt[d + c^2*d*x^2])/(f + g*x) - (2*a*c^2*Sqrt[d]*f*Log[f + g*x])/Sqrt[c^2*f^2 + g^2] + 2*a*c*Sqrt[d]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (2*a*c^2*Sqrt[d]*f*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]])/Sqrt[c^2*f^2 + g^2] + b*c*Sqrt[d + c^2*d*x^2]*((-2*g*ArcSinh[c*x])/(c*f + c*g*x) + ArcSinh[c*x]^2/Sqrt[1 + c^2*x^2] + ((2*I)*c*f*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/Sqrt[c^2*f^2 + g^2]])/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]) + (2*Log[1 + (g*x)/f])/Sqrt[1 + c^2*x^2] + (2*c*f*(2*ArcCos[((-I)*c*f)/g]*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2*I)*ArcSinh[c*x])*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] - (2*I)*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh[c*x]/2)*Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])] + (ArcCos[((-I)*c*f)/g] + (2*I)*(ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]]))*Log[((1/2 + I/2)*E^(ArcSinh[c*x]/2)*Sqrt[-(c^2*f^2) - g^2])/(Sqrt[(-I)*g]*Sqrt[c*(f + g*x)])] - (ArcCos[((-I)*c*f)/g] + (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*((-I)*c*f + g + Sqrt[-(c^2*f^2) - g^2])*(1 + I*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*(I*c*f - g + Sqrt[-(c^2*f^2) - g^2])*(I + Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(c*f - I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] + I*(PolyLog[2, ((I*c*f + Sqrt[-(c^2*f^2) - g^2])*(I*c*f + g - I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) - g^2])*(-(c*f) + I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))])))/(Sqrt[-(c^2*f^2) - g^2]*Sqrt[1 + c^2*x^2])))/(2*g^2)","C",0
39,1,779,918,3.7090455,"\int (f+g x)^3 \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{529200 a c d^{3/2} f \sqrt{c^2 x^2+1} \left(2 c^2 f^2-g^2\right) \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+5040 a d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(4 c^6 x^3 \left(35 f^3+84 f^2 g x+70 f g^2 x^2+20 g^3 x^3\right)+2 c^4 x \left(175 f^3+336 f^2 g x+245 f g^2 x^2+64 g^3 x^3\right)+c^2 g \left(336 f^2+105 f g x+16 g^2 x^2\right)-32 g^3\right)-66150 b c d f g^2 \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)+3675 b c d f g^2 \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)-37632 b c^2 d f^2 g \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)-12544 b d g^3 \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)-352800 b c^3 d f^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)-22050 b c^3 d f^3 \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)-940800 b c^2 d f^2 g \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)-256 b d g^3 \sqrt{c^2 d x^2+d} \left(c x \left(225 c^6 x^6+63 c^4 x^4-140 c^2 x^2+840\right)-105 \sqrt{c^2 x^2+1} \left(15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right) \sinh ^{-1}(c x)\right)}{2822400 c^4 \sqrt{c^2 x^2+1}}","-\frac{b c^3 d g^3 \sqrt{c^2 d x^2+d} x^7}{49 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f g^2 \sqrt{c^2 d x^2+d} x^6}{12 \sqrt{c^2 x^2+1}}-\frac{8 b c d g^3 \sqrt{c^2 d x^2+d} x^5}{175 \sqrt{c^2 x^2+1}}-\frac{3 b c^3 d f^2 g \sqrt{c^2 d x^2+d} x^5}{25 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f^3 \sqrt{c^2 d x^2+d} x^4}{16 \sqrt{c^2 x^2+1}}-\frac{7 b c d f g^2 \sqrt{c^2 d x^2+d} x^4}{32 \sqrt{c^2 x^2+1}}+\frac{3}{8} d f g^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^3+\frac{1}{2} d f g^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^3-\frac{b d g^3 \sqrt{c^2 d x^2+d} x^3}{105 c \sqrt{c^2 x^2+1}}-\frac{2 b c d f^2 g \sqrt{c^2 d x^2+d} x^3}{5 \sqrt{c^2 x^2+1}}-\frac{5 b c d f^3 \sqrt{c^2 d x^2+d} x^2}{16 \sqrt{c^2 x^2+1}}-\frac{3 b d f g^2 \sqrt{c^2 d x^2+d} x^2}{32 c \sqrt{c^2 x^2+1}}+\frac{3}{8} d f^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x+\frac{3 d f g^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x}{16 c^2}+\frac{1}{4} d f^3 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x+\frac{2 b d g^3 \sqrt{c^2 d x^2+d} x}{35 c^3 \sqrt{c^2 x^2+1}}-\frac{3 b d f^2 g \sqrt{c^2 d x^2+d} x}{5 c \sqrt{c^2 x^2+1}}+\frac{3 d f^3 \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{3 d f g^2 \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{d g^3 \left(c^2 x^2+1\right)^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{7 c^4}-\frac{d g^3 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{5 c^4}+\frac{3 d f^2 g \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{5 c^2}",1,"(5040*a*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(-32*g^3 + c^2*g*(336*f^2 + 105*f*g*x + 16*g^2*x^2) + 4*c^6*x^3*(35*f^3 + 84*f^2*g*x + 70*f*g^2*x^2 + 20*g^3*x^3) + 2*c^4*x*(175*f^3 + 336*f^2*g*x + 245*f*g^2*x^2 + 64*g^3*x^3)) - 940800*b*c^2*d*f^2*g*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]) - 37632*b*c^2*d*f^2*g*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]) - 12544*b*d*g^3*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]) - 256*b*d*g^3*Sqrt[d + c^2*d*x^2]*(c*x*(840 - 140*c^2*x^2 + 63*c^4*x^4 + 225*c^6*x^6) - 105*Sqrt[1 + c^2*x^2]*(8 - 4*c^2*x^2 + 3*c^4*x^4 + 15*c^6*x^6)*ArcSinh[c*x]) + 529200*a*c*d^(3/2)*f*(2*c^2*f^2 - g^2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 352800*b*c^3*d*f^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]])) - 22050*b*c^3*d*f^3*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]]) - 66150*b*c*d*f*g^2*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]]) + 3675*b*c*d*f*g^2*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]])))/(2822400*c^4*Sqrt[1 + c^2*x^2])","A",1
40,1,546,651,2.2181733,"\int (f+g x)^2 \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^2*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{3600 a d^{3/2} \sqrt{c^2 x^2+1} \left(6 c^2 f^2-g^2\right) \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+240 a c d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(30 c^2 f^2 x \left(2 c^2 x^2+5\right)+96 f g \left(c^2 x^2+1\right)^2+5 g^2 x \left(8 c^4 x^4+14 c^2 x^2+3\right)\right)-7200 b c^2 d f^2 \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)-450 b c^2 d f^2 \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)-450 b d g^2 \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)+25 b d g^2 \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)-512 b c d f g \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)-12800 b c d f g \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)}{57600 c^3 \sqrt{c^2 x^2+1}}","\frac{3}{8} d f^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{4} d f^2 x \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{3 d f^2 \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{2 d f g \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{5 c^2}+\frac{d g^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{6} d g^2 x^3 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)-\frac{d g^2 \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{5 b c d f^2 x^2 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{2 b d f g x \sqrt{c^2 d x^2+d}}{5 c \sqrt{c^2 x^2+1}}-\frac{4 b c d f g x^3 \sqrt{c^2 d x^2+d}}{15 \sqrt{c^2 x^2+1}}-\frac{b d g^2 x^2 \sqrt{c^2 d x^2+d}}{32 c \sqrt{c^2 x^2+1}}-\frac{7 b c d g^2 x^4 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f^2 x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d f g x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}-\frac{b c^3 d g^2 x^6 \sqrt{c^2 d x^2+d}}{36 \sqrt{c^2 x^2+1}}",1,"(240*a*c*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(96*f*g*(1 + c^2*x^2)^2 + 30*c^2*f^2*x*(5 + 2*c^2*x^2) + 5*g^2*x*(3 + 14*c^2*x^2 + 8*c^4*x^4)) - 12800*b*c*d*f*g*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]) - 512*b*c*d*f*g*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]) + 3600*a*d^(3/2)*(6*c^2*f^2 - g^2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 7200*b*c^2*d*f^2*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])) - 450*b*c^2*d*f^2*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]]) - 450*b*d*g^2*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]]) + 25*b*d*g^2*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]])))/(57600*c^3*Sqrt[1 + c^2*x^2])","A",1
41,1,392,353,1.2321728,"\int (f+g x) \left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]),x]","\frac{3600 a c d^{3/2} f \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+240 a d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(5 c^2 f x \left(2 c^2 x^2+5\right)+8 g \left(c^2 x^2+1\right)^2\right)+2400 b c d f \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(\sinh ^{-1}(c x)+\sinh \left(2 \sinh ^{-1}(c x)\right)\right)-1200 b c d f \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-75 b c d f \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)-640 b c d g x \left(c^2 x^2+3\right) \sqrt{c^2 d x^2+d}+3200 b d g \left(c^2 x^2+1\right)^{3/2} \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)+640 b d g \left(c^2 x^2+1\right)^{3/2} \left(3 c^2 x^2-2\right) \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)-128 b c^3 d g x^3 \left(3 c^2 x^2+5\right) \sqrt{c^2 d x^2+d}}{9600 c^2 \sqrt{c^2 x^2+1}}","\frac{3}{8} d f x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{1}{4} d f x \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{3 d f \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{d g \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{5 c^2}-\frac{5 b c d f x^2 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{b d g x \sqrt{c^2 d x^2+d}}{5 c \sqrt{c^2 x^2+1}}-\frac{2 b c d g x^3 \sqrt{c^2 d x^2+d}}{15 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{b c^3 d g x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}",1,"(-640*b*c*d*g*x*(3 + c^2*x^2)*Sqrt[d + c^2*d*x^2] - 128*b*c^3*d*g*x^3*(5 + 3*c^2*x^2)*Sqrt[d + c^2*d*x^2] + 240*a*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(8*g*(1 + c^2*x^2)^2 + 5*c^2*f*x*(5 + 2*c^2*x^2)) + 3200*b*d*g*(1 + c^2*x^2)^(3/2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x] + 640*b*d*g*(1 + c^2*x^2)^(3/2)*(-2 + 3*c^2*x^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x] - 1200*b*c*d*f*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] + 3600*a*c*d^(3/2)*f*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 2400*b*c*d*f*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]]) - 75*b*c*d*f*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]]))/(9600*c^2*Sqrt[1 + c^2*x^2])","A",1
42,1,2889,984,13.6188273,"\int \frac{\left(d+c^2 d x^2\right)^{3/2} \left(a+b \sinh ^{-1}(c x)\right)}{f+g x} \, dx","Integrate[((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(f + g*x),x]","\text{Result too large to show}","-\frac{b d x^3 \sqrt{c^2 d x^2+d} c^3}{9 g \sqrt{c^2 x^2+1}}+\frac{b d f x^2 \sqrt{c^2 d x^2+d} c^3}{4 g^2 \sqrt{c^2 x^2+1}}-\frac{d f x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) c^2}{2 g^2}-\frac{d \left(c^2 f^2+g^2\right) x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2 c}{2 b g^3 \sqrt{c^2 x^2+1}}-\frac{d f \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2 c}{4 b g^2 \sqrt{c^2 x^2+1}}-\frac{b d \left(c^2 f^2+g^2\right) x \sqrt{c^2 d x^2+d} c}{g^3 \sqrt{c^2 x^2+1}}-\frac{b d x \sqrt{c^2 d x^2+d} c}{3 g \sqrt{c^2 x^2+1}}+\frac{b d \left(c^2 f^2+g^2\right) \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g^3}+\frac{d \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 g}-\frac{a d \left(c^2 f^2+g^2\right)^{3/2} \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{c^2 x^2+1}}\right)}{g^4 \sqrt{c^2 x^2+1}}+\frac{b d \left(c^2 f^2+g^2\right)^{3/2} \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left(\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}+1\right)}{g^4 \sqrt{c^2 x^2+1}}-\frac{b d \left(c^2 f^2+g^2\right)^{3/2} \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left(\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}+1\right)}{g^4 \sqrt{c^2 x^2+1}}+\frac{b d \left(c^2 f^2+g^2\right)^{3/2} \sqrt{c^2 d x^2+d} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{g^4 \sqrt{c^2 x^2+1}}-\frac{b d \left(c^2 f^2+g^2\right)^{3/2} \sqrt{c^2 d x^2+d} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{g^4 \sqrt{c^2 x^2+1}}+\frac{a d \left(c^2 f^2+g^2\right) \sqrt{c^2 d x^2+d}}{g^3}+\frac{d \left(c^2 f^2+g^2\right) \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b g^2 (f+g x) c}-\frac{d \left(c^2 f^2+g^2\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b g^4 (f+g x) \sqrt{c^2 x^2+1} c}",1,"Sqrt[d*(1 + c^2*x^2)]*((a*d*(3*c^2*f^2 + 4*g^2))/(3*g^3) - (a*c^2*d*f*x)/(2*g^2) + (a*c^2*d*x^2)/(3*g)) + (a*d^(3/2)*(c^2*f^2 + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*c*d^(3/2)*f*(2*c^2*f^2 + 3*g^2)*Log[c*d*x + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/(2*g^4) - (a*d^(3/2)*(c^2*f^2 + g^2)^(3/2)*Log[d*g - c^2*d*f*x + Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Sqrt[d*(1 + c^2*x^2)]])/g^4 + (b*d*Sqrt[d*(1 + c^2*x^2)]*((-2*c*g*x)/Sqrt[1 + c^2*x^2] + 2*g*ArcSinh[c*x] - (c*f*ArcSinh[c*x]^2)/Sqrt[1 + c^2*x^2] + (2*(c^2*f^2 + g^2)*(((-I)*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/Sqrt[c^2*f^2 + g^2]])/Sqrt[c^2*f^2 + g^2] - (2*ArcCos[((-I)*c*f)/g]*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2*I)*ArcSinh[c*x])*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] - (2*I)*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh[c*x]/2)*Sqrt[(-I)*g]*Sqrt[c*f + c*g*x])] + (ArcCos[((-I)*c*f)/g] + (2*I)*(ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]]))*Log[((1/2 + I/2)*E^(ArcSinh[c*x]/2)*Sqrt[-(c^2*f^2) - g^2])/(Sqrt[(-I)*g]*Sqrt[c*f + c*g*x])] - (ArcCos[((-I)*c*f)/g] + (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*((-I)*c*f + g + Sqrt[-(c^2*f^2) - g^2])*(1 + I*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*(I*c*f - g + Sqrt[-(c^2*f^2) - g^2])*(I + Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(c*f - I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] + I*(PolyLog[2, ((I*c*f + Sqrt[-(c^2*f^2) - g^2])*(I*c*f + g - I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) - g^2])*(-(c*f) + I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))]))/Sqrt[-(c^2*f^2) - g^2]))/Sqrt[1 + c^2*x^2]))/(2*g^2) + (b*d*Sqrt[d*(1 + c^2*x^2)]*((-9*(ArcSinh[c*x]*(Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] - Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])]) + PolyLog[2, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] - PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]))/Sqrt[c^2*f^2 + g^2] + (-18*c*g*(4*c^2*f^2 + g^2)*x + 18*g*(4*c^2*f^2 + g^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 18*c*f*(2*c^2*f^2 + g^2)*ArcSinh[c*x]^2 + 9*c*f*g^2*Cosh[2*ArcSinh[c*x]] + 6*g^3*ArcSinh[c*x]*Cosh[3*ArcSinh[c*x]] + 9*(8*c^4*f^4 + 8*c^2*f^2*g^2 + g^4)*(((-I)*Pi*ArcTanh[(-g + c*f*Tanh[ArcSinh[c*x]/2])/Sqrt[c^2*f^2 + g^2]])/Sqrt[c^2*f^2 + g^2] - (2*ArcCos[((-I)*c*f)/g]*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (Pi - (2*I)*ArcSinh[c*x])*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] - (2*I)*ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((1/2 - I/2)*Sqrt[-(c^2*f^2) - g^2])/(E^(ArcSinh[c*x]/2)*Sqrt[(-I)*g]*Sqrt[c*f + c*g*x])] + (ArcCos[((-I)*c*f)/g] + (2*I)*(ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]] + ArcTanh[((c*f - I*g)*Tan[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]]))*Log[((1/2 + I/2)*E^(ArcSinh[c*x]/2)*Sqrt[-(c^2*f^2) - g^2])/(Sqrt[(-I)*g]*Sqrt[c*f + c*g*x])] - (ArcCos[((-I)*c*f)/g] + (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*((-I)*c*f + g + Sqrt[-(c^2*f^2) - g^2])*(1 + I*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - (ArcCos[((-I)*c*f)/g] - (2*I)*ArcTanh[((c*f + I*g)*Cot[(Pi + (2*I)*ArcSinh[c*x])/4])/Sqrt[-(c^2*f^2) - g^2]])*Log[((I*c*f + g)*(I*c*f - g + Sqrt[-(c^2*f^2) - g^2])*(I + Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(c*f - I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] + I*(PolyLog[2, ((I*c*f + Sqrt[-(c^2*f^2) - g^2])*(I*c*f + g - I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) - g^2])*(-(c*f) + I*g + Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))/(g*(I*c*f + g + I*Sqrt[-(c^2*f^2) - g^2]*Cot[(Pi + (2*I)*ArcSinh[c*x])/4]))]))/Sqrt[-(c^2*f^2) - g^2]) - 18*c*f*g^2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 2*g^3*Sinh[3*ArcSinh[c*x]])/g^4))/(72*Sqrt[1 + c^2*x^2])","C",0
43,1,1899,1228,7.0786186,"\int (f+g x)^3 \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{b d^2 \sqrt{d \left(c^2 x^2+1\right)} \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) f^3}{8 c \sqrt{c^2 x^2+1}}-\frac{b d^2 \sqrt{d \left(c^2 x^2+1\right)} \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) f^3}{64 c \sqrt{c^2 x^2+1}}+\frac{b d^2 \sqrt{d \left(c^2 x^2+1\right)} \left(72 \sinh ^{-1}(c x)^2-36 \sinh \left(2 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-36 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+12 \sinh \left(6 \sinh ^{-1}(c x)\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) f^3}{2304 c \sqrt{c^2 x^2+1}}+\frac{3 b d^2 g \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) f^2}{c^2}+\frac{6 b d^2 g \left(-\frac{c^3 \sqrt{d \left(c^2 x^2+1\right)} \left(3 c^2 x^2+5\right) x^3}{75 \sqrt{c^2 x^2+1}}+\frac{2 c \sqrt{d \left(c^2 x^2+1\right)} \left(c^2 x^2+3\right) x}{45 \sqrt{c^2 x^2+1}}+\frac{\left(d \left(c^2 x^2+1\right)\right)^{3/2} \left(3 c^2 x^2-2\right) \sinh ^{-1}(c x)}{15 d}\right) f^2}{c^2}+\frac{3 b d^2 g \left(-\frac{c^5 \sqrt{d \left(c^2 x^2+1\right)} \left(5 c^2 x^2+7\right) x^5}{245 \sqrt{c^2 x^2+1}}+\frac{4 c^3 \sqrt{d \left(c^2 x^2+1\right)} \left(3 c^2 x^2+5\right) x^3}{525 \sqrt{c^2 x^2+1}}-\frac{8 c \sqrt{d \left(c^2 x^2+1\right)} \left(c^2 x^2+3\right) x}{315 \sqrt{c^2 x^2+1}}+\frac{\left(d \left(c^2 x^2+1\right)\right)^{3/2} \left(15 c^4 x^4-12 c^2 x^2+8\right) \sinh ^{-1}(c x)}{105 d}\right) f^2}{c^2}+\frac{5 a d^{5/2} \left(8 c^2 f^2-3 g^2\right) \log \left(c d x+\sqrt{d} \sqrt{d \left(c^2 x^2+1\right)}\right) f}{128 c^3}-\frac{3 b d^2 g^2 \sqrt{d \left(c^2 x^2+1\right)} \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) f}{128 c^3 \sqrt{c^2 x^2+1}}+\frac{b d^2 g^2 \sqrt{d \left(c^2 x^2+1\right)} \left(72 \sinh ^{-1}(c x)^2-36 \sinh \left(2 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-36 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+12 \sinh \left(6 \sinh ^{-1}(c x)\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) f}{384 c^3 \sqrt{c^2 x^2+1}}-\frac{b d^2 g^2 \sqrt{d \left(c^2 x^2+1\right)} \left(1440 \sinh ^{-1}(c x)^2-1152 \sinh \left(2 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-576 \sinh \left(4 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+384 \sinh \left(6 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)-72 \sinh \left(8 \sinh ^{-1}(c x)\right) \sinh ^{-1}(c x)+576 \cosh \left(2 \sinh ^{-1}(c x)\right)+144 \cosh \left(4 \sinh ^{-1}(c x)\right)-64 \cosh \left(6 \sinh ^{-1}(c x)\right)+9 \cosh \left(8 \sinh ^{-1}(c x)\right)\right) f}{24576 c^3 \sqrt{c^2 x^2+1}}+\sqrt{d \left(c^2 x^2+1\right)} \left(\frac{1}{9} a c^4 d^2 g^3 x^8+\frac{3}{8} a c^4 d^2 f g^2 x^7+\frac{1}{63} a c^2 d^2 g \left(27 c^2 f^2+19 g^2\right) x^6+\frac{1}{48} a c^2 d^2 f \left(8 c^2 f^2+51 g^2\right) x^5+\frac{1}{21} a d^2 g \left(27 c^2 f^2+5 g^2\right) x^4+\frac{1}{192} a d^2 f \left(104 c^2 f^2+177 g^2\right) x^3+\frac{a d^2 g \left(81 c^2 f^2+g^2\right) x^2}{63 c^2}+\frac{a d^2 f \left(88 c^2 f^2+15 g^2\right) x}{128 c^2}-\frac{a d^2 g \left(2 g^2-27 c^2 f^2\right)}{63 c^4}\right)+\frac{b d^2 g^3 \left(-\frac{c^3 \sqrt{d \left(c^2 x^2+1\right)} \left(3 c^2 x^2+5\right) x^3}{75 \sqrt{c^2 x^2+1}}+\frac{2 c \sqrt{d \left(c^2 x^2+1\right)} \left(c^2 x^2+3\right) x}{45 \sqrt{c^2 x^2+1}}+\frac{\left(d \left(c^2 x^2+1\right)\right)^{3/2} \left(3 c^2 x^2-2\right) \sinh ^{-1}(c x)}{15 d}\right)}{c^4}+\frac{2 b d^2 g^3 \left(-\frac{c^5 \sqrt{d \left(c^2 x^2+1\right)} \left(5 c^2 x^2+7\right) x^5}{245 \sqrt{c^2 x^2+1}}+\frac{4 c^3 \sqrt{d \left(c^2 x^2+1\right)} \left(3 c^2 x^2+5\right) x^3}{525 \sqrt{c^2 x^2+1}}-\frac{8 c \sqrt{d \left(c^2 x^2+1\right)} \left(c^2 x^2+3\right) x}{315 \sqrt{c^2 x^2+1}}+\frac{\left(d \left(c^2 x^2+1\right)\right)^{3/2} \left(15 c^4 x^4-12 c^2 x^2+8\right) \sinh ^{-1}(c x)}{105 d}\right)}{c^4}+\frac{b d^2 g^3 \left(-\frac{c^7 \sqrt{d \left(c^2 x^2+1\right)} \left(7 c^2 x^2+9\right) x^7}{567 \sqrt{c^2 x^2+1}}+\frac{2 c^5 \sqrt{d \left(c^2 x^2+1\right)} \left(5 c^2 x^2+7\right) x^5}{735 \sqrt{c^2 x^2+1}}-\frac{8 c^3 \sqrt{d \left(c^2 x^2+1\right)} \left(3 c^2 x^2+5\right) x^3}{1575 \sqrt{c^2 x^2+1}}+\frac{16 c \sqrt{d \left(c^2 x^2+1\right)} \left(c^2 x^2+3\right) x}{945 \sqrt{c^2 x^2+1}}+\frac{\left(d \left(c^2 x^2+1\right)\right)^{3/2} \left(35 c^6 x^6-30 c^4 x^4+24 c^2 x^2-16\right) \sinh ^{-1}(c x)}{315 d}\right)}{c^4}","-\frac{b c^5 d^2 g^3 \sqrt{c^2 d x^2+d} x^9}{81 \sqrt{c^2 x^2+1}}-\frac{3 b c^5 d^2 f g^2 \sqrt{c^2 d x^2+d} x^8}{64 \sqrt{c^2 x^2+1}}-\frac{19 b c^3 d^2 g^3 \sqrt{c^2 d x^2+d} x^7}{441 \sqrt{c^2 x^2+1}}-\frac{3 b c^5 d^2 f^2 g \sqrt{c^2 d x^2+d} x^7}{49 \sqrt{c^2 x^2+1}}-\frac{17 b c^3 d^2 f g^2 \sqrt{c^2 d x^2+d} x^6}{96 \sqrt{c^2 x^2+1}}-\frac{b c d^2 g^3 \sqrt{c^2 d x^2+d} x^5}{21 \sqrt{c^2 x^2+1}}-\frac{9 b c^3 d^2 f^2 g \sqrt{c^2 d x^2+d} x^5}{35 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 d^2 f^3 \sqrt{c^2 d x^2+d} x^4}{96 \sqrt{c^2 x^2+1}}-\frac{59 b c d^2 f g^2 \sqrt{c^2 d x^2+d} x^4}{256 \sqrt{c^2 x^2+1}}+\frac{15}{64} d^2 f g^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^3+\frac{3}{8} d^2 f g^2 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^3+\frac{5}{16} d^2 f g^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^3-\frac{b d^2 g^3 \sqrt{c^2 d x^2+d} x^3}{189 c \sqrt{c^2 x^2+1}}-\frac{3 b c d^2 f^2 g \sqrt{c^2 d x^2+d} x^3}{7 \sqrt{c^2 x^2+1}}-\frac{25 b c d^2 f^3 \sqrt{c^2 d x^2+d} x^2}{96 \sqrt{c^2 x^2+1}}-\frac{15 b d^2 f g^2 \sqrt{c^2 d x^2+d} x^2}{256 c \sqrt{c^2 x^2+1}}+\frac{5}{16} d^2 f^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x+\frac{15 d^2 f g^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x}{128 c^2}+\frac{1}{6} d^2 f^3 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x+\frac{5}{24} d^2 f^3 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x+\frac{2 b d^2 g^3 \sqrt{c^2 d x^2+d} x}{63 c^3 \sqrt{c^2 x^2+1}}-\frac{3 b d^2 f^2 g \sqrt{c^2 d x^2+d} x}{7 c \sqrt{c^2 x^2+1}}+\frac{5 d^2 f^3 \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{15 d^2 f g^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{d^2 g^3 \left(c^2 x^2+1\right)^4 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{9 c^4}-\frac{d^2 g^3 \left(c^2 x^2+1\right)^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{7 c^4}+\frac{3 d^2 f^2 g \left(c^2 x^2+1\right)^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{7 c^2}-\frac{b d^2 f^3 \left(c^2 x^2+1\right)^{5/2} \sqrt{c^2 d x^2+d}}{36 c}",1,"Sqrt[d*(1 + c^2*x^2)]*(-1/63*(a*d^2*g*(-27*c^2*f^2 + 2*g^2))/c^4 + (a*d^2*f*(88*c^2*f^2 + 15*g^2)*x)/(128*c^2) + (a*d^2*g*(81*c^2*f^2 + g^2)*x^2)/(63*c^2) + (a*d^2*f*(104*c^2*f^2 + 177*g^2)*x^3)/192 + (a*d^2*g*(27*c^2*f^2 + 5*g^2)*x^4)/21 + (a*c^2*d^2*f*(8*c^2*f^2 + 51*g^2)*x^5)/48 + (a*c^2*d^2*g*(27*c^2*f^2 + 19*g^2)*x^6)/63 + (3*a*c^4*d^2*f*g^2*x^7)/8 + (a*c^4*d^2*g^3*x^8)/9) + (3*b*d^2*f^2*g*(-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))/c^2 + (6*b*d^2*f^2*g*((2*c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/(45*Sqrt[1 + c^2*x^2]) - (c^3*x^3*Sqrt[d*(1 + c^2*x^2)]*(5 + 3*c^2*x^2))/(75*Sqrt[1 + c^2*x^2]) + ((d*(1 + c^2*x^2))^(3/2)*(-2 + 3*c^2*x^2)*ArcSinh[c*x])/(15*d)))/c^2 + (b*d^2*g^3*((2*c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/(45*Sqrt[1 + c^2*x^2]) - (c^3*x^3*Sqrt[d*(1 + c^2*x^2)]*(5 + 3*c^2*x^2))/(75*Sqrt[1 + c^2*x^2]) + ((d*(1 + c^2*x^2))^(3/2)*(-2 + 3*c^2*x^2)*ArcSinh[c*x])/(15*d)))/c^4 + (3*b*d^2*f^2*g*((-8*c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/(315*Sqrt[1 + c^2*x^2]) + (4*c^3*x^3*Sqrt[d*(1 + c^2*x^2)]*(5 + 3*c^2*x^2))/(525*Sqrt[1 + c^2*x^2]) - (c^5*x^5*Sqrt[d*(1 + c^2*x^2)]*(7 + 5*c^2*x^2))/(245*Sqrt[1 + c^2*x^2]) + ((d*(1 + c^2*x^2))^(3/2)*(8 - 12*c^2*x^2 + 15*c^4*x^4)*ArcSinh[c*x])/(105*d)))/c^2 + (2*b*d^2*g^3*((-8*c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/(315*Sqrt[1 + c^2*x^2]) + (4*c^3*x^3*Sqrt[d*(1 + c^2*x^2)]*(5 + 3*c^2*x^2))/(525*Sqrt[1 + c^2*x^2]) - (c^5*x^5*Sqrt[d*(1 + c^2*x^2)]*(7 + 5*c^2*x^2))/(245*Sqrt[1 + c^2*x^2]) + ((d*(1 + c^2*x^2))^(3/2)*(8 - 12*c^2*x^2 + 15*c^4*x^4)*ArcSinh[c*x])/(105*d)))/c^4 + (b*d^2*g^3*((16*c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/(945*Sqrt[1 + c^2*x^2]) - (8*c^3*x^3*Sqrt[d*(1 + c^2*x^2)]*(5 + 3*c^2*x^2))/(1575*Sqrt[1 + c^2*x^2]) + (2*c^5*x^5*Sqrt[d*(1 + c^2*x^2)]*(7 + 5*c^2*x^2))/(735*Sqrt[1 + c^2*x^2]) - (c^7*x^7*Sqrt[d*(1 + c^2*x^2)]*(9 + 7*c^2*x^2))/(567*Sqrt[1 + c^2*x^2]) + ((d*(1 + c^2*x^2))^(3/2)*(-16 + 24*c^2*x^2 - 30*c^4*x^4 + 35*c^6*x^6)*ArcSinh[c*x])/(315*d)))/c^4 + (5*a*d^(5/2)*f*(8*c^2*f^2 - 3*g^2)*Log[c*d*x + Sqrt[d]*Sqrt[d*(1 + c^2*x^2)]])/(128*c^3) + (b*d^2*f^3*Sqrt[d*(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[1 + c^2*x^2]) - (b*d^2*f^3*Sqrt[d*(1 + c^2*x^2)]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(64*c*Sqrt[1 + c^2*x^2]) - (3*b*d^2*f*g^2*Sqrt[d*(1 + c^2*x^2)]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(128*c^3*Sqrt[1 + c^2*x^2]) + (b*d^2*f^3*Sqrt[d*(1 + c^2*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]] - 36*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 36*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]] + 12*ArcSinh[c*x]*Sinh[6*ArcSinh[c*x]]))/(2304*c*Sqrt[1 + c^2*x^2]) + (b*d^2*f*g^2*Sqrt[d*(1 + c^2*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]] - 36*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 36*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]] + 12*ArcSinh[c*x]*Sinh[6*ArcSinh[c*x]]))/(384*c^3*Sqrt[1 + c^2*x^2]) - (b*d^2*f*g^2*Sqrt[d*(1 + c^2*x^2)]*(1440*ArcSinh[c*x]^2 + 576*Cosh[2*ArcSinh[c*x]] + 144*Cosh[4*ArcSinh[c*x]] - 64*Cosh[6*ArcSinh[c*x]] + 9*Cosh[8*ArcSinh[c*x]] - 1152*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]] - 576*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]] + 384*ArcSinh[c*x]*Sinh[6*ArcSinh[c*x]] - 72*ArcSinh[c*x]*Sinh[8*ArcSinh[c*x]]))/(24576*c^3*Sqrt[1 + c^2*x^2])","A",1
44,1,1047,901,2.7831502,"\int (f+g x)^2 \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)^2*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{d^2 \left(-737280 b f g x^7 \sqrt{c^2 d x^2+d} c^8+2257920 a g^2 x^7 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^7+5160960 a f g x^6 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^7+3010560 a f^2 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^7-3096576 b f g x^5 \sqrt{c^2 d x^2+d} c^6+6397440 a g^2 x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^5+15482880 a f g x^4 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^5+9784320 a f^2 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^5-5160960 b f g x^3 \sqrt{c^2 d x^2+d} c^4+5550720 a g^2 x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^3+15482880 a f g x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^3+12418560 a f^2 x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c^3-211680 b f^2 \sqrt{c^2 d x^2+d} \cosh \left(4 \sinh ^{-1}(c x)\right) c^2-15680 b f^2 \sqrt{c^2 d x^2+d} \cosh \left(6 \sinh ^{-1}(c x)\right) c^2+5644800 a \sqrt{d} f^2 \sqrt{c^2 x^2+1} \log \left(c d x+\sqrt{d} \sqrt{c^2 d x^2+d}\right) c^2-5160960 b f g x \sqrt{c^2 d x^2+d} c^2+5160960 a f g \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c+705600 a g^2 x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} c+352800 b \left(8 c^2 f^2-g^2\right) \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2-141120 b \left(15 c^2 f^2-g^2\right) \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-35280 b g^2 \sqrt{c^2 d x^2+d} \cosh \left(4 \sinh ^{-1}(c x)\right)-15680 b g^2 \sqrt{c^2 d x^2+d} \cosh \left(6 \sinh ^{-1}(c x)\right)-2205 b g^2 \sqrt{c^2 d x^2+d} \cosh \left(8 \sinh ^{-1}(c x)\right)-705600 a \sqrt{d} g^2 \sqrt{c^2 x^2+1} \log \left(c d x+\sqrt{d} \sqrt{c^2 d x^2+d}\right)+840 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(6144 f g x^6 \sqrt{c^2 x^2+1} c^7+18432 f g x^4 \sqrt{c^2 x^2+1} c^5+18432 f g x^2 \sqrt{c^2 x^2+1} c^3+112 f^2 \sinh \left(6 \sinh ^{-1}(c x)\right) c^2+6144 f g \sqrt{c^2 x^2+1} c+336 \left(15 c^2 f^2-g^2\right) \sinh \left(2 \sinh ^{-1}(c x)\right)+168 \left(6 c^2 f^2+g^2\right) \sinh \left(4 \sinh ^{-1}(c x)\right)+112 g^2 \sinh \left(6 \sinh ^{-1}(c x)\right)+21 g^2 \sinh \left(8 \sinh ^{-1}(c x)\right)\right)\right)}{18063360 c^3 \sqrt{c^2 x^2+1}}","-\frac{b c^5 d^2 g^2 \sqrt{c^2 d x^2+d} x^8}{64 \sqrt{c^2 x^2+1}}-\frac{2 b c^5 d^2 f g \sqrt{c^2 d x^2+d} x^7}{49 \sqrt{c^2 x^2+1}}-\frac{17 b c^3 d^2 g^2 \sqrt{c^2 d x^2+d} x^6}{288 \sqrt{c^2 x^2+1}}-\frac{6 b c^3 d^2 f g \sqrt{c^2 d x^2+d} x^5}{35 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 d^2 f^2 \sqrt{c^2 d x^2+d} x^4}{96 \sqrt{c^2 x^2+1}}-\frac{59 b c d^2 g^2 \sqrt{c^2 d x^2+d} x^4}{768 \sqrt{c^2 x^2+1}}+\frac{5}{64} d^2 g^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^3+\frac{1}{8} d^2 g^2 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^3+\frac{5}{48} d^2 g^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x^3-\frac{2 b c d^2 f g \sqrt{c^2 d x^2+d} x^3}{7 \sqrt{c^2 x^2+1}}-\frac{25 b c d^2 f^2 \sqrt{c^2 d x^2+d} x^2}{96 \sqrt{c^2 x^2+1}}-\frac{5 b d^2 g^2 \sqrt{c^2 d x^2+d} x^2}{256 c \sqrt{c^2 x^2+1}}+\frac{5}{16} d^2 f^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x+\frac{5 d^2 g^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x}{128 c^2}+\frac{1}{6} d^2 f^2 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x+\frac{5}{24} d^2 f^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) x-\frac{2 b d^2 f g \sqrt{c^2 d x^2+d} x}{7 c \sqrt{c^2 x^2+1}}+\frac{5 d^2 f^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{5 d^2 g^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{2 d^2 f g \left(c^2 x^2+1\right)^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{7 c^2}-\frac{b d^2 f^2 \left(c^2 x^2+1\right)^{5/2} \sqrt{c^2 d x^2+d}}{36 c}",1,"(d^2*(-5160960*b*c^2*f*g*x*Sqrt[d + c^2*d*x^2] - 5160960*b*c^4*f*g*x^3*Sqrt[d + c^2*d*x^2] - 3096576*b*c^6*f*g*x^5*Sqrt[d + c^2*d*x^2] - 737280*b*c^8*f*g*x^7*Sqrt[d + c^2*d*x^2] + 5160960*a*c*f*g*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 12418560*a*c^3*f^2*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 705600*a*c*g^2*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 15482880*a*c^3*f*g*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 9784320*a*c^5*f^2*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 5550720*a*c^3*g^2*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 15482880*a*c^5*f*g*x^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 3010560*a*c^7*f^2*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 6397440*a*c^5*g^2*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 5160960*a*c^7*f*g*x^6*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 2257920*a*c^7*g^2*x^7*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 352800*b*(8*c^2*f^2 - g^2)*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2 - 141120*b*(15*c^2*f^2 - g^2)*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 211680*b*c^2*f^2*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 35280*b*g^2*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 15680*b*c^2*f^2*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 15680*b*g^2*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 2205*b*g^2*Sqrt[d + c^2*d*x^2]*Cosh[8*ArcSinh[c*x]] + 5644800*a*c^2*Sqrt[d]*f^2*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 705600*a*Sqrt[d]*g^2*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 840*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(6144*c*f*g*Sqrt[1 + c^2*x^2] + 18432*c^3*f*g*x^2*Sqrt[1 + c^2*x^2] + 18432*c^5*f*g*x^4*Sqrt[1 + c^2*x^2] + 6144*c^7*f*g*x^6*Sqrt[1 + c^2*x^2] + 336*(15*c^2*f^2 - g^2)*Sinh[2*ArcSinh[c*x]] + 168*(6*c^2*f^2 + g^2)*Sinh[4*ArcSinh[c*x]] + 112*c^2*f^2*Sinh[6*ArcSinh[c*x]] + 112*g^2*Sinh[6*ArcSinh[c*x]] + 21*g^2*Sinh[8*ArcSinh[c*x]])))/(18063360*c^3*Sqrt[1 + c^2*x^2])","A",1
45,1,656,494,1.3444942,"\int (f+g x) \left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right) \, dx","Integrate[(f + g*x)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]","\frac{d^2 \left(388080 a c^2 f x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+176400 a c \sqrt{d} f \sqrt{c^2 x^2+1} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+241920 a c^2 g x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+80640 a g \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+94080 a c^6 f x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+80640 a c^6 g x^6 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+305760 a c^4 f x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+241920 a c^4 g x^4 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+88200 b c f \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2-66150 b c f \sqrt{c^2 d x^2+d} \cosh \left(2 \sinh ^{-1}(c x)\right)-6615 b c f \sqrt{c^2 d x^2+d} \cosh \left(4 \sinh ^{-1}(c x)\right)-490 b c f \sqrt{c^2 d x^2+d} \cosh \left(6 \sinh ^{-1}(c x)\right)-80640 b c g x \sqrt{c^2 d x^2+d}-11520 b c^7 g x^7 \sqrt{c^2 d x^2+d}-48384 b c^5 g x^5 \sqrt{c^2 d x^2+d}-80640 b c^3 g x^3 \sqrt{c^2 d x^2+d}+420 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left(576 c^2 g x^2 \sqrt{c^2 x^2+1}+192 g \sqrt{c^2 x^2+1}+192 c^6 g x^6 \sqrt{c^2 x^2+1}+576 c^4 g x^4 \sqrt{c^2 x^2+1}+315 c f \sinh \left(2 \sinh ^{-1}(c x)\right)+63 c f \sinh \left(4 \sinh ^{-1}(c x)\right)+7 c f \sinh \left(6 \sinh ^{-1}(c x)\right)\right)\right)}{564480 c^2 \sqrt{c^2 x^2+1}}","\frac{1}{6} d^2 f x \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5}{16} d^2 f x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5}{24} d^2 f x \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)+\frac{5 d^2 f \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{d^2 g \left(c^2 x^2+1\right)^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{7 c^2}-\frac{25 b c d^2 f x^2 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{b d^2 f \left(c^2 x^2+1\right)^{5/2} \sqrt{c^2 d x^2+d}}{36 c}-\frac{b d^2 g x \sqrt{c^2 d x^2+d}}{7 c \sqrt{c^2 x^2+1}}-\frac{b c d^2 g x^3 \sqrt{c^2 d x^2+d}}{7 \sqrt{c^2 x^2+1}}-\frac{b c^5 d^2 g x^7 \sqrt{c^2 d x^2+d}}{49 \sqrt{c^2 x^2+1}}-\frac{5 b c^3 d^2 f x^4 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{3 b c^3 d^2 g x^5 \sqrt{c^2 d x^2+d}}{35 \sqrt{c^2 x^2+1}}",1,"(d^2*(-80640*b*c*g*x*Sqrt[d + c^2*d*x^2] - 80640*b*c^3*g*x^3*Sqrt[d + c^2*d*x^2] - 48384*b*c^5*g*x^5*Sqrt[d + c^2*d*x^2] - 11520*b*c^7*g*x^7*Sqrt[d + c^2*d*x^2] + 80640*a*g*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 388080*a*c^2*f*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 241920*a*c^2*g*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 305760*a*c^4*f*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 241920*a*c^4*g*x^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 94080*a*c^6*f*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 80640*a*c^6*g*x^6*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 88200*b*c*f*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2 - 66150*b*c*f*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 6615*b*c*f*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 490*b*c*f*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] + 176400*a*c*Sqrt[d]*f*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 420*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(192*g*Sqrt[1 + c^2*x^2] + 576*c^2*g*x^2*Sqrt[1 + c^2*x^2] + 576*c^4*g*x^4*Sqrt[1 + c^2*x^2] + 192*c^6*g*x^6*Sqrt[1 + c^2*x^2] + 315*c*f*Sinh[2*ArcSinh[c*x]] + 63*c*f*Sinh[4*ArcSinh[c*x]] + 7*c*f*Sinh[6*ArcSinh[c*x]])))/(564480*c^2*Sqrt[1 + c^2*x^2])","A",1
46,1,7163,1536,25.9371521,"\int \frac{\left(d+c^2 d x^2\right)^{5/2} \left(a+b \sinh ^{-1}(c x)\right)}{f+g x} \, dx","Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(f + g*x),x]","\text{Result too large to show}","-\frac{b d^2 x^5 \sqrt{c^2 d x^2+d} c^5}{25 g \sqrt{c^2 x^2+1}}+\frac{b d^2 f x^4 \sqrt{c^2 d x^2+d} c^5}{16 g^2 \sqrt{c^2 x^2+1}}-\frac{d^2 f x^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) c^4}{4 g^2}-\frac{b d^2 \left(c^2 f^2+2 g^2\right) x^3 \sqrt{c^2 d x^2+d} c^3}{9 g^3 \sqrt{c^2 x^2+1}}-\frac{b d^2 x^3 \sqrt{c^2 d x^2+d} c^3}{45 g \sqrt{c^2 x^2+1}}+\frac{b d^2 f \left(c^2 f^2+2 g^2\right) x^2 \sqrt{c^2 d x^2+d} c^3}{4 g^4 \sqrt{c^2 x^2+1}}+\frac{b d^2 f x^2 \sqrt{c^2 d x^2+d} c^3}{16 g^2 \sqrt{c^2 x^2+1}}-\frac{d^2 f \left(c^2 f^2+2 g^2\right) x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) c^2}{2 g^4}-\frac{d^2 f x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right) c^2}{8 g^2}-\frac{d^2 f \left(c^2 f^2+2 g^2\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2 c}{4 b g^4 \sqrt{c^2 x^2+1}}-\frac{d^2 \left(c^2 f^2+g^2\right)^2 x \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2 c}{2 b g^5 \sqrt{c^2 x^2+1}}+\frac{d^2 f \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2 c}{16 b g^2 \sqrt{c^2 x^2+1}}-\frac{b d^2 \left(c^2 f^2+g^2\right)^2 x \sqrt{c^2 d x^2+d} c}{g^5 \sqrt{c^2 x^2+1}}-\frac{b d^2 \left(c^2 f^2+2 g^2\right) x \sqrt{c^2 d x^2+d} c}{3 g^3 \sqrt{c^2 x^2+1}}+\frac{2 b d^2 x \sqrt{c^2 d x^2+d} c}{15 g \sqrt{c^2 x^2+1}}+\frac{b d^2 \left(c^2 f^2+g^2\right)^2 \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)}{g^5}+\frac{d^2 \left(c^2 x^2+1\right)^2 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{5 g}+\frac{d^2 \left(c^2 f^2+2 g^2\right) \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 g^3}-\frac{d^2 \left(c^2 x^2+1\right) \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)}{3 g}-\frac{a d^2 \left(c^2 f^2+g^2\right)^{5/2} \sqrt{c^2 d x^2+d} \tanh ^{-1}\left(\frac{g-c^2 f x}{\sqrt{c^2 f^2+g^2} \sqrt{c^2 x^2+1}}\right)}{g^6 \sqrt{c^2 x^2+1}}+\frac{b d^2 \left(c^2 f^2+g^2\right)^{5/2} \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left(\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}+1\right)}{g^6 \sqrt{c^2 x^2+1}}-\frac{b d^2 \left(c^2 f^2+g^2\right)^{5/2} \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \log \left(\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}+1\right)}{g^6 \sqrt{c^2 x^2+1}}+\frac{b d^2 \left(c^2 f^2+g^2\right)^{5/2} \sqrt{c^2 d x^2+d} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{g^6 \sqrt{c^2 x^2+1}}-\frac{b d^2 \left(c^2 f^2+g^2\right)^{5/2} \sqrt{c^2 d x^2+d} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{g^6 \sqrt{c^2 x^2+1}}+\frac{a d^2 \left(c^2 f^2+g^2\right)^2 \sqrt{c^2 d x^2+d}}{g^5}+\frac{d^2 \left(c^2 f^2+g^2\right)^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b g^4 (f+g x) c}-\frac{d^2 \left(c^2 f^2+g^2\right)^3 \sqrt{c^2 d x^2+d} \left(a+b \sinh ^{-1}(c x)\right)^2}{2 b g^6 (f+g x) \sqrt{c^2 x^2+1} c}",1,"Result too large to show","C",0
47,1,304,430,0.9507118,"\int \frac{(f+g x)^3 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[((f + g*x)^3*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]","\frac{4 \sqrt{d} g \left(3 a \left(c^2 x^2+1\right) \left(c^2 \left(18 f^2+9 f g x+2 g^2 x^2\right)-4 g^2\right)-2 b c x \sqrt{c^2 x^2+1} \left(c^2 \left(27 f^2+g^2 x^2\right)-6 g^2\right)\right)+36 a c f \sqrt{c^2 d x^2+d} \left(2 c^2 f^2-3 g^2\right) \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+12 b \sqrt{d} g \left(c^2 x^2+1\right) \sinh ^{-1}(c x) \left(c^2 \left(18 f^2+9 f g x+2 g^2 x^2\right)-4 g^2\right)+18 b c \sqrt{d} f \sqrt{c^2 x^2+1} \left(2 c^2 f^2-3 g^2\right) \sinh ^{-1}(c x)^2-27 b c \sqrt{d} f g^2 \sqrt{c^2 x^2+1} \cosh \left(2 \sinh ^{-1}(c x)\right)}{72 c^4 \sqrt{d} \sqrt{c^2 d x^2+d}}","\frac{f^3 \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}}+\frac{3 f^2 g \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^2 \sqrt{c^2 d x^2+d}}+\frac{3 f g^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{2 c^2 \sqrt{c^2 d x^2+d}}+\frac{g^3 x^2 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c^2 \sqrt{c^2 d x^2+d}}-\frac{2 g^3 \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{3 c^4 \sqrt{c^2 d x^2+d}}-\frac{3 f g^2 \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{3 b f^2 g x \sqrt{c^2 x^2+1}}{c \sqrt{c^2 d x^2+d}}-\frac{3 b f g^2 x^2 \sqrt{c^2 x^2+1}}{4 c \sqrt{c^2 d x^2+d}}-\frac{b g^3 x^3 \sqrt{c^2 x^2+1}}{9 c \sqrt{c^2 d x^2+d}}+\frac{2 b g^3 x \sqrt{c^2 x^2+1}}{3 c^3 \sqrt{c^2 d x^2+d}}",1,"(4*Sqrt[d]*g*(-2*b*c*x*Sqrt[1 + c^2*x^2]*(-6*g^2 + c^2*(27*f^2 + g^2*x^2)) + 3*a*(1 + c^2*x^2)*(-4*g^2 + c^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2))) + 12*b*Sqrt[d]*g*(1 + c^2*x^2)*(-4*g^2 + c^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2))*ArcSinh[c*x] + 18*b*c*Sqrt[d]*f*(2*c^2*f^2 - 3*g^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 - 27*b*c*Sqrt[d]*f*g^2*Sqrt[1 + c^2*x^2]*Cosh[2*ArcSinh[c*x]] + 36*a*c*f*(2*c^2*f^2 - 3*g^2)*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(72*c^4*Sqrt[d]*Sqrt[d + c^2*d*x^2])","A",1
48,1,233,258,0.6083474,"\int \frac{(f+g x)^2 \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[((f + g*x)^2*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]","\frac{4 c \sqrt{d} g \left(a \left(c^2 x^2+1\right) (4 f+g x)-4 b c f x \sqrt{c^2 x^2+1}\right)+4 a \sqrt{c^2 d x^2+d} \left(2 c^2 f^2-g^2\right) \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+2 b \sqrt{d} \sqrt{c^2 x^2+1} \left(2 c^2 f^2-g^2\right) \sinh ^{-1}(c x)^2+4 b c \sqrt{d} g \left(c^2 x^2+1\right) \sinh ^{-1}(c x) (4 f+g x)-b \sqrt{d} g^2 \sqrt{c^2 x^2+1} \cosh \left(2 \sinh ^{-1}(c x)\right)}{8 c^3 \sqrt{d} \sqrt{c^2 d x^2+d}}","\frac{f^2 \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}}+\frac{2 f g \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^2 \sqrt{c^2 d x^2+d}}+\frac{g^2 x \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{2 c^2 \sqrt{c^2 d x^2+d}}-\frac{g^2 \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{2 b f g x \sqrt{c^2 x^2+1}}{c \sqrt{c^2 d x^2+d}}-\frac{b g^2 x^2 \sqrt{c^2 x^2+1}}{4 c \sqrt{c^2 d x^2+d}}",1,"(4*c*Sqrt[d]*g*(-4*b*c*f*x*Sqrt[1 + c^2*x^2] + a*(4*f + g*x)*(1 + c^2*x^2)) + 4*b*c*Sqrt[d]*g*(4*f + g*x)*(1 + c^2*x^2)*ArcSinh[c*x] + 2*b*Sqrt[d]*(2*c^2*f^2 - g^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 - b*Sqrt[d]*g^2*Sqrt[1 + c^2*x^2]*Cosh[2*ArcSinh[c*x]] + 4*a*(2*c^2*f^2 - g^2)*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(8*c^3*Sqrt[d]*Sqrt[d + c^2*d*x^2])","A",1
49,1,158,120,0.2640453,"\int \frac{(f+g x) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{d+c^2 d x^2}} \, dx","Integrate[((f + g*x)*(a + b*ArcSinh[c*x]))/Sqrt[d + c^2*d*x^2],x]","\frac{2 \sqrt{d} g \left(a c^2 x^2+a-b c x \sqrt{c^2 x^2+1}\right)+2 a c f \sqrt{c^2 d x^2+d} \log \left(\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right)+b c \sqrt{d} f \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)^2+2 b \sqrt{d} g \left(c^2 x^2+1\right) \sinh ^{-1}(c x)}{2 c^2 \sqrt{d} \sqrt{c^2 d x^2+d}}","\frac{f \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}}+\frac{g \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{c^2 \sqrt{c^2 d x^2+d}}-\frac{b g x \sqrt{c^2 x^2+1}}{c \sqrt{c^2 d x^2+d}}",1,"(2*Sqrt[d]*g*(a + a*c^2*x^2 - b*c*x*Sqrt[1 + c^2*x^2]) + 2*b*Sqrt[d]*g*(1 + c^2*x^2)*ArcSinh[c*x] + b*c*Sqrt[d]*f*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 + 2*a*c*f*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]])/(2*c^2*Sqrt[d]*Sqrt[d + c^2*d*x^2])","A",1
50,1,48,47,0.0364009,"\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
51,1,256,325,0.6253139,"\int \frac{a+b \sinh ^{-1}(c x)}{(f+g x) \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/((f + g*x)*Sqrt[d + c^2*d*x^2]),x]","\frac{-\frac{a \log \left(\sqrt{d} \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2}+d \left(g-c^2 f x\right)\right)}{\sqrt{d}}+\frac{a \log (f+g x)}{\sqrt{d}}+\frac{b \sqrt{c^2 x^2+1} \left(\text{Li}_2\left(\frac{e^{\sinh ^{-1}(c x)} g}{\sqrt{c^2 f^2+g^2}-c f}\right)-\text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)+\sinh ^{-1}(c x) \left(\log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)-\log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)\right)\right)}{\sqrt{c^2 d x^2+d}}}{\sqrt{c^2 f^2+g^2}}","\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)}{\sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2}}-\frac{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)}{\sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2}}+\frac{b \sqrt{c^2 x^2+1} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{\sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2}}-\frac{b \sqrt{c^2 x^2+1} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{\sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2}}",1,"((a*Log[f + g*x])/Sqrt[d] - (a*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]])/Sqrt[d] + (b*Sqrt[1 + c^2*x^2]*(ArcSinh[c*x]*(Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] - Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])]) + PolyLog[2, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] - PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]))/Sqrt[d + c^2*d*x^2])/Sqrt[c^2*f^2 + g^2]","A",0
52,1,448,444,2.1716327,"\int \frac{a+b \sinh ^{-1}(c x)}{(f+g x)^2 \sqrt{d+c^2 d x^2}} \, dx","Integrate[(a + b*ArcSinh[c*x])/((f + g*x)^2*Sqrt[d + c^2*d*x^2]),x]","\frac{-a g \left(c^2 d x^2+d\right) \sqrt{c^2 f^2+g^2}-a c^2 \sqrt{d} f \sqrt{c^2 d x^2+d} (f+g x) \log \left(\sqrt{d} \sqrt{c^2 d x^2+d} \sqrt{c^2 f^2+g^2}+d \left(g-c^2 f x\right)\right)+a c^2 \sqrt{d} f \sqrt{c^2 d x^2+d} (f+g x) \log (f+g x)-b d \sqrt{c^2 x^2+1} \left(-c^2 f (f+g x) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(c x)} g}{\sqrt{c^2 f^2+g^2}-c f}\right)+c^2 f (f+g x) \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)+g \sqrt{c^2 x^2+1} \sqrt{c^2 f^2+g^2} \sinh ^{-1}(c x)-c \sqrt{c^2 f^2+g^2} (f+g x) \log (c (f+g x))+c^2 (-f) \sinh ^{-1}(c x) (f+g x) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)+c^2 f \sinh ^{-1}(c x) (f+g x) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)\right)}{d \sqrt{c^2 d x^2+d} \left(c^2 f^2+g^2\right)^{3/2} (f+g x)}","-\frac{g \left(c^2 x^2+1\right) \left(a+b \sinh ^{-1}(c x)\right)}{\sqrt{c^2 d x^2+d} \left(c^2 f^2+g^2\right) (f+g x)}+\frac{c^2 f \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)}{\sqrt{c^2 d x^2+d} \left(c^2 f^2+g^2\right)^{3/2}}-\frac{c^2 f \sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)}{\sqrt{c^2 d x^2+d} \left(c^2 f^2+g^2\right)^{3/2}}+\frac{b c^2 f \sqrt{c^2 x^2+1} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{\sqrt{c^2 d x^2+d} \left(c^2 f^2+g^2\right)^{3/2}}-\frac{b c^2 f \sqrt{c^2 x^2+1} \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{\sqrt{c^2 d x^2+d} \left(c^2 f^2+g^2\right)^{3/2}}+\frac{b c \sqrt{c^2 x^2+1} \log (f+g x)}{\sqrt{c^2 d x^2+d} \left(c^2 f^2+g^2\right)}",1,"(-(a*g*Sqrt[c^2*f^2 + g^2]*(d + c^2*d*x^2)) + a*c^2*Sqrt[d]*f*(f + g*x)*Sqrt[d + c^2*d*x^2]*Log[f + g*x] - a*c^2*Sqrt[d]*f*(f + g*x)*Sqrt[d + c^2*d*x^2]*Log[d*(g - c^2*f*x) + Sqrt[d]*Sqrt[c^2*f^2 + g^2]*Sqrt[d + c^2*d*x^2]] - b*d*Sqrt[1 + c^2*x^2]*(g*Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - c^2*f*(f + g*x)*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] + c^2*f*(f + g*x)*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])] - c*Sqrt[c^2*f^2 + g^2]*(f + g*x)*Log[c*(f + g*x)] - c^2*f*(f + g*x)*PolyLog[2, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] + c^2*f*(f + g*x)*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]))/(d*(c^2*f^2 + g^2)^(3/2)*(f + g*x)*Sqrt[d + c^2*d*x^2])","A",0
53,0,0,37,0.1381136,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^n \log \left(h (f+g x)^m\right)}{\sqrt{1+c^2 x^2}} \, dx","Integrate[((a + b*ArcSinh[c*x])^n*Log[h*(f + g*x)^m])/Sqrt[1 + c^2*x^2],x]","\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^n \log \left(h (f+g x)^m\right)}{\sqrt{1+c^2 x^2}} \, dx","\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c x)\right)^n \log \left(h (f+g x)^m\right)}{\sqrt{c^2 x^2+1}},x\right)",0,"Integrate[((a + b*ArcSinh[c*x])^n*Log[h*(f + g*x)^m])/Sqrt[1 + c^2*x^2], x]","A",-1
54,1,397,438,0.2335524,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(h (f+g x)^m\right)}{\sqrt{1+c^2 x^2}} \, dx","Integrate[((a + b*ArcSinh[c*x])^2*Log[h*(f + g*x)^m])/Sqrt[1 + c^2*x^2],x]","-\frac{3 b m \left(-2 b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_3\left(\frac{e^{\sinh ^{-1}(c x)} g}{\sqrt{c^2 f^2+g^2}-c f}\right)+\left(a+b \sinh ^{-1}(c x)\right)^2 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(c x)} g}{\sqrt{c^2 f^2+g^2}-c f}\right)+2 b^2 \text{Li}_4\left(\frac{e^{\sinh ^{-1}(c x)} g}{\sqrt{c^2 f^2+g^2}-c f}\right)\right)+3 b m \left(-2 b \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_3\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)+\left(a+b \sinh ^{-1}(c x)\right)^2 \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)+2 b^2 \text{Li}_4\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)\right)+m \left(a+b \sinh ^{-1}(c x)\right)^3 \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)+m \left(a+b \sinh ^{-1}(c x)\right)^3 \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)-\left(a+b \sinh ^{-1}(c x)\right)^3 \log \left(h (f+g x)^m\right)-\frac{m \left(a+b \sinh ^{-1}(c x)\right)^4}{4 b}}{3 b c}","\frac{m \left(a+b \sinh ^{-1}(c x)\right)^4}{12 b^2 c}-\frac{m \left(a+b \sinh ^{-1}(c x)\right)^2 \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{c}-\frac{m \left(a+b \sinh ^{-1}(c x)\right)^2 \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{c}+\frac{2 b m \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_3\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{c}+\frac{2 b m \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_3\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{c}-\frac{m \left(a+b \sinh ^{-1}(c x)\right)^3 \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)}{3 b c}-\frac{m \left(a+b \sinh ^{-1}(c x)\right)^3 \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)}{3 b c}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^3 \log \left(h (f+g x)^m\right)}{3 b c}-\frac{2 b^2 m \text{Li}_4\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{c}-\frac{2 b^2 m \text{Li}_4\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{c}",1,"-1/3*(-1/4*(m*(a + b*ArcSinh[c*x])^4)/b + m*(a + b*ArcSinh[c*x])^3*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] + m*(a + b*ArcSinh[c*x])^3*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])] - (a + b*ArcSinh[c*x])^3*Log[h*(f + g*x)^m] + 3*b*m*((a + b*ArcSinh[c*x])^2*PolyLog[2, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] - 2*b*(a + b*ArcSinh[c*x])*PolyLog[3, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] + 2*b^2*PolyLog[4, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])]) + 3*b*m*((a + b*ArcSinh[c*x])^2*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))] - 2*b*(a + b*ArcSinh[c*x])*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))] + 2*b^2*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]))/(b*c)","A",1
55,1,304,332,0.229465,"\int \frac{\left(a+b \sinh ^{-1}(c x)\right) \log \left(h (f+g x)^m\right)}{\sqrt{1+c^2 x^2}} \, dx","Integrate[((a + b*ArcSinh[c*x])*Log[h*(f + g*x)^m])/Sqrt[1 + c^2*x^2],x]","\frac{2 b m \left(b \text{Li}_3\left(\frac{e^{\sinh ^{-1}(c x)} g}{\sqrt{c^2 f^2+g^2}-c f}\right)-\left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(c x)} g}{\sqrt{c^2 f^2+g^2}-c f}\right)\right)+2 b m \left(b \text{Li}_3\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)-\left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)\right)-m \left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)-m \left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)+\left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(h (f+g x)^m\right)+\frac{m \left(a+b \sinh ^{-1}(c x)\right)^3}{3 b}}{2 b c}","\frac{m \left(a+b \sinh ^{-1}(c x)\right)^3}{6 b^2 c}-\frac{m \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{c}-\frac{m \left(a+b \sinh ^{-1}(c x)\right) \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{c}-\frac{m \left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)}{2 b c}-\frac{m \left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)}{2 b c}+\frac{\left(a+b \sinh ^{-1}(c x)\right)^2 \log \left(h (f+g x)^m\right)}{2 b c}+\frac{b m \text{Li}_3\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{c}+\frac{b m \text{Li}_3\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{c}",1,"((m*(a + b*ArcSinh[c*x])^3)/(3*b) - m*(a + b*ArcSinh[c*x])^2*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] - m*(a + b*ArcSinh[c*x])^2*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])] + (a + b*ArcSinh[c*x])^2*Log[h*(f + g*x)^m] + 2*b*m*(-((a + b*ArcSinh[c*x])*PolyLog[2, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])]) + b*PolyLog[3, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])]) + 2*b*m*(-((a + b*ArcSinh[c*x])*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]) + b*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]))/(2*b*c)","A",1
56,1,206,197,0.0183282,"\int \frac{\log \left(h (f+g x)^m\right)}{\sqrt{1+c^2 x^2}} \, dx","Integrate[Log[h*(f + g*x)^m]/Sqrt[1 + c^2*x^2],x]","-\frac{m \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{c}-\frac{m \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{c}-\frac{m \sinh ^{-1}(c x) \log \left(\frac{c g e^{\sinh ^{-1}(c x)}}{c^2 f-c \sqrt{c^2 f^2+g^2}}+1\right)}{c}-\frac{m \sinh ^{-1}(c x) \log \left(\frac{c g e^{\sinh ^{-1}(c x)}}{c \sqrt{c^2 f^2+g^2}+c^2 f}+1\right)}{c}+\frac{\sinh ^{-1}(c x) \log \left(h (f+g x)^m\right)}{c}+\frac{m \sinh ^{-1}(c x)^2}{2 c}","-\frac{m \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2+g^2}}\right)}{c}-\frac{m \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2+g^2}}\right)}{c}-\frac{m \sinh ^{-1}(c x) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{c f-\sqrt{c^2 f^2+g^2}}+1\right)}{c}-\frac{m \sinh ^{-1}(c x) \log \left(\frac{g e^{\sinh ^{-1}(c x)}}{\sqrt{c^2 f^2+g^2}+c f}+1\right)}{c}+\frac{\sinh ^{-1}(c x) \log \left(h (f+g x)^m\right)}{c}+\frac{m \sinh ^{-1}(c x)^2}{2 c}",1,"(m*ArcSinh[c*x]^2)/(2*c) - (m*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x]*g)/(c^2*f - c*Sqrt[c^2*f^2 + g^2])])/c - (m*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x]*g)/(c^2*f + c*Sqrt[c^2*f^2 + g^2])])/c + (ArcSinh[c*x]*Log[h*(f + g*x)^m])/c - (m*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/c - (m*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))])/c","A",1
57,0,0,37,0.2725782,"\int \frac{\log \left(h (f+g x)^m\right)}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]","\int \frac{\log \left(h (f+g x)^m\right)}{\sqrt{1+c^2 x^2} \left(a+b \sinh ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{\log \left(h (f+g x)^m\right)}{\sqrt{c^2 x^2+1} \left(a+b \sinh ^{-1}(c x)\right)},x\right)",0,"Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]","A",-1
58,1,95,131,0.0880143,"\int x^3 \sinh ^{-1}(a+b x) \, dx","Integrate[x^3*ArcSinh[a + b*x],x]","\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left(50 a^3-26 a^2 b x+a \left(14 b^2 x^2-55\right)-6 b^3 x^3+9 b x\right)-3 \left(8 a^4-24 a^2-8 b^4 x^4+3\right) \sinh ^{-1}(a+b x)}{96 b^4}","-\frac{\left(4 a \left(16-19 a^2\right)-\left(9-26 a^2\right) (a+b x)\right) \sqrt{(a+b x)^2+1}}{96 b^4}-\frac{\left(8 a^4-24 a^2+3\right) \sinh ^{-1}(a+b x)}{32 b^4}+\frac{7 a x^2 \sqrt{(a+b x)^2+1}}{48 b^2}+\frac{1}{4} x^4 \sinh ^{-1}(a+b x)-\frac{x^3 \sqrt{(a+b x)^2+1}}{16 b}",1,"(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(50*a^3 + 9*b*x - 26*a^2*b*x - 6*b^3*x^3 + a*(-55 + 14*b^2*x^2)) - 3*(3 - 24*a^2 + 8*a^4 - 8*b^4*x^4)*ArcSinh[a + b*x])/(96*b^4)","A",1
59,1,74,90,0.0588704,"\int x^2 \sinh ^{-1}(a+b x) \, dx","Integrate[x^2*ArcSinh[a + b*x],x]","\frac{\left(6 a^3-9 a+6 b^3 x^3\right) \sinh ^{-1}(a+b x)+\sqrt{a^2+2 a b x+b^2 x^2+1} \left(-11 a^2+5 a b x-2 b^2 x^2+4\right)}{18 b^3}","\frac{\left(-11 a^2+5 a b x+4\right) \sqrt{(a+b x)^2+1}}{18 b^3}-\frac{a \left(3-2 a^2\right) \sinh ^{-1}(a+b x)}{6 b^3}+\frac{1}{3} x^3 \sinh ^{-1}(a+b x)-\frac{x^2 \sqrt{(a+b x)^2+1}}{9 b}",1,"((4 - 11*a^2 + 5*a*b*x - 2*b^2*x^2)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (-9*a + 6*a^3 + 6*b^3*x^3)*ArcSinh[a + b*x])/(18*b^3)","A",1
60,1,60,76,0.0419583,"\int x \sinh ^{-1}(a+b x) \, dx","Integrate[x*ArcSinh[a + b*x],x]","\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} (3 a-b x)+\left(-2 a^2+2 b^2 x^2+1\right) \sinh ^{-1}(a+b x)}{4 b^2}","\frac{\left(1-2 a^2\right) \sinh ^{-1}(a+b x)}{4 b^2}+\frac{3 a \sqrt{(a+b x)^2+1}}{4 b^2}+\frac{1}{2} x^2 \sinh ^{-1}(a+b x)-\frac{x \sqrt{(a+b x)^2+1}}{4 b}",1,"((3*a - b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (1 - 2*a^2 + 2*b^2*x^2)*ArcSinh[a + b*x])/(4*b^2)","A",1
61,1,40,34,0.0280918,"\int \sinh ^{-1}(a+b x) \, dx","Integrate[ArcSinh[a + b*x],x]","\frac{(a+b x) \sinh ^{-1}(a+b x)-\sqrt{a^2+2 a b x+b^2 x^2+1}}{b}","\frac{(a+b x) \sinh ^{-1}(a+b x)}{b}-\frac{\sqrt{(a+b x)^2+1}}{b}",1,"(-Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (a + b*x)*ArcSinh[a + b*x])/b","A",1
62,1,153,131,0.0123007,"\int \frac{\sinh ^{-1}(a+b x)}{x} \, dx","Integrate[ArcSinh[a + b*x]/x,x]","\text{Li}_2\left(-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}-a}\right)+\text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)+\sinh ^{-1}(a+b x) \log \left(\frac{e^{\sinh ^{-1}(a+b x)}}{b \left(-\frac{\sqrt{a^2+1}}{b}-\frac{a}{b}\right)}+1\right)+\sinh ^{-1}(a+b x) \log \left(\frac{e^{\sinh ^{-1}(a+b x)}}{b \left(\frac{\sqrt{a^2+1}}{b}-\frac{a}{b}\right)}+1\right)-\frac{1}{2} \sinh ^{-1}(a+b x)^2","\text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+\text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)+\sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+\sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)-\frac{1}{2} \sinh ^{-1}(a+b x)^2",1,"-1/2*ArcSinh[a + b*x]^2 + ArcSinh[a + b*x]*Log[1 + E^ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b)] + ArcSinh[a + b*x]*Log[1 + E^ArcSinh[a + b*x]/((-(a/b) + Sqrt[1 + a^2]/b)*b)] + PolyLog[2, -(E^ArcSinh[a + b*x]/(-a + Sqrt[1 + a^2]))] + PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]","A",1
63,1,57,57,0.0412092,"\int \frac{\sinh ^{-1}(a+b x)}{x^2} \, dx","Integrate[ArcSinh[a + b*x]/x^2,x]","-\frac{b \tanh ^{-1}\left(\frac{a^2+a b x+1}{\sqrt{a^2+1} \sqrt{(a+b x)^2+1}}\right)}{\sqrt{a^2+1}}-\frac{\sinh ^{-1}(a+b x)}{x}","-\frac{b \tanh ^{-1}\left(\frac{a (a+b x)+1}{\sqrt{a^2+1} \sqrt{(a+b x)^2+1}}\right)}{\sqrt{a^2+1}}-\frac{\sinh ^{-1}(a+b x)}{x}",1,"-(ArcSinh[a + b*x]/x) - (b*ArcTanh[(1 + a^2 + a*b*x)/(Sqrt[1 + a^2]*Sqrt[1 + (a + b*x)^2])])/Sqrt[1 + a^2]","A",1
64,1,110,92,0.1778505,"\int \frac{\sinh ^{-1}(a+b x)}{x^3} \, dx","Integrate[ArcSinh[a + b*x]/x^3,x]","-\frac{\frac{b x \left(\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}-a b x \log \left(\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right)+a b x \log (x)\right)}{\left(a^2+1\right)^{3/2}}+\sinh ^{-1}(a+b x)}{2 x^2}","\frac{a b^2 \tanh ^{-1}\left(\frac{a (a+b x)+1}{\sqrt{a^2+1} \sqrt{(a+b x)^2+1}}\right)}{2 \left(a^2+1\right)^{3/2}}-\frac{b \sqrt{(a+b x)^2+1}}{2 \left(a^2+1\right) x}-\frac{\sinh ^{-1}(a+b x)}{2 x^2}",1,"-1/2*(ArcSinh[a + b*x] + (b*x*(Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + a*b*x*Log[x] - a*b*x*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]]))/(1 + a^2)^(3/2))/x^2","A",1
65,1,149,129,0.2350384,"\int \frac{\sinh ^{-1}(a+b x)}{x^4} \, dx","Integrate[ArcSinh[a + b*x]/x^4,x]","\frac{\left(2 a^2-1\right) b^3 x^3 \log (x)-\sqrt{a^2+1} b x \left(a^2-3 a b x+1\right) \sqrt{a^2+2 a b x+b^2 x^2+1}+\left(1-2 a^2\right) b^3 x^3 \log \left(\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right)-2 \left(a^2+1\right)^{5/2} \sinh ^{-1}(a+b x)}{6 \left(a^2+1\right)^{5/2} x^3}","\frac{\left(1-2 a^2\right) b^3 \tanh ^{-1}\left(\frac{a (a+b x)+1}{\sqrt{a^2+1} \sqrt{(a+b x)^2+1}}\right)}{6 \left(a^2+1\right)^{5/2}}+\frac{a b^2 \sqrt{(a+b x)^2+1}}{2 \left(a^2+1\right)^2 x}-\frac{b \sqrt{(a+b x)^2+1}}{6 \left(a^2+1\right) x^2}-\frac{\sinh ^{-1}(a+b x)}{3 x^3}",1,"(-(Sqrt[1 + a^2]*b*x*(1 + a^2 - 3*a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) - 2*(1 + a^2)^(5/2)*ArcSinh[a + b*x] + (-1 + 2*a^2)*b^3*x^3*Log[x] + (1 - 2*a^2)*b^3*x^3*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]])/(6*(1 + a^2)^(5/2)*x^3)","A",1
66,1,179,167,0.2140209,"\int \frac{\sinh ^{-1}(a+b x)}{x^5} \, dx","Integrate[ArcSinh[a + b*x]/x^5,x]","\frac{1}{8} \left(-\frac{a \left(2 a^2-3\right) b^4 \log (x)}{\left(a^2+1\right)^{7/2}}+\frac{a \left(2 a^2-3\right) b^4 \log \left(\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right)}{\left(a^2+1\right)^{7/2}}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2+1} \left(2 a^4-5 a^3 b x+a^2 \left(11 b^2 x^2+4\right)-5 a b x-4 b^2 x^2+2\right)}{3 \left(a^2+1\right)^3 x^3}-\frac{2 \sinh ^{-1}(a+b x)}{x^4}\right)","-\frac{a \left(3-2 a^2\right) b^4 \tanh ^{-1}\left(\frac{a (a+b x)+1}{\sqrt{a^2+1} \sqrt{(a+b x)^2+1}}\right)}{8 \left(a^2+1\right)^{7/2}}+\frac{\left(4-11 a^2\right) b^3 \sqrt{(a+b x)^2+1}}{24 \left(a^2+1\right)^3 x}+\frac{5 a b^2 \sqrt{(a+b x)^2+1}}{24 \left(a^2+1\right)^2 x^2}-\frac{b \sqrt{(a+b x)^2+1}}{12 \left(a^2+1\right) x^3}-\frac{\sinh ^{-1}(a+b x)}{4 x^4}",1,"(-1/3*(b*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(2 + 2*a^4 - 5*a*b*x - 5*a^3*b*x - 4*b^2*x^2 + a^2*(4 + 11*b^2*x^2)))/((1 + a^2)^3*x^3) - (2*ArcSinh[a + b*x])/x^4 - (a*(-3 + 2*a^2)*b^4*Log[x])/(1 + a^2)^(7/2) + (a*(-3 + 2*a^2)*b^4*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]])/(1 + a^2)^(7/2))/8","A",1
67,1,145,331,0.1913588,"\int x^3 \sinh ^{-1}(a+b x)^2 \, dx","Integrate[x^3*ArcSinh[a + b*x]^2,x]","\frac{-9 \left(8 a^4-24 a^2-8 b^4 x^4+3\right) \sinh ^{-1}(a+b x)^2+b x \left(-300 a^3+78 a^2 b x+a \left(330-28 b^2 x^2\right)+9 b x \left(b^2 x^2-3\right)\right)+6 \sqrt{a^2+2 a b x+b^2 x^2+1} \left(50 a^3-26 a^2 b x+a \left(14 b^2 x^2-55\right)-6 b^3 x^3+9 b x\right) \sinh ^{-1}(a+b x)}{288 b^4}","-\frac{a^4 \sinh ^{-1}(a+b x)^2}{4 b^4}+\frac{2 a^3 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b^4}-\frac{2 a^3 x}{b^3}+\frac{3 a^2 (a+b x)^2}{4 b^4}+\frac{3 a^2 \sinh ^{-1}(a+b x)^2}{4 b^4}-\frac{3 a^2 (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{2 b^4}-\frac{2 a (a+b x)^3}{9 b^4}+\frac{(a+b x)^4}{32 b^4}-\frac{3 (a+b x)^2}{32 b^4}+\frac{2 a (a+b x)^2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 b^4}-\frac{4 a \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 b^4}-\frac{3 \sinh ^{-1}(a+b x)^2}{32 b^4}-\frac{(a+b x)^3 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{8 b^4}+\frac{3 (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{16 b^4}+\frac{4 a x}{3 b^3}+\frac{1}{4} x^4 \sinh ^{-1}(a+b x)^2",1,"(b*x*(-300*a^3 + 78*a^2*b*x + a*(330 - 28*b^2*x^2) + 9*b*x*(-3 + b^2*x^2)) + 6*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(50*a^3 + 9*b*x - 26*a^2*b*x - 6*b^3*x^3 + a*(-55 + 14*b^2*x^2))*ArcSinh[a + b*x] - 9*(3 - 24*a^2 + 8*a^4 - 8*b^4*x^4)*ArcSinh[a + b*x]^2)/(288*b^4)","A",1
68,1,107,211,0.14332,"\int x^2 \sinh ^{-1}(a+b x)^2 \, dx","Integrate[x^2*ArcSinh[a + b*x]^2,x]","\frac{9 \left(2 a^3-3 a+2 b^3 x^3\right) \sinh ^{-1}(a+b x)^2+b x \left(66 a^2-15 a b x+4 b^2 x^2-24\right)-6 \sqrt{a^2+2 a b x+b^2 x^2+1} \left(11 a^2-5 a b x+2 b^2 x^2-4\right) \sinh ^{-1}(a+b x)}{54 b^3}","\frac{a^3 \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac{2 a^2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b^3}+\frac{2 a^2 x}{b^2}-\frac{a (a+b x)^2}{2 b^3}+\frac{2 (a+b x)^3}{27 b^3}-\frac{a \sinh ^{-1}(a+b x)^2}{2 b^3}+\frac{a (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b^3}-\frac{2 (a+b x)^2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{9 b^3}+\frac{4 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{9 b^3}+\frac{1}{3} x^3 \sinh ^{-1}(a+b x)^2-\frac{4 x}{9 b^2}",1,"(b*x*(-24 + 66*a^2 - 15*a*b*x + 4*b^2*x^2) - 6*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(-4 + 11*a^2 - 5*a*b*x + 2*b^2*x^2)*ArcSinh[a + b*x] + 9*(-3*a + 2*a^3 + 2*b^3*x^3)*ArcSinh[a + b*x]^2)/(54*b^3)","A",1
69,1,79,126,0.0827245,"\int x \sinh ^{-1}(a+b x)^2 \, dx","Integrate[x*ArcSinh[a + b*x]^2,x]","\frac{\left(-2 a^2+2 b^2 x^2+1\right) \sinh ^{-1}(a+b x)^2+2 (3 a-b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)+b x (b x-6 a)}{4 b^2}","-\frac{a^2 \sinh ^{-1}(a+b x)^2}{2 b^2}+\frac{(a+b x)^2}{4 b^2}-\frac{\sqrt{(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)}{2 b^2}+\frac{\sinh ^{-1}(a+b x)^2}{4 b^2}+\frac{2 a \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b^2}+\frac{1}{2} x^2 \sinh ^{-1}(a+b x)^2-\frac{2 a x}{b}",1,"(b*x*(-6*a + b*x) + 2*(3*a - b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x] + (1 - 2*a^2 + 2*b^2*x^2)*ArcSinh[a + b*x]^2)/(4*b^2)","A",1
70,1,47,45,0.0227028,"\int \sinh ^{-1}(a+b x)^2 \, dx","Integrate[ArcSinh[a + b*x]^2,x]","\frac{2 (a+b x)+(a+b x) \sinh ^{-1}(a+b x)^2-2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b}","\frac{(a+b x) \sinh ^{-1}(a+b x)^2}{b}-\frac{2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{b}+2 x",1,"(2*(a + b*x) - 2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x] + (a + b*x)*ArcSinh[a + b*x]^2)/b","A",1
71,1,251,205,0.0315662,"\int \frac{\sinh ^{-1}(a+b x)^2}{x} \, dx","Integrate[ArcSinh[a + b*x]^2/x,x]","2 \sinh ^{-1}(a+b x) \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(a+b x)}}{\left(-\frac{a}{b}-\frac{\sqrt{a^2+1}}{b}\right) b}\right)+2 \sinh ^{-1}(a+b x) \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(a+b x)}}{\left(\frac{\sqrt{a^2+1}}{b}-\frac{a}{b}\right) b}\right)-2 \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)-2 \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)+\sinh ^{-1}(a+b x)^2 \log \left(\frac{e^{\sinh ^{-1}(a+b x)}}{b \left(-\frac{\sqrt{a^2+1}}{b}-\frac{a}{b}\right)}+1\right)+\sinh ^{-1}(a+b x)^2 \log \left(\frac{e^{\sinh ^{-1}(a+b x)}}{b \left(\frac{\sqrt{a^2+1}}{b}-\frac{a}{b}\right)}+1\right)-\frac{1}{3} \sinh ^{-1}(a+b x)^3","2 \sinh ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+2 \sinh ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)-2 \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)-2 \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)+\sinh ^{-1}(a+b x)^2 \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+\sinh ^{-1}(a+b x)^2 \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)-\frac{1}{3} \sinh ^{-1}(a+b x)^3",1,"-1/3*ArcSinh[a + b*x]^3 + ArcSinh[a + b*x]^2*Log[1 + E^ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b)] + ArcSinh[a + b*x]^2*Log[1 + E^ArcSinh[a + b*x]/((-(a/b) + Sqrt[1 + a^2]/b)*b)] + 2*ArcSinh[a + b*x]*PolyLog[2, -(E^ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b))] + 2*ArcSinh[a + b*x]*PolyLog[2, -(E^ArcSinh[a + b*x]/((-(a/b) + Sqrt[1 + a^2]/b)*b))] - 2*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] - 2*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]","A",1
72,1,178,178,0.1201702,"\int \frac{\sinh ^{-1}(a+b x)^2}{x^2} \, dx","Integrate[ArcSinh[a + b*x]^2/x^2,x]","\frac{-2 b x \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+2 b x \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)-\sinh ^{-1}(a+b x) \left(\sqrt{a^2+1} \sinh ^{-1}(a+b x)+2 b x \left(\log \left(\frac{\sqrt{a^2+1}+e^{\sinh ^{-1}(a+b x)}-a}{\sqrt{a^2+1}-a}\right)-\log \left(\frac{\sqrt{a^2+1}-e^{\sinh ^{-1}(a+b x)}+a}{\sqrt{a^2+1}+a}\right)\right)\right)}{\sqrt{a^2+1} x}","-\frac{2 b \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\sqrt{a^2+1}}+\frac{2 b \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)}{\sqrt{a^2+1}}-\frac{2 b \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\sqrt{a^2+1}}+\frac{2 b \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)}{\sqrt{a^2+1}}-\frac{\sinh ^{-1}(a+b x)^2}{x}",1,"(-(ArcSinh[a + b*x]*(Sqrt[1 + a^2]*ArcSinh[a + b*x] + 2*b*x*(-Log[(a + Sqrt[1 + a^2] - E^ArcSinh[a + b*x])/(a + Sqrt[1 + a^2])] + Log[(-a + Sqrt[1 + a^2] + E^ArcSinh[a + b*x])/(-a + Sqrt[1 + a^2])]))) - 2*b*x*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 2*b*x*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(Sqrt[1 + a^2]*x)","A",1
73,1,279,235,0.129718,"\int \frac{\sinh ^{-1}(a+b x)^2}{x^3} \, dx","Integrate[ArcSinh[a + b*x]^2/x^3,x]","-\frac{-2 a b^2 x^2 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+2 a b^2 x^2 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)-2 \sqrt{a^2+1} b^2 x^2 \log (x)+2 a b^2 x^2 \sinh ^{-1}(a+b x) \log \left(\frac{\sqrt{a^2+1}-e^{\sinh ^{-1}(a+b x)}+a}{\sqrt{a^2+1}+a}\right)-2 a b^2 x^2 \sinh ^{-1}(a+b x) \log \left(\frac{\sqrt{a^2+1}+e^{\sinh ^{-1}(a+b x)}-a}{\sqrt{a^2+1}-a}\right)+2 \sqrt{a^2+1} b x \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)+a^2 \sqrt{a^2+1} \sinh ^{-1}(a+b x)^2+\sqrt{a^2+1} \sinh ^{-1}(a+b x)^2}{2 \left(a^2+1\right)^{3/2} x^2}","\frac{a b^2 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{3/2}}-\frac{a b^2 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{3/2}}+\frac{b^2 \log (x)}{a^2+1}+\frac{a b^2 \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{3/2}}-\frac{a b^2 \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)}{\left(a^2+1\right)^{3/2}}-\frac{b \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{\left(a^2+1\right) x}-\frac{\sinh ^{-1}(a+b x)^2}{2 x^2}",1,"-1/2*(2*Sqrt[1 + a^2]*b*x*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x] + Sqrt[1 + a^2]*ArcSinh[a + b*x]^2 + a^2*Sqrt[1 + a^2]*ArcSinh[a + b*x]^2 + 2*a*b^2*x^2*ArcSinh[a + b*x]*Log[(a + Sqrt[1 + a^2] - E^ArcSinh[a + b*x])/(a + Sqrt[1 + a^2])] - 2*a*b^2*x^2*ArcSinh[a + b*x]*Log[(-a + Sqrt[1 + a^2] + E^ArcSinh[a + b*x])/(-a + Sqrt[1 + a^2])] - 2*Sqrt[1 + a^2]*b^2*x^2*Log[x] - 2*a*b^2*x^2*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 2*a*b^2*x^2*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/((1 + a^2)^(3/2)*x^2)","A",1
74,1,1830,478,10.5226714,"\int \frac{\sinh ^{-1}(a+b x)^2}{x^4} \, dx","Integrate[ArcSinh[a + b*x]^2/x^4,x]","\frac{1}{3} \left(-\frac{2 i a^2 \pi  \tanh ^{-1}\left(\frac{-a \tanh \left(\frac{1}{2} \sinh ^{-1}(a+b x)\right)-1}{\sqrt{a^2+1}}\right) b^3}{\left(a^2+1\right)^{5/2}}+\frac{i \pi  \tanh ^{-1}\left(\frac{-a \tanh \left(\frac{1}{2} \sinh ^{-1}(a+b x)\right)-1}{\sqrt{a^2+1}}\right) b^3}{\left(a^2+1\right)^{5/2}}-\frac{3 a \log \left(-\frac{b x}{a}\right) b^3}{\left(a^2+1\right)^2}-\frac{2 a^2 \left(-2 \cos ^{-1}(i a) \tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)-\left(\pi -2 i \sinh ^{-1}(a+b x)\right) \tanh ^{-1}\left(\frac{(a+i) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)+\left(\cos ^{-1}(i a)+2 i \tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)+2 i \tanh ^{-1}\left(\frac{(a+i) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)\right) \log \left(\frac{\sqrt{-a^2-1} e^{-\frac{1}{2} \sinh ^{-1}(a+b x)}}{\sqrt{2} \sqrt{b x}}\right)+\left(\cos ^{-1}(i a)-2 i \left(\tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)+\tanh ^{-1}\left(\frac{(a+i) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)\right)\right) \log \left(\frac{i \sqrt{-a^2-1} e^{\frac{1}{2} \sinh ^{-1}(a+b x)}}{\sqrt{2} \sqrt{b x}}\right)-\left(\cos ^{-1}(i a)+2 i \tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)\right) \log \left(\frac{(a+i) \left(a+i \left(\sqrt{-a^2-1}-1\right)\right) \left(\cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)+i\right)}{a-\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)+i}\right)-\left(\cos ^{-1}(i a)-2 i \tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)\right) \log \left(\frac{(a+i) \left(a-i \left(\sqrt{-a^2-1}+1\right)\right) \left(\cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)-i\right)}{-a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)-i}\right)+i \left(\text{Li}_2\left(-\frac{\left(\sqrt{-a^2-1}-i a\right) \left(a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)+i\right)}{-a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)-i}\right)-\text{Li}_2\left(\frac{\left(i a+\sqrt{-a^2-1}\right) \left(a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)+i\right)}{-a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)-i}\right)\right)\right) b^3}{\left(-a^2-1\right)^{5/2}}+\frac{\left(-2 \cos ^{-1}(i a) \tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)-\left(\pi -2 i \sinh ^{-1}(a+b x)\right) \tanh ^{-1}\left(\frac{(a+i) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)+\left(\cos ^{-1}(i a)+2 i \tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)+2 i \tanh ^{-1}\left(\frac{(a+i) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)\right) \log \left(\frac{\sqrt{-a^2-1} e^{-\frac{1}{2} \sinh ^{-1}(a+b x)}}{\sqrt{2} \sqrt{b x}}\right)+\left(\cos ^{-1}(i a)-2 i \left(\tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)+\tanh ^{-1}\left(\frac{(a+i) \tan \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)\right)\right) \log \left(\frac{i \sqrt{-a^2-1} e^{\frac{1}{2} \sinh ^{-1}(a+b x)}}{\sqrt{2} \sqrt{b x}}\right)-\left(\cos ^{-1}(i a)+2 i \tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)\right) \log \left(\frac{(a+i) \left(a+i \left(\sqrt{-a^2-1}-1\right)\right) \left(\cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)+i\right)}{a-\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)+i}\right)-\left(\cos ^{-1}(i a)-2 i \tanh ^{-1}\left(\frac{(a-i) \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)}{\sqrt{-a^2-1}}\right)\right) \log \left(\frac{(a+i) \left(a-i \left(\sqrt{-a^2-1}+1\right)\right) \left(\cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)-i\right)}{-a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)-i}\right)+i \left(\text{Li}_2\left(-\frac{\left(\sqrt{-a^2-1}-i a\right) \left(a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)+i\right)}{-a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)-i}\right)-\text{Li}_2\left(\frac{\left(i a+\sqrt{-a^2-1}\right) \left(a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)+i\right)}{-a+\sqrt{-a^2-1} \cot \left(\frac{1}{4} \left(2 i \sinh ^{-1}(a+b x)+\pi \right)\right)-i}\right)\right)\right) b^3}{\left(-a^2-1\right)^{5/2}}-\frac{\left(a^2-3 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x) a+1\right) b^2}{\left(a^2+1\right)^2 x}-\frac{\sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x) b}{\left(a^2+1\right) x^2}-\frac{\sinh ^{-1}(a+b x)^2}{x^3}\right)","\frac{b^3 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{3 \left(a^2+1\right)^{3/2}}-\frac{a^2 b^3 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{5/2}}-\frac{b^3 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)}{3 \left(a^2+1\right)^{3/2}}+\frac{a^2 b^3 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{5/2}}-\frac{a b^3 \log (x)}{\left(a^2+1\right)^2}+\frac{b^3 \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{3 \left(a^2+1\right)^{3/2}}-\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{5/2}}-\frac{b^3 \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)}{3 \left(a^2+1\right)^{3/2}}+\frac{a^2 b^3 \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)}{\left(a^2+1\right)^{5/2}}-\frac{b^2}{3 \left(a^2+1\right) x}+\frac{a b^2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{\left(a^2+1\right)^2 x}-\frac{b \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{3 \left(a^2+1\right) x^2}-\frac{\sinh ^{-1}(a+b x)^2}{3 x^3}",1,"(-((b*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/((1 + a^2)*x^2)) - ArcSinh[a + b*x]^2/x^3 - (b^2*(1 + a^2 - 3*a*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]))/((1 + a^2)^2*x) + (I*b^3*Pi*ArcTanh[(-1 - a*Tanh[ArcSinh[a + b*x]/2])/Sqrt[1 + a^2]])/(1 + a^2)^(5/2) - ((2*I)*a^2*b^3*Pi*ArcTanh[(-1 - a*Tanh[ArcSinh[a + b*x]/2])/Sqrt[1 + a^2]])/(1 + a^2)^(5/2) - (3*a*b^3*Log[-((b*x)/a)])/(1 + a^2)^2 + (b^3*(-2*ArcCos[I*a]*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] - (Pi - (2*I)*ArcSinh[a + b*x])*ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] + (ArcCos[I*a] + (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] + (2*I)*ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[Sqrt[-1 - a^2]/(Sqrt[2]*E^(ArcSinh[a + b*x]/2)*Sqrt[b*x])] + (ArcCos[I*a] - (2*I)*(ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] + ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]]))*Log[(I*Sqrt[-1 - a^2]*E^(ArcSinh[a + b*x]/2))/(Sqrt[2]*Sqrt[b*x])] - (ArcCos[I*a] + (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[((I + a)*(a + I*(-1 + Sqrt[-1 - a^2]))*(I + Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(I + a - Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])] - (ArcCos[I*a] - (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[((I + a)*(a - I*(1 + Sqrt[-1 - a^2]))*(-I + Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])] + I*(PolyLog[2, -((((-I)*a + Sqrt[-1 - a^2])*(I + a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))] - PolyLog[2, ((I*a + Sqrt[-1 - a^2])*(I + a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])])))/(-1 - a^2)^(5/2) - (2*a^2*b^3*(-2*ArcCos[I*a]*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] - (Pi - (2*I)*ArcSinh[a + b*x])*ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] + (ArcCos[I*a] + (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] + (2*I)*ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[Sqrt[-1 - a^2]/(Sqrt[2]*E^(ArcSinh[a + b*x]/2)*Sqrt[b*x])] + (ArcCos[I*a] - (2*I)*(ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]] + ArcTanh[((I + a)*Tan[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]]))*Log[(I*Sqrt[-1 - a^2]*E^(ArcSinh[a + b*x]/2))/(Sqrt[2]*Sqrt[b*x])] - (ArcCos[I*a] + (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[((I + a)*(a + I*(-1 + Sqrt[-1 - a^2]))*(I + Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(I + a - Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])] - (ArcCos[I*a] - (2*I)*ArcTanh[((-I + a)*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])/Sqrt[-1 - a^2]])*Log[((I + a)*(a - I*(1 + Sqrt[-1 - a^2]))*(-I + Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])] + I*(PolyLog[2, -((((-I)*a + Sqrt[-1 - a^2])*(I + a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))] - PolyLog[2, ((I*a + Sqrt[-1 - a^2])*(I + a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4]))/(-I - a + Sqrt[-1 - a^2]*Cot[(Pi + (2*I)*ArcSinh[a + b*x])/4])])))/(-1 - a^2)^(5/2))/3","C",0
75,1,175,355,0.2067742,"\int x^2 \sinh ^{-1}(a+b x)^3 \, dx","Integrate[x^2*ArcSinh[a + b*x]^3,x]","\frac{18 \left(2 a^3-3 a+2 b^3 x^3\right) \sinh ^{-1}(a+b x)^3+\left(-575 a^2+65 a b x-8 b^2 x^2+160\right) \sqrt{a^2+2 a b x+b^2 x^2+1}-18 \sqrt{a^2+2 a b x+b^2 x^2+1} \left(11 a^2-5 a b x+2 b^2 x^2-4\right) \sinh ^{-1}(a+b x)^2+3 \left(170 a^3+132 a^2 b x-15 a \left(2 b^2 x^2+5\right)+8 b x \left(b^2 x^2-6\right)\right) \sinh ^{-1}(a+b x)}{108 b^3}","\frac{a^3 \sinh ^{-1}(a+b x)^3}{3 b^3}-\frac{6 a^2 \sqrt{(a+b x)^2+1}}{b^3}+\frac{6 a^2 (a+b x) \sinh ^{-1}(a+b x)}{b^3}-\frac{3 a^2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{b^3}+\frac{3 a \sqrt{(a+b x)^2+1} (a+b x)}{4 b^3}-\frac{2 \left((a+b x)^2+1\right)^{3/2}}{27 b^3}+\frac{14 \sqrt{(a+b x)^2+1}}{9 b^3}+\frac{2 (a+b x)^3 \sinh ^{-1}(a+b x)}{9 b^3}-\frac{\sqrt{(a+b x)^2+1} (a+b x)^2 \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac{3 a (a+b x)^2 \sinh ^{-1}(a+b x)}{2 b^3}+\frac{3 a \sqrt{(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)^2}{2 b^3}-\frac{4 (a+b x) \sinh ^{-1}(a+b x)}{3 b^3}-\frac{a \sinh ^{-1}(a+b x)^3}{2 b^3}+\frac{2 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac{3 a \sinh ^{-1}(a+b x)}{4 b^3}+\frac{1}{3} x^3 \sinh ^{-1}(a+b x)^3",1,"((160 - 575*a^2 + 65*a*b*x - 8*b^2*x^2)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + 3*(170*a^3 + 132*a^2*b*x + 8*b*x*(-6 + b^2*x^2) - 15*a*(5 + 2*b^2*x^2))*ArcSinh[a + b*x] - 18*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(-4 + 11*a^2 - 5*a*b*x + 2*b^2*x^2)*ArcSinh[a + b*x]^2 + 18*(-3*a + 2*a^3 + 2*b^3*x^3)*ArcSinh[a + b*x]^3)/(108*b^3)","A",1
76,1,129,203,0.1369511,"\int x \sinh ^{-1}(a+b x)^3 \, dx","Integrate[x*ArcSinh[a + b*x]^3,x]","\frac{3 (15 a-b x) \sqrt{a^2+2 a b x+b^2 x^2+1}+\left(-4 a^2+4 b^2 x^2+2\right) \sinh ^{-1}(a+b x)^3+6 (3 a-b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)^2+\left(-42 a^2-36 a b x+6 b^2 x^2+3\right) \sinh ^{-1}(a+b x)}{8 b^2}","-\frac{a^2 \sinh ^{-1}(a+b x)^3}{2 b^2}-\frac{3 (a+b x) \sqrt{(a+b x)^2+1}}{8 b^2}+\frac{6 a \sqrt{(a+b x)^2+1}}{b^2}+\frac{\sinh ^{-1}(a+b x)^3}{4 b^2}-\frac{3 (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{4 b^2}+\frac{3 a \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{b^2}+\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)}{4 b^2}-\frac{6 a (a+b x) \sinh ^{-1}(a+b x)}{b^2}+\frac{3 \sinh ^{-1}(a+b x)}{8 b^2}+\frac{1}{2} x^2 \sinh ^{-1}(a+b x)^3",1,"(3*(15*a - b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (3 - 42*a^2 - 36*a*b*x + 6*b^2*x^2)*ArcSinh[a + b*x] + 6*(3*a - b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2 + (2 - 4*a^2 + 4*b^2*x^2)*ArcSinh[a + b*x]^3)/(8*b^2)","A",1
77,1,70,78,0.0311412,"\int \sinh ^{-1}(a+b x)^3 \, dx","Integrate[ArcSinh[a + b*x]^3,x]","\frac{-6 \sqrt{(a+b x)^2+1}+(a+b x) \sinh ^{-1}(a+b x)^3-3 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2+6 (a+b x) \sinh ^{-1}(a+b x)}{b}","-\frac{6 \sqrt{(a+b x)^2+1}}{b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b}-\frac{3 \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{b}+\frac{6 (a+b x) \sinh ^{-1}(a+b x)}{b}",1,"(-6*Sqrt[1 + (a + b*x)^2] + 6*(a + b*x)*ArcSinh[a + b*x] - 3*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2 + (a + b*x)*ArcSinh[a + b*x]^3)/b","A",1
78,1,346,275,0.0330518,"\int \frac{\sinh ^{-1}(a+b x)^3}{x} \, dx","Integrate[ArcSinh[a + b*x]^3/x,x]","3 \sinh ^{-1}(a+b x)^2 \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(a+b x)}}{\left(-\frac{a}{b}-\frac{\sqrt{a^2+1}}{b}\right) b}\right)+3 \sinh ^{-1}(a+b x)^2 \text{Li}_2\left(-\frac{e^{\sinh ^{-1}(a+b x)}}{\left(\frac{\sqrt{a^2+1}}{b}-\frac{a}{b}\right) b}\right)-6 \sinh ^{-1}(a+b x) \text{Li}_3\left(-\frac{e^{\sinh ^{-1}(a+b x)}}{\left(-\frac{a}{b}-\frac{\sqrt{a^2+1}}{b}\right) b}\right)-6 \sinh ^{-1}(a+b x) \text{Li}_3\left(-\frac{e^{\sinh ^{-1}(a+b x)}}{\left(\frac{\sqrt{a^2+1}}{b}-\frac{a}{b}\right) b}\right)+6 \text{Li}_4\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+6 \text{Li}_4\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)+\sinh ^{-1}(a+b x)^3 \log \left(\frac{e^{\sinh ^{-1}(a+b x)}}{b \left(-\frac{\sqrt{a^2+1}}{b}-\frac{a}{b}\right)}+1\right)+\sinh ^{-1}(a+b x)^3 \log \left(\frac{e^{\sinh ^{-1}(a+b x)}}{b \left(\frac{\sqrt{a^2+1}}{b}-\frac{a}{b}\right)}+1\right)-\frac{1}{4} \sinh ^{-1}(a+b x)^4","3 \sinh ^{-1}(a+b x)^2 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+3 \sinh ^{-1}(a+b x)^2 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)-6 \sinh ^{-1}(a+b x) \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)-6 \sinh ^{-1}(a+b x) \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)+6 \text{Li}_4\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+6 \text{Li}_4\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)+\sinh ^{-1}(a+b x)^3 \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+\sinh ^{-1}(a+b x)^3 \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)-\frac{1}{4} \sinh ^{-1}(a+b x)^4",1,"-1/4*ArcSinh[a + b*x]^4 + ArcSinh[a + b*x]^3*Log[1 + E^ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b)] + ArcSinh[a + b*x]^3*Log[1 + E^ArcSinh[a + b*x]/((-(a/b) + Sqrt[1 + a^2]/b)*b)] + 3*ArcSinh[a + b*x]^2*PolyLog[2, -(E^ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b))] + 3*ArcSinh[a + b*x]^2*PolyLog[2, -(E^ArcSinh[a + b*x]/((-(a/b) + Sqrt[1 + a^2]/b)*b))] - 6*ArcSinh[a + b*x]*PolyLog[3, -(E^ArcSinh[a + b*x]/((-(a/b) - Sqrt[1 + a^2]/b)*b))] - 6*ArcSinh[a + b*x]*PolyLog[3, -(E^ArcSinh[a + b*x]/((-(a/b) + Sqrt[1 + a^2]/b)*b))] + 6*PolyLog[4, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 6*PolyLog[4, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])]","A",1
79,1,259,268,0.1171341,"\int \frac{\sinh ^{-1}(a+b x)^3}{x^2} \, dx","Integrate[ArcSinh[a + b*x]^3/x^2,x]","-\frac{6 b x \sinh ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)-6 b x \sinh ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)-6 b x \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+6 b x \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)+\sqrt{a^2+1} \sinh ^{-1}(a+b x)^3-3 b x \sinh ^{-1}(a+b x)^2 \log \left(\frac{\sqrt{a^2+1}-e^{\sinh ^{-1}(a+b x)}+a}{\sqrt{a^2+1}+a}\right)+3 b x \sinh ^{-1}(a+b x)^2 \log \left(\frac{\sqrt{a^2+1}+e^{\sinh ^{-1}(a+b x)}-a}{\sqrt{a^2+1}-a}\right)}{\sqrt{a^2+1} x}","-\frac{6 b \sinh ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\sqrt{a^2+1}}+\frac{6 b \sinh ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)}{\sqrt{a^2+1}}+\frac{6 b \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\sqrt{a^2+1}}-\frac{6 b \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)}{\sqrt{a^2+1}}-\frac{3 b \sinh ^{-1}(a+b x)^2 \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\sqrt{a^2+1}}+\frac{3 b \sinh ^{-1}(a+b x)^2 \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)}{\sqrt{a^2+1}}-\frac{\sinh ^{-1}(a+b x)^3}{x}",1,"-((Sqrt[1 + a^2]*ArcSinh[a + b*x]^3 - 3*b*x*ArcSinh[a + b*x]^2*Log[(a + Sqrt[1 + a^2] - E^ArcSinh[a + b*x])/(a + Sqrt[1 + a^2])] + 3*b*x*ArcSinh[a + b*x]^2*Log[(-a + Sqrt[1 + a^2] + E^ArcSinh[a + b*x])/(-a + Sqrt[1 + a^2])] + 6*b*x*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] - 6*b*x*ArcSinh[a + b*x]*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] - 6*b*x*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 6*b*x*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(Sqrt[1 + a^2]*x))","A",1
80,1,524,514,0.2387598,"\int \frac{\sinh ^{-1}(a+b x)^3}{x^3} \, dx","Integrate[ArcSinh[a + b*x]^3/x^3,x]","\frac{6 b^2 x^2 \left(\sqrt{a^2+1}+a \sinh ^{-1}(a+b x)\right) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+6 b^2 x^2 \left(\sqrt{a^2+1}-a \sinh ^{-1}(a+b x)\right) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)-6 a b^2 x^2 \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)+6 a b^2 x^2 \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)-3 \sqrt{a^2+1} b^2 x^2 \sinh ^{-1}(a+b x)^2-3 \sqrt{a^2+1} b x \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)^2-3 a b^2 x^2 \sinh ^{-1}(a+b x)^2 \log \left(\frac{\sqrt{a^2+1}-e^{\sinh ^{-1}(a+b x)}+a}{\sqrt{a^2+1}+a}\right)+3 a b^2 x^2 \sinh ^{-1}(a+b x)^2 \log \left(\frac{\sqrt{a^2+1}+e^{\sinh ^{-1}(a+b x)}-a}{\sqrt{a^2+1}-a}\right)+6 \sqrt{a^2+1} b^2 x^2 \sinh ^{-1}(a+b x) \log \left(\frac{\sqrt{a^2+1}-e^{\sinh ^{-1}(a+b x)}+a}{\sqrt{a^2+1}+a}\right)+6 \sqrt{a^2+1} b^2 x^2 \sinh ^{-1}(a+b x) \log \left(\frac{\sqrt{a^2+1}+e^{\sinh ^{-1}(a+b x)}-a}{\sqrt{a^2+1}-a}\right)-a^2 \sqrt{a^2+1} \sinh ^{-1}(a+b x)^3-\sqrt{a^2+1} \sinh ^{-1}(a+b x)^3}{2 \left(a^2+1\right)^{3/2} x^2}","\frac{3 a b^2 \sinh ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{3/2}}-\frac{3 a b^2 \sinh ^{-1}(a+b x) \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{3/2}}+\frac{3 b^2 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{a^2+1}+\frac{3 b^2 \text{Li}_2\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)}{a^2+1}-\frac{3 a b^2 \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{3/2}}+\frac{3 a b^2 \text{Li}_3\left(\frac{e^{\sinh ^{-1}(a+b x)}}{a+\sqrt{a^2+1}}\right)}{\left(a^2+1\right)^{3/2}}-\frac{3 b^2 \sinh ^{-1}(a+b x)^2}{2 \left(a^2+1\right)}+\frac{3 a b^2 \sinh ^{-1}(a+b x)^2 \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{2 \left(a^2+1\right)^{3/2}}-\frac{3 a b^2 \sinh ^{-1}(a+b x)^2 \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)}{2 \left(a^2+1\right)^{3/2}}+\frac{3 b^2 \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{a-\sqrt{a^2+1}}\right)}{a^2+1}+\frac{3 b^2 \sinh ^{-1}(a+b x) \log \left(1-\frac{e^{\sinh ^{-1}(a+b x)}}{\sqrt{a^2+1}+a}\right)}{a^2+1}-\frac{3 b \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{2 \left(a^2+1\right) x}-\frac{\sinh ^{-1}(a+b x)^3}{2 x^2}",1,"(-3*Sqrt[1 + a^2]*b^2*x^2*ArcSinh[a + b*x]^2 - 3*Sqrt[1 + a^2]*b*x*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2 - Sqrt[1 + a^2]*ArcSinh[a + b*x]^3 - a^2*Sqrt[1 + a^2]*ArcSinh[a + b*x]^3 + 6*Sqrt[1 + a^2]*b^2*x^2*ArcSinh[a + b*x]*Log[(a + Sqrt[1 + a^2] - E^ArcSinh[a + b*x])/(a + Sqrt[1 + a^2])] - 3*a*b^2*x^2*ArcSinh[a + b*x]^2*Log[(a + Sqrt[1 + a^2] - E^ArcSinh[a + b*x])/(a + Sqrt[1 + a^2])] + 6*Sqrt[1 + a^2]*b^2*x^2*ArcSinh[a + b*x]*Log[(-a + Sqrt[1 + a^2] + E^ArcSinh[a + b*x])/(-a + Sqrt[1 + a^2])] + 3*a*b^2*x^2*ArcSinh[a + b*x]^2*Log[(-a + Sqrt[1 + a^2] + E^ArcSinh[a + b*x])/(-a + Sqrt[1 + a^2])] + 6*b^2*x^2*(Sqrt[1 + a^2] + a*ArcSinh[a + b*x])*PolyLog[2, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 6*b^2*x^2*(Sqrt[1 + a^2] - a*ArcSinh[a + b*x])*PolyLog[2, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])] - 6*a*b^2*x^2*PolyLog[3, E^ArcSinh[a + b*x]/(a - Sqrt[1 + a^2])] + 6*a*b^2*x^2*PolyLog[3, E^ArcSinh[a + b*x]/(a + Sqrt[1 + a^2])])/(2*(1 + a^2)^(3/2)*x^2)","A",1
81,1,44,60,0.1465707,"\int \frac{x^2}{\sinh ^{-1}(a+b x)} \, dx","Integrate[x^2/ArcSinh[a + b*x],x]","\frac{\left(4 a^2-1\right) \text{Chi}\left(\sinh ^{-1}(a+b x)\right)+\text{Chi}\left(3 \sinh ^{-1}(a+b x)\right)-4 a \text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)}{4 b^3}","\frac{a^2 \text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{b^3}-\frac{\text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{4 b^3}+\frac{\text{Chi}\left(3 \sinh ^{-1}(a+b x)\right)}{4 b^3}-\frac{a \text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)}{b^3}",1,"((-1 + 4*a^2)*CoshIntegral[ArcSinh[a + b*x]] + CoshIntegral[3*ArcSinh[a + b*x]] - 4*a*SinhIntegral[2*ArcSinh[a + b*x]])/(4*b^3)","A",1
82,1,30,30,0.0527601,"\int \frac{x}{\sinh ^{-1}(a+b x)} \, dx","Integrate[x/ArcSinh[a + b*x],x]","\frac{\text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)}{2 b^2}-\frac{a \text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{b^2}","\frac{\text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)}{2 b^2}-\frac{a \text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{b^2}",1,"-((a*CoshIntegral[ArcSinh[a + b*x]])/b^2) + SinhIntegral[2*ArcSinh[a + b*x]]/(2*b^2)","A",1
83,1,11,11,0.0072865,"\int \frac{1}{\sinh ^{-1}(a+b x)} \, dx","Integrate[ArcSinh[a + b*x]^(-1),x]","\frac{\text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{b}","\frac{\text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{b}",1,"CoshIntegral[ArcSinh[a + b*x]]/b","A",1
84,0,0,15,0.2028094,"\int \frac{1}{x \sinh ^{-1}(a+b x)} \, dx","Integrate[1/(x*ArcSinh[a + b*x]),x]","\int \frac{1}{x \sinh ^{-1}(a+b x)} \, dx","\text{Int}\left(\frac{1}{x \sinh ^{-1}(a+b x)},x\right)",0,"Integrate[1/(x*ArcSinh[a + b*x]), x]","A",-1
85,1,83,154,0.4630786,"\int \frac{x^2}{\sinh ^{-1}(a+b x)^2} \, dx","Integrate[x^2/ArcSinh[a + b*x]^2,x]","\frac{-\frac{4 b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2+1}}{\sinh ^{-1}(a+b x)}+\left(4 a^2-1\right) \text{Shi}\left(\sinh ^{-1}(a+b x)\right)-8 a \text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)+3 \text{Shi}\left(3 \sinh ^{-1}(a+b x)\right)}{4 b^3}","\frac{a^2 \text{Shi}\left(\sinh ^{-1}(a+b x)\right)}{b^3}-\frac{a^2 \sqrt{(a+b x)^2+1}}{b^3 \sinh ^{-1}(a+b x)}-\frac{2 a \text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)}{b^3}-\frac{\text{Shi}\left(\sinh ^{-1}(a+b x)\right)}{4 b^3}+\frac{3 \text{Shi}\left(3 \sinh ^{-1}(a+b x)\right)}{4 b^3}+\frac{2 a (a+b x) \sqrt{(a+b x)^2+1}}{b^3 \sinh ^{-1}(a+b x)}-\frac{(a+b x)^2 \sqrt{(a+b x)^2+1}}{b^3 \sinh ^{-1}(a+b x)}",1,"((-4*b^2*x^2*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/ArcSinh[a + b*x] - 8*a*CoshIntegral[2*ArcSinh[a + b*x]] + (-1 + 4*a^2)*SinhIntegral[ArcSinh[a + b*x]] + 3*SinhIntegral[3*ArcSinh[a + b*x]])/(4*b^3)","A",1
86,1,62,84,0.1669634,"\int \frac{x}{\sinh ^{-1}(a+b x)^2} \, dx","Integrate[x/ArcSinh[a + b*x]^2,x]","-\frac{-\sinh ^{-1}(a+b x) \text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)+a \sinh ^{-1}(a+b x) \text{Shi}\left(\sinh ^{-1}(a+b x)\right)+b x \sqrt{(a+b x)^2+1}}{b^2 \sinh ^{-1}(a+b x)}","\frac{\text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)}{b^2}-\frac{a \text{Shi}\left(\sinh ^{-1}(a+b x)\right)}{b^2}+\frac{a \sqrt{(a+b x)^2+1}}{b^2 \sinh ^{-1}(a+b x)}-\frac{(a+b x) \sqrt{(a+b x)^2+1}}{b^2 \sinh ^{-1}(a+b x)}",1,"-((b*x*Sqrt[1 + (a + b*x)^2] - ArcSinh[a + b*x]*CoshIntegral[2*ArcSinh[a + b*x]] + a*ArcSinh[a + b*x]*SinhIntegral[ArcSinh[a + b*x]])/(b^2*ArcSinh[a + b*x]))","A",1
87,1,35,38,0.023136,"\int \frac{1}{\sinh ^{-1}(a+b x)^2} \, dx","Integrate[ArcSinh[a + b*x]^(-2),x]","\frac{\text{Shi}\left(\sinh ^{-1}(a+b x)\right)-\frac{\sqrt{(a+b x)^2+1}}{\sinh ^{-1}(a+b x)}}{b}","\frac{\text{Shi}\left(\sinh ^{-1}(a+b x)\right)}{b}-\frac{\sqrt{(a+b x)^2+1}}{b \sinh ^{-1}(a+b x)}",1,"(-(Sqrt[1 + (a + b*x)^2]/ArcSinh[a + b*x]) + SinhIntegral[ArcSinh[a + b*x]])/b","A",1
88,0,0,15,2.1403183,"\int \frac{1}{x \sinh ^{-1}(a+b x)^2} \, dx","Integrate[1/(x*ArcSinh[a + b*x]^2),x]","\int \frac{1}{x \sinh ^{-1}(a+b x)^2} \, dx","\text{Int}\left(\frac{1}{x \sinh ^{-1}(a+b x)^2},x\right)",0,"Integrate[1/(x*ArcSinh[a + b*x]^2), x]","A",-1
89,1,110,257,0.4032198,"\int \frac{x^2}{\sinh ^{-1}(a+b x)^3} \, dx","Integrate[x^2/ArcSinh[a + b*x]^3,x]","\frac{-\frac{4 b x \left(b x \sqrt{a^2+2 a b x+b^2 x^2+1}+\left(2 a^2+5 a b x+3 b^2 x^2+2\right) \sinh ^{-1}(a+b x)\right)}{\sinh ^{-1}(a+b x)^2}+\left(4 a^2-1\right) \text{Chi}\left(\sinh ^{-1}(a+b x)\right)+9 \text{Chi}\left(3 \sinh ^{-1}(a+b x)\right)-16 a \text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)}{8 b^3}","\frac{a^2 \text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{2 b^3}-\frac{a^2 (a+b x)}{2 b^3 \sinh ^{-1}(a+b x)}-\frac{a^2 \sqrt{(a+b x)^2+1}}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac{\text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{8 b^3}+\frac{9 \text{Chi}\left(3 \sinh ^{-1}(a+b x)\right)}{8 b^3}-\frac{2 a \text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)}{b^3}-\frac{3 (a+b x)^3}{2 b^3 \sinh ^{-1}(a+b x)}+\frac{2 a (a+b x)^2}{b^3 \sinh ^{-1}(a+b x)}-\frac{\sqrt{(a+b x)^2+1} (a+b x)^2}{2 b^3 \sinh ^{-1}(a+b x)^2}-\frac{a+b x}{b^3 \sinh ^{-1}(a+b x)}+\frac{a \sqrt{(a+b x)^2+1} (a+b x)}{b^3 \sinh ^{-1}(a+b x)^2}+\frac{a}{b^3 \sinh ^{-1}(a+b x)}",1,"((-4*b*x*(b*x*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (2 + 2*a^2 + 5*a*b*x + 3*b^2*x^2)*ArcSinh[a + b*x]))/ArcSinh[a + b*x]^2 + (-1 + 4*a^2)*CoshIntegral[ArcSinh[a + b*x]] + 9*CoshIntegral[3*ArcSinh[a + b*x]] - 16*a*SinhIntegral[2*ArcSinh[a + b*x]])/(8*b^3)","A",1
90,1,117,147,0.1089402,"\int \frac{x}{\sinh ^{-1}(a+b x)^3} \, dx","Integrate[x/ArcSinh[a + b*x]^3,x]","-\frac{b x \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2 \sinh ^{-1}(a+b x)+2 b^2 x^2 \sinh ^{-1}(a+b x)+a \sinh ^{-1}(a+b x)^2 \text{Chi}\left(\sinh ^{-1}(a+b x)\right)-2 \sinh ^{-1}(a+b x)^2 \text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)+3 a b x \sinh ^{-1}(a+b x)+\sinh ^{-1}(a+b x)}{2 b^2 \sinh ^{-1}(a+b x)^2}","-\frac{a \text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{2 b^2}+\frac{\text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)}{b^2}-\frac{(a+b x)^2}{b^2 \sinh ^{-1}(a+b x)}+\frac{a (a+b x)}{2 b^2 \sinh ^{-1}(a+b x)}-\frac{\sqrt{(a+b x)^2+1} (a+b x)}{2 b^2 \sinh ^{-1}(a+b x)^2}-\frac{1}{2 b^2 \sinh ^{-1}(a+b x)}+\frac{a \sqrt{(a+b x)^2+1}}{2 b^2 \sinh ^{-1}(a+b x)^2}",1,"-1/2*(b*x*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + ArcSinh[a + b*x] + a^2*ArcSinh[a + b*x] + 3*a*b*x*ArcSinh[a + b*x] + 2*b^2*x^2*ArcSinh[a + b*x] + a*ArcSinh[a + b*x]^2*CoshIntegral[ArcSinh[a + b*x]] - 2*ArcSinh[a + b*x]^2*SinhIntegral[2*ArcSinh[a + b*x]])/(b^2*ArcSinh[a + b*x]^2)","A",1
91,1,53,63,0.0424581,"\int \frac{1}{\sinh ^{-1}(a+b x)^3} \, dx","Integrate[ArcSinh[a + b*x]^(-3),x]","\frac{\text{Chi}\left(\sinh ^{-1}(a+b x)\right)-\frac{a+b x}{\sinh ^{-1}(a+b x)}-\frac{\sqrt{(a+b x)^2+1}}{\sinh ^{-1}(a+b x)^2}}{2 b}","\frac{\text{Chi}\left(\sinh ^{-1}(a+b x)\right)}{2 b}-\frac{a+b x}{2 b \sinh ^{-1}(a+b x)}-\frac{\sqrt{(a+b x)^2+1}}{2 b \sinh ^{-1}(a+b x)^2}",1,"(-(Sqrt[1 + (a + b*x)^2]/ArcSinh[a + b*x]^2) - (a + b*x)/ArcSinh[a + b*x] + CoshIntegral[ArcSinh[a + b*x]])/(2*b)","A",1
92,0,0,15,2.1418151,"\int \frac{1}{x \sinh ^{-1}(a+b x)^3} \, dx","Integrate[1/(x*ArcSinh[a + b*x]^3),x]","\int \frac{1}{x \sinh ^{-1}(a+b x)^3} \, dx","\text{Int}\left(\frac{1}{x \sinh ^{-1}(a+b x)^3},x\right)",0,"Integrate[1/(x*ArcSinh[a + b*x]^3), x]","A",-1
93,0,0,19,0.444742,"\int x^m \left(a+b \sinh ^{-1}(c+d x)\right)^n \, dx","Integrate[x^m*(a + b*ArcSinh[c + d*x])^n,x]","\int x^m \left(a+b \sinh ^{-1}(c+d x)\right)^n \, dx","\text{Int}\left(x^m \left(a+b \sinh ^{-1}(c+d x)\right)^n,x\right)",0,"Integrate[x^m*(a + b*ArcSinh[c + d*x])^n, x]","A",-1
94,1,345,545,1.0281333,"\int x^2 \left(a+b \sinh ^{-1}(c+d x)\right)^n \, dx","Integrate[x^2*(a + b*ArcSinh[c + d*x])^n,x]","\frac{2^{-n-3} 3^{-n-1} e^{-\frac{3 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}\right)^{-n} \left(\left(4 c^2-1\right) 2^n 3^{n+1} e^{\frac{2 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)^n \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-\left(4 c^2-1\right) 2^n 3^{n+1} e^{\frac{4 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^n \Gamma \left(n+1,\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+2^n \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)^n \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-2 c 3^{n+1} e^{a/b} \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-e^{\frac{5 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^n \left(2 c 3^{n+1} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+2^n e^{a/b} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{d^3}","\frac{c^2 e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{2 d^3}-\frac{c^2 e^{a/b} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{2 d^3}+\frac{3^{-n-1} e^{-\frac{3 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{8 d^3}-\frac{c 2^{-n-2} e^{-\frac{2 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{d^3}-\frac{e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{8 d^3}+\frac{e^{a/b} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{8 d^3}-\frac{c 2^{-n-2} e^{\frac{2 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{d^3}-\frac{3^{-n-1} e^{\frac{3 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{8 d^3}",1,"(2^(-3 - n)*3^(-1 - n)*(a + b*ArcSinh[c + d*x])^n*(-(2^n*3^(1 + n)*(-1 + 4*c^2)*E^((4*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*Gamma[1 + n, a/b + ArcSinh[c + d*x]]) + 2^n*(a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, (-3*(a + b*ArcSinh[c + d*x]))/b] - 2*3^(1 + n)*c*E^(a/b)*(a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c + d*x]))/b] + 2^n*3^(1 + n)*(-1 + 4*c^2)*E^((2*a)/b)*(a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)] - E^((5*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*(2*3^(1 + n)*c*Gamma[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b] + 2^n*E^(a/b)*Gamma[1 + n, (3*(a + b*ArcSinh[c + d*x]))/b])))/(d^3*E^((3*a)/b)*(-((a + b*ArcSinh[c + d*x])^2/b^2))^n)","A",1
95,1,228,267,0.189328,"\int x \left(a+b \sinh ^{-1}(c+d x)\right)^n \, dx","Integrate[x*(a + b*ArcSinh[c + d*x])^n,x]","\frac{2^{-n-3} e^{-\frac{2 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}\right)^{-n} \left(\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)^n \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-c 2^{n+2} e^{a/b} \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)^n \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+c 2^{n+2} e^{\frac{3 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^n \Gamma \left(n+1,\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+e^{\frac{4 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^n \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{d^2}","\frac{2^{-n-3} e^{-\frac{2 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{d^2}-\frac{c e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{2 d^2}+\frac{c e^{a/b} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{2 d^2}+\frac{2^{-n-3} e^{\frac{2 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{d^2}",1,"(2^(-3 - n)*(a + b*ArcSinh[c + d*x])^n*(2^(2 + n)*c*E^((3*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*Gamma[1 + n, a/b + ArcSinh[c + d*x]] + (a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, (-2*(a + b*ArcSinh[c + d*x]))/b] - 2^(2 + n)*c*E^(a/b)*(a/b + ArcSinh[c + d*x])^n*Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)] + E^((4*a)/b)*(-((a + b*ArcSinh[c + d*x])/b))^n*Gamma[1 + n, (2*(a + b*ArcSinh[c + d*x]))/b]))/(d^2*E^((2*a)/b)*(-((a + b*ArcSinh[c + d*x])^2/b^2))^n)","A",1
96,1,109,128,0.1187595,"\int \left(a+b \sinh ^{-1}(c+d x)\right)^n \, dx","Integrate[(a + b*ArcSinh[c + d*x])^n,x]","\frac{e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(\left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-e^{\frac{2 a}{b}} \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)^{-n} \Gamma \left(n+1,\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)}{2 d}","\frac{e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{2 d}-\frac{e^{a/b} \left(a+b \sinh ^{-1}(c+d x)\right)^n \left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{-n} \Gamma \left(n+1,\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{2 d}",1,"((a + b*ArcSinh[c + d*x])^n*(-((E^((2*a)/b)*Gamma[1 + n, a/b + ArcSinh[c + d*x]])/(a/b + ArcSinh[c + d*x])^n) + Gamma[1 + n, -((a + b*ArcSinh[c + d*x])/b)]/(-((a + b*ArcSinh[c + d*x])/b))^n))/(2*d*E^(a/b))","A",1
97,0,0,19,0.1721859,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^n}{x} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^n/x,x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^n}{x} \, dx","\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^n}{x},x\right)",0,"Integrate[(a + b*ArcSinh[c + d*x])^n/x, x]","A",-1
98,1,656,496,1.7748888,"\int x^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[x^2*Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{9 \sqrt{\pi } \sqrt{b} \left(4 c^2-1\right) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+36 \sqrt{\pi } \sqrt{b} c^2 \sinh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-36 \sqrt{\pi } \sqrt{b} c^2 \cosh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+144 c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}+\sqrt{3 \pi } \sqrt{b} \sinh \left(\frac{3 a}{b}\right) \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+9 \sqrt{2 \pi } \sqrt{b} c \left(\sinh \left(\frac{2 a}{b}\right)+\cosh \left(\frac{2 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{3 \pi } \sqrt{b} \cosh \left(\frac{3 a}{b}\right) \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-9 \sqrt{2 \pi } \sqrt{b} c \sinh \left(\frac{2 a}{b}\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-9 \sqrt{\pi } \sqrt{b} \sinh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{3 \pi } \sqrt{b} \sinh \left(\frac{3 a}{b}\right) \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+9 \sqrt{2 \pi } \sqrt{b} c \cosh \left(\frac{2 a}{b}\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+9 \sqrt{\pi } \sqrt{b} \cosh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-\sqrt{3 \pi } \sqrt{b} \cosh \left(\frac{3 a}{b}\right) \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+12 \sinh \left(3 \sinh ^{-1}(c+d x)\right) \sqrt{a+b \sinh ^{-1}(c+d x)}-36 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}-72 c \cosh \left(2 \sinh ^{-1}(c+d x)\right) \sqrt{a+b \sinh ^{-1}(c+d x)}}{144 d^3}","\frac{\sqrt{\pi } \sqrt{b} c^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d^3}-\frac{\sqrt{\pi } \sqrt{b} c^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d^3}+\frac{c^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^3}-\frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d^3}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{48 d^3}+\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d^3}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} c e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d^3}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{48 d^3}+\frac{(c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d^3}-\frac{c \cosh \left(2 \sinh ^{-1}(c+d x)\right) \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d^3}",1,"(-36*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]] + 144*c^2*(c + d*x)*Sqrt[a + b*ArcSinh[c + d*x]] - 72*c*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + Sqrt[b]*Sqrt[3*Pi]*Cosh[(3*a)/b]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] + 9*Sqrt[b]*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - 36*Sqrt[b]*c^2*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] + 9*Sqrt[b]*c*Sqrt[2*Pi]*Cosh[(2*a)/b]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - Sqrt[b]*Sqrt[3*Pi]*Cosh[(3*a)/b]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - 9*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 36*Sqrt[b]*c^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 9*Sqrt[b]*(-1 + 4*c^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) - 9*Sqrt[b]*c*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] + 9*Sqrt[b]*c*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + Sqrt[b]*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(3*a)/b] + Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(3*a)/b] + 12*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[3*ArcSinh[c + d*x]])/(144*d^3)","A",1
99,1,251,259,1.8218428,"\int x \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[x*Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{-\sqrt{2 \pi } \sqrt{b} \left(\sinh \left(\frac{2 a}{b}\right)+\cosh \left(\frac{2 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{2 \pi } \sqrt{b} \left(\sinh \left(\frac{2 a}{b}\right)-\cosh \left(\frac{2 a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+8 \cosh \left(2 \sinh ^{-1}(c+d x)\right) \sqrt{a+b \sinh ^{-1}(c+d x)}-16 c e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right)}{32 d^2}","-\frac{\sqrt{\pi } \sqrt{b} c e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d^2}+\frac{\sqrt{\pi } \sqrt{b} c e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d^2}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d^2}-\frac{c (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d^2}+\frac{\cosh \left(2 \sinh ^{-1}(c+d x)\right) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d^2}",1,"(8*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] - (16*c*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) + Sqrt[b]*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(-Cosh[(2*a)/b] + Sinh[(2*a)/b]) - Sqrt[b]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]))/(32*d^2)","A",0
100,1,111,115,0.079082,"\int \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right)}{2 d}","\frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}-\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}+\frac{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d}",1,"(Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/(2*d*E^(a/b))","A",0
101,1,582,326,5.1933868,"\int x \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[x*(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{-16 \sqrt{b} c \left(\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+d 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+d x)}}{\sqrt{b}}\right)+4 \sqrt{b} \left(2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt{(c+d x)^2+1}\right) \sqrt{a+b \sinh ^{-1}(c+d x)}\right)+4 a \left(-\sqrt{2 \pi } \sqrt{b} \left(\sinh \left(\frac{2 a}{b}\right)+\cosh \left(\frac{2 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{2 \pi } \sqrt{b} \left(\sinh \left(\frac{2 a}{b}\right)-\cosh \left(\frac{2 a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+8 \cosh \left(2 \sinh ^{-1}(c+d x)\right) \sqrt{a+b \sinh ^{-1}(c+d x)}\right)+\sqrt{b} \left(\sqrt{2 \pi } (4 a-3 b) \left(\sinh \left(\frac{2 a}{b}\right)+\cosh \left(\frac{2 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{2 \pi } (4 a+3 b) \left(\cosh \left(\frac{2 a}{b}\right)-\sinh \left(\frac{2 a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+8 \sqrt{b} \left(4 \sinh ^{-1}(c+d x) \cosh \left(2 \sinh ^{-1}(c+d x)\right)-3 \sinh \left(2 \sinh ^{-1}(c+d x)\right)\right) \sqrt{a+b \sinh ^{-1}(c+d x)}\right)-64 a c e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right)}{128 d^2}","-\frac{3 \sqrt{\pi } b^{3/2} c e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d^2}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d^2}-\frac{3 \sqrt{\pi } b^{3/2} c e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d^2}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d^2}-\frac{3 b \sinh \left(2 \sinh ^{-1}(c+d x)\right) \sqrt{a+b \sinh ^{-1}(c+d x)}}{16 d^2}+\frac{3 b c \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d^2}-\frac{c (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{d^2}+\frac{\cosh \left(2 \sinh ^{-1}(c+d x)\right) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{4 d^2}",1,"((-64*a*c*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) - 16*Sqrt[b]*c*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(-3*Sqrt[1 + (c + d*x)^2] + 2*(c + d*x)*ArcSinh[c + d*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + 4*a*(8*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + Sqrt[b]*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(-Cosh[(2*a)/b] + Sinh[(2*a)/b]) - Sqrt[b]*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b])) + Sqrt[b]*((4*a + 3*b)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) + (4*a - 3*b)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + 8*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(4*ArcSinh[c + d*x]*Cosh[2*ArcSinh[c + d*x]] - 3*Sinh[2*ArcSinh[c + d*x]])))/(128*d^2)","A",0
102,1,272,150,0.1476948,"\int \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{\sqrt{b} \left(\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+d 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+d x)}}{\sqrt{b}}\right)+4 \sqrt{b} \left(2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt{(c+d x)^2+1}\right) \sqrt{a+b \sinh ^{-1}(c+d x)}\right)}{8 d}+\frac{a e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right)}{2 d}","\frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}-\frac{3 b \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{d}",1,"(a*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/(2*d*E^(a/b)) + (Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(-3*Sqrt[1 + (c + d*x)^2] + 2*(c + d*x)*ArcSinh[c + d*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(8*d)","A",0
103,1,939,389,10.1908328,"\int x \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[x*(a + b*ArcSinh[c + d*x])^(5/2),x]","\frac{480 c \sqrt{\pi } \cosh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right) b^{5/2}-15 \sqrt{2 \pi } \cosh \left(\frac{2 a}{b}\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right) b^{5/2}-480 c \sqrt{\pi } \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right) \sinh \left(\frac{a}{b}\right) b^{5/2}+15 \sqrt{2 \pi } \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right) \sinh \left(\frac{2 a}{b}\right) b^{5/2}-15 \sqrt{2 \pi } \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right) \left(\cosh \left(\frac{2 a}{b}\right)+\sinh \left(\frac{2 a}{b}\right)\right) b^{5/2}+128 \sinh ^{-1}(c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left(2 \sinh ^{-1}(c+d x)\right) b^2+120 \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left(2 \sinh ^{-1}(c+d x)\right) b^2-160 \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} \sinh \left(2 \sinh ^{-1}(c+d x)\right) b^2-1920 c^2 \sqrt{a+b \sinh ^{-1}(c+d x)} b^2-512 c^2 \sinh ^{-1}(c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)} b^2-512 c d x \sinh ^{-1}(c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)} b^2-1920 c d x \sqrt{a+b \sinh ^{-1}(c+d x)} b^2+1280 c \sqrt{c^2+2 d x c+d^2 x^2+1} \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} b^2+256 a \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left(2 \sinh ^{-1}(c+d x)\right) b+\frac{256 a^2 c e^{a/b} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right) b}{\sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{256 a^2 c e^{-\frac{a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right) b}{\sqrt{a+b \sinh ^{-1}(c+d x)}}-160 a \sqrt{a+b \sinh ^{-1}(c+d x)} \sinh \left(2 \sinh ^{-1}(c+d x)\right) b-1024 a c^2 \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} b-1024 a c d x \sinh ^{-1}(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)} b+1280 a c \sqrt{c^2+2 d x c+d^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)} b-128 a^2 c \sqrt{\pi } \cosh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right) \sqrt{b}+128 a^2 c \sqrt{\pi } \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right) \sinh \left(\frac{a}{b}\right) \sqrt{b}+32 \left(4 a^2-15 b^2\right) c \sqrt{\pi } \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right) \left(\cosh \left(\frac{a}{b}\right)+\sinh \left(\frac{a}{b}\right)\right) \sqrt{b}+128 a^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \cosh \left(2 \sinh ^{-1}(c+d x)\right)}{512 d^2}","-\frac{15 \sqrt{\pi } b^{5/2} c e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{256 d^2}+\frac{15 \sqrt{\pi } b^{5/2} c e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d^2}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{256 d^2}-\frac{15 b^2 c (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac{15 b^2 \cosh \left(2 \sinh ^{-1}(c+d x)\right) \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d^2}+\frac{5 b c \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{2 d^2}-\frac{5 b \sinh \left(2 \sinh ^{-1}(c+d x)\right) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{16 d^2}-\frac{c (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{d^2}+\frac{\cosh \left(2 \sinh ^{-1}(c+d x)\right) \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{4 d^2}",1,"(-1920*b^2*c^2*Sqrt[a + b*ArcSinh[c + d*x]] - 1920*b^2*c*d*x*Sqrt[a + b*ArcSinh[c + d*x]] + 1280*a*b*c*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]*Sqrt[a + b*ArcSinh[c + d*x]] - 1024*a*b*c^2*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c + d*x]] - 1024*a*b*c*d*x*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c + d*x]] + 1280*b^2*c*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c + d*x]] - 512*b^2*c^2*ArcSinh[c + d*x]^2*Sqrt[a + b*ArcSinh[c + d*x]] - 512*b^2*c*d*x*ArcSinh[c + d*x]^2*Sqrt[a + b*ArcSinh[c + d*x]] + 128*a^2*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + 120*b^2*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + 256*a*b*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] + 128*b^2*ArcSinh[c + d*x]^2*Sqrt[a + b*ArcSinh[c + d*x]]*Cosh[2*ArcSinh[c + d*x]] - 128*a^2*Sqrt[b]*c*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] + 480*b^(5/2)*c*Sqrt[Pi]*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - 15*b^(5/2)*Sqrt[2*Pi]*Cosh[(2*a)/b]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] + (256*a^2*b*c*E^(a/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a + b*ArcSinh[c + d*x]] + (256*a^2*b*c*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)])/(E^(a/b)*Sqrt[a + b*ArcSinh[c + d*x]]) + 128*a^2*Sqrt[b]*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] - 480*b^(5/2)*c*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 32*Sqrt[b]*(4*a^2 - 15*b^2)*c*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) + 15*b^(5/2)*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] - 15*b^(5/2)*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) - 160*a*b*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[2*ArcSinh[c + d*x]] - 160*b^2*ArcSinh[c + d*x]*Sqrt[a + b*ArcSinh[c + d*x]]*Sinh[2*ArcSinh[c + d*x]])/(512*d^2)","B",0
104,1,458,179,2.4458388,"\int \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(5/2),x]","\frac{\sqrt{b} \left(\sqrt{\pi } \left(4 a^2-12 a b+15 b^2\right) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{\pi } \left(4 a^2+12 a b+15 b^2\right) \left(\sinh \left(\frac{a}{b}\right)-\cosh \left(\frac{a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+4 \sqrt{b} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(2 \sqrt{(c+d x)^2+1} \left(a-5 b \sinh ^{-1}(c+d x)\right)+b (c+d x) \left(4 \sinh ^{-1}(c+d x)^2+15\right)\right)\right)+8 a^2 e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right)+4 a \sqrt{b} \left(\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+d 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+d x)}}{\sqrt{b}}\right)+4 \sqrt{b} \left(2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt{(c+d x)^2+1}\right) \sqrt{a+b \sinh ^{-1}(c+d x)}\right)}{16 d}","\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}+\frac{15 b^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{2 d}+\frac{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{d}",1,"((8*a^2*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) + 4*a*Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(-3*Sqrt[1 + (c + d*x)^2] + 2*(c + d*x)*ArcSinh[c + d*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(2*Sqrt[1 + (c + d*x)^2]*(a - 5*b*ArcSinh[c + d*x]) + b*(c + d*x)*(15 + 4*ArcSinh[c + d*x]^2)) + (4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(-Cosh[a/b] + Sinh[a/b]) + (4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(16*d)","B",0
105,1,471,411,1.0989199,"\int \frac{x^2}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","Integrate[x^2/Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{\sqrt{\pi } \left(3 \left(4 c^2-1\right) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-12 c^2 \sinh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+12 c^2 \cosh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{3} \sinh \left(\frac{3 a}{b}\right) \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+6 \sqrt{2} c \left(\sinh \left(\frac{2 a}{b}\right)+\cosh \left(\frac{2 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{3} \cosh \left(\frac{3 a}{b}\right) \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+6 \sqrt{2} c \sinh \left(\frac{2 a}{b}\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+3 \sinh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-\sqrt{3} \sinh \left(\frac{3 a}{b}\right) \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-6 \sqrt{2} c \cosh \left(\frac{2 a}{b}\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-3 \cosh \left(\frac{a}{b}\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{3} \cosh \left(\frac{3 a}{b}\right) \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)\right)}{24 \sqrt{b} d^3}","\frac{\sqrt{\pi } c^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^3}+\frac{\sqrt{\pi } c^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} c e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^3}-\frac{\sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{3}} e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{2}} c e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^3}-\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{3}} e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d^3}",1,"(Sqrt[Pi]*(Sqrt[3]*Cosh[(3*a)/b]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - 3*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] + 12*c^2*Cosh[a/b]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]] - 6*Sqrt[2]*c*Cosh[(2*a)/b]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] + Sqrt[3]*Cosh[(3*a)/b]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] + 3*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] - 12*c^2*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 3*(-1 + 4*c^2)*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]) + 6*Sqrt[2]*c*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] + 6*Sqrt[2]*c*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + Sqrt[3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(3*a)/b] - Sqrt[3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*Sinh[(3*a)/b]))/(24*Sqrt[b]*d^3)","A",1
106,1,217,204,1.0200958,"\int \frac{x}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","Integrate[x/Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{\frac{e^{-\frac{a}{b}} \left(4 c e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-4 c \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)\right)}{\sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{\sqrt{2 \pi } \left(\left(\sinh \left(\frac{2 a}{b}\right)+\cosh \left(\frac{2 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\left(\sinh \left(\frac{2 a}{b}\right)-\cosh \left(\frac{2 a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)\right)}{\sqrt{b}}}{8 d^2}","-\frac{\sqrt{\pi } c e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^2}-\frac{\sqrt{\frac{\pi }{2}} e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 \sqrt{b} d^2}-\frac{\sqrt{\pi } c e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d^2}+\frac{\sqrt{\frac{\pi }{2}} e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 \sqrt{b} d^2}",1,"((4*c*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]] - 4*c*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/(E^(a/b)*Sqrt[a + b*ArcSinh[c + d*x]]) - (Sqrt[2*Pi]*(Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(-Cosh[(2*a)/b] + Sinh[(2*a)/b]) + Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b])))/Sqrt[b])/(8*d^2)","A",0
107,1,111,92,0.1134718,"\int \frac{1}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^{-\frac{a}{b}} \left(\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)}{2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}","\frac{\sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d}",1,"(-(E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]]) + Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/(2*d*E^(a/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
108,1,301,269,2.694233,"\int \frac{x}{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[x/(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{\sqrt{2 \pi } \left(\sinh \left(\frac{2 a}{b}\right)+\cosh \left(\frac{2 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{2 \pi } \left(\cosh \left(\frac{2 a}{b}\right)-\sinh \left(\frac{2 a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-\frac{2 \sqrt{b} \sinh \left(2 \sinh ^{-1}(c+d x)\right)}{\sqrt{a+b \sinh ^{-1}(c+d x)}}}{2 b^{3/2} d^2}-\frac{c e^{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(-e^{a/b} \left(e^{2 \sinh ^{-1}(c+d x)}+1\right)+e^{\frac{2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+e^{\sinh ^{-1}(c+d x)} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)\right)}{b d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}","\frac{\sqrt{\pi } c e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d^2}+\frac{\sqrt{\frac{\pi }{2}} e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d^2}-\frac{\sqrt{\pi } c e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d^2}+\frac{\sqrt{\frac{\pi }{2}} e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d^2}+\frac{2 c \sqrt{(c+d x)^2+1}}{b d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 (c+d x) \sqrt{(c+d x)^2+1}}{b d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}",1,"-((c*(-(E^(a/b)*(1 + E^(2*ArcSinh[c + d*x]))) + E^((2*a)/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]] + E^ArcSinh[c + d*x]*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)]))/(b*d^2*E^((a + b*ArcSinh[c + d*x])/b)*Sqrt[a + b*ArcSinh[c + d*x]])) + (Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) + Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) - (2*Sqrt[b]*Sinh[2*ArcSinh[c + d*x]])/Sqrt[a + b*ArcSinh[c + d*x]])/(2*b^(3/2)*d^2)","A",0
109,1,155,122,0.0965751,"\int \frac{1}{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-3/2),x]","\frac{e^{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(-e^{a/b} \left(e^{2 \sinh ^{-1}(c+d x)}+1\right)+e^{\frac{2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+e^{\sinh ^{-1}(c+d x)} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)\right)}{b d \sqrt{a+b \sinh ^{-1}(c+d x)}}","-\frac{\sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{2 \sqrt{(c+d x)^2+1}}{b d \sqrt{a+b \sinh ^{-1}(c+d x)}}",1,"(-(E^(a/b)*(1 + E^(2*ArcSinh[c + d*x]))) + E^((2*a)/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]] + E^ArcSinh[c + d*x]*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/(b*d*E^((a + b*ArcSinh[c + d*x])/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
110,1,375,365,2.7309728,"\int \frac{x}{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[x/(a + b*ArcSinh[c + d*x])^(5/2),x]","\frac{-2 \sqrt{2 \pi } \left(\sinh \left(\frac{2 a}{b}\right)+\cosh \left(\frac{2 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+2 \sqrt{2 \pi } \left(\cosh \left(\frac{2 a}{b}\right)-\sinh \left(\frac{2 a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)-\frac{\sqrt{b} \left(4 \cosh \left(2 \sinh ^{-1}(c+d x)\right) \left(a+b \sinh ^{-1}(c+d x)\right)+b \sinh \left(2 \sinh ^{-1}(c+d x)\right)\right)}{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}+\frac{\sqrt{b} c e^{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(e^{a/b} \left(2 a e^{2 \sinh ^{-1}(c+d x)}-2 a+b e^{2 \sinh ^{-1}(c+d x)}+2 b \left(e^{2 \sinh ^{-1}(c+d x)}-1\right) \sinh ^{-1}(c+d x)+b\right)+2 b e^{\sinh ^{-1}(c+d x)} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+2 e^{\frac{2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)}{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}}{3 b^{5/2} d^2}","-\frac{2 \sqrt{\pi } c e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d^2}-\frac{2 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d^2}-\frac{2 \sqrt{\pi } c e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d^2}+\frac{2 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d^2}-\frac{8 (c+d x)^2}{3 b^2 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{4 c (c+d x)}{3 b^2 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4}{3 b^2 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 \sqrt{(c+d x)^2+1} (c+d x)}{3 b d^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}+\frac{2 c \sqrt{(c+d x)^2+1}}{3 b d^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}",1,"((Sqrt[b]*c*(E^(a/b)*(-2*a + b + 2*a*E^(2*ArcSinh[c + d*x]) + b*E^(2*ArcSinh[c + d*x]) + 2*b*(-1 + E^(2*ArcSinh[c + d*x]))*ArcSinh[c + d*x]) + 2*E^((2*a)/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, a/b + ArcSinh[c + d*x]] + 2*b*E^ArcSinh[c + d*x]*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)]))/(E^((a + b*ArcSinh[c + d*x])/b)*(a + b*ArcSinh[c + d*x])^(3/2)) + 2*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) - 2*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) - (Sqrt[b]*(4*(a + b*ArcSinh[c + d*x])*Cosh[2*ArcSinh[c + d*x]] + b*Sinh[2*ArcSinh[c + d*x]]))/(a + b*ArcSinh[c + d*x])^(3/2))/(3*b^(5/2)*d^2)","A",0
111,1,207,158,0.5569307,"\int \frac{1}{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-5/2),x]","\frac{e^{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(-e^{a/b} \left(2 a \left(e^{2 \sinh ^{-1}(c+d x)}-1\right)-2 b \sinh ^{-1}(c+d x)+b e^{2 \sinh ^{-1}(c+d x)} \left(2 \sinh ^{-1}(c+d x)+1\right)+b\right)-2 b e^{\sinh ^{-1}(c+d x)} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-2 e^{\frac{2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)}{3 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}","\frac{2 \sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{2 \sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}-\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 \sqrt{(c+d x)^2+1}}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}",1,"(-(E^(a/b)*(b + 2*a*(-1 + E^(2*ArcSinh[c + d*x])) - 2*b*ArcSinh[c + d*x] + b*E^(2*ArcSinh[c + d*x])*(1 + 2*ArcSinh[c + d*x]))) - 2*E^((2*a)/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, a/b + ArcSinh[c + d*x]] - 2*b*E^ArcSinh[c + d*x]*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/(3*b^2*d*E^((a + b*ArcSinh[c + d*x])/b)*(a + b*ArcSinh[c + d*x])^(3/2))","A",0
112,1,429,445,2.5915132,"\int \frac{x}{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[x/(a + b*ArcSinh[c + d*x])^(7/2),x]","\frac{\frac{\sqrt{b} \left(-\sinh \left(2 \sinh ^{-1}(c+d x)\right) \left(16 \left(a+b \sinh ^{-1}(c+d x)\right)^2+3 b^2\right)-4 b \cosh \left(2 \sinh ^{-1}(c+d x)\right) \left(a+b \sinh ^{-1}(c+d x)\right)\right)}{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}+8 \sqrt{2 \pi } \left(\sinh \left(\frac{2 a}{b}\right)+\cosh \left(\frac{2 a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+8 \sqrt{2 \pi } \left(\cosh \left(\frac{2 a}{b}\right)-\sinh \left(\frac{2 a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d^2}-\frac{c \left(-2 e^{-\sinh ^{-1}(c+d x)} \left(4 a^2+2 a b \left(4 \sinh ^{-1}(c+d x)-1\right)+b^2 \left(4 \sinh ^{-1}(c+d x)^2-2 \sinh ^{-1}(c+d x)+3\right)\right)+8 e^{a/b} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^2 \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-4 e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(2 a+2 b \sinh ^{-1}(c+d x)+b\right)+2 b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)\right)-6 b^2 e^{\sinh ^{-1}(c+d x)}\right)}{30 b^3 d^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}","\frac{4 \sqrt{\pi } c e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d^2}+\frac{8 \sqrt{2 \pi } e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d^2}-\frac{4 \sqrt{\pi } c e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d^2}+\frac{8 \sqrt{2 \pi } e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d^2}-\frac{32 \sqrt{(c+d x)^2+1} (c+d x)}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{8 c \sqrt{(c+d x)^2+1}}{15 b^3 d^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 (c+d x)^2}{15 b^2 d^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}+\frac{4 c (c+d x)}{15 b^2 d^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{4}{15 b^2 d^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{2 \sqrt{(c+d x)^2+1} (c+d x)}{5 b d^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}+\frac{2 c \sqrt{(c+d x)^2+1}}{5 b d^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}",1,"-1/30*(c*(-6*b^2*E^ArcSinh[c + d*x] - (2*(4*a^2 + 2*a*b*(-1 + 4*ArcSinh[c + d*x]) + b^2*(3 - 2*ArcSinh[c + d*x] + 4*ArcSinh[c + d*x]^2)))/E^ArcSinh[c + d*x] + 8*E^(a/b)*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])^2*Gamma[1/2, a/b + ArcSinh[c + d*x]] - (4*(a + b*ArcSinh[c + d*x])*(E^(a/b + ArcSinh[c + d*x])*(2*a + b + 2*b*ArcSinh[c + d*x]) + 2*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b)))/(b^3*d^2*(a + b*ArcSinh[c + d*x])^(5/2)) + (8*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] - Sinh[(2*a)/b]) + 8*Sqrt[2*Pi]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]]*(Cosh[(2*a)/b] + Sinh[(2*a)/b]) + (Sqrt[b]*(-4*b*(a + b*ArcSinh[c + d*x])*Cosh[2*ArcSinh[c + d*x]] - (3*b^2 + 16*(a + b*ArcSinh[c + d*x])^2)*Sinh[2*ArcSinh[c + d*x]]))/(a + b*ArcSinh[c + d*x])^(5/2))/(15*b^(7/2)*d^2)","A",0
113,1,238,195,0.5437652,"\int \frac{1}{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-7/2),x]","\frac{-2 e^{-\sinh ^{-1}(c+d x)} \left(4 a^2+2 a b \left(4 \sinh ^{-1}(c+d x)-1\right)+b^2 \left(4 \sinh ^{-1}(c+d x)^2-2 \sinh ^{-1}(c+d x)+3\right)\right)+8 e^{a/b} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^2 \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-4 e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(2 a+2 b \sinh ^{-1}(c+d x)+b\right)+2 b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)\right)-6 b^2 e^{\sinh ^{-1}(c+d x)}}{30 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}","-\frac{4 \sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{4 \sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}-\frac{8 \sqrt{(c+d x)^2+1}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4 (c+d x)}{15 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{2 \sqrt{(c+d x)^2+1}}{5 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}",1,"(-6*b^2*E^ArcSinh[c + d*x] - (2*(4*a^2 + 2*a*b*(-1 + 4*ArcSinh[c + d*x]) + b^2*(3 - 2*ArcSinh[c + d*x] + 4*ArcSinh[c + d*x]^2)))/E^ArcSinh[c + d*x] + 8*E^(a/b)*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])^2*Gamma[1/2, a/b + ArcSinh[c + d*x]] - (4*(a + b*ArcSinh[c + d*x])*(E^(a/b + ArcSinh[c + d*x])*(2*a + b + 2*b*ArcSinh[c + d*x]) + 2*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b))/(30*b^3*d*(a + b*ArcSinh[c + d*x])^(5/2))","A",0
114,1,79,91,0.0475394,"\int (c e+d e x)^m \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x]),x]","-\frac{(c+d x) (e (c+d x))^m \left(b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-(c+d x)^2\right)-(m+2) \left(a+b \sinh ^{-1}(c+d x)\right)\right)}{d (m+1) (m+2)}","\frac{(e (c+d x))^{m+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{d e (m+1)}-\frac{b (e (c+d x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-(c+d x)^2\right)}{d e^2 (m+1) (m+2)}",1,"-(((c + d*x)*(e*(c + d*x))^m*(-((2 + m)*(a + b*ArcSinh[c + d*x])) + b*(c + d*x)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c + d*x)^2]))/(d*(1 + m)*(2 + m)))","A",1
115,1,71,100,0.1027794,"\int (c e+d e x)^4 \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x]),x]","\frac{e^4 \left(\frac{1}{5} (c+d x)^5 \left(a+b \sinh ^{-1}(c+d x)\right)-\frac{1}{75} b \sqrt{(c+d x)^2+1} \left(-10 (c+d x)^2+3 \left((c+d x)^2+1\right)^2+5\right)\right)}{d}","\frac{e^4 (c+d x)^5 \left(a+b \sinh ^{-1}(c+d x)\right)}{5 d}-\frac{b e^4 \left((c+d x)^2+1\right)^{5/2}}{25 d}+\frac{2 b e^4 \left((c+d x)^2+1\right)^{3/2}}{15 d}-\frac{b e^4 \sqrt{(c+d x)^2+1}}{5 d}",1,"(e^4*(-1/75*(b*Sqrt[1 + (c + d*x)^2]*(5 - 10*(c + d*x)^2 + 3*(1 + (c + d*x)^2)^2)) + ((c + d*x)^5*(a + b*ArcSinh[c + d*x]))/5))/d","A",1
116,1,83,105,0.0758118,"\int (c e+d e x)^3 \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x]),x]","\frac{e^3 \left(8 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)-2 b \sqrt{(c+d x)^2+1} (c+d x)^3+3 b \sqrt{(c+d x)^2+1} (c+d x)-3 b \sinh ^{-1}(c+d x)\right)}{32 d}","\frac{e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)}{4 d}-\frac{b e^3 \sqrt{(c+d x)^2+1} (c+d x)^3}{16 d}+\frac{3 b e^3 \sqrt{(c+d x)^2+1} (c+d x)}{32 d}-\frac{3 b e^3 \sinh ^{-1}(c+d x)}{32 d}",1,"(e^3*(3*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] - 2*b*(c + d*x)^3*Sqrt[1 + (c + d*x)^2] - 3*b*ArcSinh[c + d*x] + 8*(c + d*x)^4*(a + b*ArcSinh[c + d*x])))/(32*d)","A",1
117,1,64,76,0.0455419,"\int (c e+d e x)^2 \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x]),x]","\frac{e^2 \left(\frac{1}{3} (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)-\frac{1}{9} b \left(c^2+2 c d x+d^2 x^2-2\right) \sqrt{(c+d x)^2+1}\right)}{d}","\frac{e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)}{3 d}-\frac{b e^2 \left((c+d x)^2+1\right)^{3/2}}{9 d}+\frac{b e^2 \sqrt{(c+d x)^2+1}}{3 d}",1,"(e^2*(-1/9*(b*(-2 + c^2 + 2*c*d*x + d^2*x^2)*Sqrt[1 + (c + d*x)^2]) + ((c + d*x)^3*(a + b*ArcSinh[c + d*x]))/3))/d","A",1
118,1,57,68,0.0638027,"\int (c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x]),x]","\frac{e \left(2 (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)-b \sqrt{(c+d x)^2+1} (c+d x)+b \sinh ^{-1}(c+d x)\right)}{4 d}","\frac{e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)}{2 d}-\frac{b e \sqrt{(c+d x)^2+1} (c+d x)}{4 d}+\frac{b e \sinh ^{-1}(c+d x)}{4 d}",1,"(e*(-(b*(c + d*x)*Sqrt[1 + (c + d*x)^2]) + b*ArcSinh[c + d*x] + 2*(c + d*x)^2*(a + b*ArcSinh[c + d*x])))/(4*d)","A",1
119,1,50,39,0.0305389,"\int \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[a + b*ArcSinh[c + d*x],x]","a x-\frac{b \left(\sqrt{c^2+2 c d x+d^2 x^2+1}-c \sinh ^{-1}(c+d x)\right)}{d}+b x \sinh ^{-1}(c+d x)","a x-\frac{b \sqrt{(c+d x)^2+1}}{d}+\frac{b (c+d x) \sinh ^{-1}(c+d x)}{d}",1,"a*x + b*x*ArcSinh[c + d*x] - (b*(Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2] - c*ArcSinh[c + d*x]))/d","A",1
120,1,70,81,0.0259251,"\int \frac{a+b \sinh ^{-1}(c+d x)}{c e+d e x} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x),x]","\frac{b^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c+d x)}\right)-\left(a+b \sinh ^{-1}(c+d x)\right) \left(a+b \sinh ^{-1}(c+d x)-2 b \log \left(1-e^{2 \sinh ^{-1}(c+d x)}\right)\right)}{2 b d e}","\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{2 b d e}+\frac{\log \left(1-e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e}-\frac{b \text{Li}_2\left(e^{-2 \sinh ^{-1}(c+d x)}\right)}{2 d e}",1,"(-((a + b*ArcSinh[c + d*x])*(a + b*ArcSinh[c + d*x] - 2*b*Log[1 - E^(2*ArcSinh[c + d*x])])) + b^2*PolyLog[2, E^(2*ArcSinh[c + d*x])])/(2*b*d*e)","A",0
121,1,43,49,0.0394595,"\int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^2,x]","-\frac{\frac{a+b \sinh ^{-1}(c+d x)}{c+d x}+b \tanh ^{-1}\left(\sqrt{(c+d x)^2+1}\right)}{d e^2}","-\frac{a+b \sinh ^{-1}(c+d x)}{d e^2 (c+d x)}-\frac{b \tanh ^{-1}\left(\sqrt{(c+d x)^2+1}\right)}{d e^2}",1,"-(((a + b*ArcSinh[c + d*x])/(c + d*x) + b*ArcTanh[Sqrt[1 + (c + d*x)^2]])/(d*e^2))","A",1
122,1,57,59,0.0577868,"\int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^3,x]","\frac{\frac{-a-b \sinh ^{-1}(c+d x)}{2 (c+d x)^2}-\frac{b \sqrt{(c+d x)^2+1}}{2 (c+d x)}}{d e^3}","-\frac{a+b \sinh ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac{b \sqrt{(c+d x)^2+1}}{2 d e^3 (c+d x)}",1,"(-1/2*(b*Sqrt[1 + (c + d*x)^2])/(c + d*x) + (-a - b*ArcSinh[c + d*x])/(2*(c + d*x)^2))/(d*e^3)","A",1
123,1,82,84,0.085601,"\int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^4,x]","-\frac{2 a+b \sqrt{c^2+2 c d x+d^2 x^2+1} (c+d x)+2 b \sinh ^{-1}(c+d x)-b (c+d x)^3 \tanh ^{-1}\left(\sqrt{(c+d x)^2+1}\right)}{6 d e^4 (c+d x)^3}","-\frac{a+b \sinh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}-\frac{b \sqrt{(c+d x)^2+1}}{6 d e^4 (c+d x)^2}+\frac{b \tanh ^{-1}\left(\sqrt{(c+d x)^2+1}\right)}{6 d e^4}",1,"-1/6*(2*a + b*(c + d*x)*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2] + 2*b*ArcSinh[c + d*x] - b*(c + d*x)^3*ArcTanh[Sqrt[1 + (c + d*x)^2]])/(d*e^4*(c + d*x)^3)","A",1
124,1,61,90,0.0753137,"\int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^5} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^5,x]","-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)+b (c+d x) \sqrt{(c+d x)^2+1} \left(1-2 (c+d x)^2\right)}{12 d e^5 (c+d x)^4}","-\frac{a+b \sinh ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}+\frac{b \sqrt{(c+d x)^2+1}}{6 d e^5 (c+d x)}-\frac{b \sqrt{(c+d x)^2+1}}{12 d e^5 (c+d x)^3}",1,"-1/12*(b*(c + d*x)*(1 - 2*(c + d*x)^2)*Sqrt[1 + (c + d*x)^2] + 3*(a + b*ArcSinh[c + d*x]))/(d*e^5*(c + d*x)^4)","A",1
125,1,61,115,0.0819803,"\int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^6} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^6,x]","-\frac{\frac{a+b \sinh ^{-1}(c+d x)}{(c+d x)^5}+b \sqrt{(c+d x)^2+1} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};(c+d x)^2+1\right)}{5 d e^6}","-\frac{a+b \sinh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac{3 b \sqrt{(c+d x)^2+1}}{40 d e^6 (c+d x)^2}-\frac{b \sqrt{(c+d x)^2+1}}{20 d e^6 (c+d x)^4}-\frac{3 b \tanh ^{-1}\left(\sqrt{(c+d x)^2+1}\right)}{40 d e^6}",1,"-1/5*((a + b*ArcSinh[c + d*x])/(c + d*x)^5 + b*Sqrt[1 + (c + d*x)^2]*Hypergeometric2F1[1/2, 3, 3/2, 1 + (c + d*x)^2])/(d*e^6)","C",1
126,1,155,187,0.1195655,"\int (c e+d e x)^m \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^2,x]","\frac{(c+d x) (e (c+d x))^m \left(\frac{2 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2};\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2};-(c+d x)^2\right)}{(m+2) (m+3)}-\frac{2 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{m+2}+\left(a+b \sinh ^{-1}(c+d x)\right)^2\right)}{d (m+1)}","\frac{2 b^2 (e (c+d x))^{m+3} \, _3F_2\left(1,\frac{m}{2}+\frac{3}{2},\frac{m}{2}+\frac{3}{2};\frac{m}{2}+2,\frac{m}{2}+\frac{5}{2};-(c+d x)^2\right)}{d e^3 (m+1) (m+2) (m+3)}-\frac{2 b (e (c+d x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^2 (m+1) (m+2)}+\frac{(e (c+d x))^{m+1} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e (m+1)}",1,"((c + d*x)*(e*(c + d*x))^m*((a + b*ArcSinh[c + d*x])^2 - (2*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, -(c + d*x)^2])/(2 + m) + (2*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 3/2 + m/2, 3/2 + m/2}, {2 + m/2, 5/2 + m/2}, -(c + d*x)^2])/((2 + m)*(3 + m))))/(d*(1 + m))","A",1
127,1,192,197,0.2595633,"\int (c e+d e x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^2,x]","\frac{e^4 \left(9 \left(25 a^2+2 b^2\right) (c+d x)^5+30 a b \sqrt{(c+d x)^2+1} \left(-3 (c+d x)^4+4 (c+d x)^2-8\right)+30 b \sinh ^{-1}(c+d x) \left(15 a (c+d x)^5-3 b \sqrt{(c+d x)^2+1} (c+d x)^4+4 b \sqrt{(c+d x)^2+1} (c+d x)^2-8 b \sqrt{(c+d x)^2+1}\right)-40 b^2 (c+d x)^3+240 b^2 (c+d x)+225 b^2 (c+d x)^5 \sinh ^{-1}(c+d x)^2\right)}{1125 d}","\frac{e^4 (c+d x)^5 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{5 d}-\frac{2 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)}{25 d}+\frac{8 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)}{75 d}-\frac{16 b e^4 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{75 d}+\frac{2 b^2 e^4 (c+d x)^5}{125 d}-\frac{8 b^2 e^4 (c+d x)^3}{225 d}+\frac{16}{75} b^2 e^4 x",1,"(e^4*(240*b^2*(c + d*x) - 40*b^2*(c + d*x)^3 + 9*(25*a^2 + 2*b^2)*(c + d*x)^5 + 30*a*b*Sqrt[1 + (c + d*x)^2]*(-8 + 4*(c + d*x)^2 - 3*(c + d*x)^4) + 30*b*(15*a*(c + d*x)^5 - 8*b*Sqrt[1 + (c + d*x)^2] + 4*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] - 3*b*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 225*b^2*(c + d*x)^5*ArcSinh[c + d*x]^2))/(1125*d)","A",1
128,1,170,172,0.2030975,"\int (c e+d e x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^2,x]","\frac{e^3 \left(\left(8 a^2+b^2\right) (c+d x)^4+2 a b \left(3-2 (c+d x)^2\right) \sqrt{(c+d x)^2+1} (c+d x)+2 b (c+d x) \sinh ^{-1}(c+d x) \left(8 a (c+d x)^3-2 b \sqrt{(c+d x)^2+1} (c+d x)^2+3 b \sqrt{(c+d x)^2+1}\right)-6 a b \sinh ^{-1}(c+d x)-3 b^2 (c+d x)^2+b^2 \left(8 (c+d x)^4-3\right) \sinh ^{-1}(c+d x)^2\right)}{32 d}","\frac{e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{4 d}-\frac{b e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)}{8 d}+\frac{3 b e^3 \sqrt{(c+d x)^2+1} (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)}{16 d}-\frac{3 e^3 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{32 d}+\frac{b^2 e^3 (c+d x)^4}{32 d}-\frac{3 b^2 e^3 (c+d x)^2}{32 d}",1,"(e^3*(-3*b^2*(c + d*x)^2 + (8*a^2 + b^2)*(c + d*x)^4 + 2*a*b*(c + d*x)*(3 - 2*(c + d*x)^2)*Sqrt[1 + (c + d*x)^2] - 6*a*b*ArcSinh[c + d*x] + 2*b*(c + d*x)*(8*a*(c + d*x)^3 + 3*b*Sqrt[1 + (c + d*x)^2] - 2*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + b^2*(-3 + 8*(c + d*x)^4)*ArcSinh[c + d*x]^2))/(32*d)","A",1
129,1,147,136,0.1757714,"\int (c e+d e x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^2,x]","\frac{e^2 \left(\left(9 a^2+2 b^2\right) (c+d x)^3+6 a b \left(2-(c+d x)^2\right) \sqrt{(c+d x)^2+1}+6 b \sinh ^{-1}(c+d x) \left(3 a (c+d x)^3-b \sqrt{(c+d x)^2+1} (c+d x)^2+2 b \sqrt{(c+d x)^2+1}\right)-12 b^2 (c+d x)+9 b^2 (c+d x)^3 \sinh ^{-1}(c+d x)^2\right)}{27 d}","\frac{e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{3 d}-\frac{2 b e^2 \sqrt{(c+d x)^2+1} (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)}{9 d}+\frac{4 b e^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{9 d}+\frac{2 b^2 e^2 (c+d x)^3}{27 d}-\frac{4}{9} b^2 e^2 x",1,"(e^2*(-12*b^2*(c + d*x) + (9*a^2 + 2*b^2)*(c + d*x)^3 + 6*a*b*(2 - (c + d*x)^2)*Sqrt[1 + (c + d*x)^2] + 6*b*(3*a*(c + d*x)^3 + 2*b*Sqrt[1 + (c + d*x)^2] - b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 9*b^2*(c + d*x)^3*ArcSinh[c + d*x]^2))/(27*d)","A",1
130,1,120,103,0.1849841,"\int (c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^2,x]","\frac{e \left(\left(2 a^2+b^2\right) (c+d x)^2-2 a b \sqrt{(c+d x)^2+1} (c+d x)+2 b (c+d x) \sinh ^{-1}(c+d x) \left(2 a (c+d x)-b \sqrt{(c+d x)^2+1}\right)+2 a b \sinh ^{-1}(c+d x)+b^2 \left(2 (c+d x)^2+1\right) \sinh ^{-1}(c+d x)^2\right)}{4 d}","\frac{e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{2 d}-\frac{b e \sqrt{(c+d x)^2+1} (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)}{2 d}+\frac{e \left(a+b \sinh ^{-1}(c+d x)\right)^2}{4 d}+\frac{b^2 e (c+d x)^2}{4 d}",1,"(e*((2*a^2 + b^2)*(c + d*x)^2 - 2*a*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] + 2*a*b*ArcSinh[c + d*x] + 2*b*(c + d*x)*(2*a*(c + d*x) - b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + b^2*(1 + 2*(c + d*x)^2)*ArcSinh[c + d*x]^2))/(4*d)","A",1
131,1,87,57,0.1009097,"\int \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(a + b*ArcSinh[c + d*x])^2,x]","\frac{\left(a^2+2 b^2\right) (c+d x)-2 a b \sqrt{(c+d x)^2+1}+2 b \sinh ^{-1}(c+d x) \left(a c+a d x+b \left(-\sqrt{(c+d x)^2+1}\right)\right)+b^2 (c+d x) \sinh ^{-1}(c+d x)^2}{d}","-\frac{2 b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{d}+\frac{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d}+2 b^2 x",1,"((a^2 + 2*b^2)*(c + d*x) - 2*a*b*Sqrt[1 + (c + d*x)^2] + 2*b*(a*c + a*d*x - b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + b^2*(c + d*x)*ArcSinh[c + d*x]^2)/d","A",1
132,1,100,116,0.0488148,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{c e+d e x} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x),x]","\frac{6 b^2 \text{Li}_2\left(e^{2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)-2 \left(a+b \sinh ^{-1}(c+d x)\right)^2 \left(a+b \sinh ^{-1}(c+d x)-3 b \log \left(1-e^{2 \sinh ^{-1}(c+d x)}\right)\right)-3 b^3 \text{Li}_3\left(e^{2 \sinh ^{-1}(c+d x)}\right)}{6 b d e}","-\frac{b \text{Li}_2\left(e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e}+\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{3 b d e}+\frac{\log \left(1-e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e}-\frac{b^2 \text{Li}_3\left(e^{-2 \sinh ^{-1}(c+d x)}\right)}{2 d e}",1,"(-2*(a + b*ArcSinh[c + d*x])^2*(a + b*ArcSinh[c + d*x] - 3*b*Log[1 - E^(2*ArcSinh[c + d*x])]) + 6*b^2*(a + b*ArcSinh[c + d*x])*PolyLog[2, E^(2*ArcSinh[c + d*x])] - 3*b^3*PolyLog[3, E^(2*ArcSinh[c + d*x])])/(6*b*d*e)","A",0
133,1,154,100,0.604983,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^2,x]","\frac{-\frac{a^2}{c+d x}+a b \left(2 \log \left(\frac{2 \sinh ^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)}{c+d x}\right)-\frac{2 \sinh ^{-1}(c+d x)}{c+d x}\right)+b^2 \left(2 \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)-2 \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)+\sinh ^{-1}(c+d x) \left(-\frac{\sinh ^{-1}(c+d x)}{c+d x}+2 \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)-2 \log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)\right)\right)}{d e^2}","-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e^2 (c+d x)}-\frac{4 b \tanh ^{-1}\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^2}-\frac{2 b^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c+d x)}\right)}{d e^2}+\frac{2 b^2 \text{Li}_2\left(e^{\sinh ^{-1}(c+d x)}\right)}{d e^2}",1,"(-(a^2/(c + d*x)) + a*b*((-2*ArcSinh[c + d*x])/(c + d*x) + 2*Log[(2*Sinh[ArcSinh[c + d*x]/2]^2)/(c + d*x)]) + b^2*(ArcSinh[c + d*x]*(-(ArcSinh[c + d*x]/(c + d*x)) + 2*Log[1 - E^(-ArcSinh[c + d*x])] - 2*Log[1 + E^(-ArcSinh[c + d*x])]) + 2*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 2*PolyLog[2, E^(-ArcSinh[c + d*x])]))/(d*e^2)","A",0
134,1,120,85,0.2509934,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^3,x]","-\frac{a \left(a+2 b (c+d x) \sqrt{c^2+2 c d x+d^2 x^2+1}\right)+2 b \sinh ^{-1}(c+d x) \left(a+b (c+d x) \sqrt{c^2+2 c d x+d^2 x^2+1}\right)-2 b^2 (c+d x)^2 \log (c+d x)+b^2 \sinh ^{-1}(c+d x)^2}{2 d e^3 (c+d x)^2}","-\frac{b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^3 (c+d x)}-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{2 d e^3 (c+d x)^2}+\frac{b^2 \log (c+d x)}{d e^3}",1,"-1/2*(a*(a + 2*b*(c + d*x)*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]) + 2*b*(a + b*(c + d*x)*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2])*ArcSinh[c + d*x] + b^2*ArcSinh[c + d*x]^2 - 2*b^2*(c + d*x)^2*Log[c + d*x])/(d*e^3*(c + d*x)^2)","A",1
135,1,212,169,1.7204442,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^4,x]","-\frac{4 a^2+a b \left(8 \sinh ^{-1}(c+d x)+2 \sinh \left(2 \sinh ^{-1}(c+d x)\right)+\left(\sinh \left(3 \sinh ^{-1}(c+d x)\right)-3 (c+d x)\right) \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)\right)\right)+b^2 \left(4 (c+d x)^3 \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)-4 (c+d x)^3 \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)+4 (c+d x)^2+4 \sinh ^{-1}(c+d x)^2+\sinh ^{-1}(c+d x) \left(2 \sinh \left(2 \sinh ^{-1}(c+d x)\right)+\left(\sinh \left(3 \sinh ^{-1}(c+d x)\right)-3 (c+d x)\right) \left(\log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)-\log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)\right)\right)\right)}{12 d e^4 (c+d x)^3}","-\frac{b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{3 d e^4 (c+d x)^2}-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{3 d e^4 (c+d x)^3}+\frac{2 b \tanh ^{-1}\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{3 d e^4}+\frac{b^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c+d x)}\right)}{3 d e^4}-\frac{b^2 \text{Li}_2\left(e^{\sinh ^{-1}(c+d x)}\right)}{3 d e^4}-\frac{b^2}{3 d e^4 (c+d x)}",1,"-1/12*(4*a^2 + a*b*(8*ArcSinh[c + d*x] + 2*Sinh[2*ArcSinh[c + d*x]] + Log[Tanh[ArcSinh[c + d*x]/2]]*(-3*(c + d*x) + Sinh[3*ArcSinh[c + d*x]])) + b^2*(4*(c + d*x)^2 + 4*ArcSinh[c + d*x]^2 + 4*(c + d*x)^3*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 4*(c + d*x)^3*PolyLog[2, E^(-ArcSinh[c + d*x])] + ArcSinh[c + d*x]*(2*Sinh[2*ArcSinh[c + d*x]] + (Log[1 - E^(-ArcSinh[c + d*x])] - Log[1 + E^(-ArcSinh[c + d*x])])*(-3*(c + d*x) + Sinh[3*ArcSinh[c + d*x]]))))/(d*e^4*(c + d*x)^3)","A",0
136,0,0,87,3.7488845,"\int (c e+d e x)^m \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^3,x]","\int (c e+d e x)^m \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","\frac{(e (c+d x))^{m+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d e (m+1)}-\frac{3 b \text{Int}\left(\frac{(e (c+d x))^{m+1} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{(c+d x)^2+1}},x\right)}{e (m+1)}",0,"Integrate[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^3, x]","A",-1
137,1,355,326,0.4764302,"\int (c e+d e x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^3,x]","\frac{e^4 \left(3 a \left(25 a^2+6 b^2\right) (c+d x)^5+\frac{1}{15} b \sqrt{(c+d x)^2+1} \left(-27 \left(25 a^2+2 b^2\right) (c+d x)^4+4 \left(225 a^2+68 b^2\right) (c+d x)^2-8 \left(225 a^2+518 b^2\right)\right)-b \sinh ^{-1}(c+d x) \left(-225 a^2 (c+d x)^5+90 a b \sqrt{(c+d x)^2+1} (c+d x)^4-120 a b \sqrt{(c+d x)^2+1} (c+d x)^2+240 a b \sqrt{(c+d x)^2+1}-18 b^2 (c+d x)^5+40 b^2 (c+d x)^3-240 b^2 (c+d x)\right)-40 a b^2 (c+d x)^3+240 a b^2 (c+d x)-15 b^2 \sinh ^{-1}(c+d x)^2 \left(-15 a (c+d x)^5+3 b \sqrt{(c+d x)^2+1} (c+d x)^4-4 b \sqrt{(c+d x)^2+1} (c+d x)^2+8 b \sqrt{(c+d x)^2+1}\right)+75 b^3 (c+d x)^5 \sinh ^{-1}(c+d x)^3\right)}{375 d}","\frac{6 b^2 e^4 (c+d x)^5 \left(a+b \sinh ^{-1}(c+d x)\right)}{125 d}-\frac{8 b^2 e^4 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)}{75 d}+\frac{16}{25} a b^2 e^4 x+\frac{e^4 (c+d x)^5 \left(a+b \sinh ^{-1}(c+d x)\right)^3}{5 d}-\frac{3 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{25 d}+\frac{4 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{25 d}-\frac{8 b e^4 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{25 d}-\frac{6 b^3 e^4 \left((c+d x)^2+1\right)^{5/2}}{625 d}+\frac{76 b^3 e^4 \left((c+d x)^2+1\right)^{3/2}}{1125 d}-\frac{298 b^3 e^4 \sqrt{(c+d x)^2+1}}{375 d}+\frac{16 b^3 e^4 (c+d x) \sinh ^{-1}(c+d x)}{25 d}",1,"(e^4*(240*a*b^2*(c + d*x) - 40*a*b^2*(c + d*x)^3 + 3*a*(25*a^2 + 6*b^2)*(c + d*x)^5 + (b*Sqrt[1 + (c + d*x)^2]*(-8*(225*a^2 + 518*b^2) + 4*(225*a^2 + 68*b^2)*(c + d*x)^2 - 27*(25*a^2 + 2*b^2)*(c + d*x)^4))/15 - b*(-240*b^2*(c + d*x) + 40*b^2*(c + d*x)^3 - 225*a^2*(c + d*x)^5 - 18*b^2*(c + d*x)^5 + 240*a*b*Sqrt[1 + (c + d*x)^2] - 120*a*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] + 90*a*b*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] - 15*b^2*(-15*a*(c + d*x)^5 + 8*b*Sqrt[1 + (c + d*x)^2] - 4*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] + 3*b*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 75*b^3*(c + d*x)^5*ArcSinh[c + d*x]^3))/(375*d)","A",1
138,1,303,279,0.4020733,"\int (c e+d e x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^3,x]","\frac{e^3 \left(8 a \left(8 a^2+3 b^2\right) (c+d x)^4+3 b \sqrt{(c+d x)^2+1} (c+d x) \left(3 \left(8 a^2+5 b^2\right)-2 \left(8 a^2+b^2\right) (c+d x)^2\right)-24 b (c+d x) \sinh ^{-1}(c+d x) \left(-8 a^2 (c+d x)^3+4 a b \sqrt{(c+d x)^2+1} (c+d x)^2-6 a b \sqrt{(c+d x)^2+1}-b^2 (c+d x)^3+3 b^2 (c+d x)\right)-9 b \left(8 a^2+5 b^2\right) \sinh ^{-1}(c+d x)-72 a b^2 (c+d x)^2+24 b^2 \sinh ^{-1}(c+d x)^2 \left(8 a (c+d x)^4-3 a-2 b \sqrt{(c+d x)^2+1} (c+d x)^3+3 b \sqrt{(c+d x)^2+1} (c+d x)\right)+8 b^3 \left(8 (c+d x)^4-3\right) \sinh ^{-1}(c+d x)^3\right)}{256 d}","\frac{3 b^2 e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)}{32 d}-\frac{9 b^2 e^3 (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)}{32 d}+\frac{e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^3}{4 d}-\frac{3 b e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{16 d}+\frac{9 b e^3 \sqrt{(c+d x)^2+1} (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{32 d}-\frac{3 e^3 \left(a+b \sinh ^{-1}(c+d x)\right)^3}{32 d}-\frac{3 b^3 e^3 \sqrt{(c+d x)^2+1} (c+d x)^3}{128 d}+\frac{45 b^3 e^3 \sqrt{(c+d x)^2+1} (c+d x)}{256 d}-\frac{45 b^3 e^3 \sinh ^{-1}(c+d x)}{256 d}",1,"(e^3*(-72*a*b^2*(c + d*x)^2 + 8*a*(8*a^2 + 3*b^2)*(c + d*x)^4 + 3*b*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(3*(8*a^2 + 5*b^2) - 2*(8*a^2 + b^2)*(c + d*x)^2) - 9*b*(8*a^2 + 5*b^2)*ArcSinh[c + d*x] - 24*b*(c + d*x)*(3*b^2*(c + d*x) - 8*a^2*(c + d*x)^3 - b^2*(c + d*x)^3 - 6*a*b*Sqrt[1 + (c + d*x)^2] + 4*a*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 24*b^2*(-3*a + 8*a*(c + d*x)^4 + 3*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] - 2*b*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 8*b^3*(-3 + 8*(c + d*x)^4)*ArcSinh[c + d*x]^3))/(256*d)","A",1
139,1,258,227,0.3275683,"\int (c e+d e x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^3,x]","\frac{e^2 \left(a \left(3 a^2+2 b^2\right) (c+d x)^3+\frac{1}{3} b \sqrt{(c+d x)^2+1} \left(-\left(9 a^2+2 b^2\right) (c+d x)^2+18 a^2+40 b^2\right)-b \sinh ^{-1}(c+d x) \left(-9 a^2 (c+d x)^3+6 a b \sqrt{(c+d x)^2+1} (c+d x)^2-12 a b \sqrt{(c+d x)^2+1}-2 b^2 (c+d x)^3+12 b^2 (c+d x)\right)-12 a b^2 (c+d x)-3 b^2 \sinh ^{-1}(c+d x)^2 \left(-3 a (c+d x)^3+b \sqrt{(c+d x)^2+1} (c+d x)^2-2 b \sqrt{(c+d x)^2+1}\right)+3 b^3 (c+d x)^3 \sinh ^{-1}(c+d x)^3\right)}{9 d}","\frac{2 b^2 e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)}{9 d}-\frac{4}{3} a b^2 e^2 x+\frac{2 b e^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{3 d}-\frac{b e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{3 d}+\frac{e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^3}{3 d}-\frac{2 b^3 e^2 \left((c+d x)^2+1\right)^{3/2}}{27 d}+\frac{14 b^3 e^2 \sqrt{(c+d x)^2+1}}{9 d}-\frac{4 b^3 e^2 (c+d x) \sinh ^{-1}(c+d x)}{3 d}",1,"(e^2*(-12*a*b^2*(c + d*x) + a*(3*a^2 + 2*b^2)*(c + d*x)^3 + (b*Sqrt[1 + (c + d*x)^2]*(18*a^2 + 40*b^2 - (9*a^2 + 2*b^2)*(c + d*x)^2))/3 - b*(12*b^2*(c + d*x) - 9*a^2*(c + d*x)^3 - 2*b^2*(c + d*x)^3 - 12*a*b*Sqrt[1 + (c + d*x)^2] + 6*a*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] - 3*b^2*(-3*a*(c + d*x)^3 - 2*b*Sqrt[1 + (c + d*x)^2] + b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 3*b^3*(c + d*x)^3*ArcSinh[c + d*x]^3))/(9*d)","A",1
140,1,200,161,0.2114268,"\int (c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^3,x]","\frac{e \left(2 a \left(2 a^2+3 b^2\right) (c+d x)^2-3 b \left(2 a^2+b^2\right) (c+d x) \sqrt{(c+d x)^2+1}+3 b \left(2 a^2+b^2\right) \sinh ^{-1}(c+d x)-6 b (c+d x) \sinh ^{-1}(c+d x) \left(-2 a^2 (c+d x)+2 a b \sqrt{(c+d x)^2+1}-b^2 (c+d x)\right)+6 b^2 \sinh ^{-1}(c+d x)^2 \left(2 a (c+d x)^2+a-b \sqrt{(c+d x)^2+1} (c+d x)\right)+2 b^3 \left(2 (c+d x)^2+1\right) \sinh ^{-1}(c+d x)^3\right)}{8 d}","\frac{3 b^2 e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)}{4 d}-\frac{3 b e (c+d x) \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{4 d}+\frac{e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^3}{2 d}+\frac{e \left(a+b \sinh ^{-1}(c+d x)\right)^3}{4 d}-\frac{3 b^3 e (c+d x) \sqrt{(c+d x)^2+1}}{8 d}+\frac{3 b^3 e \sinh ^{-1}(c+d x)}{8 d}",1,"(e*(2*a*(2*a^2 + 3*b^2)*(c + d*x)^2 - 3*b*(2*a^2 + b^2)*(c + d*x)*Sqrt[1 + (c + d*x)^2] + 3*b*(2*a^2 + b^2)*ArcSinh[c + d*x] - 6*b*(c + d*x)*(-2*a^2*(c + d*x) - b^2*(c + d*x) + 2*a*b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 6*b^2*(a + 2*a*(c + d*x)^2 - b*(c + d*x)*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 2*b^3*(1 + 2*(c + d*x)^2)*ArcSinh[c + d*x]^3))/(8*d)","A",1
141,1,147,100,0.1671418,"\int \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(a + b*ArcSinh[c + d*x])^3,x]","\frac{a \left(a^2+6 b^2\right) (c+d x)-3 b \left(a^2+2 b^2\right) \sqrt{(c+d x)^2+1}-3 b \sinh ^{-1}(c+d x) \left(-\left(a^2 (c+d x)\right)+2 a b \sqrt{(c+d x)^2+1}-2 b^2 (c+d x)\right)-3 b^2 \sinh ^{-1}(c+d x)^2 \left(b \sqrt{(c+d x)^2+1}-a (c+d x)\right)+b^3 (c+d x) \sinh ^{-1}(c+d x)^3}{d}","6 a b^2 x-\frac{3 b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d}+\frac{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d}-\frac{6 b^3 \sqrt{(c+d x)^2+1}}{d}+\frac{6 b^3 (c+d x) \sinh ^{-1}(c+d x)}{d}",1,"(a*(a^2 + 6*b^2)*(c + d*x) - 3*b*(a^2 + 2*b^2)*Sqrt[1 + (c + d*x)^2] - 3*b*(-(a^2*(c + d*x)) - 2*b^2*(c + d*x) + 2*a*b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] - 3*b^2*(-(a*(c + d*x)) + b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + b^3*(c + d*x)*ArcSinh[c + d*x]^3)/d","A",1
142,1,128,155,0.0490595,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{c e+d e x} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x),x]","\frac{-6 b^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)+6 b \text{Li}_2\left(e^{2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{b}+4 \log \left(1-e^{2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^3+3 b^3 \text{Li}_4\left(e^{2 \sinh ^{-1}(c+d x)}\right)}{4 d e}","-\frac{3 b^2 \text{Li}_3\left(e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{2 d e}-\frac{3 b \text{Li}_2\left(e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{2 d e}+\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{4 b d e}+\frac{\log \left(1-e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d e}-\frac{3 b^3 \text{Li}_4\left(e^{-2 \sinh ^{-1}(c+d x)}\right)}{4 d e}",1,"(-((a + b*ArcSinh[c + d*x])^4/b) + 4*(a + b*ArcSinh[c + d*x])^3*Log[1 - E^(2*ArcSinh[c + d*x])] + 6*b*(a + b*ArcSinh[c + d*x])^2*PolyLog[2, E^(2*ArcSinh[c + d*x])] - 6*b^2*(a + b*ArcSinh[c + d*x])*PolyLog[3, E^(2*ArcSinh[c + d*x])] + 3*b^3*PolyLog[4, E^(2*ArcSinh[c + d*x])])/(4*d*e)","A",0
143,1,315,166,0.7228868,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^2,x]","\frac{-\frac{a^3}{c+d x}-3 a^2 b \log \left(\sqrt{c^2+2 c d x+d^2 x^2+1}+1\right)+3 a^2 b \log (c+d x)-\frac{3 a^2 b \sinh ^{-1}(c+d x)}{c+d x}+3 a b^2 \left(2 \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)-2 \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)+\sinh ^{-1}(c+d x) \left(-\frac{\sinh ^{-1}(c+d x)}{c+d x}+2 \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)-2 \log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)\right)\right)+b^3 \left(6 \sinh ^{-1}(c+d x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)-6 \sinh ^{-1}(c+d x) \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)+6 \text{Li}_3\left(-e^{-\sinh ^{-1}(c+d x)}\right)-6 \text{Li}_3\left(e^{-\sinh ^{-1}(c+d x)}\right)-\frac{\sinh ^{-1}(c+d x)^3}{c+d x}+3 \sinh ^{-1}(c+d x)^2 \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)-3 \sinh ^{-1}(c+d x)^2 \log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)\right)}{d e^2}","-\frac{6 b^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^2}+\frac{6 b^2 \text{Li}_2\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^2}-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{d e^2 (c+d x)}-\frac{6 b \tanh ^{-1}\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e^2}+\frac{6 b^3 \text{Li}_3\left(-e^{\sinh ^{-1}(c+d x)}\right)}{d e^2}-\frac{6 b^3 \text{Li}_3\left(e^{\sinh ^{-1}(c+d x)}\right)}{d e^2}",1,"(-(a^3/(c + d*x)) - (3*a^2*b*ArcSinh[c + d*x])/(c + d*x) + 3*a^2*b*Log[c + d*x] - 3*a^2*b*Log[1 + Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]] + 3*a*b^2*(ArcSinh[c + d*x]*(-(ArcSinh[c + d*x]/(c + d*x)) + 2*Log[1 - E^(-ArcSinh[c + d*x])] - 2*Log[1 + E^(-ArcSinh[c + d*x])]) + 2*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 2*PolyLog[2, E^(-ArcSinh[c + d*x])]) + b^3*(-(ArcSinh[c + d*x]^3/(c + d*x)) + 3*ArcSinh[c + d*x]^2*Log[1 - E^(-ArcSinh[c + d*x])] - 3*ArcSinh[c + d*x]^2*Log[1 + E^(-ArcSinh[c + d*x])] + 6*ArcSinh[c + d*x]*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 6*ArcSinh[c + d*x]*PolyLog[2, E^(-ArcSinh[c + d*x])] + 6*PolyLog[3, -E^(-ArcSinh[c + d*x])] - 6*PolyLog[3, E^(-ArcSinh[c + d*x])]))/(d*e^2)","A",0
144,1,229,157,0.8115853,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^3,x]","-\frac{a \left(a \left(a+3 b (c+d x) \sqrt{c^2+2 c d x+d^2 x^2+1}\right)-6 b^2 (c+d x)^2 \log (c+d x)\right)+3 b^2 \sinh ^{-1}(c+d x)^2 \left(a+b (c+d x) \left(\sqrt{c^2+2 c d x+d^2 x^2+1}-c-d x\right)\right)+3 b \sinh ^{-1}(c+d x) \left(a \left(a+2 b (c+d x) \sqrt{c^2+2 c d x+d^2 x^2+1}\right)-2 b^2 (c+d x)^2 \log \left(1-e^{-2 \sinh ^{-1}(c+d x)}\right)\right)+3 b^3 (c+d x)^2 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c+d x)}\right)+b^3 \sinh ^{-1}(c+d x)^3}{2 d e^3 (c+d x)^2}","\frac{3 b^2 \log \left(1-e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^3}-\frac{3 b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{2 d e^3 (c+d x)}+\frac{3 b \left(a+b \sinh ^{-1}(c+d x)\right)^2}{2 d e^3}-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{2 d e^3 (c+d x)^2}-\frac{3 b^3 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c+d x)}\right)}{2 d e^3}",1,"-1/2*(3*b^2*(a + b*(c + d*x)*(-c - d*x + Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]))*ArcSinh[c + d*x]^2 + b^3*ArcSinh[c + d*x]^3 + 3*b*ArcSinh[c + d*x]*(a*(a + 2*b*(c + d*x)*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]) - 2*b^2*(c + d*x)^2*Log[1 - E^(-2*ArcSinh[c + d*x])]) + a*(a*(a + 3*b*(c + d*x)*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]) - 6*b^2*(c + d*x)^2*Log[c + d*x]) + 3*b^3*(c + d*x)^2*PolyLog[2, E^(-2*ArcSinh[c + d*x])])/(d*e^3*(c + d*x)^2)","A",0
145,1,694,261,7.1414249,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^4,x]","-\frac{a^3}{3 d e^4 (c+d x)^3}-\frac{a^2 b \sqrt{c^2+2 c d x+d^2 x^2+1}}{2 d e^4 (c+d x)^2}+\frac{a^2 b \log \left(\sqrt{c^2+2 c d x+d^2 x^2+1}+1\right)}{2 d e^4}-\frac{a^2 b \log (c+d x)}{2 d e^4}-\frac{a^2 b \sinh ^{-1}(c+d x)}{d e^4 (c+d x)^3}+\frac{a b^2 \left(-8 \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)-\frac{2 \left(-4 (c+d x)^3 \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)+4 \sinh ^{-1}(c+d x)^2+2 \sinh ^{-1}(c+d x) \sinh \left(2 \sinh ^{-1}(c+d x)\right)-3 (c+d x) \sinh ^{-1}(c+d x) \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)+3 (c+d x) \sinh ^{-1}(c+d x) \log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)+\sinh ^{-1}(c+d x) \sinh \left(3 \sinh ^{-1}(c+d x)\right) \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)-\sinh ^{-1}(c+d x) \sinh \left(3 \sinh ^{-1}(c+d x)\right) \log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)+2 \cosh \left(2 \sinh ^{-1}(c+d x)\right)-2\right)}{(c+d x)^3}\right)}{8 d e^4}+\frac{b^3 \left(-48 \sinh ^{-1}(c+d x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)+48 \sinh ^{-1}(c+d x) \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)-48 \text{Li}_3\left(-e^{-\sinh ^{-1}(c+d x)}\right)+48 \text{Li}_3\left(e^{-\sinh ^{-1}(c+d x)}\right)-\frac{16 \sinh ^{-1}(c+d x)^3 \sinh ^4\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)}{(c+d x)^3}-24 \sinh ^{-1}(c+d x)^2 \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)+24 \sinh ^{-1}(c+d x)^2 \log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)-4 \sinh ^{-1}(c+d x)^3 \tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)+24 \sinh ^{-1}(c+d x) \tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)+4 \sinh ^{-1}(c+d x)^3 \coth \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-24 \sinh ^{-1}(c+d x) \coth \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-\left((c+d x) \sinh ^{-1}(c+d x)^3 \text{csch}^4\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)\right)-6 \sinh ^{-1}(c+d x)^2 \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-6 \sinh ^{-1}(c+d x)^2 \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)+48 \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)\right)\right)}{48 d e^4}","\frac{b^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^4}-\frac{b^2 \text{Li}_2\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^4}-\frac{b^2 \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^4 (c+d x)}-\frac{b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{2 d e^4 (c+d x)^2}-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{3 d e^4 (c+d x)^3}+\frac{b \tanh ^{-1}\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e^4}-\frac{b^3 \text{Li}_3\left(-e^{\sinh ^{-1}(c+d x)}\right)}{d e^4}+\frac{b^3 \text{Li}_3\left(e^{\sinh ^{-1}(c+d x)}\right)}{d e^4}-\frac{b^3 \tanh ^{-1}\left(\sqrt{(c+d x)^2+1}\right)}{d e^4}",1,"-1/3*a^3/(d*e^4*(c + d*x)^3) - (a^2*b*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*d*e^4*(c + d*x)^2) - (a^2*b*ArcSinh[c + d*x])/(d*e^4*(c + d*x)^3) - (a^2*b*Log[c + d*x])/(2*d*e^4) + (a^2*b*Log[1 + Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2]])/(2*d*e^4) + (a*b^2*(-8*PolyLog[2, -E^(-ArcSinh[c + d*x])] - (2*(-2 + 4*ArcSinh[c + d*x]^2 + 2*Cosh[2*ArcSinh[c + d*x]] - 3*(c + d*x)*ArcSinh[c + d*x]*Log[1 - E^(-ArcSinh[c + d*x])] + 3*(c + d*x)*ArcSinh[c + d*x]*Log[1 + E^(-ArcSinh[c + d*x])] - 4*(c + d*x)^3*PolyLog[2, E^(-ArcSinh[c + d*x])] + 2*ArcSinh[c + d*x]*Sinh[2*ArcSinh[c + d*x]] + ArcSinh[c + d*x]*Log[1 - E^(-ArcSinh[c + d*x])]*Sinh[3*ArcSinh[c + d*x]] - ArcSinh[c + d*x]*Log[1 + E^(-ArcSinh[c + d*x])]*Sinh[3*ArcSinh[c + d*x]]))/(c + d*x)^3))/(8*d*e^4) + (b^3*(-24*ArcSinh[c + d*x]*Coth[ArcSinh[c + d*x]/2] + 4*ArcSinh[c + d*x]^3*Coth[ArcSinh[c + d*x]/2] - 6*ArcSinh[c + d*x]^2*Csch[ArcSinh[c + d*x]/2]^2 - (c + d*x)*ArcSinh[c + d*x]^3*Csch[ArcSinh[c + d*x]/2]^4 - 24*ArcSinh[c + d*x]^2*Log[1 - E^(-ArcSinh[c + d*x])] + 24*ArcSinh[c + d*x]^2*Log[1 + E^(-ArcSinh[c + d*x])] + 48*Log[Tanh[ArcSinh[c + d*x]/2]] - 48*ArcSinh[c + d*x]*PolyLog[2, -E^(-ArcSinh[c + d*x])] + 48*ArcSinh[c + d*x]*PolyLog[2, E^(-ArcSinh[c + d*x])] - 48*PolyLog[3, -E^(-ArcSinh[c + d*x])] + 48*PolyLog[3, E^(-ArcSinh[c + d*x])] - 6*ArcSinh[c + d*x]^2*Sech[ArcSinh[c + d*x]/2]^2 - (16*ArcSinh[c + d*x]^3*Sinh[ArcSinh[c + d*x]/2]^4)/(c + d*x)^3 + 24*ArcSinh[c + d*x]*Tanh[ArcSinh[c + d*x]/2] - 4*ArcSinh[c + d*x]^3*Tanh[ArcSinh[c + d*x]/2]))/(48*d*e^4)","B",0
146,0,0,87,1.9196578,"\int (c e+d e x)^m \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^4,x]","\int (c e+d e x)^m \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","\frac{(e (c+d x))^{m+1} \left(a+b \sinh ^{-1}(c+d x)\right)^4}{d e (m+1)}-\frac{4 b \text{Int}\left(\frac{(e (c+d x))^{m+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{(c+d x)^2+1}},x\right)}{e (m+1)}",0,"Integrate[(c*e + d*e*x)^m*(a + b*ArcSinh[c + d*x])^4, x]","A",-1
147,1,475,349,0.6381248,"\int (c e+d e x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^4,x]","\frac{e^3 \left(-9 b^2 \left(8 a^2+5 b^2\right) (c+d x)^2+2 a b \sqrt{(c+d x)^2+1} (c+d x) \left(-2 \left(8 a^2+3 b^2\right) (c+d x)^2+24 a^2+45 b^2\right)+3 b^2 \sinh ^{-1}(c+d x)^2 \left(64 a^2 (c+d x)^4-24 a^2-32 a b \sqrt{(c+d x)^2+1} (c+d x)^3+48 a b \sqrt{(c+d x)^2+1} (c+d x)+8 b^2 (c+d x)^4-24 b^2 (c+d x)^2-15 b^2\right)-6 a b \left(8 a^2+15 b^2\right) \sinh ^{-1}(c+d x)+\left(32 a^4+24 a^2 b^2+3 b^4\right) (c+d x)^4+2 b (c+d x) \sinh ^{-1}(c+d x) \left(64 a^3 (c+d x)^3+72 a^2 b \sqrt{(c+d x)^2+1}-48 a^2 b (c+d x)^2 \sqrt{(c+d x)^2+1}+24 a b^2 (c+d x)^3-72 a b^2 (c+d x)-6 b^3 (c+d x)^2 \sqrt{(c+d x)^2+1}+45 b^3 \sqrt{(c+d x)^2+1}\right)+16 b^3 \sinh ^{-1}(c+d x)^3 \left(8 a (c+d x)^4-3 a-2 b \sqrt{(c+d x)^2+1} (c+d x)^3+3 b \sqrt{(c+d x)^2+1} (c+d x)\right)+4 b^4 \left(8 (c+d x)^4-3\right) \sinh ^{-1}(c+d x)^4\right)}{128 d}","-\frac{3 b^3 e^3 (c+d x)^3 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{32 d}+\frac{45 b^3 e^3 (c+d x) \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{64 d}+\frac{3 b^2 e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{16 d}-\frac{45 b^2 e^3 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{128 d}-\frac{9 b^2 e^3 (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{16 d}-\frac{b e^3 (c+d x)^3 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{4 d}+\frac{3 b e^3 (c+d x) \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{8 d}+\frac{e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^4}{4 d}-\frac{3 e^3 \left(a+b \sinh ^{-1}(c+d x)\right)^4}{32 d}+\frac{3 b^4 e^3 (c+d x)^4}{128 d}-\frac{45 b^4 e^3 (c+d x)^2}{128 d}",1,"(e^3*(-9*b^2*(8*a^2 + 5*b^2)*(c + d*x)^2 + (32*a^4 + 24*a^2*b^2 + 3*b^4)*(c + d*x)^4 + 2*a*b*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(24*a^2 + 45*b^2 - 2*(8*a^2 + 3*b^2)*(c + d*x)^2) - 6*a*b*(8*a^2 + 15*b^2)*ArcSinh[c + d*x] + 2*b*(c + d*x)*(-72*a*b^2*(c + d*x) + 64*a^3*(c + d*x)^3 + 24*a*b^2*(c + d*x)^3 + 72*a^2*b*Sqrt[1 + (c + d*x)^2] + 45*b^3*Sqrt[1 + (c + d*x)^2] - 48*a^2*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] - 6*b^3*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 3*b^2*(-24*a^2 - 15*b^2 - 24*b^2*(c + d*x)^2 + 64*a^2*(c + d*x)^4 + 8*b^2*(c + d*x)^4 + 48*a*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] - 32*a*b*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 16*b^3*(-3*a + 8*a*(c + d*x)^4 + 3*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] - 2*b*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + 4*b^4*(-3 + 8*(c + d*x)^4)*ArcSinh[c + d*x]^4))/(128*d)","A",1
148,1,412,281,0.4521705,"\int (c e+d e x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^4,x]","\frac{e^2 \left(-24 b^2 \left(9 a^2+20 b^2\right) (c+d x)+12 a b \sqrt{(c+d x)^2+1} \left(-\left(3 a^2+2 b^2\right) (c+d x)^2+6 a^2+40 b^2\right)+18 b^2 \sinh ^{-1}(c+d x)^2 \left(9 a^2 (c+d x)^3-6 a b \sqrt{(c+d x)^2+1} (c+d x)^2+12 a b \sqrt{(c+d x)^2+1}+2 b^2 (c+d x)^3-12 b^2 (c+d x)\right)+\left(27 a^4+36 a^2 b^2+8 b^4\right) (c+d x)^3+12 b \sinh ^{-1}(c+d x) \left(9 a^3 (c+d x)^3+18 a^2 b \sqrt{(c+d x)^2+1}-9 a^2 b (c+d x)^2 \sqrt{(c+d x)^2+1}+6 a b^2 (c+d x)^3-36 a b^2 (c+d x)-2 b^3 (c+d x)^2 \sqrt{(c+d x)^2+1}+40 b^3 \sqrt{(c+d x)^2+1}\right)-36 b^3 \sinh ^{-1}(c+d x)^3 \left(-3 a (c+d x)^3+b \sqrt{(c+d x)^2+1} (c+d x)^2-2 b \sqrt{(c+d x)^2+1}\right)+27 b^4 (c+d x)^3 \sinh ^{-1}(c+d x)^4\right)}{81 d}","\frac{160 b^3 e^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{27 d}-\frac{8 b^3 e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{27 d}+\frac{4 b^2 e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{9 d}-\frac{8 b^2 e^2 (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{3 d}+\frac{8 b e^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{9 d}-\frac{4 b e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{9 d}+\frac{e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^4}{3 d}+\frac{8 b^4 e^2 (c+d x)^3}{81 d}-\frac{160}{27} b^4 e^2 x",1,"(e^2*(-24*b^2*(9*a^2 + 20*b^2)*(c + d*x) + (27*a^4 + 36*a^2*b^2 + 8*b^4)*(c + d*x)^3 + 12*a*b*Sqrt[1 + (c + d*x)^2]*(6*a^2 + 40*b^2 - (3*a^2 + 2*b^2)*(c + d*x)^2) + 12*b*(-36*a*b^2*(c + d*x) + 9*a^3*(c + d*x)^3 + 6*a*b^2*(c + d*x)^3 + 18*a^2*b*Sqrt[1 + (c + d*x)^2] + 40*b^3*Sqrt[1 + (c + d*x)^2] - 9*a^2*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] - 2*b^3*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 18*b^2*(-12*b^2*(c + d*x) + 9*a^2*(c + d*x)^3 + 2*b^2*(c + d*x)^3 + 12*a*b*Sqrt[1 + (c + d*x)^2] - 6*a*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 - 36*b^3*(-3*a*(c + d*x)^3 - 2*b*Sqrt[1 + (c + d*x)^2] + b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + 27*b^4*(c + d*x)^3*ArcSinh[c + d*x]^4))/(81*d)","A",1
149,1,300,195,0.3232732,"\int (c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4,x]","\frac{e \left(-2 a b \left(2 a^2+3 b^2\right) (c+d x) \sqrt{(c+d x)^2+1}+3 b^2 \sinh ^{-1}(c+d x)^2 \left(4 a^2 (c+d x)^2+2 a^2-4 a b (c+d x) \sqrt{(c+d x)^2+1}+2 b^2 (c+d x)^2+b^2\right)+2 a b \left(2 a^2+3 b^2\right) \sinh ^{-1}(c+d x)+\left(2 a^4+6 a^2 b^2+3 b^4\right) (c+d x)^2-2 b (c+d x) \sinh ^{-1}(c+d x) \left(-4 a^3 (c+d x)+6 a^2 b \sqrt{(c+d x)^2+1}-6 a b^2 (c+d x)+3 b^3 \sqrt{(c+d x)^2+1}\right)+4 b^3 \sinh ^{-1}(c+d x)^3 \left(2 a (c+d x)^2+a-b \sqrt{(c+d x)^2+1} (c+d x)\right)+b^4 \left(2 (c+d x)^2+1\right) \sinh ^{-1}(c+d x)^4\right)}{4 d}","-\frac{3 b^3 e (c+d x) \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{2 d}+\frac{3 b^2 e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{2 d}+\frac{3 b^2 e \left(a+b \sinh ^{-1}(c+d x)\right)^2}{4 d}-\frac{b e (c+d x) \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d}+\frac{e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^4}{2 d}+\frac{e \left(a+b \sinh ^{-1}(c+d x)\right)^4}{4 d}+\frac{3 b^4 e (c+d x)^2}{4 d}",1,"(e*((2*a^4 + 6*a^2*b^2 + 3*b^4)*(c + d*x)^2 - 2*a*b*(2*a^2 + 3*b^2)*(c + d*x)*Sqrt[1 + (c + d*x)^2] + 2*a*b*(2*a^2 + 3*b^2)*ArcSinh[c + d*x] - 2*b*(c + d*x)*(-4*a^3*(c + d*x) - 6*a*b^2*(c + d*x) + 6*a^2*b*Sqrt[1 + (c + d*x)^2] + 3*b^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 3*b^2*(2*a^2 + b^2 + 4*a^2*(c + d*x)^2 + 2*b^2*(c + d*x)^2 - 4*a*b*(c + d*x)*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 4*b^3*(a + 2*a*(c + d*x)^2 - b*(c + d*x)*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + b^4*(1 + 2*(c + d*x)^2)*ArcSinh[c + d*x]^4))/(4*d)","A",1
150,1,226,115,0.2636735,"\int \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","Integrate[(a + b*ArcSinh[c + d*x])^4,x]","\frac{-4 a b \left(a^2+6 b^2\right) \sqrt{(c+d x)^2+1}+6 b^2 \sinh ^{-1}(c+d x)^2 \left(a^2 (c+d x)-2 a b \sqrt{(c+d x)^2+1}+2 b^2 (c+d x)\right)+\left(a^4+12 a^2 b^2+24 b^4\right) (c+d x)-4 b \sinh ^{-1}(c+d x) \left(-\left(a^3 (c+d x)\right)+3 a^2 b \sqrt{(c+d x)^2+1}-6 a b^2 (c+d x)+6 b^3 \sqrt{(c+d x)^2+1}\right)-4 b^3 \sinh ^{-1}(c+d x)^3 \left(b \sqrt{(c+d x)^2+1}-a (c+d x)\right)+b^4 (c+d x) \sinh ^{-1}(c+d x)^4}{d}","-\frac{24 b^3 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)}{d}+\frac{12 b^2 (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d}-\frac{4 b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d}+\frac{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^4}{d}+24 b^4 x",1,"((a^4 + 12*a^2*b^2 + 24*b^4)*(c + d*x) - 4*a*b*(a^2 + 6*b^2)*Sqrt[1 + (c + d*x)^2] - 4*b*(-(a^3*(c + d*x)) - 6*a*b^2*(c + d*x) + 3*a^2*b*Sqrt[1 + (c + d*x)^2] + 6*b^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 6*b^2*(a^2*(c + d*x) + 2*b^2*(c + d*x) - 2*a*b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 - 4*b^3*(-(a*(c + d*x)) + b*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + b^4*(c + d*x)*ArcSinh[c + d*x]^4)/d","A",1
151,1,157,186,0.0653847,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{c e+d e x} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x),x]","\frac{3 b^3 \text{Li}_4\left(e^{2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)-3 b^2 \text{Li}_3\left(e^{2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2+2 b \text{Li}_2\left(e^{2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^3-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^5}{5 b}+\log \left(1-e^{2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^4-\frac{3}{2} b^4 \text{Li}_5\left(e^{2 \sinh ^{-1}(c+d x)}\right)}{d e}","-\frac{3 b^3 \text{Li}_4\left(e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e}-\frac{3 b^2 \text{Li}_3\left(e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e}-\frac{2 b \text{Li}_2\left(e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d e}+\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^5}{5 b d e}+\frac{\log \left(1-e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^4}{d e}-\frac{3 b^4 \text{Li}_5\left(e^{-2 \sinh ^{-1}(c+d x)}\right)}{2 d e}",1,"(-1/5*(a + b*ArcSinh[c + d*x])^5/b + (a + b*ArcSinh[c + d*x])^4*Log[1 - E^(2*ArcSinh[c + d*x])] + 2*b*(a + b*ArcSinh[c + d*x])^3*PolyLog[2, E^(2*ArcSinh[c + d*x])] - 3*b^2*(a + b*ArcSinh[c + d*x])^2*PolyLog[3, E^(2*ArcSinh[c + d*x])] + 3*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[4, E^(2*ArcSinh[c + d*x])] - (3*b^4*PolyLog[5, E^(2*ArcSinh[c + d*x])])/2)/(d*e)","A",0
152,1,501,234,1.5957318,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{(c e+d e x)^2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^2,x]","\frac{-\frac{2 a^4}{c+d x}+4 a^3 b \left(2 \log \left(\frac{2 \sinh ^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)}{c+d x}\right)-\frac{2 \sinh ^{-1}(c+d x)}{c+d x}\right)+12 a^2 b^2 \left(2 \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)-2 \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)+\sinh ^{-1}(c+d x) \left(-\frac{\sinh ^{-1}(c+d x)}{c+d x}+2 \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)-2 \log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)\right)\right)+8 a b^3 \left(6 \sinh ^{-1}(c+d x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)-6 \sinh ^{-1}(c+d x) \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)+6 \text{Li}_3\left(-e^{-\sinh ^{-1}(c+d x)}\right)-6 \text{Li}_3\left(e^{-\sinh ^{-1}(c+d x)}\right)-\frac{\sinh ^{-1}(c+d x)^3}{c+d x}+3 \sinh ^{-1}(c+d x)^2 \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)-3 \sinh ^{-1}(c+d x)^2 \log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)\right)+b^4 \left(24 \sinh ^{-1}(c+d x)^2 \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)+24 \sinh ^{-1}(c+d x)^2 \text{Li}_2\left(e^{\sinh ^{-1}(c+d x)}\right)+48 \sinh ^{-1}(c+d x) \text{Li}_3\left(-e^{-\sinh ^{-1}(c+d x)}\right)-48 \sinh ^{-1}(c+d x) \text{Li}_3\left(e^{\sinh ^{-1}(c+d x)}\right)+48 \text{Li}_4\left(-e^{-\sinh ^{-1}(c+d x)}\right)+48 \text{Li}_4\left(e^{\sinh ^{-1}(c+d x)}\right)-\frac{2 \sinh ^{-1}(c+d x)^4}{c+d x}-2 \sinh ^{-1}(c+d x)^4-8 \sinh ^{-1}(c+d x)^3 \log \left(e^{-\sinh ^{-1}(c+d x)}+1\right)+8 \sinh ^{-1}(c+d x)^3 \log \left(1-e^{\sinh ^{-1}(c+d x)}\right)+\pi ^4\right)}{2 d e^2}","\frac{24 b^3 \text{Li}_3\left(-e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^2}-\frac{24 b^3 \text{Li}_3\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^2}-\frac{12 b^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e^2}+\frac{12 b^2 \text{Li}_2\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e^2}-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{d e^2 (c+d x)}-\frac{8 b \tanh ^{-1}\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d e^2}-\frac{24 b^4 \text{Li}_4\left(-e^{\sinh ^{-1}(c+d x)}\right)}{d e^2}+\frac{24 b^4 \text{Li}_4\left(e^{\sinh ^{-1}(c+d x)}\right)}{d e^2}",1,"((-2*a^4)/(c + d*x) + 4*a^3*b*((-2*ArcSinh[c + d*x])/(c + d*x) + 2*Log[(2*Sinh[ArcSinh[c + d*x]/2]^2)/(c + d*x)]) + 12*a^2*b^2*(ArcSinh[c + d*x]*(-(ArcSinh[c + d*x]/(c + d*x)) + 2*Log[1 - E^(-ArcSinh[c + d*x])] - 2*Log[1 + E^(-ArcSinh[c + d*x])]) + 2*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 2*PolyLog[2, E^(-ArcSinh[c + d*x])]) + 8*a*b^3*(-(ArcSinh[c + d*x]^3/(c + d*x)) + 3*ArcSinh[c + d*x]^2*Log[1 - E^(-ArcSinh[c + d*x])] - 3*ArcSinh[c + d*x]^2*Log[1 + E^(-ArcSinh[c + d*x])] + 6*ArcSinh[c + d*x]*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 6*ArcSinh[c + d*x]*PolyLog[2, E^(-ArcSinh[c + d*x])] + 6*PolyLog[3, -E^(-ArcSinh[c + d*x])] - 6*PolyLog[3, E^(-ArcSinh[c + d*x])]) + b^4*(Pi^4 - 2*ArcSinh[c + d*x]^4 - (2*ArcSinh[c + d*x]^4)/(c + d*x) - 8*ArcSinh[c + d*x]^3*Log[1 + E^(-ArcSinh[c + d*x])] + 8*ArcSinh[c + d*x]^3*Log[1 - E^ArcSinh[c + d*x]] + 24*ArcSinh[c + d*x]^2*PolyLog[2, -E^(-ArcSinh[c + d*x])] + 24*ArcSinh[c + d*x]^2*PolyLog[2, E^ArcSinh[c + d*x]] + 48*ArcSinh[c + d*x]*PolyLog[3, -E^(-ArcSinh[c + d*x])] - 48*ArcSinh[c + d*x]*PolyLog[3, E^ArcSinh[c + d*x]] + 48*PolyLog[4, -E^(-ArcSinh[c + d*x])] + 48*PolyLog[4, E^ArcSinh[c + d*x]]))/(2*d*e^2)","B",0
153,1,360,186,1.2700851,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{(c e+d e x)^3} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^3,x]","\frac{-\frac{2 a^4}{(c+d x)^2}-\frac{8 a^3 b \sqrt{(c+d x)^2+1}}{c+d x}-\frac{8 a^3 b \sinh ^{-1}(c+d x)}{(c+d x)^2}+24 a^2 b^2 \left(\log (c+d x)-\frac{\sinh ^{-1}(c+d x)^2}{2 (c+d x)^2}-\frac{\sqrt{(c+d x)^2+1} \sinh ^{-1}(c+d x)}{c+d x}\right)+8 a b^3 \left(\sinh ^{-1}(c+d x) \left(-\frac{\sinh ^{-1}(c+d x)^2}{(c+d x)^2}-\frac{3 \sqrt{(c+d x)^2+1} \sinh ^{-1}(c+d x)}{c+d x}+3 \sinh ^{-1}(c+d x)+6 \log \left(1-e^{-2 \sinh ^{-1}(c+d x)}\right)\right)-3 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c+d x)}\right)\right)+b^4 \left(24 \sinh ^{-1}(c+d x) \text{Li}_2\left(e^{2 \sinh ^{-1}(c+d x)}\right)-12 \text{Li}_3\left(e^{2 \sinh ^{-1}(c+d x)}\right)-\frac{8 \sqrt{(c+d x)^2+1} \sinh ^{-1}(c+d x)^3}{c+d x}-8 \sinh ^{-1}(c+d x)^3+24 \sinh ^{-1}(c+d x)^2 \log \left(1-e^{2 \sinh ^{-1}(c+d x)}\right)+i \pi ^3\right)-\frac{2 b^4 \sinh ^{-1}(c+d x)^4}{(c+d x)^2}}{4 d e^3}","-\frac{6 b^3 \text{Li}_2\left(e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^3}+\frac{6 b^2 \log \left(1-e^{-2 \sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e^3}-\frac{2 b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d e^3 (c+d x)}+\frac{2 b \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d e^3}-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{2 d e^3 (c+d x)^2}-\frac{3 b^4 \text{Li}_3\left(e^{-2 \sinh ^{-1}(c+d x)}\right)}{d e^3}",1,"((-2*a^4)/(c + d*x)^2 - (8*a^3*b*Sqrt[1 + (c + d*x)^2])/(c + d*x) - (8*a^3*b*ArcSinh[c + d*x])/(c + d*x)^2 - (2*b^4*ArcSinh[c + d*x]^4)/(c + d*x)^2 + 24*a^2*b^2*(-((Sqrt[1 + (c + d*x)^2]*ArcSinh[c + d*x])/(c + d*x)) - ArcSinh[c + d*x]^2/(2*(c + d*x)^2) + Log[c + d*x]) + 8*a*b^3*(ArcSinh[c + d*x]*(3*ArcSinh[c + d*x] - (3*Sqrt[1 + (c + d*x)^2]*ArcSinh[c + d*x])/(c + d*x) - ArcSinh[c + d*x]^2/(c + d*x)^2 + 6*Log[1 - E^(-2*ArcSinh[c + d*x])]) - 3*PolyLog[2, E^(-2*ArcSinh[c + d*x])]) + b^4*(I*Pi^3 - 8*ArcSinh[c + d*x]^3 - (8*Sqrt[1 + (c + d*x)^2]*ArcSinh[c + d*x]^3)/(c + d*x) + 24*ArcSinh[c + d*x]^2*Log[1 - E^(2*ArcSinh[c + d*x])] + 24*ArcSinh[c + d*x]*PolyLog[2, E^(2*ArcSinh[c + d*x])] - 12*PolyLog[3, E^(2*ArcSinh[c + d*x])]))/(4*d*e^3)","C",0
154,1,1182,385,8.5930197,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{(c e+d e x)^4} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^4,x]","-\frac{a^4}{3 d e^4 (c+d x)^3}+\frac{4 b \left(-\frac{1}{24} \sinh ^{-1}(c+d x) \coth \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-\frac{1}{24} \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-\frac{1}{24} \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)+\frac{1}{12} \sinh ^{-1}(c+d x) \coth \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-\frac{1}{6} \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)\right)-\frac{1}{24} \sinh ^{-1}(c+d x) \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-\frac{1}{12} \sinh ^{-1}(c+d x) \tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)\right) a^3}{d e^4}+\frac{b^2 \left(-8 \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)-\frac{2 \left(-4 \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right) (c+d x)^3-3 \sinh ^{-1}(c+d x) \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right) (c+d x)+3 \sinh ^{-1}(c+d x) \log \left(1+e^{-\sinh ^{-1}(c+d x)}\right) (c+d x)+4 \sinh ^{-1}(c+d x)^2+2 \cosh \left(2 \sinh ^{-1}(c+d x)\right)+2 \sinh ^{-1}(c+d x) \sinh \left(2 \sinh ^{-1}(c+d x)\right)+\sinh ^{-1}(c+d x) \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right) \sinh \left(3 \sinh ^{-1}(c+d x)\right)-\sinh ^{-1}(c+d x) \log \left(1+e^{-\sinh ^{-1}(c+d x)}\right) \sinh \left(3 \sinh ^{-1}(c+d x)\right)-2\right)}{(c+d x)^3}\right) a^2}{4 d e^4}+\frac{b^3 \left(-\left((c+d x) \sinh ^{-1}(c+d x)^3 \text{csch}^4\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)\right)-6 \sinh ^{-1}(c+d x)^2 \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-\frac{16 \sinh ^{-1}(c+d x)^3 \sinh ^4\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)}{(c+d x)^3}-6 \sinh ^{-1}(c+d x)^2 \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)+4 \sinh ^{-1}(c+d x)^3 \coth \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-24 \sinh ^{-1}(c+d x) \coth \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)-24 \sinh ^{-1}(c+d x)^2 \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right)+24 \sinh ^{-1}(c+d x)^2 \log \left(1+e^{-\sinh ^{-1}(c+d x)}\right)+48 \log \left(\tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)\right)-48 \sinh ^{-1}(c+d x) \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)+48 \sinh ^{-1}(c+d x) \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)-48 \text{Li}_3\left(-e^{-\sinh ^{-1}(c+d x)}\right)+48 \text{Li}_3\left(e^{-\sinh ^{-1}(c+d x)}\right)-4 \sinh ^{-1}(c+d x)^3 \tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)+24 \sinh ^{-1}(c+d x) \tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right)\right) a}{12 d e^4}+\frac{b^4 \left(-\frac{1}{2} (c+d x) \text{csch}^4\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \sinh ^{-1}(c+d x)^4-\frac{8 \sinh ^4\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \sinh ^{-1}(c+d x)^4}{(c+d x)^3}+2 \coth \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \sinh ^{-1}(c+d x)^4-2 \tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \sinh ^{-1}(c+d x)^4+4 \sinh ^{-1}(c+d x)^4-4 \text{csch}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \sinh ^{-1}(c+d x)^3-4 \text{sech}^2\left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \sinh ^{-1}(c+d x)^3+16 \log \left(1+e^{-\sinh ^{-1}(c+d x)}\right) \sinh ^{-1}(c+d x)^3-16 \log \left(1-e^{\sinh ^{-1}(c+d x)}\right) \sinh ^{-1}(c+d x)^3-24 \coth \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \sinh ^{-1}(c+d x)^2-48 \text{Li}_2\left(e^{\sinh ^{-1}(c+d x)}\right) \sinh ^{-1}(c+d x)^2+24 \tanh \left(\frac{1}{2} \sinh ^{-1}(c+d x)\right) \sinh ^{-1}(c+d x)^2+96 \log \left(1-e^{-\sinh ^{-1}(c+d x)}\right) \sinh ^{-1}(c+d x)-96 \log \left(1+e^{-\sinh ^{-1}(c+d x)}\right) \sinh ^{-1}(c+d x)-96 \text{Li}_3\left(-e^{-\sinh ^{-1}(c+d x)}\right) \sinh ^{-1}(c+d x)+96 \text{Li}_3\left(e^{\sinh ^{-1}(c+d x)}\right) \sinh ^{-1}(c+d x)-48 \left(\sinh ^{-1}(c+d x)^2-2\right) \text{Li}_2\left(-e^{-\sinh ^{-1}(c+d x)}\right)-96 \text{Li}_2\left(e^{-\sinh ^{-1}(c+d x)}\right)-96 \text{Li}_4\left(-e^{-\sinh ^{-1}(c+d x)}\right)-96 \text{Li}_4\left(e^{\sinh ^{-1}(c+d x)}\right)-2 \pi ^4\right)}{24 d e^4}","-\frac{4 b^3 \text{Li}_3\left(-e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^4}+\frac{4 b^3 \text{Li}_3\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^4}-\frac{8 b^3 \tanh ^{-1}\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^4}+\frac{2 b^2 \text{Li}_2\left(-e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e^4}-\frac{2 b^2 \text{Li}_2\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e^4}-\frac{2 b^2 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e^4 (c+d x)}-\frac{2 b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{3 d e^4 (c+d x)^2}-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{3 d e^4 (c+d x)^3}+\frac{4 b \tanh ^{-1}\left(e^{\sinh ^{-1}(c+d x)}\right) \left(a+b \sinh ^{-1}(c+d x)\right)^3}{3 d e^4}-\frac{4 b^4 \text{Li}_2\left(-e^{\sinh ^{-1}(c+d x)}\right)}{d e^4}+\frac{4 b^4 \text{Li}_2\left(e^{\sinh ^{-1}(c+d x)}\right)}{d e^4}+\frac{4 b^4 \text{Li}_4\left(-e^{\sinh ^{-1}(c+d x)}\right)}{d e^4}-\frac{4 b^4 \text{Li}_4\left(e^{\sinh ^{-1}(c+d x)}\right)}{d e^4}",1,"-1/3*a^4/(d*e^4*(c + d*x)^3) + (a^2*b^2*(-8*PolyLog[2, -E^(-ArcSinh[c + d*x])] - (2*(-2 + 4*ArcSinh[c + d*x]^2 + 2*Cosh[2*ArcSinh[c + d*x]] - 3*(c + d*x)*ArcSinh[c + d*x]*Log[1 - E^(-ArcSinh[c + d*x])] + 3*(c + d*x)*ArcSinh[c + d*x]*Log[1 + E^(-ArcSinh[c + d*x])] - 4*(c + d*x)^3*PolyLog[2, E^(-ArcSinh[c + d*x])] + 2*ArcSinh[c + d*x]*Sinh[2*ArcSinh[c + d*x]] + ArcSinh[c + d*x]*Log[1 - E^(-ArcSinh[c + d*x])]*Sinh[3*ArcSinh[c + d*x]] - ArcSinh[c + d*x]*Log[1 + E^(-ArcSinh[c + d*x])]*Sinh[3*ArcSinh[c + d*x]]))/(c + d*x)^3))/(4*d*e^4) + (a*b^3*(-24*ArcSinh[c + d*x]*Coth[ArcSinh[c + d*x]/2] + 4*ArcSinh[c + d*x]^3*Coth[ArcSinh[c + d*x]/2] - 6*ArcSinh[c + d*x]^2*Csch[ArcSinh[c + d*x]/2]^2 - (c + d*x)*ArcSinh[c + d*x]^3*Csch[ArcSinh[c + d*x]/2]^4 - 24*ArcSinh[c + d*x]^2*Log[1 - E^(-ArcSinh[c + d*x])] + 24*ArcSinh[c + d*x]^2*Log[1 + E^(-ArcSinh[c + d*x])] + 48*Log[Tanh[ArcSinh[c + d*x]/2]] - 48*ArcSinh[c + d*x]*PolyLog[2, -E^(-ArcSinh[c + d*x])] + 48*ArcSinh[c + d*x]*PolyLog[2, E^(-ArcSinh[c + d*x])] - 48*PolyLog[3, -E^(-ArcSinh[c + d*x])] + 48*PolyLog[3, E^(-ArcSinh[c + d*x])] - 6*ArcSinh[c + d*x]^2*Sech[ArcSinh[c + d*x]/2]^2 - (16*ArcSinh[c + d*x]^3*Sinh[ArcSinh[c + d*x]/2]^4)/(c + d*x)^3 + 24*ArcSinh[c + d*x]*Tanh[ArcSinh[c + d*x]/2] - 4*ArcSinh[c + d*x]^3*Tanh[ArcSinh[c + d*x]/2]))/(12*d*e^4) + (b^4*(-2*Pi^4 + 4*ArcSinh[c + d*x]^4 - 24*ArcSinh[c + d*x]^2*Coth[ArcSinh[c + d*x]/2] + 2*ArcSinh[c + d*x]^4*Coth[ArcSinh[c + d*x]/2] - 4*ArcSinh[c + d*x]^3*Csch[ArcSinh[c + d*x]/2]^2 - ((c + d*x)*ArcSinh[c + d*x]^4*Csch[ArcSinh[c + d*x]/2]^4)/2 + 96*ArcSinh[c + d*x]*Log[1 - E^(-ArcSinh[c + d*x])] - 96*ArcSinh[c + d*x]*Log[1 + E^(-ArcSinh[c + d*x])] + 16*ArcSinh[c + d*x]^3*Log[1 + E^(-ArcSinh[c + d*x])] - 16*ArcSinh[c + d*x]^3*Log[1 - E^ArcSinh[c + d*x]] - 48*(-2 + ArcSinh[c + d*x]^2)*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 96*PolyLog[2, E^(-ArcSinh[c + d*x])] - 48*ArcSinh[c + d*x]^2*PolyLog[2, E^ArcSinh[c + d*x]] - 96*ArcSinh[c + d*x]*PolyLog[3, -E^(-ArcSinh[c + d*x])] + 96*ArcSinh[c + d*x]*PolyLog[3, E^ArcSinh[c + d*x]] - 96*PolyLog[4, -E^(-ArcSinh[c + d*x])] - 96*PolyLog[4, E^ArcSinh[c + d*x]] - 4*ArcSinh[c + d*x]^3*Sech[ArcSinh[c + d*x]/2]^2 - (8*ArcSinh[c + d*x]^4*Sinh[ArcSinh[c + d*x]/2]^4)/(c + d*x)^3 + 24*ArcSinh[c + d*x]^2*Tanh[ArcSinh[c + d*x]/2] - 2*ArcSinh[c + d*x]^4*Tanh[ArcSinh[c + d*x]/2]))/(24*d*e^4) + (4*a^3*b*((ArcSinh[c + d*x]*Coth[ArcSinh[c + d*x]/2])/12 - Csch[ArcSinh[c + d*x]/2]^2/24 - (ArcSinh[c + d*x]*Coth[ArcSinh[c + d*x]/2]*Csch[ArcSinh[c + d*x]/2]^2)/24 - Log[Tanh[ArcSinh[c + d*x]/2]]/6 - Sech[ArcSinh[c + d*x]/2]^2/24 - (ArcSinh[c + d*x]*Tanh[ArcSinh[c + d*x]/2])/12 - (ArcSinh[c + d*x]*Sech[ArcSinh[c + d*x]/2]^2*Tanh[ArcSinh[c + d*x]/2])/24))/(d*e^4)","B",0
155,0,0,25,1.1501274,"\int \frac{(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^m/(a + b*ArcSinh[c + d*x]),x]","\int \frac{(c e+d e x)^m}{a+b \sinh ^{-1}(c+d x)} \, dx","\text{Int}\left(\frac{(e (c+d x))^m}{a+b \sinh ^{-1}(c+d x)},x\right)",0,"Integrate[(c*e + d*e*x)^m/(a + b*ArcSinh[c + d*x]), x]","A",-1
156,1,151,213,0.3073172,"\int \frac{(c e+d e x)^4}{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x]),x]","\frac{e^4 \left(2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)}{16 b d}","\frac{e^4 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{8 b d}-\frac{3 e^4 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{16 b d}+\frac{e^4 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{16 b d}-\frac{e^4 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{8 b d}+\frac{3 e^4 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{16 b d}-\frac{e^4 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{16 b d}",1,"(e^4*(2*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]] - 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c + d*x])] + Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c + d*x])] - 2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])] - Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c + d*x])]))/(16*b*d)","A",1
157,1,109,145,0.2156911,"\int \frac{(c e+d e x)^3}{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x]),x]","\frac{e^3 \left(2 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-2 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)}{8 b d}","\frac{e^3 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{4 b d}-\frac{e^3 \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{8 b d}-\frac{e^3 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{4 b d}+\frac{e^3 \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{8 b d}",1,"(e^3*(2*CoshIntegral[2*(a/b + ArcSinh[c + d*x])]*Sinh[(2*a)/b] - CoshIntegral[4*(a/b + ArcSinh[c + d*x])]*Sinh[(4*a)/b] - 2*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])] + Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c + d*x])]))/(8*b*d)","A",1
158,1,102,141,0.1840383,"\int \frac{(c e+d e x)^2}{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x]),x]","\frac{e^2 \left(-\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)}{4 b d}","-\frac{e^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{4 b d}+\frac{e^2 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{4 b d}+\frac{e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{4 b d}-\frac{e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{4 b d}",1,"(e^2*(-(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]]) + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c + d*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])]))/(4*b*d)","A",1
159,1,61,69,0.0902859,"\int \frac{c e+d e x}{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSinh[c + d*x]),x]","-\frac{e \left(\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c+d x)\right)-\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 a}{b}+2 \sinh ^{-1}(c+d x)\right)\right)}{2 b d}","\frac{e \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{2 b d}-\frac{e \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{2 b d}",1,"-1/2*(e*(CoshIntegral[(2*a)/b + 2*ArcSinh[c + d*x]]*Sinh[(2*a)/b] - Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcSinh[c + d*x]]))/(b*d)","A",1
160,1,49,58,0.0218049,"\int \frac{1}{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-1),x]","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{b d}","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{b d}-\frac{\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{b d}",1,"(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]] - Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]])/(b*d)","A",1
161,0,0,27,0.8400286,"\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])),x]","\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])), x]","A",-1
162,1,281,256,1.0963692,"\int \frac{(c e+d e x)^4}{\left(a+b \sinh ^{-1}(c+d x)\right)^2} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^2,x]","\frac{e^4 \left(16 \left(3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)-5 \left(10 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-5 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-10 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+5 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)-\frac{16 b \sqrt{(c+d x)^2+1} (c+d x)^4}{a+b \sinh ^{-1}(c+d x)}\right)}{16 b^2 d}","-\frac{e^4 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{8 b^2 d}+\frac{9 e^4 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{16 b^2 d}-\frac{5 e^4 \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{16 b^2 d}+\frac{e^4 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{8 b^2 d}-\frac{9 e^4 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{16 b^2 d}+\frac{5 e^4 \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{16 b^2 d}-\frac{e^4 (c+d x)^4 \sqrt{(c+d x)^2+1}}{b d \left(a+b \sinh ^{-1}(c+d x)\right)}",1,"(e^4*((-16*b*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x]) + 16*(3*CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] - CoshIntegral[3*(a/b + ArcSinh[c + d*x])]*Sinh[(3*a)/b] - 3*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])]) - 5*(10*CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] - 5*CoshIntegral[3*(a/b + ArcSinh[c + d*x])]*Sinh[(3*a)/b] + CoshIntegral[5*(a/b + ArcSinh[c + d*x])]*Sinh[(5*a)/b] - 10*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + 5*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])] - Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c + d*x])])))/(16*b^2*d)","A",1
163,1,193,188,0.8799914,"\int \frac{(c e+d e x)^3}{\left(a+b \sinh ^{-1}(c+d x)\right)^2} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^2,x]","-\frac{e^3 \left(\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-4 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+3 \left(\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\log \left(a+b \sinh ^{-1}(c+d x)\right)\right)+\frac{2 b \sqrt{(c+d x)^2+1} (c+d x)^3}{a+b \sinh ^{-1}(c+d x)}-3 \log \left(a+b \sinh ^{-1}(c+d x)\right)\right)}{2 b^2 d}","-\frac{e^3 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{2 b^2 d}+\frac{e^3 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{2 b^2 d}+\frac{e^3 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{2 b^2 d}-\frac{e^3 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{2 b^2 d}-\frac{e^3 (c+d x)^3 \sqrt{(c+d x)^2+1}}{b d \left(a+b \sinh ^{-1}(c+d x)\right)}",1,"-1/2*(e^3*((2*b*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x]) + Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c + d*x])] - Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c + d*x])] - 3*Log[a + b*ArcSinh[c + d*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])] + 3*(Log[a + b*ArcSinh[c + d*x]] + Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])]) + Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c + d*x])]))/(b^2*d)","A",1
164,1,138,184,0.7384132,"\int \frac{(c e+d e x)^2}{\left(a+b \sinh ^{-1}(c+d x)\right)^2} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^2,x]","\frac{e^2 \left(\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-3 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+3 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\frac{4 b \sqrt{(c+d x)^2+1} (c+d x)^2}{a+b \sinh ^{-1}(c+d x)}\right)}{4 b^2 d}","\frac{e^2 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{4 b^2 d}-\frac{3 e^2 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{4 b^2 d}-\frac{e^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{4 b^2 d}+\frac{3 e^2 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{4 b^2 d}-\frac{e^2 (c+d x)^2 \sqrt{(c+d x)^2+1}}{b d \left(a+b \sinh ^{-1}(c+d x)\right)}",1,"(e^2*((-4*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x]) + CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] - 3*CoshIntegral[3*(a/b + ArcSinh[c + d*x])]*Sinh[(3*a)/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + 3*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])]))/(4*b^2*d)","A",1
165,1,97,103,0.2988376,"\int \frac{c e+d e x}{\left(a+b \sinh ^{-1}(c+d x)\right)^2} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^2,x]","\frac{e \left(-\frac{b \sqrt{c^2+2 c d x+d^2 x^2+1} (c+d x)}{a+b \sinh ^{-1}(c+d x)}+\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)}{b^2 d}","\frac{e \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{b^2 d}-\frac{e \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{b^2 d}-\frac{e \sqrt{(c+d x)^2+1} (c+d x)}{b d \left(a+b \sinh ^{-1}(c+d x)\right)}",1,"(e*(-((b*(c + d*x)*Sqrt[1 + c^2 + 2*c*d*x + d^2*x^2])/(a + b*ArcSinh[c + d*x])) + Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c + d*x])] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])]))/(b^2*d)","A",1
166,1,77,91,0.089332,"\int \frac{1}{\left(a+b \sinh ^{-1}(c+d x)\right)^2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-2),x]","\frac{-\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\frac{b \sqrt{(c+d x)^2+1}}{a+b \sinh ^{-1}(c+d x)}}{b^2 d}","-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{b^2 d}+\frac{\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{b^2 d}-\frac{\sqrt{(c+d x)^2+1}}{b d \left(a+b \sinh ^{-1}(c+d x)\right)}",1,"(-((b*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])) - CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] + Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]])/(b^2*d)","A",1
167,0,0,27,1.4434755,"\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^2} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^2),x]","\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^2} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^2},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^2), x]","A",-1
168,1,316,320,1.2502776,"\int \frac{(c e+d e x)^4}{\left(a+b \sinh ^{-1}(c+d x)\right)^3} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^3,x]","\frac{e^4 \left(-\frac{16 b^2 \sqrt{(c+d x)^2+1} (c+d x)^4}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}+48 \left(-\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)+25 \left(2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-3 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+3 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)+\frac{16 b \left(-5 (c+d x)^5-4 (c+d x)^3\right)}{a+b \sinh ^{-1}(c+d x)}\right)}{32 b^3 d}","\frac{e^4 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{16 b^3 d}-\frac{27 e^4 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{32 b^3 d}+\frac{25 e^4 \cosh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{32 b^3 d}-\frac{e^4 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{16 b^3 d}+\frac{27 e^4 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{32 b^3 d}-\frac{25 e^4 \sinh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{32 b^3 d}-\frac{5 e^4 (c+d x)^5}{2 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{2 e^4 (c+d x)^3}{b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e^4 \sqrt{(c+d x)^2+1} (c+d x)^4}{2 b d \left(a+b \sinh ^{-1}(c+d x)\right)^2}",1,"(e^4*((-16*b^2*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2 + (16*b*(-4*(c + d*x)^3 - 5*(c + d*x)^5))/(a + b*ArcSinh[c + d*x]) + 48*(-(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]]) + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c + d*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])]) + 25*(2*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]] - 3*Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c + d*x])] + Cosh[(5*a)/b]*CoshIntegral[5*(a/b + ArcSinh[c + d*x])] - 2*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + 3*Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])] - Sinh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c + d*x])])))/(32*b^3*d)","A",1
169,1,179,247,0.6580464,"\int \frac{(c e+d e x)^3}{\left(a+b \sinh ^{-1}(c+d x)\right)^3} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^3,x]","\frac{e^3 \left(-\frac{b^2 \sqrt{(c+d x)^2+1} (c+d x)^3}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}+\sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-2 \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+2 \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\frac{b \left(-4 (c+d x)^4-3 (c+d x)^2\right)}{a+b \sinh ^{-1}(c+d x)}\right)}{2 b^3 d}","\frac{e^3 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{2 b^3 d}-\frac{e^3 \sinh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{b^3 d}-\frac{e^3 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{2 b^3 d}+\frac{e^3 \cosh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{b^3 d}-\frac{2 e^3 (c+d x)^4}{b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{3 e^3 (c+d x)^2}{2 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e^3 \sqrt{(c+d x)^2+1} (c+d x)^3}{2 b d \left(a+b \sinh ^{-1}(c+d x)\right)^2}",1,"(e^3*(-((b^2*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2) + (b*(-3*(c + d*x)^2 - 4*(c + d*x)^4))/(a + b*ArcSinh[c + d*x]) + CoshIntegral[2*(a/b + ArcSinh[c + d*x])]*Sinh[(2*a)/b] - 2*CoshIntegral[4*(a/b + ArcSinh[c + d*x])]*Sinh[(4*a)/b] - Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])] + 2*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c + d*x])]))/(2*b^3*d)","A",1
170,1,216,246,0.7252477,"\int \frac{(c e+d e x)^2}{\left(a+b \sinh ^{-1}(c+d x)\right)^3} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^3,x]","\frac{e^2 \left(-\frac{4 b^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}+9 \left(-\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)+8 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-8 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\frac{4 b \left(-3 (c+d x)^3-2 (c+d x)\right)}{a+b \sinh ^{-1}(c+d x)}\right)}{8 b^3 d}","-\frac{e^2 \cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{8 b^3 d}+\frac{9 e^2 \cosh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{8 b^3 d}+\frac{e^2 \sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{8 b^3 d}-\frac{9 e^2 \sinh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{8 b^3 d}-\frac{3 e^2 (c+d x)^3}{2 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e^2 (c+d x)}{b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{2 b d \left(a+b \sinh ^{-1}(c+d x)\right)^2}",1,"(e^2*((-4*b^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2 + (4*b*(-2*(c + d*x) - 3*(c + d*x)^3))/(a + b*ArcSinh[c + d*x]) + 8*Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]] - 8*Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + 9*(-(Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]]) + Cosh[(3*a)/b]*CoshIntegral[3*(a/b + ArcSinh[c + d*x])] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] - Sinh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])])))/(8*b^3*d)","A",1
171,1,120,156,0.3149251,"\int \frac{c e+d e x}{\left(a+b \sinh ^{-1}(c+d x)\right)^3} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^3,x]","\frac{e \left(-\frac{b^2 (c+d x) \sqrt{(c+d x)^2+1}}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}-2 \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+2 \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\frac{b \left(-2 (c+d x)^2-1\right)}{a+b \sinh ^{-1}(c+d x)}\right)}{2 b^3 d}","-\frac{e \sinh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{b^3 d}+\frac{e \cosh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{b^3 d}-\frac{e (c+d x)^2}{b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e}{2 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e \sqrt{(c+d x)^2+1} (c+d x)}{2 b d \left(a+b \sinh ^{-1}(c+d x)\right)^2}",1,"(e*(-((b^2*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2) + (b*(-1 - 2*(c + d*x)^2))/(a + b*ArcSinh[c + d*x]) - 2*CoshIntegral[2*(a/b + ArcSinh[c + d*x])]*Sinh[(2*a)/b] + 2*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])]))/(2*b^3*d)","A",1
172,1,100,125,0.2806507,"\int \frac{1}{\left(a+b \sinh ^{-1}(c+d x)\right)^3} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-3),x]","-\frac{\frac{b^2 \sqrt{(c+d x)^2+1}}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}-\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\frac{b (c+d x)}{a+b \sinh ^{-1}(c+d x)}}{2 b^3 d}","\frac{\cosh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{2 b^3 d}-\frac{\sinh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{2 b^3 d}-\frac{c+d x}{2 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{\sqrt{(c+d x)^2+1}}{2 b d \left(a+b \sinh ^{-1}(c+d x)\right)^2}",1,"-1/2*((b^2*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^2 + (b*(c + d*x))/(a + b*ArcSinh[c + d*x]) - Cosh[a/b]*CoshIntegral[a/b + ArcSinh[c + d*x]] + Sinh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]])/(b^3*d)","A",1
173,0,0,27,1.0583298,"\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^3} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^3),x]","\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^3} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^3},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^3), x]","A",-1
174,1,410,410,1.9040156,"\int \frac{(c e+d e x)^4}{\left(a+b \sinh ^{-1}(c+d x)\right)^4} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^4,x]","-\frac{e^4 \left(\frac{32 b^3 \sqrt{(c+d x)^2+1} (c+d x)^4}{\left(a+b \sinh ^{-1}(c+d x)\right)^3}-\frac{16 b^2 \left(-5 (c+d x)^5-4 (c+d x)^3\right)}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}+384 \left(\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+544 \left(-3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)+125 \left(10 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-5 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-10 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+5 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)+\frac{16 b \sqrt{(c+d x)^2+1} \left(25 (c+d x)^4+12 (c+d x)^2\right)}{a+b \sinh ^{-1}(c+d x)}\right)}{96 b^4 d}","-\frac{e^4 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{48 b^4 d}+\frac{27 e^4 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{32 b^4 d}-\frac{125 e^4 \sinh \left(\frac{5 a}{b}\right) \text{Chi}\left(\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{96 b^4 d}+\frac{e^4 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{48 b^4 d}-\frac{27 e^4 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{32 b^4 d}+\frac{125 e^4 \cosh \left(\frac{5 a}{b}\right) \text{Shi}\left(\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{96 b^4 d}-\frac{25 e^4 \sqrt{(c+d x)^2+1} (c+d x)^4}{6 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{2 e^4 \sqrt{(c+d x)^2+1} (c+d x)^2}{b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{5 e^4 (c+d x)^5}{6 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^2}-\frac{2 e^4 (c+d x)^3}{3 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^2}-\frac{e^4 \sqrt{(c+d x)^2+1} (c+d x)^4}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^3}",1,"-1/96*(e^4*((32*b^3*(c + d*x)^4*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^3 - (16*b^2*(-4*(c + d*x)^3 - 5*(c + d*x)^5))/(a + b*ArcSinh[c + d*x])^2 + (16*b*Sqrt[1 + (c + d*x)^2]*(12*(c + d*x)^2 + 25*(c + d*x)^4))/(a + b*ArcSinh[c + d*x]) + 384*(CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]]) + 544*(-3*CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] + CoshIntegral[3*(a/b + ArcSinh[c + d*x])]*Sinh[(3*a)/b] + 3*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] - Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])]) + 125*(10*CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] - 5*CoshIntegral[3*(a/b + ArcSinh[c + d*x])]*Sinh[(3*a)/b] + CoshIntegral[5*(a/b + ArcSinh[c + d*x])]*Sinh[(5*a)/b] - 10*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + 5*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])] - Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcSinh[c + d*x])])))/(b^4*d)","A",1
175,1,318,340,1.1117834,"\int \frac{(c e+d e x)^3}{\left(a+b \sinh ^{-1}(c+d x)\right)^4} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^4,x]","\frac{e^3 \left(-\frac{2 b^3 \sqrt{(c+d x)^2+1} (c+d x)^3}{\left(a+b \sinh ^{-1}(c+d x)\right)^3}+\frac{b^2 \left(-4 (c+d x)^4-3 (c+d x)^2\right)}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}+30 \left(\cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\log \left(a+b \sinh ^{-1}(c+d x)\right)\right)+8 \left(-4 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+4 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-\sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+3 \log \left(a+b \sinh ^{-1}(c+d x)\right)\right)-\frac{2 b \sqrt{(c+d x)^2+1} \left(8 (c+d x)^3+3 (c+d x)\right)}{a+b \sinh ^{-1}(c+d x)}+6 \log \left(a+b \sinh ^{-1}(c+d x)\right)\right)}{6 b^4 d}","-\frac{e^3 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}+\frac{4 e^3 \cosh \left(\frac{4 a}{b}\right) \text{Chi}\left(\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}+\frac{e^3 \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}-\frac{4 e^3 \sinh \left(\frac{4 a}{b}\right) \text{Shi}\left(\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}-\frac{8 e^3 \sqrt{(c+d x)^2+1} (c+d x)^3}{3 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e^3 \sqrt{(c+d x)^2+1} (c+d x)}{b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{2 e^3 (c+d x)^4}{3 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^2}-\frac{e^3 (c+d x)^2}{2 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^2}-\frac{e^3 \sqrt{(c+d x)^2+1} (c+d x)^3}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^3}",1,"(e^3*((-2*b^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^3 + (b^2*(-3*(c + d*x)^2 - 4*(c + d*x)^4))/(a + b*ArcSinh[c + d*x])^2 - (2*b*Sqrt[1 + (c + d*x)^2]*(3*(c + d*x) + 8*(c + d*x)^3))/(a + b*ArcSinh[c + d*x]) + 6*Log[a + b*ArcSinh[c + d*x]] + 30*(Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c + d*x])] - Log[a + b*ArcSinh[c + d*x]] - Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])]) + 8*(-4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c + d*x])] + Cosh[(4*a)/b]*CoshIntegral[4*(a/b + ArcSinh[c + d*x])] + 3*Log[a + b*ArcSinh[c + d*x]] + 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])] - Sinh[(4*a)/b]*SinhIntegral[4*(a/b + ArcSinh[c + d*x])])))/(6*b^4*d)","A",1
176,1,258,331,0.8425365,"\int \frac{(c e+d e x)^2}{\left(a+b \sinh ^{-1}(c+d x)\right)^4} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^4,x]","\frac{e^2 \left(-\frac{8 b^3 (c+d x)^2 \sqrt{(c+d x)^2+1}}{\left(a+b \sinh ^{-1}(c+d x)\right)^3}+\frac{4 b^2 \left(-3 (c+d x)^3-2 (c+d x)\right)}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}+27 \left(3 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-3 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)\right)-80 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+80 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\frac{4 b \sqrt{(c+d x)^2+1} \left(9 (c+d x)^2+2\right)}{a+b \sinh ^{-1}(c+d x)}\right)}{24 b^4 d}","\frac{e^2 \sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{24 b^4 d}-\frac{9 e^2 \sinh \left(\frac{3 a}{b}\right) \text{Chi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{8 b^4 d}-\frac{e^2 \cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{24 b^4 d}+\frac{9 e^2 \cosh \left(\frac{3 a}{b}\right) \text{Shi}\left(\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{8 b^4 d}-\frac{3 e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{2 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e^2 \sqrt{(c+d x)^2+1}}{3 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e^2 (c+d x)^3}{2 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^2}-\frac{e^2 (c+d x)}{3 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^2}-\frac{e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^3}",1,"(e^2*((-8*b^3*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^3 + (4*b^2*(-2*(c + d*x) - 3*(c + d*x)^3))/(a + b*ArcSinh[c + d*x])^2 - (4*b*Sqrt[1 + (c + d*x)^2]*(2 + 9*(c + d*x)^2))/(a + b*ArcSinh[c + d*x]) - 80*CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] + 80*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + 27*(3*CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] - CoshIntegral[3*(a/b + ArcSinh[c + d*x])]*Sinh[(3*a)/b] - 3*Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]] + Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcSinh[c + d*x])])))/(24*b^4*d)","A",1
177,1,181,204,0.8080224,"\int \frac{c e+d e x}{\left(a+b \sinh ^{-1}(c+d x)\right)^4} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^4,x]","\frac{e \left(-\frac{2 b^3 (c+d x) \sqrt{(c+d x)^2+1}}{\left(a+b \sinh ^{-1}(c+d x)\right)^3}+\frac{b^2 \left(-2 (c+d x)^2-1\right)}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}+4 \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)-4 \left(\sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)+\log \left(a+b \sinh ^{-1}(c+d x)\right)\right)-\frac{4 b (c+d x) \sqrt{(c+d x)^2+1}}{a+b \sinh ^{-1}(c+d x)}+4 \log \left(a+b \sinh ^{-1}(c+d x)\right)\right)}{6 b^4 d}","\frac{2 e \cosh \left(\frac{2 a}{b}\right) \text{Chi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}-\frac{2 e \sinh \left(\frac{2 a}{b}\right) \text{Shi}\left(\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)}{3 b^4 d}-\frac{2 e \sqrt{(c+d x)^2+1} (c+d x)}{3 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{e (c+d x)^2}{3 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^2}-\frac{e}{6 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^2}-\frac{e \sqrt{(c+d x)^2+1} (c+d x)}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^3}",1,"(e*((-2*b^3*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^3 + (b^2*(-1 - 2*(c + d*x)^2))/(a + b*ArcSinh[c + d*x])^2 - (4*b*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x]) + 4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh[c + d*x])] + 4*Log[a + b*ArcSinh[c + d*x]] - 4*(Log[a + b*ArcSinh[c + d*x]] + Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcSinh[c + d*x])])))/(6*b^4*d)","A",1
178,1,130,160,0.4394654,"\int \frac{1}{\left(a+b \sinh ^{-1}(c+d x)\right)^4} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-4),x]","-\frac{\frac{2 b^3 \sqrt{(c+d x)^2+1}}{\left(a+b \sinh ^{-1}(c+d x)\right)^3}+\frac{b^2 (c+d x)}{\left(a+b \sinh ^{-1}(c+d x)\right)^2}+\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\frac{b \sqrt{(c+d x)^2+1}}{a+b \sinh ^{-1}(c+d x)}}{6 b^4 d}","-\frac{\sinh \left(\frac{a}{b}\right) \text{Chi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{6 b^4 d}+\frac{\cosh \left(\frac{a}{b}\right) \text{Shi}\left(\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{6 b^4 d}-\frac{\sqrt{(c+d x)^2+1}}{6 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)}-\frac{c+d x}{6 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^2}-\frac{\sqrt{(c+d x)^2+1}}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^3}",1,"-1/6*((2*b^3*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x])^3 + (b^2*(c + d*x))/(a + b*ArcSinh[c + d*x])^2 + (b*Sqrt[1 + (c + d*x)^2])/(a + b*ArcSinh[c + d*x]) + CoshIntegral[a/b + ArcSinh[c + d*x]]*Sinh[a/b] - Cosh[a/b]*SinhIntegral[a/b + ArcSinh[c + d*x]])/(b^4*d)","A",1
179,0,0,27,3.6296992,"\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^4} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4),x]","\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^4} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^4},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^4), x]","A",-1
180,1,342,361,0.6530769,"\int (c e+d e x)^4 \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^4*Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^4 e^{-\frac{5 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(-150 e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+3 \sqrt{5} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{3}{2},-\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-25 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{3}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+150 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+25 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{3}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-3 \sqrt{5} e^{\frac{10 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{3}{2},\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{2400 d \sqrt{-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}}}","\frac{\sqrt{\pi } \sqrt{b} e^4 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}-\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^4 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}+\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^4 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{320 d}-\frac{\sqrt{\pi } \sqrt{b} e^4 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{b} e^4 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}-\frac{\sqrt{\frac{\pi }{5}} \sqrt{b} e^4 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{320 d}+\frac{e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}",1,"(e^4*Sqrt[a + b*ArcSinh[c + d*x]]*(-150*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, a/b + ArcSinh[c + d*x]] + 3*Sqrt[5]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 25*Sqrt[3]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 150*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)] + 25*Sqrt[3]*E^((8*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, (3*(a + b*ArcSinh[c + d*x]))/b] - 3*Sqrt[5]*E^((10*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(2400*d*E^((5*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])","A",0
181,1,223,272,0.2904736,"\int (c e+d e x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^3*Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^3 e^{-\frac{4 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{3}{2},-\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-4 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{3}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-4 \sqrt{2} \Gamma \left(\frac{3}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{128 d \sqrt{-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}}}","-\frac{\sqrt{\pi } \sqrt{b} e^3 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{256 d}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^3 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}-\frac{\sqrt{\pi } \sqrt{b} e^3 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{256 d}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e^3 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}+\frac{e^3 (c+d x)^4 \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{3 e^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}",1,"(e^3*Sqrt[a + b*ArcSinh[c + d*x]]*(Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, (-4*(a + b*ArcSinh[c + d*x]))/b] - 4*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*(-4*Sqrt[2]*Gamma[3/2, (2*(a + b*ArcSinh[c + d*x]))/b] + E^((2*a)/b)*Gamma[3/2, (4*(a + b*ArcSinh[c + d*x]))/b])))/(128*d*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])","A",0
182,1,238,245,0.4080739,"\int (c e+d e x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)^2*Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^2 e^{-\frac{3 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{3}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{3}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{72 d \sqrt{-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}}}","-\frac{\sqrt{\pi } \sqrt{b} e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}+\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+d x)}}{\sqrt{b}}\right)}{48 d}+\frac{\sqrt{\pi } \sqrt{b} e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}-\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+d x)}}{\sqrt{b}}\right)}{48 d}+\frac{e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d}",1,"(e^2*Sqrt[a + b*ArcSinh[c + d*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, a/b + ArcSinh[c + d*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, (-3*(a + b*ArcSinh[c + d*x]))/b] - 9*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, (3*(a + b*ArcSinh[c + d*x]))/b]))/(72*d*E^((3*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])","A",0
183,1,140,164,0.1009313,"\int (c e+d e x) \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[(c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e e^{-\frac{2 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{3}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{3}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{8 \sqrt{2} d \sqrt{-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}}}","-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{b} e e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}+\frac{e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{e \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}",1,"(e*Sqrt[a + b*ArcSinh[c + d*x]]*(Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[3/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[3/2, (2*(a + b*ArcSinh[c + d*x]))/b]))/(8*Sqrt[2]*d*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])","A",0
184,1,111,115,0.1218298,"\int \sqrt{a+b \sinh ^{-1}(c+d x)} \, dx","Integrate[Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right)}{2 d}","\frac{\sqrt{\pi } \sqrt{b} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}-\frac{\sqrt{\pi } \sqrt{b} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 d}+\frac{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{d}",1,"(Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/(2*d*E^(a/b))","A",0
185,0,0,29,2.164463,"\int \frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{c e+d e x} \, dx","Integrate[Sqrt[a + b*ArcSinh[c + d*x]]/(c*e + d*e*x),x]","\int \frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{c e+d e x} \, dx","\frac{\text{Int}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{c+d x},x\right)}{e}",0,"Integrate[Sqrt[a + b*ArcSinh[c + d*x]]/(c*e + d*e*x), x]","A",-1
186,1,343,601,0.476502,"\int (c e+d e x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(3/2),x]","-\frac{b e^4 e^{-\frac{5 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(2250 e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{5}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+9 \sqrt{5} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{5}{2},-\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-125 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{5}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+2250 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{5}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-125 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{5}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+9 \sqrt{5} e^{\frac{10 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{5}{2},\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{36000 d \sqrt{-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}}}","\frac{3 \sqrt{\pi } b^{3/2} e^4 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}-\frac{3 \sqrt{3 \pi } b^{3/2} e^4 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3200 d}-\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^4 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{200 d}+\frac{3 \sqrt{\frac{\pi }{5}} b^{3/2} e^4 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3200 d}+\frac{3 \sqrt{\pi } b^{3/2} e^4 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}-\frac{3 \sqrt{3 \pi } b^{3/2} e^4 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3200 d}-\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^4 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{200 d}+\frac{3 \sqrt{\frac{\pi }{5}} b^{3/2} e^4 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3200 d}+\frac{e^4 (c+d x)^5 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{5 d}-\frac{3 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^4 \sqrt{a+b \sinh ^{-1}(c+d x)}}{50 d}+\frac{2 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}-\frac{4 b e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{25 d}",1,"-1/36000*(b*e^4*Sqrt[a + b*ArcSinh[c + d*x]]*(2250*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[5/2, a/b + ArcSinh[c + d*x]] + 9*Sqrt[5]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 125*Sqrt[3]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 2250*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, -((a + b*ArcSinh[c + d*x])/b)] - 125*Sqrt[3]*E^((8*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[5/2, (3*(a + b*ArcSinh[c + d*x]))/b] + 9*Sqrt[5]*E^((10*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[5/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(d*E^((5*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])","A",0
187,1,225,360,0.2881007,"\int (c e+d e x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{b e^3 e^{-\frac{4 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(-\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{5}{2},-\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+8 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{5}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(e^{\frac{2 a}{b}} \Gamma \left(\frac{5}{2},\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-8 \sqrt{2} \Gamma \left(\frac{5}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{512 d \sqrt{-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}}}","-\frac{3 \sqrt{\pi } b^{3/2} e^3 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2048 d}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^3 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{128 d}+\frac{3 \sqrt{\pi } b^{3/2} e^3 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2048 d}-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e^3 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{128 d}+\frac{e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{4 d}-\frac{3 b e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}+\frac{9 b e^3 \sqrt{(c+d x)^2+1} (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d}-\frac{3 e^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{32 d}",1,"(b*e^3*Sqrt[a + b*ArcSinh[c + d*x]]*(-(Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, (-4*(a + b*ArcSinh[c + d*x]))/b]) + 8*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*(-8*Sqrt[2]*Gamma[5/2, (2*(a + b*ArcSinh[c + d*x]))/b] + E^((2*a)/b)*Gamma[5/2, (4*(a + b*ArcSinh[c + d*x]))/b])))/(512*d*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])","A",0
188,1,238,328,0.2985064,"\int (c e+d e x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^(3/2),x]","-\frac{b e^2 e^{-\frac{3 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(-27 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{5}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{5}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-27 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{5}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{5}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{216 d \sqrt{-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}}}","-\frac{3 \sqrt{\pi } b^{3/2} e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}+\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{96 d}-\frac{3 \sqrt{\pi } b^{3/2} e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}+\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{96 d}+\frac{e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{3 d}-\frac{b e^2 \sqrt{(c+d x)^2+1} (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac{b e^2 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{3 d}",1,"-1/216*(b*e^2*Sqrt[a + b*ArcSinh[c + d*x]]*(-27*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[5/2, a/b + ArcSinh[c + d*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, (-3*(a + b*ArcSinh[c + d*x]))/b] - 27*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, -((a + b*ArcSinh[c + d*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[5/2, (3*(a + b*ArcSinh[c + d*x]))/b]))/(d*E^((3*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])","A",0
189,1,142,205,0.108094,"\int (c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{b e e^{-\frac{2 a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{5}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{5}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{16 \sqrt{2} d \sqrt{-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}}}","-\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} e e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}+\frac{e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{2 d}-\frac{3 b e \sqrt{(c+d x)^2+1} (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac{e \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{4 d}",1,"(b*e*Sqrt[a + b*ArcSinh[c + d*x]]*(-(Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, (-2*(a + b*ArcSinh[c + d*x]))/b]) + E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[5/2, (2*(a + b*ArcSinh[c + d*x]))/b]))/(16*Sqrt[2]*d*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])","A",0
190,1,272,150,0.2268096,"\int \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{\sqrt{b} \left(\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+d 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+d x)}}{\sqrt{b}}\right)+4 \sqrt{b} \left(2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt{(c+d x)^2+1}\right) \sqrt{a+b \sinh ^{-1}(c+d x)}\right)}{8 d}+\frac{a e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right)}{2 d}","\frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 d}-\frac{3 b \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{d}",1,"(a*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/(2*d*E^(a/b)) + (Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(-3*Sqrt[1 + (c + d*x)^2] + 2*(c + d*x)*ArcSinh[c + d*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(8*d)","A",0
191,0,0,29,1.4952185,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{c e+d e x} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(3/2)/(c*e + d*e*x),x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{c e+d e x} \, dx","\frac{\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{c+d x},x\right)}{e}",0,"Integrate[(a + b*ArcSinh[c + d*x])^(3/2)/(c*e + d*e*x), x]","A",-1
192,1,342,701,0.7004806,"\int (c e+d e x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(5/2),x]","-\frac{e^4 e^{-\frac{5 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \left(-33750 e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{7}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+27 \sqrt{5} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{7}{2},-\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-625 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{7}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+33750 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{7}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+625 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{7}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-27 \sqrt{5} e^{\frac{10 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{7}{2},\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{540000 d \left(-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}\right)^{3/2}}","\frac{15 \sqrt{\pi } b^{5/2} e^4 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{128 d}-\frac{\sqrt{3 \pi } b^{5/2} e^4 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{1280 d}-\frac{\sqrt{\frac{\pi }{3}} b^{5/2} e^4 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{240 d}+\frac{3 \sqrt{\frac{\pi }{5}} b^{5/2} e^4 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{6400 d}-\frac{15 \sqrt{\pi } b^{5/2} e^4 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{128 d}+\frac{\sqrt{3 \pi } b^{5/2} e^4 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{1280 d}+\frac{\sqrt{\frac{\pi }{3}} b^{5/2} e^4 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{240 d}-\frac{3 \sqrt{\frac{\pi }{5}} b^{5/2} e^4 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{6400 d}+\frac{3 b^2 e^4 (c+d x)^5 \sqrt{a+b \sinh ^{-1}(c+d x)}}{100 d}-\frac{b^2 e^4 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{15 d}+\frac{2 b^2 e^4 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{5 d}+\frac{e^4 (c+d x)^5 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{5 d}-\frac{b e^4 \sqrt{(c+d x)^2+1} (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{10 d}+\frac{2 b e^4 \sqrt{(c+d x)^2+1} (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{15 d}-\frac{4 b e^4 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{15 d}",1,"-1/540000*(e^4*(a + b*ArcSinh[c + d*x])^(5/2)*(-33750*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, a/b + ArcSinh[c + d*x]] + 27*Sqrt[5]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 625*Sqrt[3]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 33750*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, -((a + b*ArcSinh[c + d*x])/b)] + 625*Sqrt[3]*E^((8*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, (3*(a + b*ArcSinh[c + d*x]))/b] - 27*Sqrt[5]*E^((10*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(d*E^((5*a)/b)*(-((a + b*ArcSinh[c + d*x])^2/b^2))^(3/2))","A",0
193,1,223,455,0.3214417,"\int (c e+d e x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^(5/2),x]","-\frac{e^3 e^{-\frac{4 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \left(\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{7}{2},-\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-16 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{7}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(e^{\frac{2 a}{b}} \Gamma \left(\frac{7}{2},\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-16 \sqrt{2} \Gamma \left(\frac{7}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{2048 d \left(-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}\right)^{3/2}}","-\frac{15 \sqrt{\pi } b^{5/2} e^3 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16384 d}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^3 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{512 d}-\frac{15 \sqrt{\pi } b^{5/2} e^3 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16384 d}+\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e^3 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{512 d}+\frac{15 b^2 e^3 (c+d x)^4 \sqrt{a+b \sinh ^{-1}(c+d x)}}{256 d}-\frac{45 b^2 e^3 (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{256 d}-\frac{225 b^2 e^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}+\frac{e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{4 d}-\frac{5 b e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{32 d}+\frac{15 b e^3 \sqrt{(c+d x)^2+1} (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{64 d}-\frac{3 e^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{32 d}",1,"-1/2048*(e^3*(a + b*ArcSinh[c + d*x])^(5/2)*(Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (-4*(a + b*ArcSinh[c + d*x]))/b] - 16*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*(-16*Sqrt[2]*Gamma[7/2, (2*(a + b*ArcSinh[c + d*x]))/b] + E^((2*a)/b)*Gamma[7/2, (4*(a + b*ArcSinh[c + d*x]))/b])))/(d*E^((4*a)/b)*(-((a + b*ArcSinh[c + d*x])^2/b^2))^(3/2))","A",0
194,1,238,394,0.4245174,"\int (c e+d e x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^(5/2),x]","-\frac{e^2 e^{-\frac{3 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \left(81 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{7}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{7}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-81 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{7}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{7}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{648 d \left(-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}\right)^{3/2}}","-\frac{15 \sqrt{\pi } b^{5/2} e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}+\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{576 d}+\frac{15 \sqrt{\pi } b^{5/2} e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64 d}-\frac{5 \sqrt{\frac{\pi }{3}} b^{5/2} e^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{576 d}+\frac{5 b^2 e^2 (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{36 d}-\frac{5 b^2 e^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac{e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{3 d}-\frac{5 b e^2 \sqrt{(c+d x)^2+1} (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{18 d}+\frac{5 b e^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{9 d}",1,"-1/648*(e^2*(a + b*ArcSinh[c + d*x])^(5/2)*(81*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, a/b + ArcSinh[c + d*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (-3*(a + b*ArcSinh[c + d*x]))/b] - 81*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, -((a + b*ArcSinh[c + d*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, (3*(a + b*ArcSinh[c + d*x]))/b]))/(d*E^((3*a)/b)*(-((a + b*ArcSinh[c + d*x])^2/b^2))^(3/2))","A",0
195,1,126,262,0.0766834,"\int (c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2),x]","\frac{e e^{-\frac{2 a}{b}} \left(b^3 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{7}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-b^3 \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{7}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{32 \sqrt{2} d \sqrt{a+b \sinh ^{-1}(c+d x)}}","-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{256 d}-\frac{15 \sqrt{\frac{\pi }{2}} b^{5/2} e e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{256 d}+\frac{15 b^2 e (c+d x)^2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{32 d}+\frac{15 b^2 e \sqrt{a+b \sinh ^{-1}(c+d x)}}{64 d}-\frac{5 b e (c+d x) \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{8 d}+\frac{e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{2 d}+\frac{e \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{4 d}",1,"(e*(-(b^3*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[7/2, (-2*(a + b*ArcSinh[c + d*x]))/b]) + b^3*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (2*(a + b*ArcSinh[c + d*x]))/b]))/(32*Sqrt[2]*d*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
196,1,458,179,1.4979403,"\int \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(5/2),x]","\frac{\sqrt{b} \left(\sqrt{\pi } \left(4 a^2-12 a b+15 b^2\right) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{\pi } \left(4 a^2+12 a b+15 b^2\right) \left(\sinh \left(\frac{a}{b}\right)-\cosh \left(\frac{a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+4 \sqrt{b} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(2 \sqrt{(c+d x)^2+1} \left(a-5 b \sinh ^{-1}(c+d x)\right)+b (c+d x) \left(4 \sinh ^{-1}(c+d x)^2+15\right)\right)\right)+8 a^2 e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right)+4 a \sqrt{b} \left(\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+d 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+d x)}}{\sqrt{b}}\right)+4 \sqrt{b} \left(2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt{(c+d x)^2+1}\right) \sqrt{a+b \sinh ^{-1}(c+d x)}\right)}{16 d}","\frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 d}+\frac{15 b^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{2 d}+\frac{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{d}",1,"((8*a^2*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) + 4*a*Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(-3*Sqrt[1 + (c + d*x)^2] + 2*(c + d*x)*ArcSinh[c + d*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(2*Sqrt[1 + (c + d*x)^2]*(a - 5*b*ArcSinh[c + d*x]) + b*(c + d*x)*(15 + 4*ArcSinh[c + d*x]^2)) + (4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(-Cosh[a/b] + Sinh[a/b]) + (4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(16*d)","B",0
197,0,0,29,1.1310689,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{c e+d e x} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(5/2)/(c*e + d*e*x),x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{c e+d e x} \, dx","\frac{\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{c+d x},x\right)}{e}",0,"Integrate[(a + b*ArcSinh[c + d*x])^(5/2)/(c*e + d*e*x), x]","A",-1
198,1,343,835,0.517311,"\int (c e+d e x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2} \, dx","Integrate[(c*e + d*e*x)^4*(a + b*ArcSinh[c + d*x])^(7/2),x]","\frac{b e^4 e^{-\frac{5 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \left(506250 e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{9}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+81 \sqrt{5} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{9}{2},-\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-3125 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{9}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+506250 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{9}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-3125 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{9}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+81 \sqrt{5} e^{\frac{10 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{9}{2},\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{8100000 d \left(-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}\right)^{3/2}}","\frac{e^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2} (c+d x)^5}{5 d}+\frac{7 b^2 e^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} (c+d x)^5}{100 d}-\frac{7 b e^4 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} (c+d x)^4}{50 d}-\frac{21 b^3 e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)} (c+d x)^4}{1000 d}-\frac{7 b^2 e^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} (c+d x)^3}{45 d}+\frac{14 b e^4 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} (c+d x)^2}{75 d}+\frac{119 b^3 e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)} (c+d x)^2}{1125 d}+\frac{14 b^2 e^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2} (c+d x)}{15 d}-\frac{28 b e^4 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{75 d}+\frac{105 b^{7/2} e^4 e^{a/b} \sqrt{\pi } \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{256 d}-\frac{21 b^{7/2} e^4 e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64000 d}-\frac{119 b^{7/2} e^4 e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{18000 d}+\frac{21 b^{7/2} e^4 e^{\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64000 d}+\frac{105 b^{7/2} e^4 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{256 d}-\frac{21 b^{7/2} e^4 e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64000 d}-\frac{119 b^{7/2} e^4 e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{18000 d}+\frac{21 b^{7/2} e^4 e^{-\frac{5 a}{b}} \sqrt{\frac{\pi }{5}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{64000 d}-\frac{1813 b^3 e^4 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{1125 d}",1,"(b*e^4*(a + b*ArcSinh[c + d*x])^(5/2)*(506250*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, a/b + ArcSinh[c + d*x]] + 81*Sqrt[5]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 3125*Sqrt[3]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 506250*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, -((a + b*ArcSinh[c + d*x])/b)] - 3125*Sqrt[3]*E^((8*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (3*(a + b*ArcSinh[c + d*x]))/b] + 81*Sqrt[5]*E^((10*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(8100000*d*E^((5*a)/b)*(-((a + b*ArcSinh[c + d*x])^2/b^2))^(3/2))","A",0
199,1,225,547,0.3246426,"\int (c e+d e x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2} \, dx","Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^(7/2),x]","-\frac{b e^3 e^{-\frac{4 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \left(-\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{9}{2},-\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+32 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{9}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(e^{\frac{2 a}{b}} \Gamma \left(\frac{9}{2},\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-32 \sqrt{2} \Gamma \left(\frac{9}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{8192 d \left(-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}\right)^{3/2}}","-\frac{105 \sqrt{\pi } b^{7/2} e^3 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{131072 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e^3 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2048 d}+\frac{105 \sqrt{\pi } b^{7/2} e^3 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{131072 d}-\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e^3 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2048 d}-\frac{105 b^3 e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \sqrt{a+b \sinh ^{-1}(c+d x)}}{2048 d}+\frac{1575 b^3 e^3 \sqrt{(c+d x)^2+1} (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4096 d}+\frac{35 b^2 e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{256 d}-\frac{105 b^2 e^3 (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{256 d}-\frac{525 b^2 e^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{2048 d}+\frac{e^3 (c+d x)^4 \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}}{4 d}-\frac{7 b e^3 \sqrt{(c+d x)^2+1} (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{32 d}+\frac{21 b e^3 \sqrt{(c+d x)^2+1} (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{64 d}-\frac{3 e^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}}{32 d}",1,"-1/8192*(b*e^3*(a + b*ArcSinh[c + d*x])^(5/2)*(-(Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (-4*(a + b*ArcSinh[c + d*x]))/b]) + 32*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*(-32*Sqrt[2]*Gamma[9/2, (2*(a + b*ArcSinh[c + d*x]))/b] + E^((2*a)/b)*Gamma[9/2, (4*(a + b*ArcSinh[c + d*x]))/b])))/(d*E^((4*a)/b)*(-((a + b*ArcSinh[c + d*x])^2/b^2))^(3/2))","A",0
200,1,238,481,0.3356612,"\int (c e+d e x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2} \, dx","Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^(7/2),x]","\frac{b e^2 e^{-\frac{3 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2} \left(-243 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{9}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{9}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-243 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{9}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{9}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{1944 d \left(-\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{b^2}\right)^{3/2}}","-\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{128 d}+\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3456 d}-\frac{105 \sqrt{\pi } b^{7/2} e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{128 d}+\frac{35 \sqrt{\frac{\pi }{3}} b^{7/2} e^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3456 d}+\frac{175 b^3 e^2 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{54 d}-\frac{35 b^3 e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{216 d}+\frac{35 b^2 e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{108 d}-\frac{35 b^2 e^2 (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{18 d}+\frac{7 b e^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{9 d}-\frac{7 b e^2 (c+d x)^2 \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{18 d}+\frac{e^2 (c+d x)^3 \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}}{3 d}",1,"(b*e^2*(a + b*ArcSinh[c + d*x])^(5/2)*(-243*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, a/b + ArcSinh[c + d*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (-3*(a + b*ArcSinh[c + d*x]))/b] - 243*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, -((a + b*ArcSinh[c + d*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (3*(a + b*ArcSinh[c + d*x]))/b]))/(1944*d*E^((3*a)/b)*(-((a + b*ArcSinh[c + d*x])^2/b^2))^(3/2))","A",0
201,1,125,305,0.0751611,"\int (c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2} \, dx","Integrate[(c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(7/2),x]","\frac{e e^{-\frac{2 a}{b}} \left(b^4 \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{9}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+b^4 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{9}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{64 \sqrt{2} d \sqrt{a+b \sinh ^{-1}(c+d x)}}","-\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{1024 d}+\frac{105 \sqrt{\frac{\pi }{2}} b^{7/2} e e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{1024 d}-\frac{105 b^3 e (c+d x) \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{128 d}+\frac{35 b^2 e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{32 d}+\frac{35 b^2 e \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{64 d}-\frac{7 b e (c+d x) \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{8 d}+\frac{e (c+d x)^2 \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}}{2 d}+\frac{e \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}}{4 d}",1,"(e*(b^4*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[9/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + b^4*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[9/2, (2*(a + b*ArcSinh[c + d*x]))/b]))/(64*Sqrt[2]*d*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
202,1,698,216,4.6819445,"\int \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(7/2),x]","\frac{16 a^3 e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\frac{\Gamma \left(\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \Gamma \left(\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right)+6 a \sqrt{b} \left(\sqrt{\pi } \left(4 a^2-12 a b+15 b^2\right) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{\pi } \left(4 a^2+12 a b+15 b^2\right) \left(\sinh \left(\frac{a}{b}\right)-\cosh \left(\frac{a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+4 \sqrt{b} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(2 \sqrt{(c+d x)^2+1} \left(a-5 b \sinh ^{-1}(c+d x)\right)+b (c+d x) \left(4 \sinh ^{-1}(c+d x)^2+15\right)\right)\right)+12 a^2 \sqrt{b} \left(\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+d 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+d x)}}{\sqrt{b}}\right)+4 \sqrt{b} \left(2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt{(c+d x)^2+1}\right) \sqrt{a+b \sinh ^{-1}(c+d x)}\right)+\sqrt{b} \left(4 \sqrt{b} \sqrt{a+b \sinh ^{-1}(c+d x)} \left(\sqrt{(c+d x)^2+1} \left(-4 a^2+4 a b \sinh ^{-1}(c+d x)-7 b^2 \left(4 \sinh ^{-1}(c+d x)^2+15\right)\right)+2 b (c+d x) \left(-10 a+4 b \sinh ^{-1}(c+d x)^3+35 b \sinh ^{-1}(c+d x)\right)\right)+\sqrt{\pi } \left(-8 a^3+36 a^2 b-90 a b^2+105 b^3\right) \left(\sinh \left(\frac{a}{b}\right)+\cosh \left(\frac{a}{b}\right)\right) \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)+\sqrt{\pi } \left(8 a^3+36 a^2 b+90 a b^2+105 b^3\right) \left(\cosh \left(\frac{a}{b}\right)-\sinh \left(\frac{a}{b}\right)\right) \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)\right)}{32 d}","\frac{105 \sqrt{\pi } b^{7/2} e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}+\frac{105 \sqrt{\pi } b^{7/2} e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 d}-\frac{105 b^3 \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac{35 b^2 (c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}{4 d}-\frac{7 b \sqrt{(c+d x)^2+1} \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}{2 d}+\frac{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}}{d}",1,"((16*a^3*Sqrt[a + b*ArcSinh[c + d*x]]*(-((E^((2*a)/b)*Gamma[3/2, a/b + ArcSinh[c + d*x]])/Sqrt[a/b + ArcSinh[c + d*x]]) + Gamma[3/2, -((a + b*ArcSinh[c + d*x])/b)]/Sqrt[-((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) + 12*a^2*Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(-3*Sqrt[1 + (c + d*x)^2] + 2*(c + d*x)*ArcSinh[c + d*x]) + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-2*a + 3*b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + 6*a*Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(2*Sqrt[1 + (c + d*x)^2]*(a - 5*b*ArcSinh[c + d*x]) + b*(c + d*x)*(15 + 4*ArcSinh[c + d*x]^2)) + (4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(-Cosh[a/b] + Sinh[a/b]) + (4*a^2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + Sqrt[b]*(4*Sqrt[b]*Sqrt[a + b*ArcSinh[c + d*x]]*(2*b*(c + d*x)*(-10*a + 35*b*ArcSinh[c + d*x] + 4*b*ArcSinh[c + d*x]^3) + Sqrt[1 + (c + d*x)^2]*(-4*a^2 + 4*a*b*ArcSinh[c + d*x] - 7*b^2*(15 + 4*ArcSinh[c + d*x]^2))) + (8*a^3 + 36*a^2*b + 90*a*b^2 + 105*b^3)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (-8*a^3 + 36*a^2*b - 90*a*b^2 + 105*b^3)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])))/(32*d)","B",0
203,0,0,29,1.1379313,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}}{c e+d e x} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(7/2)/(c*e + d*e*x),x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}}{c e+d e x} \, dx","\frac{\text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}}{c+d x},x\right)}{e}",0,"Integrate[(a + b*ArcSinh[c + d*x])^(7/2)/(c*e + d*e*x), x]","A",-1
204,1,320,326,0.3697089,"\int \frac{(c e+d e x)^4}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","Integrate[(c*e + d*e*x)^4/Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^4 e^{-\frac{5 a}{b}} \left(-10 e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sqrt{5} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-5 \sqrt{3} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+10 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+5 \sqrt{3} e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-\sqrt{5} e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{160 d \sqrt{a+b \sinh ^{-1}(c+d x)}}","\frac{\sqrt{\pi } e^4 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 \sqrt{b} d}-\frac{\sqrt{3 \pi } e^4 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{5}} e^4 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 \sqrt{b} d}+\frac{\sqrt{\pi } e^4 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 \sqrt{b} d}-\frac{\sqrt{3 \pi } e^4 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{5}} e^4 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 \sqrt{b} d}",1,"(e^4*(-10*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]] + Sqrt[5]*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 5*Sqrt[3]*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 10*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)] + 5*Sqrt[3]*E^((8*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b] - Sqrt[5]*E^((10*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(160*d*E^((5*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
205,1,205,217,0.2292098,"\int \frac{(c e+d e x)^3}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","Integrate[(c*e + d*e*x)^3/Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^3 e^{-\frac{4 a}{b}} \left(\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-2 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(e^{\frac{2 a}{b}} \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-2 \sqrt{2} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{32 d \sqrt{a+b \sinh ^{-1}(c+d x)}}","-\frac{\sqrt{\pi } e^3 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{2}} e^3 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}+\frac{\sqrt{\pi } e^3 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{32 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{2}} e^3 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}",1,"(e^3*(Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c + d*x]))/b] - 2*Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((6*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*(-2*Sqrt[2]*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b] + E^((2*a)/b)*Gamma[1/2, (4*(a + b*ArcSinh[c + d*x]))/b])))/(32*d*E^((4*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
206,1,217,214,0.2399468,"\int \frac{(c e+d e x)^2}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","Integrate[(c*e + d*e*x)^2/Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^2 e^{-\frac{3 a}{b}} \left(3 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sqrt{3} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-3 e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{24 d \sqrt{a+b \sinh ^{-1}(c+d x)}}","-\frac{\sqrt{\pi } e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{3}} e^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}-\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}+\frac{\sqrt{\frac{\pi }{3}} e^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 \sqrt{b} d}",1,"(e^2*(3*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]] + Sqrt[3]*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b] - 3*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)] - Sqrt[3]*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b]))/(24*d*E^((3*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
207,1,119,113,0.0665356,"\int \frac{c e+d e x}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","Integrate[(c*e + d*e*x)/Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e e^{-\frac{2 a}{b}} \left(\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{4 \sqrt{2} d \sqrt{a+b \sinh ^{-1}(c+d x)}}","\frac{\sqrt{\frac{\pi }{2}} e e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 \sqrt{b} d}-\frac{\sqrt{\frac{\pi }{2}} e e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 \sqrt{b} d}",1,"(e*(Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b]))/(4*Sqrt[2]*d*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
208,1,111,92,0.066315,"\int \frac{1}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*ArcSinh[c + d*x]],x]","\frac{e^{-\frac{a}{b}} \left(\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)}{2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}","\frac{\sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 \sqrt{b} d}",1,"(-(E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]]) + Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/(2*d*E^(a/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
209,0,0,29,0.0701204,"\int \frac{1}{(c e+d e x) \sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","Integrate[1/((c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]]),x]","\int \frac{1}{(c e+d e x) \sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*Sqrt[a + b*ArcSinh[c + d*x]]), x]","A",-1
210,1,490,367,0.6682952,"\int \frac{(c e+d e x)^4}{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{e^4 e^{-5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(3 e^{\frac{5 a}{b}+2 \sinh ^{-1}(c+d x)}-2 e^{\frac{5 a}{b}+4 \sinh ^{-1}(c+d x)}-2 e^{\frac{5 a}{b}+6 \sinh ^{-1}(c+d x)}+3 e^{\frac{5 a}{b}+8 \sinh ^{-1}(c+d x)}-e^{\frac{5 a}{b}+10 \sinh ^{-1}(c+d x)}+2 e^{\frac{6 a}{b}+5 \sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sqrt{5} e^{5 \sinh ^{-1}(c+d x)} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-3 \sqrt{3} e^{\frac{2 a}{b}+5 \sinh ^{-1}(c+d x)} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+2 e^{\frac{4 a}{b}+5 \sinh ^{-1}(c+d x)} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-3 \sqrt{3} e^{\frac{8 a}{b}+5 \sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+\sqrt{5} e^{5 \left(\frac{2 a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-e^{\frac{5 a}{b}}\right)}{16 b d \sqrt{a+b \sinh ^{-1}(c+d x)}}","-\frac{\sqrt{\pi } e^4 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 b^{3/2} d}+\frac{3 \sqrt{3 \pi } e^4 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 b^{3/2} d}-\frac{\sqrt{5 \pi } e^4 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 b^{3/2} d}+\frac{\sqrt{\pi } e^4 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 b^{3/2} d}-\frac{3 \sqrt{3 \pi } e^4 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 b^{3/2} d}+\frac{\sqrt{5 \pi } e^4 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{16 b^{3/2} d}-\frac{2 e^4 (c+d x)^4 \sqrt{(c+d x)^2+1}}{b d \sqrt{a+b \sinh ^{-1}(c+d x)}}",1,"(e^4*(-E^((5*a)/b) + 3*E^((5*a)/b + 2*ArcSinh[c + d*x]) - 2*E^((5*a)/b + 4*ArcSinh[c + d*x]) - 2*E^((5*a)/b + 6*ArcSinh[c + d*x]) + 3*E^((5*a)/b + 8*ArcSinh[c + d*x]) - E^((5*a)/b + 10*ArcSinh[c + d*x]) + 2*E^((6*a)/b + 5*ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]] + Sqrt[5]*E^(5*ArcSinh[c + d*x])*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-5*(a + b*ArcSinh[c + d*x]))/b] - 3*Sqrt[3]*E^((2*a)/b + 5*ArcSinh[c + d*x])*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 2*E^((4*a)/b + 5*ArcSinh[c + d*x])*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)] - 3*Sqrt[3]*E^((8*a)/b + 5*ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b] + Sqrt[5]*E^(5*((2*a)/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/(16*b*d*E^(5*(a/b + ArcSinh[c + d*x]))*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
211,1,253,262,0.4522932,"\int \frac{(c e+d e x)^3}{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{e^3 e^{-\frac{4 a}{b}} \left(\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-\sqrt{2} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-e^{\frac{4 a}{b}} \left(-\sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-2 \sinh \left(2 \sinh ^{-1}(c+d x)\right)+\sinh \left(4 \sinh ^{-1}(c+d x)\right)\right)\right)}{4 b d \sqrt{a+b \sinh ^{-1}(c+d x)}}","\frac{\sqrt{\pi } e^3 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}-\frac{\sqrt{\frac{\pi }{2}} e^3 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 b^{3/2} d}+\frac{\sqrt{\pi } e^3 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}-\frac{\sqrt{\frac{\pi }{2}} e^3 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 b^{3/2} d}-\frac{2 e^3 (c+d x)^3 \sqrt{(c+d x)^2+1}}{b d \sqrt{a+b \sinh ^{-1}(c+d x)}}",1,"(e^3*(Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-4*(a + b*ArcSinh[c + d*x]))/b] - Sqrt[2]*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b] - E^((4*a)/b)*(-(Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b]) + E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (4*(a + b*ArcSinh[c + d*x]))/b] - 2*Sinh[2*ArcSinh[c + d*x]] + Sinh[4*ArcSinh[c + d*x]])))/(4*b*d*E^((4*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
212,1,327,255,0.3717948,"\int \frac{(c e+d e x)^2}{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{e^2 e^{-3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(e^{\frac{3 a}{b}+2 \sinh ^{-1}(c+d x)}+e^{\frac{3 a}{b}+4 \sinh ^{-1}(c+d x)}-e^{\frac{3 a}{b}+6 \sinh ^{-1}(c+d x)}-e^{\frac{4 a}{b}+3 \sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+\sqrt{3} e^{3 \sinh ^{-1}(c+d x)} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-e^{\frac{2 a}{b}+3 \sinh ^{-1}(c+d x)} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+\sqrt{3} e^{\frac{6 a}{b}+3 \sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-e^{\frac{3 a}{b}}\right)}{4 b d \sqrt{a+b \sinh ^{-1}(c+d x)}}","\frac{\sqrt{\pi } e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}-\frac{\sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}-\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}+\frac{\sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{4 b^{3/2} d}-\frac{2 e^2 (c+d x)^2 \sqrt{(c+d x)^2+1}}{b d \sqrt{a+b \sinh ^{-1}(c+d x)}}",1,"(e^2*(-E^((3*a)/b) + E^((3*a)/b + 2*ArcSinh[c + d*x]) + E^((3*a)/b + 4*ArcSinh[c + d*x]) - E^((3*a)/b + 6*ArcSinh[c + d*x]) - E^((4*a)/b + 3*ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]] + Sqrt[3]*E^(3*ArcSinh[c + d*x])*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b] - E^((2*a)/b + 3*ArcSinh[c + d*x])*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)] + Sqrt[3]*E^((6*a)/b + 3*ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b]))/(4*b*d*E^(3*(a/b + ArcSinh[c + d*x]))*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
213,1,147,148,0.1171788,"\int \frac{c e+d e x}{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(3/2),x]","\frac{e e^{-\frac{2 a}{b}} \left(-2 e^{\frac{2 a}{b}} \sinh \left(2 \sinh ^{-1}(c+d x)\right)+\sqrt{2} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-\sqrt{2} e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{2 b d \sqrt{a+b \sinh ^{-1}(c+d x)}}","\frac{\sqrt{\frac{\pi }{2}} e e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}+\frac{\sqrt{\frac{\pi }{2}} e e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{2 e \sqrt{(c+d x)^2+1} (c+d x)}{b d \sqrt{a+b \sinh ^{-1}(c+d x)}}",1,"(e*(Sqrt[2]*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b] - Sqrt[2]*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b] - 2*E^((2*a)/b)*Sinh[2*ArcSinh[c + d*x]]))/(2*b*d*E^((2*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
214,1,155,122,0.1133608,"\int \frac{1}{\left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-3/2),x]","\frac{e^{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(-e^{a/b} \left(e^{2 \sinh ^{-1}(c+d x)}+1\right)+e^{\frac{2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+e^{\sinh ^{-1}(c+d x)} \sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)\right)}{b d \sqrt{a+b \sinh ^{-1}(c+d x)}}","-\frac{\sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}+\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{b^{3/2} d}-\frac{2 \sqrt{(c+d x)^2+1}}{b d \sqrt{a+b \sinh ^{-1}(c+d x)}}",1,"(-(E^(a/b)*(1 + E^(2*ArcSinh[c + d*x]))) + E^((2*a)/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[1/2, a/b + ArcSinh[c + d*x]] + E^ArcSinh[c + d*x]*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/(b*d*E^((a + b*ArcSinh[c + d*x])/b)*Sqrt[a + b*ArcSinh[c + d*x]])","A",0
215,0,0,29,0.0763129,"\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(3/2)),x]","\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(3/2)), x]","A",-1
216,1,551,437,3.453747,"\int \frac{(c e+d e x)^4}{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^(5/2),x]","\frac{e^4 \left(-2 e^{\sinh ^{-1}(c+d x)} \left(2 a+2 b \sinh ^{-1}(c+d x)+b\right)-4 b e^{-\frac{a}{b}} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+e^{-\sinh ^{-1}(c+d x)} \left(-4 e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+4 a+4 b \sinh ^{-1}(c+d x)-2 b\right)+e^{-\frac{5 a}{b}} \left(-e^{5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(10 a+10 b \sinh ^{-1}(c+d x)+b\right)-10 \sqrt{5} b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)+e^{-\frac{3 a}{b}} \left(3 e^{3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(6 a+6 b \sinh ^{-1}(c+d x)+b\right)+18 \sqrt{3} b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)+3 e^{-3 \sinh ^{-1}(c+d x)} \left(6 \sqrt{3} e^{3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-6 a-6 b \sinh ^{-1}(c+d x)+b\right)+e^{-5 \sinh ^{-1}(c+d x)} \left(-10 \sqrt{5} e^{5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+10 a+10 b \sinh ^{-1}(c+d x)-b\right)\right)}{48 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}","\frac{\sqrt{\pi } e^4 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{12 b^{5/2} d}-\frac{3 \sqrt{3 \pi } e^4 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 b^{5/2} d}+\frac{5 \sqrt{5 \pi } e^4 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{24 b^{5/2} d}+\frac{\sqrt{\pi } e^4 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{12 b^{5/2} d}-\frac{3 \sqrt{3 \pi } e^4 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{8 b^{5/2} d}+\frac{5 \sqrt{5 \pi } e^4 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{24 b^{5/2} d}-\frac{20 e^4 (c+d x)^5}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{16 e^4 (c+d x)^3}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 e^4 \sqrt{(c+d x)^2+1} (c+d x)^4}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}",1,"(e^4*(-2*E^ArcSinh[c + d*x]*(2*a + b + 2*b*ArcSinh[c + d*x]) + (4*a - 2*b + 4*b*ArcSinh[c + d*x] - 4*E^(a/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, a/b + ArcSinh[c + d*x]])/E^ArcSinh[c + d*x] + (-(E^(5*(a/b + ArcSinh[c + d*x]))*(10*a + b + 10*b*ArcSinh[c + d*x])) - 10*Sqrt[5]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-5*(a + b*ArcSinh[c + d*x]))/b])/E^((5*a)/b) + (3*E^(3*(a/b + ArcSinh[c + d*x]))*(6*a + b + 6*b*ArcSinh[c + d*x]) + 18*Sqrt[3]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b])/E^((3*a)/b) - (4*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/E^(a/b) + (3*(-6*a + b - 6*b*ArcSinh[c + d*x] + 6*Sqrt[3]*E^(3*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b]))/E^(3*ArcSinh[c + d*x]) + (10*a - b + 10*b*ArcSinh[c + d*x] - 10*Sqrt[5]*E^(5*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (5*(a + b*ArcSinh[c + d*x]))/b])/E^(5*ArcSinh[c + d*x])))/(48*b^2*d*(a + b*ArcSinh[c + d*x])^(3/2))","A",0
217,1,390,326,1.7768486,"\int \frac{(c e+d e x)^3}{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^(5/2),x]","\frac{e^3 e^{-4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(-8 b e^{4 \sinh ^{-1}(c+d x)} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+4 \sqrt{2} b e^{\frac{2 a}{b}+4 \sinh ^{-1}(c+d x)} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+\frac{1}{2} e^{\frac{4 a}{b}} \left(-\left(\left(e^{2 \sinh ^{-1}(c+d x)}-1\right)^2 \left(8 a \left(e^{2 \sinh ^{-1}(c+d x)}+e^{4 \sinh ^{-1}(c+d x)}+1\right)+b \left(e^{4 \sinh ^{-1}(c+d x)}-1\right)+8 b \left(e^{2 \sinh ^{-1}(c+d x)}+e^{4 \sinh ^{-1}(c+d x)}+1\right) \sinh ^{-1}(c+d x)\right)\right)-8 \sqrt{2} e^{\frac{2 a}{b}+4 \sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+16 e^{4 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{12 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}","-\frac{2 \sqrt{\pi } e^3 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{\sqrt{2 \pi } e^3 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{2 \sqrt{\pi } e^3 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}-\frac{\sqrt{2 \pi } e^3 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}-\frac{16 e^3 (c+d x)^4}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4 e^3 (c+d x)^2}{b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 e^3 \sqrt{(c+d x)^2+1} (c+d x)^3}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}",1,"(e^3*(-8*b*E^(4*ArcSinh[c + d*x])*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-4*(a + b*ArcSinh[c + d*x]))/b] + 4*Sqrt[2]*b*E^((2*a)/b + 4*ArcSinh[c + d*x])*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + (E^((4*a)/b)*(-((-1 + E^(2*ArcSinh[c + d*x]))^2*(b*(-1 + E^(4*ArcSinh[c + d*x])) + 8*a*(1 + E^(2*ArcSinh[c + d*x]) + E^(4*ArcSinh[c + d*x])) + 8*b*(1 + E^(2*ArcSinh[c + d*x]) + E^(4*ArcSinh[c + d*x]))*ArcSinh[c + d*x])) - 8*Sqrt[2]*E^((2*a)/b + 4*ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b] + 16*E^(4*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (4*(a + b*ArcSinh[c + d*x]))/b]))/2))/(12*b^2*d*E^(4*(a/b + ArcSinh[c + d*x]))*(a + b*ArcSinh[c + d*x])^(3/2))","A",0
218,1,389,321,1.5484246,"\int \frac{(c e+d e x)^2}{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^(5/2),x]","\frac{e^2 e^{-3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(-6 \sqrt{3} b e^{3 \sinh ^{-1}(c+d x)} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+2 b e^{\frac{2 a}{b}+3 \sinh ^{-1}(c+d x)} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)+2 e^{\frac{4 a}{b}+3 \sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-e^{\frac{3 a}{b}} \left(\left(e^{2 \sinh ^{-1}(c+d x)}-1\right) \left(a \left(4 e^{2 \sinh ^{-1}(c+d x)}+6 e^{4 \sinh ^{-1}(c+d x)}+6\right)+b \left(e^{4 \sinh ^{-1}(c+d x)}-1\right)+2 b \left(2 e^{2 \sinh ^{-1}(c+d x)}+3 e^{4 \sinh ^{-1}(c+d x)}+3\right) \sinh ^{-1}(c+d x)\right)+6 \sqrt{3} e^{3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)\right)}{12 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}","-\frac{\sqrt{\pi } e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{6 b^{5/2} d}+\frac{\sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 b^{5/2} d}-\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{6 b^{5/2} d}+\frac{\sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{2 b^{5/2} d}-\frac{4 e^2 (c+d x)^3}{b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e^2 (c+d x)}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}",1,"(e^2*(2*E^((4*a)/b + 3*ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, a/b + ArcSinh[c + d*x]] - 6*Sqrt[3]*b*E^(3*ArcSinh[c + d*x])*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b] + 2*b*E^((2*a)/b + 3*ArcSinh[c + d*x])*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)] - E^((3*a)/b)*((-1 + E^(2*ArcSinh[c + d*x]))*(b*(-1 + E^(4*ArcSinh[c + d*x])) + a*(6 + 4*E^(2*ArcSinh[c + d*x]) + 6*E^(4*ArcSinh[c + d*x])) + 2*b*(3 + 2*E^(2*ArcSinh[c + d*x]) + 3*E^(4*ArcSinh[c + d*x]))*ArcSinh[c + d*x]) + 6*Sqrt[3]*E^(3*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b])))/(12*b^2*d*E^(3*(a/b + ArcSinh[c + d*x]))*(a + b*ArcSinh[c + d*x])^(3/2))","A",0
219,1,227,209,0.6955499,"\int \frac{c e+d e x}{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(5/2),x]","\frac{e e^{-2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(e^{\frac{2 a}{b}} \left(4 \sqrt{2} e^{2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-4 a e^{4 \sinh ^{-1}(c+d x)}-4 a-b e^{4 \sinh ^{-1}(c+d x)}-4 b \left(e^{4 \sinh ^{-1}(c+d x)}+1\right) \sinh ^{-1}(c+d x)+b\right)-4 \sqrt{2} b e^{2 \sinh ^{-1}(c+d x)} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)}{6 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}","-\frac{2 \sqrt{2 \pi } e e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{2 \sqrt{2 \pi } e e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 e \sqrt{(c+d x)^2+1} (c+d x)}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}",1,"(e*(-4*Sqrt[2]*b*E^(2*ArcSinh[c + d*x])*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b] + E^((2*a)/b)*(-4*a + b - 4*a*E^(4*ArcSinh[c + d*x]) - b*E^(4*ArcSinh[c + d*x]) - 4*b*(1 + E^(4*ArcSinh[c + d*x]))*ArcSinh[c + d*x] + 4*Sqrt[2]*E^(2*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b])))/(6*b^2*d*E^(2*(a/b + ArcSinh[c + d*x]))*(a + b*ArcSinh[c + d*x])^(3/2))","A",0
220,1,207,158,0.2935843,"\int \frac{1}{\left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-5/2),x]","\frac{e^{-\frac{a+b \sinh ^{-1}(c+d x)}{b}} \left(-e^{a/b} \left(2 a \left(e^{2 \sinh ^{-1}(c+d x)}-1\right)-2 b \sinh ^{-1}(c+d x)+b e^{2 \sinh ^{-1}(c+d x)} \left(2 \sinh ^{-1}(c+d x)+1\right)+b\right)-2 b e^{\sinh ^{-1}(c+d x)} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)-2 e^{\frac{2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)\right)}{3 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}","\frac{2 \sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}+\frac{2 \sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{3 b^{5/2} d}-\frac{4 (c+d x)}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 \sqrt{(c+d x)^2+1}}{3 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}",1,"(-(E^(a/b)*(b + 2*a*(-1 + E^(2*ArcSinh[c + d*x])) - 2*b*ArcSinh[c + d*x] + b*E^(2*ArcSinh[c + d*x])*(1 + 2*ArcSinh[c + d*x]))) - 2*E^((2*a)/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, a/b + ArcSinh[c + d*x]] - 2*b*E^ArcSinh[c + d*x]*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)])/(3*b^2*d*E^((a + b*ArcSinh[c + d*x])/b)*(a + b*ArcSinh[c + d*x])^(3/2))","A",0
221,0,0,29,0.0804827,"\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2)),x]","\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(5/2)), x]","A",-1
222,1,701,531,2.7170116,"\int \frac{(c e+d e x)^4}{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(c*e + d*e*x)^4/(a + b*ArcSinh[c + d*x])^(7/2),x]","\frac{e^4 \left(e^{-\sinh ^{-1}(c+d x)} \left(-8 a^2-4 b (4 a-b) \sinh ^{-1}(c+d x)+8 e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^2 \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)+4 a b-8 b^2 \sinh ^{-1}(c+d x)^2-6 b^2\right)+9 \left(2 e^{-\frac{3 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(6 a+6 b \sinh ^{-1}(c+d x)+b\right)+6 \sqrt{3} b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)+b^2 e^{3 \sinh ^{-1}(c+d x)}\right)+9 e^{-3 \sinh ^{-1}(c+d x)} \left(2 \left(a+b \sinh ^{-1}(c+d x)\right) \left(-6 \sqrt{3} e^{3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+6 a+6 b \sinh ^{-1}(c+d x)-b\right)+b^2\right)-e^{-5 \sinh ^{-1}(c+d x)} \left(10 \left(a+b \sinh ^{-1}(c+d x)\right) \left(-10 \sqrt{5} e^{5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+10 a+10 b \sinh ^{-1}(c+d x)-b\right)+3 b^2\right)-10 e^{-\frac{5 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{5 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(10 a+10 b \sinh ^{-1}(c+d x)+b\right)+10 \sqrt{5} b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{5 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)-4 e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(2 a+2 b \sinh ^{-1}(c+d x)+b\right)+2 b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)\right)-6 b^2 e^{\sinh ^{-1}(c+d x)}-3 b^2 e^{5 \sinh ^{-1}(c+d x)}\right)}{240 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}","-\frac{\sqrt{\pi } e^4 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{30 b^{7/2} d}+\frac{9 \sqrt{3 \pi } e^4 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{20 b^{7/2} d}-\frac{5 \sqrt{5 \pi } e^4 e^{\frac{5 a}{b}} \text{erf}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{12 b^{7/2} d}+\frac{\sqrt{\pi } e^4 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{30 b^{7/2} d}-\frac{9 \sqrt{3 \pi } e^4 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{20 b^{7/2} d}+\frac{5 \sqrt{5 \pi } e^4 e^{-\frac{5 a}{b}} \text{erfi}\left(\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{12 b^{7/2} d}-\frac{40 e^4 \sqrt{(c+d x)^2+1} (c+d x)^4}{3 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{32 e^4 \sqrt{(c+d x)^2+1} (c+d x)^2}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4 e^4 (c+d x)^5}{3 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{16 e^4 (c+d x)^3}{15 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{2 e^4 \sqrt{(c+d x)^2+1} (c+d x)^4}{5 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}",1,"(e^4*(-6*b^2*E^ArcSinh[c + d*x] - 3*b^2*E^(5*ArcSinh[c + d*x]) + (-8*a^2 + 4*a*b - 6*b^2 - 4*(4*a - b)*b*ArcSinh[c + d*x] - 8*b^2*ArcSinh[c + d*x]^2 + 8*E^(a/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])^2*Gamma[1/2, a/b + ArcSinh[c + d*x]])/E^ArcSinh[c + d*x] - (10*(a + b*ArcSinh[c + d*x])*(E^(5*(a/b + ArcSinh[c + d*x]))*(10*a + b + 10*b*ArcSinh[c + d*x]) + 10*Sqrt[5]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-5*(a + b*ArcSinh[c + d*x]))/b]))/E^((5*a)/b) + 9*(b^2*E^(3*ArcSinh[c + d*x]) + (2*(a + b*ArcSinh[c + d*x])*(E^(3*(a/b + ArcSinh[c + d*x]))*(6*a + b + 6*b*ArcSinh[c + d*x]) + 6*Sqrt[3]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b]))/E^((3*a)/b)) - (4*(a + b*ArcSinh[c + d*x])*(E^(a/b + ArcSinh[c + d*x])*(2*a + b + 2*b*ArcSinh[c + d*x]) + 2*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) + (9*(b^2 + 2*(a + b*ArcSinh[c + d*x])*(6*a - b + 6*b*ArcSinh[c + d*x] - 6*Sqrt[3]*E^(3*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b])))/E^(3*ArcSinh[c + d*x]) - (3*b^2 + 10*(a + b*ArcSinh[c + d*x])*(10*a - b + 10*b*ArcSinh[c + d*x] - 10*Sqrt[5]*E^(5*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (5*(a + b*ArcSinh[c + d*x]))/b]))/E^(5*ArcSinh[c + d*x])))/(240*b^3*d*(a + b*ArcSinh[c + d*x])^(5/2))","A",0
223,1,429,420,2.1060716,"\int \frac{(c e+d e x)^3}{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(c*e + d*e*x)^3/(a + b*ArcSinh[c + d*x])^(7/2),x]","\frac{e^3 \left(4 \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{2 \sinh ^{-1}(c+d x)} \left(4 a+4 b \sinh ^{-1}(c+d x)+b\right)+4 \sqrt{2} b e^{-\frac{2 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+4 \sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-4 a e^{-2 \sinh ^{-1}(c+d x)}+b e^{-2 \sinh ^{-1}(c+d x)}-4 b e^{-2 \sinh ^{-1}(c+d x)} \sinh ^{-1}(c+d x)\right)-4 \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{4 \sinh ^{-1}(c+d x)} \left(8 a+8 b \sinh ^{-1}(c+d x)+b\right)+16 b e^{-\frac{4 a}{b}} \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+16 e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{4 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-8 a e^{-4 \sinh ^{-1}(c+d x)}+b e^{-4 \sinh ^{-1}(c+d x)} \left(1-8 \sinh ^{-1}(c+d x)\right)\right)+6 b^2 \sinh \left(2 \sinh ^{-1}(c+d x)\right)-3 b^2 \sinh \left(4 \sinh ^{-1}(c+d x)\right)\right)}{60 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}","\frac{16 \sqrt{\pi } e^3 e^{\frac{4 a}{b}} \text{erf}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}-\frac{4 \sqrt{2 \pi } e^3 e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{16 \sqrt{\pi } e^3 e^{-\frac{4 a}{b}} \text{erfi}\left(\frac{2 \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}-\frac{4 \sqrt{2 \pi } e^3 e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}-\frac{128 e^3 \sqrt{(c+d x)^2+1} (c+d x)^3}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{16 e^3 \sqrt{(c+d x)^2+1} (c+d x)}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{16 e^3 (c+d x)^4}{15 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{4 e^3 (c+d x)^2}{5 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{2 e^3 \sqrt{(c+d x)^2+1} (c+d x)^3}{5 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}",1,"(e^3*(4*(a + b*ArcSinh[c + d*x])*((-4*a)/E^(2*ArcSinh[c + d*x]) + b/E^(2*ArcSinh[c + d*x]) - (4*b*ArcSinh[c + d*x])/E^(2*ArcSinh[c + d*x]) + E^(2*ArcSinh[c + d*x])*(4*a + b + 4*b*ArcSinh[c + d*x]) + (4*Sqrt[2]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b])/E^((2*a)/b) + 4*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b]) - 4*(a + b*ArcSinh[c + d*x])*((-8*a)/E^(4*ArcSinh[c + d*x]) + (b*(1 - 8*ArcSinh[c + d*x]))/E^(4*ArcSinh[c + d*x]) + E^(4*ArcSinh[c + d*x])*(8*a + b + 8*b*ArcSinh[c + d*x]) + (16*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-4*(a + b*ArcSinh[c + d*x]))/b])/E^((4*a)/b) + 16*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (4*(a + b*ArcSinh[c + d*x]))/b]) + 6*b^2*Sinh[2*ArcSinh[c + d*x]] - 3*b^2*Sinh[4*ArcSinh[c + d*x]]))/(60*b^3*d*(a + b*ArcSinh[c + d*x])^(5/2))","A",0
224,1,474,410,1.4943943,"\int \frac{(c e+d e x)^2}{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(c*e + d*e*x)^2/(a + b*ArcSinh[c + d*x])^(7/2),x]","\frac{e^2 \left(e^{-\sinh ^{-1}(c+d x)} \left(4 a^2+2 b (4 a-b) \sinh ^{-1}(c+d x)-4 e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^2 \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-2 a b+4 b^2 \sinh ^{-1}(c+d x)^2+3 b^2\right)-3 \left(2 e^{-\frac{3 a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(6 a+6 b \sinh ^{-1}(c+d x)+b\right)+6 \sqrt{3} b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)+b^2 e^{3 \sinh ^{-1}(c+d x)}\right)-3 e^{-3 \sinh ^{-1}(c+d x)} \left(2 \left(a+b \sinh ^{-1}(c+d x)\right) \left(-6 \sqrt{3} e^{3 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{3 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)+6 a+6 b \sinh ^{-1}(c+d x)-b\right)+b^2\right)+2 e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(2 a+2 b \sinh ^{-1}(c+d x)+b\right)+2 b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)\right)+3 b^2 e^{\sinh ^{-1}(c+d x)}\right)}{60 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}","\frac{\sqrt{\pi } e^2 e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}-\frac{3 \sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{erf}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{5 b^{7/2} d}-\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{3 \sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{erfi}\left(\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{5 b^{7/2} d}-\frac{24 e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{5 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{16 e^2 \sqrt{(c+d x)^2+1}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4 e^2 (c+d x)^3}{5 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{8 e^2 (c+d x)}{15 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{2 e^2 \sqrt{(c+d x)^2+1} (c+d x)^2}{5 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}",1,"(e^2*(3*b^2*E^ArcSinh[c + d*x] + (4*a^2 - 2*a*b + 3*b^2 + 2*(4*a - b)*b*ArcSinh[c + d*x] + 4*b^2*ArcSinh[c + d*x]^2 - 4*E^(a/b + ArcSinh[c + d*x])*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])^2*Gamma[1/2, a/b + ArcSinh[c + d*x]])/E^ArcSinh[c + d*x] - 3*(b^2*E^(3*ArcSinh[c + d*x]) + (2*(a + b*ArcSinh[c + d*x])*(E^(3*(a/b + ArcSinh[c + d*x]))*(6*a + b + 6*b*ArcSinh[c + d*x]) + 6*Sqrt[3]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcSinh[c + d*x]))/b]))/E^((3*a)/b)) + (2*(a + b*ArcSinh[c + d*x])*(E^(a/b + ArcSinh[c + d*x])*(2*a + b + 2*b*ArcSinh[c + d*x]) + 2*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b) - (3*(b^2 + 2*(a + b*ArcSinh[c + d*x])*(6*a - b + 6*b*ArcSinh[c + d*x] - 6*Sqrt[3]*E^(3*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (3*(a + b*ArcSinh[c + d*x]))/b])))/E^(3*ArcSinh[c + d*x])))/(60*b^3*d*(a + b*ArcSinh[c + d*x])^(5/2))","A",0
225,1,235,252,0.8807754,"\int \frac{c e+d e x}{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(c*e + d*e*x)/(a + b*ArcSinh[c + d*x])^(7/2),x]","-\frac{e \left(\left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{-\frac{2 a}{b}} \left(2 e^{2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \left(4 a+4 b \sinh ^{-1}(c+d x)+b\right)+8 \sqrt{2} b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)\right)+e^{-2 \sinh ^{-1}(c+d x)} \left(8 \sqrt{2} e^{2 \left(\frac{a}{b}+\sinh ^{-1}(c+d x)\right)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right) \Gamma \left(\frac{1}{2},\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{b}\right)-8 a-8 b \sinh ^{-1}(c+d x)+2 b\right)\right)+3 b^2 \sinh \left(2 \sinh ^{-1}(c+d x)\right)\right)}{15 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}","\frac{8 \sqrt{2 \pi } e e^{\frac{2 a}{b}} \text{erf}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{8 \sqrt{2 \pi } e e^{-\frac{2 a}{b}} \text{erfi}\left(\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}-\frac{32 e \sqrt{(c+d x)^2+1} (c+d x)}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e (c+d x)^2}{15 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{4 e}{15 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{2 e \sqrt{(c+d x)^2+1} (c+d x)}{5 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}",1,"-1/15*(e*((a + b*ArcSinh[c + d*x])*((2*E^(2*(a/b + ArcSinh[c + d*x]))*(4*a + b + 4*b*ArcSinh[c + d*x]) + 8*Sqrt[2]*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, (-2*(a + b*ArcSinh[c + d*x]))/b])/E^((2*a)/b) + (-8*a + 2*b - 8*b*ArcSinh[c + d*x] + 8*Sqrt[2]*E^(2*(a/b + ArcSinh[c + d*x]))*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])*Gamma[1/2, (2*(a + b*ArcSinh[c + d*x]))/b])/E^(2*ArcSinh[c + d*x])) + 3*b^2*Sinh[2*ArcSinh[c + d*x]]))/(b^3*d*(a + b*ArcSinh[c + d*x])^(5/2))","A",0
226,1,238,195,0.186334,"\int \frac{1}{\left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^(-7/2),x]","\frac{-2 e^{-\sinh ^{-1}(c+d x)} \left(4 a^2+2 a b \left(4 \sinh ^{-1}(c+d x)-1\right)+b^2 \left(4 \sinh ^{-1}(c+d x)^2-2 \sinh ^{-1}(c+d x)+3\right)\right)+8 e^{a/b} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^2 \Gamma \left(\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right)-4 e^{-\frac{a}{b}} \left(a+b \sinh ^{-1}(c+d x)\right) \left(e^{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left(2 a+2 b \sinh ^{-1}(c+d x)+b\right)+2 b \left(-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)^{3/2} \Gamma \left(\frac{1}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right)\right)-6 b^2 e^{\sinh ^{-1}(c+d x)}}{30 b^3 d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}","-\frac{4 \sqrt{\pi } e^{a/b} \text{erf}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}+\frac{4 \sqrt{\pi } e^{-\frac{a}{b}} \text{erfi}\left(\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right)}{15 b^{7/2} d}-\frac{8 \sqrt{(c+d x)^2+1}}{15 b^3 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4 (c+d x)}{15 b^2 d \left(a+b \sinh ^{-1}(c+d x)\right)^{3/2}}-\frac{2 \sqrt{(c+d x)^2+1}}{5 b d \left(a+b \sinh ^{-1}(c+d x)\right)^{5/2}}",1,"(-6*b^2*E^ArcSinh[c + d*x] - (2*(4*a^2 + 2*a*b*(-1 + 4*ArcSinh[c + d*x]) + b^2*(3 - 2*ArcSinh[c + d*x] + 4*ArcSinh[c + d*x]^2)))/E^ArcSinh[c + d*x] + 8*E^(a/b)*Sqrt[a/b + ArcSinh[c + d*x]]*(a + b*ArcSinh[c + d*x])^2*Gamma[1/2, a/b + ArcSinh[c + d*x]] - (4*(a + b*ArcSinh[c + d*x])*(E^(a/b + ArcSinh[c + d*x])*(2*a + b + 2*b*ArcSinh[c + d*x]) + 2*b*(-((a + b*ArcSinh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c + d*x])/b)]))/E^(a/b))/(30*b^3*d*(a + b*ArcSinh[c + d*x])^(5/2))","A",0
227,0,0,29,0.0824601,"\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}} \, dx","Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(7/2)),x]","\int \frac{1}{(c e+d e x) \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}} \, dx","\frac{\text{Int}\left(\frac{1}{(c+d x) \left(a+b \sinh ^{-1}(c+d x)\right)^{7/2}},x\right)}{e}",0,"Integrate[1/((c*e + d*e*x)*(a + b*ArcSinh[c + d*x])^(7/2)), x]","A",-1
228,1,113,298,0.213305,"\int (c e+d e x)^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x]),x]","\frac{2 (e (c+d x))^{7/2} \left(45 a (c+d x)^3-14 b \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-(c+d x)^2\right)-10 b \sqrt{(c+d x)^2+1} (c+d x)^2+14 b \sqrt{(c+d x)^2+1}+45 b (c+d x)^3 \sinh ^{-1}(c+d x)\right)}{405 d (c+d x)^2}","\frac{2 (e (c+d x))^{9/2} \left(a+b \sinh ^{-1}(c+d x)\right)}{9 d e}-\frac{14 b e^{7/2} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{135 d \sqrt{(c+d x)^2+1}}+\frac{28 b e^{7/2} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{135 d \sqrt{(c+d x)^2+1}}-\frac{28 b e^3 \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{135 d (c+d x+1)}+\frac{28 b e^2 \sqrt{(c+d x)^2+1} (e (c+d x))^{3/2}}{405 d}-\frac{4 b \sqrt{(c+d x)^2+1} (e (c+d x))^{7/2}}{81 d}",1,"(2*(e*(c + d*x))^(7/2)*(45*a*(c + d*x)^3 + 14*b*Sqrt[1 + (c + d*x)^2] - 10*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] + 45*b*(c + d*x)^3*ArcSinh[c + d*x] - 14*b*Hypergeometric2F1[1/2, 3/4, 7/4, -(c + d*x)^2]))/(405*d*(c + d*x)^2)","C",1
229,1,113,177,0.1862432,"\int (c e+d e x)^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x]),x]","\frac{2 (e (c+d x))^{5/2} \left(21 a (c+d x)^3-10 b \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-(c+d x)^2\right)-6 b \sqrt{(c+d x)^2+1} (c+d x)^2+10 b \sqrt{(c+d x)^2+1}+21 b (c+d x)^3 \sinh ^{-1}(c+d x)\right)}{147 d (c+d x)^2}","\frac{2 (e (c+d x))^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)}{7 d e}-\frac{10 b e^{5/2} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{147 d \sqrt{(c+d x)^2+1}}+\frac{20 b e^2 \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{147 d}-\frac{4 b \sqrt{(c+d x)^2+1} (e (c+d x))^{5/2}}{49 d}",1,"(2*(e*(c + d*x))^(5/2)*(21*a*(c + d*x)^3 + 10*b*Sqrt[1 + (c + d*x)^2] - 6*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] + 21*b*(c + d*x)^3*ArcSinh[c + d*x] - 10*b*Hypergeometric2F1[1/4, 1/2, 5/4, -(c + d*x)^2]))/(147*d*(c + d*x)^2)","C",1
230,1,87,261,0.0465115,"\int (c e+d e x)^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x]),x]","\frac{2 (e (c+d x))^{3/2} \left(5 a c+5 a d x+2 b \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-(c+d x)^2\right)-2 b \sqrt{(c+d x)^2+1}+5 b c \sinh ^{-1}(c+d x)+5 b d x \sinh ^{-1}(c+d x)\right)}{25 d}","\frac{2 (e (c+d x))^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)}{5 d e}+\frac{6 b e^{3/2} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{25 d \sqrt{(c+d x)^2+1}}-\frac{12 b e^{3/2} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{25 d \sqrt{(c+d x)^2+1}}-\frac{4 b \sqrt{(c+d x)^2+1} (e (c+d x))^{3/2}}{25 d}+\frac{12 b e \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{25 d (c+d x+1)}",1,"(2*(e*(c + d*x))^(3/2)*(5*a*c + 5*a*d*x - 2*b*Sqrt[1 + (c + d*x)^2] + 5*b*c*ArcSinh[c + d*x] + 5*b*d*x*ArcSinh[c + d*x] + 2*b*Hypergeometric2F1[1/2, 3/4, 7/4, -(c + d*x)^2]))/(25*d)","C",1
231,1,87,142,0.0321098,"\int \sqrt{c e+d e x} \left(a+b \sinh ^{-1}(c+d x)\right) \, dx","Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x]),x]","\frac{2 \sqrt{e (c+d x)} \left(3 a c+3 a d x+2 b \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-(c+d x)^2\right)-2 b \sqrt{(c+d x)^2+1}+3 b c \sinh ^{-1}(c+d x)+3 b d x \sinh ^{-1}(c+d x)\right)}{9 d}","\frac{2 (e (c+d x))^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)}{3 d e}-\frac{4 b \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{9 d}+\frac{2 b \sqrt{e} (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{9 d \sqrt{(c+d x)^2+1}}",1,"(2*Sqrt[e*(c + d*x)]*(3*a*c + 3*a*d*x - 2*b*Sqrt[1 + (c + d*x)^2] + 3*b*c*ArcSinh[c + d*x] + 3*b*d*x*ArcSinh[c + d*x] + 2*b*Hypergeometric2F1[1/4, 1/2, 5/4, -(c + d*x)^2]))/(9*d)","C",1
232,1,61,223,0.0318186,"\int \frac{a+b \sinh ^{-1}(c+d x)}{\sqrt{c e+d e x}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/Sqrt[c*e + d*e*x],x]","-\frac{2 \sqrt{e (c+d x)} \left(2 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-(c+d x)^2\right)-3 \left(a+b \sinh ^{-1}(c+d x)\right)\right)}{3 d e}","\frac{2 \sqrt{e (c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)}{d e}-\frac{4 b \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{d e (c+d x+1)}-\frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{d \sqrt{e} \sqrt{(c+d x)^2+1}}+\frac{4 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{d \sqrt{e} \sqrt{(c+d x)^2+1}}",1,"(-2*Sqrt[e*(c + d*x)]*(-3*(a + b*ArcSinh[c + d*x]) + 2*b*(c + d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(c + d*x)^2]))/(3*d*e)","C",1
233,1,56,106,0.025594,"\int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^(3/2),x]","-\frac{2 \left(a-2 b (c+d x) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-(c+d x)^2\right)+b \sinh ^{-1}(c+d x)\right)}{d e \sqrt{e (c+d x)}}","\frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{d e^{3/2} \sqrt{(c+d x)^2+1}}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{d e \sqrt{e (c+d x)}}",1,"(-2*(a + b*ArcSinh[c + d*x] - 2*b*(c + d*x)*Hypergeometric2F1[1/4, 1/2, 5/4, -(c + d*x)^2]))/(d*e*Sqrt[e*(c + d*x)])","C",1
234,1,58,266,0.0312936,"\int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^(5/2),x]","-\frac{2 \left(a+2 b (c+d x) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-(c+d x)^2\right)+b \sinh ^{-1}(c+d x)\right)}{3 d e (e (c+d x))^{3/2}}","-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{3 d e (e (c+d x))^{3/2}}+\frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{3 d e^{5/2} \sqrt{(c+d x)^2+1}}-\frac{4 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{3 d e^{5/2} \sqrt{(c+d x)^2+1}}+\frac{4 b \sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}}{3 d e^3 (c+d x+1)}-\frac{4 b \sqrt{(c+d x)^2+1}}{3 d e^2 \sqrt{e (c+d x)}}",1,"(-2*(a + b*ArcSinh[c + d*x] + 2*b*(c + d*x)*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c + d*x)^2]))/(3*d*e*(e*(c + d*x))^(3/2))","C",1
235,1,61,145,0.0358926,"\int \frac{a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^{7/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])/(c*e + d*e*x)^(7/2),x]","\frac{-6 \left(a+b \sinh ^{-1}(c+d x)\right)-4 b (c+d x) \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-(c+d x)^2\right)}{15 d e (e (c+d x))^{5/2}}","-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)}{5 d e (e (c+d x))^{5/2}}-\frac{2 b (c+d x+1) \sqrt{\frac{(c+d x)^2+1}{(c+d x+1)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt{e (c+d x)}}{\sqrt{e}}\right)|\frac{1}{2}\right)}{15 d e^{7/2} \sqrt{(c+d x)^2+1}}-\frac{4 b \sqrt{(c+d x)^2+1}}{15 d e^2 (e (c+d x))^{3/2}}",1,"(-6*(a + b*ArcSinh[c + d*x]) - 4*b*(c + d*x)*Hypergeometric2F1[-3/4, 1/2, 1/4, -(c + d*x)^2])/(15*d*e*(e*(c + d*x))^(5/2))","C",1
236,1,110,134,0.1327479,"\int (c e+d e x)^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^2,x]","\frac{2 (e (c+d x))^{9/2} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{13}{4},\frac{13}{4};\frac{15}{4},\frac{17}{4};-(c+d x)^2\right)-52 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{4};\frac{15}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)+143 \left(a+b \sinh ^{-1}(c+d x)\right)^2\right)}{1287 d e}","\frac{16 b^2 (e (c+d x))^{13/2} \, _3F_2\left(1,\frac{13}{4},\frac{13}{4};\frac{15}{4},\frac{17}{4};-(c+d x)^2\right)}{1287 d e^3}-\frac{8 b (e (c+d x))^{11/2} \, _2F_1\left(\frac{1}{2},\frac{11}{4};\frac{15}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{99 d e^2}+\frac{2 (e (c+d x))^{9/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{9 d e}",1,"(2*(e*(c + d*x))^(9/2)*(143*(a + b*ArcSinh[c + d*x])^2 - 52*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 11/4, 15/4, -(c + d*x)^2] + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 13/4, 13/4}, {15/4, 17/4}, -(c + d*x)^2]))/(1287*d*e)","A",1
237,1,110,134,0.1192722,"\int (c e+d e x)^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^2,x]","\frac{2 (e (c+d x))^{7/2} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{11}{4},\frac{11}{4};\frac{13}{4},\frac{15}{4};-(c+d x)^2\right)-44 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{9}{4};\frac{13}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)+99 \left(a+b \sinh ^{-1}(c+d x)\right)^2\right)}{693 d e}","\frac{16 b^2 (e (c+d x))^{11/2} \, _3F_2\left(1,\frac{11}{4},\frac{11}{4};\frac{13}{4},\frac{15}{4};-(c+d x)^2\right)}{693 d e^3}-\frac{8 b (e (c+d x))^{9/2} \, _2F_1\left(\frac{1}{2},\frac{9}{4};\frac{13}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{63 d e^2}+\frac{2 (e (c+d x))^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{7 d e}",1,"(2*(e*(c + d*x))^(7/2)*(99*(a + b*ArcSinh[c + d*x])^2 - 44*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 9/4, 13/4, -(c + d*x)^2] + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 11/4, 11/4}, {13/4, 15/4}, -(c + d*x)^2]))/(693*d*e)","A",1
238,1,110,134,0.1017209,"\int (c e+d e x)^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x])^2,x]","\frac{2 (e (c+d x))^{5/2} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{9}{4},\frac{9}{4};\frac{11}{4},\frac{13}{4};-(c+d x)^2\right)-36 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{4};\frac{11}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)+63 \left(a+b \sinh ^{-1}(c+d x)\right)^2\right)}{315 d e}","\frac{16 b^2 (e (c+d x))^{9/2} \, _3F_2\left(1,\frac{9}{4},\frac{9}{4};\frac{11}{4},\frac{13}{4};-(c+d x)^2\right)}{315 d e^3}-\frac{8 b (e (c+d x))^{7/2} \, _2F_1\left(\frac{1}{2},\frac{7}{4};\frac{11}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{35 d e^2}+\frac{2 (e (c+d x))^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{5 d e}",1,"(2*(e*(c + d*x))^(5/2)*(63*(a + b*ArcSinh[c + d*x])^2 - 36*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 7/4, 11/4, -(c + d*x)^2] + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 9/4, 9/4}, {11/4, 13/4}, -(c + d*x)^2]))/(315*d*e)","A",1
239,1,110,134,0.0838975,"\int \sqrt{c e+d e x} \left(a+b \sinh ^{-1}(c+d x)\right)^2 \, dx","Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^2,x]","\frac{2 (e (c+d x))^{3/2} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};-(c+d x)^2\right)-28 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)+35 \left(a+b \sinh ^{-1}(c+d x)\right)^2\right)}{105 d e}","\frac{16 b^2 (e (c+d x))^{7/2} \, _3F_2\left(1,\frac{7}{4},\frac{7}{4};\frac{9}{4},\frac{11}{4};-(c+d x)^2\right)}{105 d e^3}-\frac{8 b (e (c+d x))^{5/2} \, _2F_1\left(\frac{1}{2},\frac{5}{4};\frac{9}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{15 d e^2}+\frac{2 (e (c+d x))^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{3 d e}",1,"(2*(e*(c + d*x))^(3/2)*(35*(a + b*ArcSinh[c + d*x])^2 - 28*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 5/4, 9/4, -(c + d*x)^2] + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 7/4, 7/4}, {9/4, 11/4}, -(c + d*x)^2]))/(105*d*e)","A",1
240,1,110,132,0.0625657,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{c e+d e x}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^2/Sqrt[c*e + d*e*x],x]","\frac{2 \sqrt{e (c+d x)} \left(8 b^2 (c+d x)^2 \, _3F_2\left(1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};-(c+d x)^2\right)-20 b (c+d x) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)+15 \left(a+b \sinh ^{-1}(c+d x)\right)^2\right)}{15 d e}","\frac{16 b^2 (e (c+d x))^{5/2} \, _3F_2\left(1,\frac{5}{4},\frac{5}{4};\frac{7}{4},\frac{9}{4};-(c+d x)^2\right)}{15 d e^3}-\frac{8 b (e (c+d x))^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{3 d e^2}+\frac{2 \sqrt{e (c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e}",1,"(2*Sqrt[e*(c + d*x)]*(15*(a + b*ArcSinh[c + d*x])^2 - 20*b*(c + d*x)*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, -(c + d*x)^2] + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{1, 5/4, 5/4}, {7/4, 9/4}, -(c + d*x)^2]))/(15*d*e)","A",1
241,1,109,130,0.0989055,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{(c e+d e x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^(3/2),x]","\frac{2 \left(-4 b (c+d x) \left(2 b (c+d x) \, _3F_2\left(\frac{3}{4},\frac{3}{4},1;\frac{5}{4},\frac{7}{4};-(c+d x)^2\right)-3 \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)\right)-3 \left(a+b \sinh ^{-1}(c+d x)\right)^2\right)}{3 d e \sqrt{e (c+d x)}}","-\frac{16 b^2 (e (c+d x))^{3/2} \, _3F_2\left(\frac{3}{4},\frac{3}{4},1;\frac{5}{4},\frac{7}{4};-(c+d x)^2\right)}{3 d e^3}+\frac{8 b \sqrt{e (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{d e^2}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{d e \sqrt{e (c+d x)}}",1,"(2*(-3*(a + b*ArcSinh[c + d*x])^2 - 4*b*(c + d*x)*(-3*(a + b*ArcSinh[c + d*x])*Hypergeometric2F1[1/4, 1/2, 5/4, -(c + d*x)^2] + 2*b*(c + d*x)*HypergeometricPFQ[{3/4, 3/4, 1}, {5/4, 7/4}, -(c + d*x)^2])))/(3*d*e*Sqrt[e*(c + d*x)])","A",1
242,1,106,134,0.0766635,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{(c e+d e x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^(5/2),x]","-\frac{2 \left(4 b (c+d x) \left(\, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)-2 b (c+d x) \, _3F_2\left(\frac{1}{4},\frac{1}{4},1;\frac{3}{4},\frac{5}{4};-(c+d x)^2\right)\right)+\left(a+b \sinh ^{-1}(c+d x)\right)^2\right)}{3 d e (e (c+d x))^{3/2}}","\frac{16 b^2 \sqrt{e (c+d x)} \, _3F_2\left(\frac{1}{4},\frac{1}{4},1;\frac{3}{4},\frac{5}{4};-(c+d x)^2\right)}{3 d e^3}-\frac{8 b \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{3 d e^2 \sqrt{e (c+d x)}}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{3 d e (e (c+d x))^{3/2}}",1,"(-2*((a + b*ArcSinh[c + d*x])^2 + 4*b*(c + d*x)*((a + b*ArcSinh[c + d*x])*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c + d*x)^2] - 2*b*(c + d*x)*HypergeometricPFQ[{1/4, 1/4, 1}, {3/4, 5/4}, -(c + d*x)^2])))/(3*d*e*(e*(c + d*x))^(3/2))","A",1
243,1,110,134,0.0872796,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{(c e+d e x)^{7/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^2/(c*e + d*e*x)^(7/2),x]","-\frac{2 \left(8 b^2 (c+d x)^2 \, _3F_2\left(-\frac{1}{4},-\frac{1}{4},1;\frac{1}{4},\frac{3}{4};-(c+d x)^2\right)+\left(a+b \sinh ^{-1}(c+d x)\right) \left(3 \left(a+b \sinh ^{-1}(c+d x)\right)+4 b (c+d x) \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-(c+d x)^2\right)\right)\right)}{15 d e (e (c+d x))^{5/2}}","-\frac{16 b^2 \, _3F_2\left(-\frac{1}{4},-\frac{1}{4},1;\frac{1}{4},\frac{3}{4};-(c+d x)^2\right)}{15 d e^3 \sqrt{e (c+d x)}}-\frac{8 b \, _2F_1\left(-\frac{3}{4},\frac{1}{2};\frac{1}{4};-(c+d x)^2\right) \left(a+b \sinh ^{-1}(c+d x)\right)}{15 d e^2 (e (c+d x))^{3/2}}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)^2}{5 d e (e (c+d x))^{5/2}}",1,"(-2*((a + b*ArcSinh[c + d*x])*(3*(a + b*ArcSinh[c + d*x]) + 4*b*(c + d*x)*Hypergeometric2F1[-3/4, 1/2, 1/4, -(c + d*x)^2]) + 8*b^2*(c + d*x)^2*HypergeometricPFQ[{-1/4, -1/4, 1}, {1/4, 3/4}, -(c + d*x)^2]))/(15*d*e*(e*(c + d*x))^(5/2))","A",1
244,0,0,82,89.803215,"\int (c e+d e x)^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^3,x]","\int (c e+d e x)^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","\frac{2 (e (c+d x))^{9/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{9 d e}-\frac{2 b \text{Int}\left(\frac{(e (c+d x))^{9/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{(c+d x)^2+1}},x\right)}{3 e}",0,"Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^3, x]","A",-1
245,0,0,82,108.2309694,"\int (c e+d e x)^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^3,x]","\int (c e+d e x)^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","\frac{2 (e (c+d x))^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{7 d e}-\frac{6 b \text{Int}\left(\frac{(e (c+d x))^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{(c+d x)^2+1}},x\right)}{7 e}",0,"Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^3, x]","A",-1
246,0,0,82,73.2086102,"\int (c e+d e x)^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x])^3,x]","\int (c e+d e x)^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","\frac{2 (e (c+d x))^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{5 d e}-\frac{6 b \text{Int}\left(\frac{(e (c+d x))^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{(c+d x)^2+1}},x\right)}{5 e}",0,"Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x])^3, x]","A",-1
247,0,0,80,93.8307812,"\int \sqrt{c e+d e x} \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^3,x]","\int \sqrt{c e+d e x} \left(a+b \sinh ^{-1}(c+d x)\right)^3 \, dx","\frac{2 (e (c+d x))^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{3 d e}-\frac{2 b \text{Int}\left(\frac{(e (c+d x))^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{(c+d x)^2+1}},x\right)}{e}",0,"Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^3, x]","A",-1
248,0,0,78,9.4393203,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{c e+d e x}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^3/Sqrt[c*e + d*e*x],x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{c e+d e x}} \, dx","\frac{2 \sqrt{e (c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d e}-\frac{6 b \text{Int}\left(\frac{\sqrt{e (c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{(c+d x)^2+1}},x\right)}{e}",0,"Integrate[(a + b*ArcSinh[c + d*x])^3/Sqrt[c*e + d*e*x], x]","A",-1
249,0,0,78,19.9338968,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{(c e+d e x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(3/2),x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{(c e+d e x)^{3/2}} \, dx","\frac{6 b \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}},x\right)}{e}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)^3}{d e \sqrt{e (c+d x)}}",0,"Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(3/2), x]","A",-1
250,0,0,80,23.0697164,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{(c e+d e x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(5/2),x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{(c e+d e x)^{5/2}} \, dx","\frac{2 b \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{(c+d x)^2+1} (e (c+d x))^{3/2}},x\right)}{e}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)^3}{3 d e (e (c+d x))^{3/2}}",0,"Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(5/2), x]","A",-1
251,0,0,82,73.7107933,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{(c e+d e x)^{7/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(7/2),x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{(c e+d e x)^{7/2}} \, dx","\frac{6 b \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^2}{\sqrt{(c+d x)^2+1} (e (c+d x))^{5/2}},x\right)}{5 e}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)^3}{5 d e (e (c+d x))^{5/2}}",0,"Integrate[(a + b*ArcSinh[c + d*x])^3/(c*e + d*e*x)^(7/2), x]","A",-1
252,0,0,82,98.2275537,"\int (c e+d e x)^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^4,x]","\int (c e+d e x)^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","\frac{2 (e (c+d x))^{9/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4}{9 d e}-\frac{8 b \text{Int}\left(\frac{(e (c+d x))^{9/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{(c+d x)^2+1}},x\right)}{9 e}",0,"Integrate[(c*e + d*e*x)^(7/2)*(a + b*ArcSinh[c + d*x])^4, x]","A",-1
253,0,0,82,128.4696771,"\int (c e+d e x)^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^4,x]","\int (c e+d e x)^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","\frac{2 (e (c+d x))^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4}{7 d e}-\frac{8 b \text{Int}\left(\frac{(e (c+d x))^{7/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{(c+d x)^2+1}},x\right)}{7 e}",0,"Integrate[(c*e + d*e*x)^(5/2)*(a + b*ArcSinh[c + d*x])^4, x]","A",-1
254,0,0,82,83.6686535,"\int (c e+d e x)^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x])^4,x]","\int (c e+d e x)^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","\frac{2 (e (c+d x))^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4}{5 d e}-\frac{8 b \text{Int}\left(\frac{(e (c+d x))^{5/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{(c+d x)^2+1}},x\right)}{5 e}",0,"Integrate[(c*e + d*e*x)^(3/2)*(a + b*ArcSinh[c + d*x])^4, x]","A",-1
255,0,0,82,120.9154383,"\int \sqrt{c e+d e x} \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^4,x]","\int \sqrt{c e+d e x} \left(a+b \sinh ^{-1}(c+d x)\right)^4 \, dx","\frac{2 (e (c+d x))^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^4}{3 d e}-\frac{8 b \text{Int}\left(\frac{(e (c+d x))^{3/2} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{(c+d x)^2+1}},x\right)}{3 e}",0,"Integrate[Sqrt[c*e + d*e*x]*(a + b*ArcSinh[c + d*x])^4, x]","A",-1
256,0,0,78,9.3313079,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{\sqrt{c e+d e x}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^4/Sqrt[c*e + d*e*x],x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{\sqrt{c e+d e x}} \, dx","\frac{2 \sqrt{e (c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^4}{d e}-\frac{8 b \text{Int}\left(\frac{\sqrt{e (c+d x)} \left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{(c+d x)^2+1}},x\right)}{e}",0,"Integrate[(a + b*ArcSinh[c + d*x])^4/Sqrt[c*e + d*e*x], x]","A",-1
257,0,0,78,37.7556157,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{(c e+d e x)^{3/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(3/2),x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{(c e+d e x)^{3/2}} \, dx","\frac{8 b \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}},x\right)}{e}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)^4}{d e \sqrt{e (c+d x)}}",0,"Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(3/2), x]","A",-1
258,0,0,82,43.2179244,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{(c e+d e x)^{5/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(5/2),x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{(c e+d e x)^{5/2}} \, dx","\frac{8 b \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{(c+d x)^2+1} (e (c+d x))^{3/2}},x\right)}{3 e}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)^4}{3 d e (e (c+d x))^{3/2}}",0,"Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(5/2), x]","A",-1
259,0,0,82,112.4833178,"\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{(c e+d e x)^{7/2}} \, dx","Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(7/2),x]","\int \frac{\left(a+b \sinh ^{-1}(c+d x)\right)^4}{(c e+d e x)^{7/2}} \, dx","\frac{8 b \text{Int}\left(\frac{\left(a+b \sinh ^{-1}(c+d x)\right)^3}{\sqrt{(c+d x)^2+1} (e (c+d x))^{5/2}},x\right)}{5 e}-\frac{2 \left(a+b \sinh ^{-1}(c+d x)\right)^4}{5 d e (e (c+d x))^{5/2}}",0,"Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x)^(7/2), x]","A",-1
260,1,127,131,0.1217993,"\int \sqrt{1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)^3 \, dx","Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^3,x]","\frac{4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)^3-3 \left(2 a^2+4 a b x+2 b^2 x^2+1\right) \sinh ^{-1}(a+b x)^2+6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)-3 b x (2 a+b x)+\sinh ^{-1}(a+b x)^4}{8 b}","-\frac{3 (a+b x)^2}{8 b}+\frac{\sinh ^{-1}(a+b x)^4}{8 b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^3}{2 b}-\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{4 b}-\frac{3 \sinh ^{-1}(a+b x)^2}{8 b}+\frac{3 (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)}{4 b}",1,"(-3*b*x*(2*a + b*x) + 6*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x] - 3*(1 + 2*a^2 + 4*a*b*x + 2*b^2*x^2)*ArcSinh[a + b*x]^2 + 4*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^3 + ArcSinh[a + b*x]^4)/(8*b)","A",1
261,1,110,107,0.0939433,"\int \sqrt{1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)^2 \, dx","Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2,x]","\frac{3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1}+6 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)^2-3 \left(2 a^2+4 a b x+2 b^2 x^2+1\right) \sinh ^{-1}(a+b x)+2 \sinh ^{-1}(a+b x)^3}{12 b}","\frac{(a+b x) \sqrt{(a+b x)^2+1}}{4 b}+\frac{\sinh ^{-1}(a+b x)^3}{6 b}+\frac{(a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{2 b}-\frac{(a+b x)^2 \sinh ^{-1}(a+b x)}{2 b}-\frac{\sinh ^{-1}(a+b x)}{4 b}",1,"(3*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] - 3*(1 + 2*a^2 + 4*a*b*x + 2*b^2*x^2)*ArcSinh[a + b*x] + 6*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2 + 2*ArcSinh[a + b*x]^3)/(12*b)","A",1
262,1,61,61,0.0590019,"\int \sqrt{1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x) \, dx","Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x],x]","\frac{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)-b x (2 a+b x)+\sinh ^{-1}(a+b x)^2}{4 b}","-\frac{(a+b x)^2}{4 b}+\frac{\sqrt{(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)}{2 b}+\frac{\sinh ^{-1}(a+b x)^2}{4 b}",1,"(-(b*x*(2*a + b*x)) + 2*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x] + ArcSinh[a + b*x]^2)/(4*b)","A",1
263,1,24,31,0.066231,"\int \frac{\sqrt{1+a^2+2 a b x+b^2 x^2}}{\sinh ^{-1}(a+b x)} \, dx","Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/ArcSinh[a + b*x],x]","\frac{\text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)+\log \left(\sinh ^{-1}(a+b x)\right)}{2 b}","\frac{\text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)}{2 b}+\frac{\log \left(\sinh ^{-1}(a+b x)\right)}{2 b}",1,"(CoshIntegral[2*ArcSinh[a + b*x]] + Log[ArcSinh[a + b*x]])/(2*b)","A",1
264,1,47,36,0.0640033,"\int \frac{\sqrt{1+a^2+2 a b x+b^2 x^2}}{\sinh ^{-1}(a+b x)^2} \, dx","Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/ArcSinh[a + b*x]^2,x]","-\frac{a^2-\sinh ^{-1}(a+b x) \text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)+2 a b x+b^2 x^2+1}{b \sinh ^{-1}(a+b x)}","\frac{\text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)}{b}-\frac{(a+b x)^2+1}{b \sinh ^{-1}(a+b x)}",1,"-((1 + a^2 + 2*a*b*x + b^2*x^2 - ArcSinh[a + b*x]*SinhIntegral[2*ArcSinh[a + b*x]])/(b*ArcSinh[a + b*x]))","A",1
265,1,85,71,0.2942095,"\int \frac{\sqrt{1+a^2+2 a b x+b^2 x^2}}{\sinh ^{-1}(a+b x)^3} \, dx","Integrate[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]/ArcSinh[a + b*x]^3,x]","-\frac{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)+a^2-2 \sinh ^{-1}(a+b x)^2 \text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)+2 a b x+b^2 x^2+1}{2 b \sinh ^{-1}(a+b x)^2}","\frac{\text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)}{b}-\frac{\sqrt{(a+b x)^2+1} (a+b x)}{b \sinh ^{-1}(a+b x)}+\frac{-(a+b x)^2-1}{2 b \sinh ^{-1}(a+b x)^2}",1,"-1/2*(1 + a^2 + 2*a*b*x + b^2*x^2 + 2*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x] - 2*ArcSinh[a + b*x]^2*CoshIntegral[2*ArcSinh[a + b*x]])/(b*ArcSinh[a + b*x]^2)","A",1
266,1,266,235,0.2438237,"\int \left(1+a^2+2 a b x+b^2 x^2\right)^{3/2} \sinh ^{-1}(a+b x)^3 \, dx","Integrate[(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]^3,x]","-\frac{3 \left(6 a^2+17\right) b^2 x^2+6 a \left(2 a^2+17\right) b x-16 \sqrt{a^2+2 a b x+b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2+5 a+2 b^3 x^3+5 b x\right) \sinh ^{-1}(a+b x)^3-6 \sqrt{a^2+2 a b x+b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2+17 a+2 b^3 x^3+17 b x\right) \sinh ^{-1}(a+b x)+3 \left(8 a^4+32 a^3 b x+8 a^2 \left(6 b^2 x^2+5\right)+16 a b x \left(2 b^2 x^2+5\right)+8 b^4 x^4+40 b^2 x^2+17\right) \sinh ^{-1}(a+b x)^2+12 a b^3 x^3-12 \sinh ^{-1}(a+b x)^4+3 b^4 x^4}{128 b}","-\frac{3 (a+b x)^4}{128 b}-\frac{51 (a+b x)^2}{128 b}-\frac{9 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{16 b}+\frac{\left((a+b x)^2+1\right)^{3/2} (a+b x) \sinh ^{-1}(a+b x)^3}{4 b}+\frac{3 \sqrt{(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)^3}{8 b}+\frac{3 \left((a+b x)^2+1\right)^{3/2} (a+b x) \sinh ^{-1}(a+b x)}{32 b}+\frac{45 \sqrt{(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)}{64 b}+\frac{3 \sinh ^{-1}(a+b x)^4}{32 b}-\frac{3 \left((a+b x)^2+1\right)^2 \sinh ^{-1}(a+b x)^2}{16 b}-\frac{27 \sinh ^{-1}(a+b x)^2}{128 b}",1,"-1/128*(6*a*(17 + 2*a^2)*b*x + 3*(17 + 6*a^2)*b^2*x^2 + 12*a*b^3*x^3 + 3*b^4*x^4 - 6*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(17*a + 2*a^3 + 17*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSinh[a + b*x] + 3*(17 + 8*a^4 + 32*a^3*b*x + 40*b^2*x^2 + 8*b^4*x^4 + 16*a*b*x*(5 + 2*b^2*x^2) + 8*a^2*(5 + 6*b^2*x^2))*ArcSinh[a + b*x]^2 - 16*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(5*a + 2*a^3 + 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSinh[a + b*x]^3 - 12*ArcSinh[a + b*x]^4)/b","A",1
267,1,211,189,0.1720887,"\int \left(1+a^2+2 a b x+b^2 x^2\right)^{3/2} \sinh ^{-1}(a+b x)^2 \, dx","Integrate[(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]^2,x]","\frac{-\left(8 a^4+40 a^2+17\right) \sinh ^{-1}(a+b x)+\sqrt{a^2+2 a b x+b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2+17 a+2 b^3 x^3+17 b x\right)+8 \sqrt{a^2+2 a b x+b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2+5 a+2 b^3 x^3+5 b x\right) \sinh ^{-1}(a+b x)^2-8 b x \left(4 a^3+6 a^2 b x+4 a b^2 x^2+10 a+b^3 x^3+5 b x\right) \sinh ^{-1}(a+b x)+8 \sinh ^{-1}(a+b x)^3}{64 b}","\frac{(a+b x) \left((a+b x)^2+1\right)^{3/2}}{32 b}+\frac{15 (a+b x) \sqrt{(a+b x)^2+1}}{64 b}+\frac{\sinh ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left((a+b x)^2+1\right)^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac{3 (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{8 b}-\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac{\left((a+b x)^2+1\right)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac{9 \sinh ^{-1}(a+b x)}{64 b}",1,"(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(17*a + 2*a^3 + 17*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3) - (17 + 40*a^2 + 8*a^4)*ArcSinh[a + b*x] - 8*b*x*(10*a + 4*a^3 + 5*b*x + 6*a^2*b*x + 4*a*b^2*x^2 + b^3*x^3)*ArcSinh[a + b*x] + 8*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(5*a + 2*a^3 + 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSinh[a + b*x]^2 + 8*ArcSinh[a + b*x]^3)/(64*b)","A",1
268,1,124,106,0.1000126,"\int \left(1+a^2+2 a b x+b^2 x^2\right)^{3/2} \sinh ^{-1}(a+b x) \, dx","Integrate[(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x],x]","\frac{-b x \left(4 a^3+6 a^2 b x+4 a b^2 x^2+10 a+b^3 x^3+5 b x\right)+2 \sqrt{a^2+2 a b x+b^2 x^2+1} \left(2 a^3+6 a^2 b x+6 a b^2 x^2+5 a+2 b^3 x^3+5 b x\right) \sinh ^{-1}(a+b x)+3 \sinh ^{-1}(a+b x)^2}{16 b}","-\frac{(a+b x)^4}{16 b}-\frac{5 (a+b x)^2}{16 b}+\frac{\left((a+b x)^2+1\right)^{3/2} (a+b x) \sinh ^{-1}(a+b x)}{4 b}+\frac{3 \sqrt{(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)}{8 b}+\frac{3 \sinh ^{-1}(a+b x)^2}{16 b}",1,"(-(b*x*(10*a + 4*a^3 + 5*b*x + 6*a^2*b*x + 4*a*b^2*x^2 + b^3*x^3)) + 2*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(5*a + 2*a^3 + 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSinh[a + b*x] + 3*ArcSinh[a + b*x]^2)/(16*b)","A",1
269,1,37,47,0.3197264,"\int \frac{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2}}{\sinh ^{-1}(a+b x)} \, dx","Integrate[(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)/ArcSinh[a + b*x],x]","\frac{4 \text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)+\text{Chi}\left(4 \sinh ^{-1}(a+b x)\right)+3 \log \left(\sinh ^{-1}(a+b x)\right)}{8 b}","\frac{\text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)}{2 b}+\frac{\text{Chi}\left(4 \sinh ^{-1}(a+b x)\right)}{8 b}+\frac{3 \log \left(\sinh ^{-1}(a+b x)\right)}{8 b}",1,"(4*CoshIntegral[2*ArcSinh[a + b*x]] + CoshIntegral[4*ArcSinh[a + b*x]] + 3*Log[ArcSinh[a + b*x]])/(8*b)","A",1
270,1,70,54,0.2887699,"\int \frac{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2}}{\sinh ^{-1}(a+b x)^2} \, dx","Integrate[(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)/ArcSinh[a + b*x]^2,x]","\frac{-2 \left(a^2+2 a b x+b^2 x^2+1\right)^2+2 \sinh ^{-1}(a+b x) \text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)+\sinh ^{-1}(a+b x) \text{Shi}\left(4 \sinh ^{-1}(a+b x)\right)}{2 b \sinh ^{-1}(a+b x)}","\frac{\text{Shi}\left(2 \sinh ^{-1}(a+b x)\right)}{b}+\frac{\text{Shi}\left(4 \sinh ^{-1}(a+b x)\right)}{2 b}-\frac{\left((a+b x)^2+1\right)^2}{b \sinh ^{-1}(a+b x)}",1,"(-2*(1 + a^2 + 2*a*b*x + b^2*x^2)^2 + 2*ArcSinh[a + b*x]*SinhIntegral[2*ArcSinh[a + b*x]] + ArcSinh[a + b*x]*SinhIntegral[4*ArcSinh[a + b*x]])/(2*b*ArcSinh[a + b*x])","A",1
271,1,108,84,0.4040323,"\int \frac{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2}}{\sinh ^{-1}(a+b x)^3} \, dx","Integrate[(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)/ArcSinh[a + b*x]^3,x]","\frac{-\frac{\left(a^2+2 a b x+b^2 x^2+1\right) \left(4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2+1} \sinh ^{-1}(a+b x)+a^2+2 a b x+b^2 x^2+1\right)}{\sinh ^{-1}(a+b x)^2}+2 \text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)+2 \text{Chi}\left(4 \sinh ^{-1}(a+b x)\right)}{2 b}","\frac{\text{Chi}\left(2 \sinh ^{-1}(a+b x)\right)}{b}+\frac{\text{Chi}\left(4 \sinh ^{-1}(a+b x)\right)}{b}-\frac{\left((a+b x)^2+1\right)^2}{2 b \sinh ^{-1}(a+b x)^2}-\frac{2 (a+b x) \left((a+b x)^2+1\right)^{3/2}}{b \sinh ^{-1}(a+b x)}",1,"(-(((1 + a^2 + 2*a*b*x + b^2*x^2)*(1 + a^2 + 2*a*b*x + b^2*x^2 + 4*(a + b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]))/ArcSinh[a + b*x]^2) + 2*CoshIntegral[2*ArcSinh[a + b*x]] + 2*CoshIntegral[4*ArcSinh[a + b*x]])/(2*b)","A",1
272,1,15,15,0.0221605,"\int \frac{\sinh ^{-1}(a+b x)^3}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[ArcSinh[a + b*x]^3/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2],x]","\frac{\sinh ^{-1}(a+b x)^4}{4 b}","\frac{\sinh ^{-1}(a+b x)^4}{4 b}",1,"ArcSinh[a + b*x]^4/(4*b)","A",1
273,1,15,15,0.0195952,"\int \frac{\sinh ^{-1}(a+b x)^2}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[ArcSinh[a + b*x]^2/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2],x]","\frac{\sinh ^{-1}(a+b x)^3}{3 b}","\frac{\sinh ^{-1}(a+b x)^3}{3 b}",1,"ArcSinh[a + b*x]^3/(3*b)","A",1
274,1,15,15,0.016326,"\int \frac{\sinh ^{-1}(a+b x)}{\sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx","Integrate[ArcSinh[a + b*x]/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2],x]","\frac{\sinh ^{-1}(a+b x)^2}{2 b}","\frac{\sinh ^{-1}(a+b x)^2}{2 b}",1,"ArcSinh[a + b*x]^2/(2*b)","A",1
275,1,11,11,0.0303517,"\int \frac{1}{\sqrt{1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)} \, dx","Integrate[1/(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]),x]","\frac{\log \left(\sinh ^{-1}(a+b x)\right)}{b}","\frac{\log \left(\sinh ^{-1}(a+b x)\right)}{b}",1,"Log[ArcSinh[a + b*x]]/b","A",1
276,1,13,13,0.0153143,"\int \frac{1}{\sqrt{1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)^2} \, dx","Integrate[1/(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^2),x]","-\frac{1}{b \sinh ^{-1}(a+b x)}","-\frac{1}{b \sinh ^{-1}(a+b x)}",1,"-(1/(b*ArcSinh[a + b*x]))","A",1
277,1,15,15,0.014544,"\int \frac{1}{\sqrt{1+a^2+2 a b x+b^2 x^2} \sinh ^{-1}(a+b x)^3} \, dx","Integrate[1/(Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*ArcSinh[a + b*x]^3),x]","-\frac{1}{2 b \sinh ^{-1}(a+b x)^2}","-\frac{1}{2 b \sinh ^{-1}(a+b x)^2}",1,"-1/2*1/(b*ArcSinh[a + b*x]^2)","A",1
278,1,128,115,0.6288879,"\int \frac{\sinh ^{-1}(a+b x)^3}{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[ArcSinh[a + b*x]^3/(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{2 \sinh ^{-1}(a+b x)^2 \left(\frac{\left(-\sqrt{a^2+2 a b x+b^2 x^2+1}+a+b x\right) \sinh ^{-1}(a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2+1}}-3 \log \left(e^{-2 \sinh ^{-1}(a+b x)}+1\right)\right)+6 \sinh ^{-1}(a+b x) \text{Li}_2\left(-e^{-2 \sinh ^{-1}(a+b x)}\right)+3 \text{Li}_3\left(-e^{-2 \sinh ^{-1}(a+b x)}\right)}{2 b}","-\frac{3 \sinh ^{-1}(a+b x) \text{Li}_2\left(-e^{2 \sinh ^{-1}(a+b x)}\right)}{b}+\frac{3 \text{Li}_3\left(-e^{2 \sinh ^{-1}(a+b x)}\right)}{2 b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^3}{b \sqrt{(a+b x)^2+1}}+\frac{\sinh ^{-1}(a+b x)^3}{b}-\frac{3 \sinh ^{-1}(a+b x)^2 \log \left(e^{2 \sinh ^{-1}(a+b x)}+1\right)}{b}",1,"(2*ArcSinh[a + b*x]^2*(((a + b*x - Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])*ArcSinh[a + b*x])/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] - 3*Log[1 + E^(-2*ArcSinh[a + b*x])]) + 6*ArcSinh[a + b*x]*PolyLog[2, -E^(-2*ArcSinh[a + b*x])] + 3*PolyLog[3, -E^(-2*ArcSinh[a + b*x])])/(2*b)","A",0
279,1,98,86,0.3998589,"\int \frac{\sinh ^{-1}(a+b x)^2}{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[ArcSinh[a + b*x]^2/(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{\sinh ^{-1}(a+b x) \left(\frac{\left(-\sqrt{a^2+2 a b x+b^2 x^2+1}+a+b x\right) \sinh ^{-1}(a+b x)}{\sqrt{a^2+2 a b x+b^2 x^2+1}}-2 \log \left(e^{-2 \sinh ^{-1}(a+b x)}+1\right)\right)+\text{Li}_2\left(-e^{-2 \sinh ^{-1}(a+b x)}\right)}{b}","-\frac{\text{Li}_2\left(-e^{2 \sinh ^{-1}(a+b x)}\right)}{b}+\frac{(a+b x) \sinh ^{-1}(a+b x)^2}{b \sqrt{(a+b x)^2+1}}+\frac{\sinh ^{-1}(a+b x)^2}{b}-\frac{2 \sinh ^{-1}(a+b x) \log \left(e^{2 \sinh ^{-1}(a+b x)}+1\right)}{b}",1,"(ArcSinh[a + b*x]*(((a + b*x - Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])*ArcSinh[a + b*x])/Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] - 2*Log[1 + E^(-2*ArcSinh[a + b*x])]) + PolyLog[2, -E^(-2*ArcSinh[a + b*x])])/b","A",0
280,1,62,46,0.0979571,"\int \frac{\sinh ^{-1}(a+b x)}{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2}} \, dx","Integrate[ArcSinh[a + b*x]/(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]","\frac{(a+b x) \sinh ^{-1}(a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2+1}}-\frac{\log \left(a^2+2 a b x+b^2 x^2+1\right)}{2 b}","\frac{(a+b x) \sinh ^{-1}(a+b x)}{b \sqrt{(a+b x)^2+1}}-\frac{\log \left((a+b x)^2+1\right)}{2 b}",1,"((a + b*x)*ArcSinh[a + b*x])/(b*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) - Log[1 + a^2 + 2*a*b*x + b^2*x^2]/(2*b)","A",1
281,0,0,25,0.6011863,"\int \frac{1}{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2} \sinh ^{-1}(a+b x)} \, dx","Integrate[1/((1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]),x]","\int \frac{1}{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2} \sinh ^{-1}(a+b x)} \, dx","\text{Int}\left(\frac{1}{\left((a+b x)^2+1\right)^{3/2} \sinh ^{-1}(a+b x)},x\right)",0,"Integrate[1/((1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]), x]","A",-1
282,0,0,55,2.9180906,"\int \frac{1}{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2} \sinh ^{-1}(a+b x)^2} \, dx","Integrate[1/((1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]^2),x]","\int \frac{1}{\left(1+a^2+2 a b x+b^2 x^2\right)^{3/2} \sinh ^{-1}(a+b x)^2} \, dx","-2 \text{Int}\left(\frac{a+b x}{\left((a+b x)^2+1\right)^2 \sinh ^{-1}(a+b x)},x\right)-\frac{1}{b \left((a+b x)^2+1\right) \sinh ^{-1}(a+b x)}",0,"Integrate[1/((1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]^2), x]","A",-1
283,1,44,50,0.0187609,"\int x^3 \sinh ^{-1}\left(a x^2\right) \, dx","Integrate[x^3*ArcSinh[a*x^2],x]","\frac{\left(2 a^2 x^4+1\right) \sinh ^{-1}\left(a x^2\right)-a x^2 \sqrt{a^2 x^4+1}}{8 a^2}","\frac{\sinh ^{-1}\left(a x^2\right)}{8 a^2}-\frac{x^2 \sqrt{a^2 x^4+1}}{8 a}+\frac{1}{4} x^4 \sinh ^{-1}\left(a x^2\right)",1,"(-(a*x^2*Sqrt[1 + a^2*x^4]) + (1 + 2*a^2*x^4)*ArcSinh[a*x^2])/(8*a^2)","A",1
284,1,75,101,0.1293387,"\int x^2 \sinh ^{-1}\left(a x^2\right) \, dx","Integrate[x^2*ArcSinh[a*x^2],x]","\frac{1}{9} \left(-\frac{2 \left(a^2 x^5+x\right)}{a \sqrt{a^2 x^4+1}}-\frac{2 \sqrt{i a} F\left(\left.i \sinh ^{-1}\left(\sqrt{i a} x\right)\right|-1\right)}{a^2}+3 x^3 \sinh ^{-1}\left(a x^2\right)\right)","-\frac{2 x \sqrt{a^2 x^4+1}}{9 a}+\frac{\left(a x^2+1\right) \sqrt{\frac{a^2 x^4+1}{\left(a x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt{a} x\right)|\frac{1}{2}\right)}{9 a^{3/2} \sqrt{a^2 x^4+1}}+\frac{1}{3} x^3 \sinh ^{-1}\left(a x^2\right)",1,"((-2*(x + a^2*x^5))/(a*Sqrt[1 + a^2*x^4]) + 3*x^3*ArcSinh[a*x^2] - (2*Sqrt[I*a]*EllipticF[I*ArcSinh[Sqrt[I*a]*x], -1])/a^2)/9","C",1
285,1,34,34,0.015277,"\int x \sinh ^{-1}\left(a x^2\right) \, dx","Integrate[x*ArcSinh[a*x^2],x]","\frac{1}{2} x^2 \sinh ^{-1}\left(a x^2\right)-\frac{\sqrt{a^2 x^4+1}}{2 a}","\frac{1}{2} x^2 \sinh ^{-1}\left(a x^2\right)-\frac{\sqrt{a^2 x^4+1}}{2 a}",1,"-1/2*Sqrt[1 + a^2*x^4]/a + (x^2*ArcSinh[a*x^2])/2","A",1
286,1,35,162,0.0052574,"\int \sinh ^{-1}\left(a x^2\right) \, dx","Integrate[ArcSinh[a*x^2],x]","x \sinh ^{-1}\left(a x^2\right)-\frac{2}{3} a x^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-a^2 x^4\right)","-\frac{2 x \sqrt{a^2 x^4+1}}{a x^2+1}-\frac{\left(a x^2+1\right) \sqrt{\frac{a^2 x^4+1}{\left(a x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt{a} x\right)|\frac{1}{2}\right)}{\sqrt{a} \sqrt{a^2 x^4+1}}+\frac{2 \left(a x^2+1\right) \sqrt{\frac{a^2 x^4+1}{\left(a x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt{a} x\right)|\frac{1}{2}\right)}{\sqrt{a} \sqrt{a^2 x^4+1}}+x \sinh ^{-1}\left(a x^2\right)",1,"x*ArcSinh[a*x^2] - (2*a*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, -(a^2*x^4)])/3","C",1
287,1,54,54,0.0076578,"\int \frac{\sinh ^{-1}\left(a x^2\right)}{x} \, dx","Integrate[ArcSinh[a*x^2]/x,x]","\frac{1}{4} \text{Li}_2\left(e^{2 \sinh ^{-1}\left(a x^2\right)}\right)-\frac{1}{4} \sinh ^{-1}\left(a x^2\right)^2+\frac{1}{2} \sinh ^{-1}\left(a x^2\right) \log \left(1-e^{2 \sinh ^{-1}\left(a x^2\right)}\right)","\frac{1}{4} \text{Li}_2\left(e^{2 \sinh ^{-1}\left(a x^2\right)}\right)-\frac{1}{4} \sinh ^{-1}\left(a x^2\right)^2+\frac{1}{2} \sinh ^{-1}\left(a x^2\right) \log \left(1-e^{2 \sinh ^{-1}\left(a x^2\right)}\right)",1,"-1/4*ArcSinh[a*x^2]^2 + (ArcSinh[a*x^2]*Log[1 - E^(2*ArcSinh[a*x^2])])/2 + PolyLog[2, E^(2*ArcSinh[a*x^2])]/4","A",1
288,1,42,75,0.0416266,"\int \frac{\sinh ^{-1}\left(a x^2\right)}{x^2} \, dx","Integrate[ArcSinh[a*x^2]/x^2,x]","-\frac{\sinh ^{-1}\left(a x^2\right)+2 \sqrt{i a} x F\left(\left.i \sinh ^{-1}\left(\sqrt{i a} x\right)\right|-1\right)}{x}","\frac{\sqrt{a} \left(a x^2+1\right) \sqrt{\frac{a^2 x^4+1}{\left(a x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt{a} x\right)|\frac{1}{2}\right)}{\sqrt{a^2 x^4+1}}-\frac{\sinh ^{-1}\left(a x^2\right)}{x}",1,"-((ArcSinh[a*x^2] + 2*Sqrt[I*a]*x*EllipticF[I*ArcSinh[Sqrt[I*a]*x], -1])/x)","C",1
289,1,33,33,0.0066103,"\int \frac{\sinh ^{-1}\left(a x^2\right)}{x^3} \, dx","Integrate[ArcSinh[a*x^2]/x^3,x]","-\frac{1}{2} a \tanh ^{-1}\left(\sqrt{a^2 x^4+1}\right)-\frac{\sinh ^{-1}\left(a x^2\right)}{2 x^2}","-\frac{1}{2} a \tanh ^{-1}\left(\sqrt{a^2 x^4+1}\right)-\frac{\sinh ^{-1}\left(a x^2\right)}{2 x^2}",1,"-1/2*ArcSinh[a*x^2]/x^2 - (a*ArcTanh[Sqrt[1 + a^2*x^4]])/2","A",1
290,1,88,197,0.1693798,"\int \frac{\sinh ^{-1}\left(a x^2\right)}{x^4} \, dx","Integrate[ArcSinh[a*x^2]/x^4,x]","\frac{1}{3} \left(-\frac{2 a \sqrt{a^2 x^4+1}}{x}+\frac{2 a^2 \left(E\left(\left.i \sinh ^{-1}\left(\sqrt{i a} x\right)\right|-1\right)-F\left(\left.i \sinh ^{-1}\left(\sqrt{i a} x\right)\right|-1\right)\right)}{\sqrt{i a}}-\frac{\sinh ^{-1}\left(a x^2\right)}{x^3}\right)","-\frac{2 a \sqrt{a^2 x^4+1}}{3 x}+\frac{2 a^2 x \sqrt{a^2 x^4+1}}{3 \left(a x^2+1\right)}+\frac{a^{3/2} \left(a x^2+1\right) \sqrt{\frac{a^2 x^4+1}{\left(a x^2+1\right)^2}} F\left(2 \tan ^{-1}\left(\sqrt{a} x\right)|\frac{1}{2}\right)}{3 \sqrt{a^2 x^4+1}}-\frac{2 a^{3/2} \left(a x^2+1\right) \sqrt{\frac{a^2 x^4+1}{\left(a x^2+1\right)^2}} E\left(2 \tan ^{-1}\left(\sqrt{a} x\right)|\frac{1}{2}\right)}{3 \sqrt{a^2 x^4+1}}-\frac{\sinh ^{-1}\left(a x^2\right)}{3 x^3}",1,"((-2*a*Sqrt[1 + a^2*x^4])/x - ArcSinh[a*x^2]/x^3 + (2*a^2*(EllipticE[I*ArcSinh[Sqrt[I*a]*x], -1] - EllipticF[I*ArcSinh[Sqrt[I*a]*x], -1]))/Sqrt[I*a])/3","C",1
291,1,54,54,0.0081901,"\int \frac{\sinh ^{-1}\left(a x^5\right)}{x} \, dx","Integrate[ArcSinh[a*x^5]/x,x]","\frac{1}{10} \text{Li}_2\left(e^{2 \sinh ^{-1}\left(a x^5\right)}\right)-\frac{1}{10} \sinh ^{-1}\left(a x^5\right)^2+\frac{1}{5} \sinh ^{-1}\left(a x^5\right) \log \left(1-e^{2 \sinh ^{-1}\left(a x^5\right)}\right)","\frac{1}{10} \text{Li}_2\left(e^{2 \sinh ^{-1}\left(a x^5\right)}\right)-\frac{1}{10} \sinh ^{-1}\left(a x^5\right)^2+\frac{1}{5} \sinh ^{-1}\left(a x^5\right) \log \left(1-e^{2 \sinh ^{-1}\left(a x^5\right)}\right)",1,"-1/10*ArcSinh[a*x^5]^2 + (ArcSinh[a*x^5]*Log[1 - E^(2*ArcSinh[a*x^5])])/5 + PolyLog[2, E^(2*ArcSinh[a*x^5])]/10","A",1
292,1,43,72,0.0241037,"\int x^2 \sinh ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x^2*ArcSinh[Sqrt[x]],x]","\frac{1}{144} \left(3 \left(16 x^3+5\right) \sinh ^{-1}\left(\sqrt{x}\right)+\sqrt{x} \sqrt{x+1} \left(-8 x^2+10 x-15\right)\right)","-\frac{1}{18} \sqrt{x+1} x^{5/2}+\frac{5}{72} \sqrt{x+1} x^{3/2}+\frac{1}{3} x^3 \sinh ^{-1}\left(\sqrt{x}\right)-\frac{5}{48} \sqrt{x+1} \sqrt{x}+\frac{5}{48} \sinh ^{-1}\left(\sqrt{x}\right)",1,"(Sqrt[x]*Sqrt[1 + x]*(-15 + 10*x - 8*x^2) + 3*(5 + 16*x^3)*ArcSinh[Sqrt[x]])/144","A",1
293,1,37,56,0.0181376,"\int x \sinh ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[x*ArcSinh[Sqrt[x]],x]","\frac{1}{16} \left(\left(8 x^2-3\right) \sinh ^{-1}\left(\sqrt{x}\right)+\sqrt{x} \sqrt{x+1} (3-2 x)\right)","-\frac{1}{8} \sqrt{x+1} x^{3/2}+\frac{1}{2} x^2 \sinh ^{-1}\left(\sqrt{x}\right)+\frac{3}{16} \sqrt{x+1} \sqrt{x}-\frac{3}{16} \sinh ^{-1}\left(\sqrt{x}\right)",1,"((3 - 2*x)*Sqrt[x]*Sqrt[1 + x] + (-3 + 8*x^2)*ArcSinh[Sqrt[x]])/16","A",1
294,1,33,35,0.0410117,"\int \sinh ^{-1}\left(\sqrt{x}\right) \, dx","Integrate[ArcSinh[Sqrt[x]],x]","\frac{1}{2} \left((2 x+1) \sinh ^{-1}\left(\sqrt{x}\right)-\sqrt{\frac{x}{x+1}} (x+1)\right)","-\frac{1}{2} \sqrt{x} \sqrt{x+1}+x \sinh ^{-1}\left(\sqrt{x}\right)+\frac{1}{2} \sinh ^{-1}\left(\sqrt{x}\right)",1,"(-(Sqrt[x/(1 + x)]*(1 + x)) + (1 + 2*x)*ArcSinh[Sqrt[x]])/2","A",1
295,1,46,46,0.0093376,"\int \frac{\sinh ^{-1}\left(\sqrt{x}\right)}{x} \, dx","Integrate[ArcSinh[Sqrt[x]]/x,x]","\text{Li}_2\left(e^{2 \sinh ^{-1}\left(\sqrt{x}\right)}\right)-\sinh ^{-1}\left(\sqrt{x}\right)^2+2 \sinh ^{-1}\left(\sqrt{x}\right) \log \left(1-e^{2 \sinh ^{-1}\left(\sqrt{x}\right)}\right)","\text{Li}_2\left(e^{2 \sinh ^{-1}\left(\sqrt{x}\right)}\right)-\sinh ^{-1}\left(\sqrt{x}\right)^2+2 \sinh ^{-1}\left(\sqrt{x}\right) \log \left(1-e^{2 \sinh ^{-1}\left(\sqrt{x}\right)}\right)",1,"-ArcSinh[Sqrt[x]]^2 + 2*ArcSinh[Sqrt[x]]*Log[1 - E^(2*ArcSinh[Sqrt[x]])] + PolyLog[2, E^(2*ArcSinh[Sqrt[x]])]","A",1
296,1,26,26,0.0114978,"\int \frac{\sinh ^{-1}\left(\sqrt{x}\right)}{x^2} \, dx","Integrate[ArcSinh[Sqrt[x]]/x^2,x]","-\frac{\sqrt{x+1}}{\sqrt{x}}-\frac{\sinh ^{-1}\left(\sqrt{x}\right)}{x}","-\frac{\sqrt{x+1}}{\sqrt{x}}-\frac{\sinh ^{-1}\left(\sqrt{x}\right)}{x}",1,"-(Sqrt[1 + x]/Sqrt[x]) - ArcSinh[Sqrt[x]]/x","A",1
297,1,34,46,0.015283,"\int \frac{\sinh ^{-1}\left(\sqrt{x}\right)}{x^3} \, dx","Integrate[ArcSinh[Sqrt[x]]/x^3,x]","\frac{\sqrt{x} \sqrt{x+1} (2 x-1)-3 \sinh ^{-1}\left(\sqrt{x}\right)}{6 x^2}","-\frac{\sqrt{x+1}}{6 x^{3/2}}-\frac{\sinh ^{-1}\left(\sqrt{x}\right)}{2 x^2}+\frac{\sqrt{x+1}}{3 \sqrt{x}}",1,"(Sqrt[x]*Sqrt[1 + x]*(-1 + 2*x) - 3*ArcSinh[Sqrt[x]])/(6*x^2)","A",1
298,1,39,62,0.0165106,"\int \frac{\sinh ^{-1}\left(\sqrt{x}\right)}{x^4} \, dx","Integrate[ArcSinh[Sqrt[x]]/x^4,x]","\frac{\sqrt{x} \sqrt{x+1} \left(-8 x^2+4 x-3\right)-15 \sinh ^{-1}\left(\sqrt{x}\right)}{45 x^3}","\frac{4 \sqrt{x+1}}{45 x^{3/2}}-\frac{\sqrt{x+1}}{15 x^{5/2}}-\frac{\sinh ^{-1}\left(\sqrt{x}\right)}{3 x^3}-\frac{8 \sqrt{x+1}}{45 \sqrt{x}}",1,"(Sqrt[x]*Sqrt[1 + x]*(-3 + 4*x - 8*x^2) - 15*ArcSinh[Sqrt[x]])/(45*x^3)","A",1
299,1,44,78,0.0187774,"\int \frac{\sinh ^{-1}\left(\sqrt{x}\right)}{x^5} \, dx","Integrate[ArcSinh[Sqrt[x]]/x^5,x]","\frac{\sqrt{x} \sqrt{x+1} \left(16 x^3-8 x^2+6 x-5\right)-35 \sinh ^{-1}\left(\sqrt{x}\right)}{140 x^4}","-\frac{2 \sqrt{x+1}}{35 x^{3/2}}+\frac{3 \sqrt{x+1}}{70 x^{5/2}}-\frac{\sqrt{x+1}}{28 x^{7/2}}-\frac{\sinh ^{-1}\left(\sqrt{x}\right)}{4 x^4}+\frac{4 \sqrt{x+1}}{35 \sqrt{x}}",1,"(Sqrt[x]*Sqrt[1 + x]*(-5 + 6*x - 8*x^2 + 16*x^3) - 35*ArcSinh[Sqrt[x]])/(140*x^4)","A",1
300,1,57,56,0.0437577,"\int x^2 \sinh ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[x^2*ArcSinh[a/x],x]","\frac{1}{6} \left(a x^2 \sqrt{\frac{a^2}{x^2}+1}+a^3 \left(-\log \left(x \left(\sqrt{\frac{a^2}{x^2}+1}+1\right)\right)\right)+2 x^3 \sinh ^{-1}\left(\frac{a}{x}\right)\right)","\frac{1}{6} a x^2 \sqrt{\frac{a^2}{x^2}+1}-\frac{1}{6} a^3 \tanh ^{-1}\left(\sqrt{\frac{a^2}{x^2}+1}\right)+\frac{1}{3} x^3 \text{csch}^{-1}\left(\frac{x}{a}\right)",1,"(a*Sqrt[1 + a^2/x^2]*x^2 + 2*x^3*ArcSinh[a/x] - a^3*Log[(1 + Sqrt[1 + a^2/x^2])*x])/6","A",1
301,1,29,33,0.0243791,"\int x \sinh ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[x*ArcSinh[a/x],x]","\frac{1}{2} x \left(a \sqrt{\frac{a^2}{x^2}+1}+x \sinh ^{-1}\left(\frac{a}{x}\right)\right)","\frac{1}{2} a x \sqrt{\frac{a^2}{x^2}+1}+\frac{1}{2} x^2 \text{csch}^{-1}\left(\frac{x}{a}\right)",1,"(x*(a*Sqrt[1 + a^2/x^2] + x*ArcSinh[a/x]))/2","A",1
302,1,77,25,0.0955831,"\int \sinh ^{-1}\left(\frac{a}{x}\right) \, dx","Integrate[ArcSinh[a/x],x]","\frac{a \sqrt{a^2+x^2} \left(\log \left(\frac{x}{\sqrt{a^2+x^2}}+1\right)-\log \left(1-\frac{x}{\sqrt{a^2+x^2}}\right)\right)}{2 x \sqrt{\frac{a^2}{x^2}+1}}+x \sinh ^{-1}\left(\frac{a}{x}\right)","a \tanh ^{-1}\left(\sqrt{\frac{a^2}{x^2}+1}\right)+x \text{csch}^{-1}\left(\frac{x}{a}\right)",1,"x*ArcSinh[a/x] + (a*Sqrt[a^2 + x^2]*(-Log[1 - x/Sqrt[a^2 + x^2]] + Log[1 + x/Sqrt[a^2 + x^2]]))/(2*Sqrt[1 + a^2/x^2]*x)","B",1
303,1,52,52,0.0078655,"\int \frac{\sinh ^{-1}\left(\frac{a}{x}\right)}{x} \, dx","Integrate[ArcSinh[a/x]/x,x]","-\frac{1}{2} \text{Li}_2\left(e^{2 \sinh ^{-1}\left(\frac{a}{x}\right)}\right)+\frac{1}{2} \sinh ^{-1}\left(\frac{a}{x}\right)^2-\sinh ^{-1}\left(\frac{a}{x}\right) \log \left(1-e^{2 \sinh ^{-1}\left(\frac{a}{x}\right)}\right)","-\frac{1}{2} \text{Li}_2\left(e^{2 \sinh ^{-1}\left(\frac{a}{x}\right)}\right)+\frac{1}{2} \sinh ^{-1}\left(\frac{a}{x}\right)^2-\sinh ^{-1}\left(\frac{a}{x}\right) \log \left(1-e^{2 \sinh ^{-1}\left(\frac{a}{x}\right)}\right)",1,"ArcSinh[a/x]^2/2 - ArcSinh[a/x]*Log[1 - E^(2*ArcSinh[a/x])] - PolyLog[2, E^(2*ArcSinh[a/x])]/2","A",1
304,1,29,29,0.0171969,"\int \frac{\sinh ^{-1}\left(\frac{a}{x}\right)}{x^2} \, dx","Integrate[ArcSinh[a/x]/x^2,x]","\frac{\sqrt{\frac{a^2}{x^2}+1}}{a}-\frac{\sinh ^{-1}\left(\frac{a}{x}\right)}{x}","\frac{\sqrt{\frac{a^2}{x^2}+1}}{a}-\frac{\text{csch}^{-1}\left(\frac{x}{a}\right)}{x}",1,"Sqrt[1 + a^2/x^2]/a - ArcSinh[a/x]/x","A",1
305,1,44,50,0.0238169,"\int \frac{\sinh ^{-1}\left(\frac{a}{x}\right)}{x^3} \, dx","Integrate[ArcSinh[a/x]/x^3,x]","\frac{a x \sqrt{\frac{a^2}{x^2}+1}-\left(2 a^2+x^2\right) \sinh ^{-1}\left(\frac{a}{x}\right)}{4 a^2 x^2}","\frac{\sqrt{\frac{a^2}{x^2}+1}}{4 a x}-\frac{\text{csch}^{-1}\left(\frac{x}{a}\right)}{4 a^2}-\frac{\text{csch}^{-1}\left(\frac{x}{a}\right)}{2 x^2}",1,"(a*Sqrt[1 + a^2/x^2]*x - (2*a^2 + x^2)*ArcSinh[a/x])/(4*a^2*x^2)","A",1
306,1,48,54,0.0313901,"\int \frac{\sinh ^{-1}\left(\frac{a}{x}\right)}{x^4} \, dx","Integrate[ArcSinh[a/x]/x^4,x]","\left(\frac{1}{9 a x^2}-\frac{2}{9 a^3}\right) \sqrt{\frac{a^2+x^2}{x^2}}-\frac{\sinh ^{-1}\left(\frac{a}{x}\right)}{3 x^3}","\frac{\left(\frac{a^2}{x^2}+1\right)^{3/2}}{9 a^3}-\frac{\sqrt{\frac{a^2}{x^2}+1}}{3 a^3}-\frac{\text{csch}^{-1}\left(\frac{x}{a}\right)}{3 x^3}",1,"(-2/(9*a^3) + 1/(9*a*x^2))*Sqrt[(a^2 + x^2)/x^2] - ArcSinh[a/x]/(3*x^3)","A",1
307,1,74,77,0.08087,"\int x^m \sinh ^{-1}\left(a x^n\right) \, dx","Integrate[x^m*ArcSinh[a*x^n],x]","\frac{x^{m+1} \left((m+n+1) \sinh ^{-1}\left(a x^n\right)-a n x^n \, _2F_1\left(\frac{1}{2},\frac{m+n+1}{2 n};\frac{m+3 n+1}{2 n};-a^2 x^{2 n}\right)\right)}{(m+1) (m+n+1)}","\frac{x^{m+1} \sinh ^{-1}\left(a x^n\right)}{m+1}-\frac{a n x^{m+n+1} \, _2F_1\left(\frac{1}{2},\frac{m+n+1}{2 n};\frac{m+3 n+1}{2 n};-a^2 x^{2 n}\right)}{(m+1) (m+n+1)}",1,"(x^(1 + m)*((1 + m + n)*ArcSinh[a*x^n] - a*n*x^n*Hypergeometric2F1[1/2, (1 + m + n)/(2*n), (1 + m + 3*n)/(2*n), -(a^2*x^(2*n))]))/((1 + m)*(1 + m + n))","A",1
308,1,66,64,0.0527891,"\int x^2 \sinh ^{-1}\left(a x^n\right) \, dx","Integrate[x^2*ArcSinh[a*x^n],x]","\frac{1}{3} x^3 \sinh ^{-1}\left(a x^n\right)-\frac{a n x^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2 n};\frac{n+3}{2 n}+1;-a^2 x^{2 n}\right)}{3 (n+3)}","\frac{1}{3} x^3 \sinh ^{-1}\left(a x^n\right)-\frac{a n x^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2 n};\frac{3 (n+1)}{2 n};-a^2 x^{2 n}\right)}{3 (n+3)}",1,"(x^3*ArcSinh[a*x^n])/3 - (a*n*x^(3 + n)*Hypergeometric2F1[1/2, (3 + n)/(2*n), 1 + (3 + n)/(2*n), -(a^2*x^(2*n))])/(3*(3 + n))","A",1
309,1,58,65,0.052841,"\int x \sinh ^{-1}\left(a x^n\right) \, dx","Integrate[x*ArcSinh[a*x^n],x]","\frac{x^2 \left((n+2) \sinh ^{-1}\left(a x^n\right)-a n x^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}+\frac{1}{n};\frac{3}{2}+\frac{1}{n};-a^2 x^{2 n}\right)\right)}{2 (n+2)}","\frac{1}{2} x^2 \sinh ^{-1}\left(a x^n\right)-\frac{a n x^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2 n};\frac{1}{2} \left(3+\frac{2}{n}\right);-a^2 x^{2 n}\right)}{2 (n+2)}",1,"(x^2*((2 + n)*ArcSinh[a*x^n] - a*n*x^n*Hypergeometric2F1[1/2, 1/2 + n^(-1), 3/2 + n^(-1), -(a^2*x^(2*n))]))/(2*(2 + n))","A",1
310,1,56,56,0.0276023,"\int \sinh ^{-1}\left(a x^n\right) \, dx","Integrate[ArcSinh[a*x^n],x]","x \sinh ^{-1}\left(a x^n\right)-\frac{a n x^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-a^2 x^{2 n}\right)}{n+1}","x \sinh ^{-1}\left(a x^n\right)-\frac{a n x^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2 n};\frac{1}{2} \left(3+\frac{1}{n}\right);-a^2 x^{2 n}\right)}{n+1}",1,"x*ArcSinh[a*x^n] - (a*n*x^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/(2*n), (3 + n^(-1))/2, -(a^2*x^(2*n))])/(1 + n)","A",1
311,1,60,60,0.0083619,"\int \frac{\sinh ^{-1}\left(a x^n\right)}{x} \, dx","Integrate[ArcSinh[a*x^n]/x,x]","\frac{\text{Li}_2\left(e^{2 \sinh ^{-1}\left(a x^n\right)}\right)}{2 n}-\frac{\sinh ^{-1}\left(a x^n\right)^2}{2 n}+\frac{\sinh ^{-1}\left(a x^n\right) \log \left(1-e^{2 \sinh ^{-1}\left(a x^n\right)}\right)}{n}","\frac{\text{Li}_2\left(e^{2 \sinh ^{-1}\left(a x^n\right)}\right)}{2 n}-\frac{\sinh ^{-1}\left(a x^n\right)^2}{2 n}+\frac{\sinh ^{-1}\left(a x^n\right) \log \left(1-e^{2 \sinh ^{-1}\left(a x^n\right)}\right)}{n}",1,"-1/2*ArcSinh[a*x^n]^2/n + (ArcSinh[a*x^n]*Log[1 - E^(2*ArcSinh[a*x^n])])/n + PolyLog[2, E^(2*ArcSinh[a*x^n])]/(2*n)","A",1
312,1,61,65,0.0572681,"\int \frac{\sinh ^{-1}\left(a x^n\right)}{x^2} \, dx","Integrate[ArcSinh[a*x^n]/x^2,x]","\frac{a n x^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2 n};\frac{n-1}{2 n}+1;-a^2 x^{2 n}\right)}{n-1}-\frac{\sinh ^{-1}\left(a x^n\right)}{x}","-\frac{a n x^{n-1} \, _2F_1\left(\frac{1}{2},-\frac{1-n}{2 n};\frac{1}{2} \left(3-\frac{1}{n}\right);-a^2 x^{2 n}\right)}{1-n}-\frac{\sinh ^{-1}\left(a x^n\right)}{x}",1,"-(ArcSinh[a*x^n]/x) + (a*n*x^(-1 + n)*Hypergeometric2F1[1/2, (-1 + n)/(2*n), 1 + (-1 + n)/(2*n), -(a^2*x^(2*n))])/(-1 + n)","A",1
313,1,62,68,0.0467586,"\int \frac{\sinh ^{-1}\left(a x^n\right)}{x^3} \, dx","Integrate[ArcSinh[a*x^n]/x^3,x]","\frac{a n x^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-\frac{1}{n};\frac{3}{2}-\frac{1}{n};-a^2 x^{2 n}\right)-(n-2) \sinh ^{-1}\left(a x^n\right)}{2 (n-2) x^2}","-\frac{a n x^{n-2} \, _2F_1\left(\frac{1}{2},\frac{1}{2} \left(1-\frac{2}{n}\right);\frac{1}{2} \left(3-\frac{2}{n}\right);-a^2 x^{2 n}\right)}{2 (2-n)}-\frac{\sinh ^{-1}\left(a x^n\right)}{2 x^2}",1,"(-((-2 + n)*ArcSinh[a*x^n]) + a*n*x^n*Hypergeometric2F1[1/2, 1/2 - n^(-1), 3/2 - n^(-1), -(a^2*x^(2*n))])/(2*(-2 + n)*x^2)","A",1
314,1,149,153,0.1251691,"\int \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^4 \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^4,x]","48 b^2 \left(-\frac{4 b \sqrt{d x^2 \left(d x^2+2 i\right)} \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}{d x}+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^2+8 b^2 x\right)+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^4-\frac{8 b \sqrt{d x^2 \left(d x^2+2 i\right)} \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^3}{d x}","-\frac{192 b^3 \sqrt{d^2 x^4+2 i d x^2} \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}{d x}+48 b^2 x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^2-\frac{8 b \sqrt{d^2 x^4+2 i d x^2} \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^3}{d x}+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^4+384 b^4 x",1,"(-8*b*Sqrt[d*x^2*(2*I + d*x^2)]*(a + I*b*ArcSin[1 - I*d*x^2])^3)/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^4 + 48*b^2*(8*b^2*x - (4*b*Sqrt[d*x^2*(2*I + d*x^2)]*(a + I*b*ArcSin[1 - I*d*x^2]))/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^2)","A",1
315,1,180,129,0.1466377,"\int \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^3 \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^3,x]","\frac{a d x^2 \left(a^2+24 b^2\right)-6 b \left(a^2+8 b^2\right) \sqrt{d x^2 \left(d x^2+2 i\right)}+3 i b \sin ^{-1}\left(1-i d x^2\right) \left(a^2 d x^2-4 a b \sqrt{d x^2 \left(d x^2+2 i\right)}+8 b^2 d x^2\right)+3 b^2 \sin ^{-1}\left(1-i d x^2\right)^2 \left(-a d x^2+2 b \sqrt{d x^2 \left(d x^2+2 i\right)}\right)-i b^3 d x^2 \sin ^{-1}\left(1-i d x^2\right)^3}{d x}","24 a b^2 x-\frac{6 b \sqrt{d^2 x^4+2 i d x^2} \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^2}{d x}+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^3-\frac{48 b^3 \sqrt{d^2 x^4+2 i d x^2}}{d x}+24 i b^3 x \sin ^{-1}\left(1-i d x^2\right)",1,"(a*(a^2 + 24*b^2)*d*x^2 - 6*b*(a^2 + 8*b^2)*Sqrt[d*x^2*(2*I + d*x^2)] + (3*I)*b*(a^2*d*x^2 + 8*b^2*d*x^2 - 4*a*b*Sqrt[d*x^2*(2*I + d*x^2)])*ArcSin[1 - I*d*x^2] + 3*b^2*(-(a*d*x^2) + 2*b*Sqrt[d*x^2*(2*I + d*x^2)])*ArcSin[1 - I*d*x^2]^2 - I*b^3*d*x^2*ArcSin[1 - I*d*x^2]^3)/(d*x)","A",1
316,1,76,76,0.0277584,"\int \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^2 \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^2,x]","-\frac{4 b \sqrt{d^2 x^4+2 i d x^2} \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}{d x}+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^2+8 b^2 x","-\frac{4 b \sqrt{d^2 x^4+2 i d x^2} \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}{d x}+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^2+8 b^2 x",1,"8*b^2*x - (4*b*Sqrt[(2*I)*d*x^2 + d^2*x^4]*(a + I*b*ArcSin[1 - I*d*x^2]))/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^2","A",1
317,1,48,50,0.0259687,"\int \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right) \, dx","Integrate[a + I*b*ArcSin[1 - I*d*x^2],x]","a x-\frac{2 b \sqrt{d x^2 \left(d x^2+2 i\right)}}{d x}+i b x \sin ^{-1}\left(1-i d x^2\right)","a x-\frac{2 b \sqrt{d^2 x^4+2 i d x^2}}{d x}+i b x \sin ^{-1}\left(1-i d x^2\right)",1,"a*x - (2*b*Sqrt[d*x^2*(2*I + d*x^2)])/(d*x) + I*b*x*ArcSin[1 - I*d*x^2]","A",1
318,1,150,194,0.7911784,"\int \frac{1}{a+i b \sin ^{-1}\left(1-i d x^2\right)} \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(-1),x]","\frac{x \left(\left(-\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\sin ^{-1}\left(1-i d x^2\right)-\frac{i a}{b}\right)\right)+\left(-\sinh \left(\frac{a}{2 b}\right)-i \cosh \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{i a}{2 b}-\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}","\frac{x \left(-\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(-\frac{i \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}{2 b}\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{i a}{2 b}-\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}",1,"(x*(CosIntegral[(((-I)*a)/b + ArcSin[1 - I*d*x^2])/2]*(I*Cosh[a/(2*b)] - Sinh[a/(2*b)]) + ((-I)*Cosh[a/(2*b)] - Sinh[a/(2*b)])*SinIntegral[((I/2)*a)/b - ArcSin[1 - I*d*x^2]/2]))/(2*b*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))","A",1
319,1,197,245,1.4465805,"\int \frac{1}{\left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^2} \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(-2),x]","\frac{\frac{x^2 \left(\left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\sin ^{-1}\left(1-i d x^2\right)-\frac{i a}{b}\right)\right)+\left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{i a}{2 b}-\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}-\frac{2 b \sqrt{d x^2 \left(d x^2+2 i\right)}}{d \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}}{4 b^2 x}","\frac{x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(-\frac{i \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}{2 b}\right)}{4 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}+\frac{x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{i a}{2 b}-\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}{4 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{\sqrt{d^2 x^4+2 i d x^2}}{2 b d x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}",1,"((-2*b*Sqrt[d*x^2*(2*I + d*x^2)])/(d*(a + I*b*ArcSin[1 - I*d*x^2])) + (x^2*(CosIntegral[(((-I)*a)/b + ArcSin[1 - I*d*x^2])/2]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]) + (Cosh[a/(2*b)] + I*Sinh[a/(2*b)])*SinIntegral[((I/2)*a)/b - ArcSin[1 - I*d*x^2]/2]))/(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))/(4*b^2*x)","A",1
320,1,229,275,0.5806006,"\int \frac{1}{\left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^3} \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(-3),x]","\frac{-\frac{8 b^2 \sqrt{d x^2 \left(d x^2+2 i\right)}}{d \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^2}+\frac{2 i x^2 \left(\left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\sin ^{-1}\left(1-i d x^2\right)-\frac{i a}{b}\right)\right)-\left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{i a}{2 b}-\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}-\frac{4 b x^2}{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{32 b^3 x}","\frac{x \left(-\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(-\frac{i \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}{2 b}\right)}{16 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{i a}{2 b}-\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}{16 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{x}{8 b^2 \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)}-\frac{\sqrt{d^2 x^4+2 i d x^2}}{4 b d x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^2}",1,"((-8*b^2*Sqrt[d*x^2*(2*I + d*x^2)])/(d*(a + I*b*ArcSin[1 - I*d*x^2])^2) - (4*b*x^2)/(a + I*b*ArcSin[1 - I*d*x^2]) + ((2*I)*x^2*(CosIntegral[(((-I)*a)/b + ArcSin[1 - I*d*x^2])/2]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]) - (Cosh[a/(2*b)] - I*Sinh[a/(2*b)])*SinIntegral[((I/2)*a)/b - ArcSin[1 - I*d*x^2]/2]))/(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))/(32*b^3*x)","A",1
321,1,149,153,0.1246684,"\int \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^4 \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^4,x]","48 b^2 \left(-\frac{4 b \sqrt{d x^2 \left(d x^2-2 i\right)} \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)}{d x}+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^2+8 b^2 x\right)+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^4-\frac{8 b \sqrt{d x^2 \left(d x^2-2 i\right)} \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^3}{d x}","-\frac{192 b^3 \sqrt{d^2 x^4-2 i d x^2} \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)}{d x}+48 b^2 x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^2-\frac{8 b \sqrt{d^2 x^4-2 i d x^2} \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^3}{d x}+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^4+384 b^4 x",1,"(-8*b*Sqrt[d*x^2*(-2*I + d*x^2)]*(a - I*b*ArcSin[1 + I*d*x^2])^3)/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^4 + 48*b^2*(8*b^2*x - (4*b*Sqrt[d*x^2*(-2*I + d*x^2)]*(a - I*b*ArcSin[1 + I*d*x^2]))/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^2)","A",1
322,1,180,129,0.1460466,"\int \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^3 \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^3,x]","\frac{a d x^2 \left(a^2+24 b^2\right)-6 b \left(a^2+8 b^2\right) \sqrt{d x^2 \left(d x^2-2 i\right)}-3 i b \sin ^{-1}\left(1+i d x^2\right) \left(a^2 d x^2-4 a b \sqrt{d x^2 \left(d x^2-2 i\right)}+8 b^2 d x^2\right)+3 b^2 \sin ^{-1}\left(1+i d x^2\right)^2 \left(-a d x^2+2 b \sqrt{d x^2 \left(d x^2-2 i\right)}\right)+i b^3 d x^2 \sin ^{-1}\left(1+i d x^2\right)^3}{d x}","24 a b^2 x-\frac{6 b \sqrt{d^2 x^4-2 i d x^2} \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^2}{d x}+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^3-\frac{48 b^3 \sqrt{d^2 x^4-2 i d x^2}}{d x}-24 i b^3 x \sin ^{-1}\left(1+i d x^2\right)",1,"(a*(a^2 + 24*b^2)*d*x^2 - 6*b*(a^2 + 8*b^2)*Sqrt[d*x^2*(-2*I + d*x^2)] - (3*I)*b*(a^2*d*x^2 + 8*b^2*d*x^2 - 4*a*b*Sqrt[d*x^2*(-2*I + d*x^2)])*ArcSin[1 + I*d*x^2] + 3*b^2*(-(a*d*x^2) + 2*b*Sqrt[d*x^2*(-2*I + d*x^2)])*ArcSin[1 + I*d*x^2]^2 + I*b^3*d*x^2*ArcSin[1 + I*d*x^2]^3)/(d*x)","A",1
323,1,76,76,0.0198034,"\int \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^2 \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^2,x]","-\frac{4 b \sqrt{d^2 x^4-2 i d x^2} \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)}{d x}+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^2+8 b^2 x","-\frac{4 b \sqrt{d^2 x^4-2 i d x^2} \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)}{d x}+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^2+8 b^2 x",1,"8*b^2*x - (4*b*Sqrt[(-2*I)*d*x^2 + d^2*x^4]*(a - I*b*ArcSin[1 + I*d*x^2]))/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^2","A",1
324,1,48,50,0.0271524,"\int \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right) \, dx","Integrate[a - I*b*ArcSin[1 + I*d*x^2],x]","a x-\frac{2 b \sqrt{d x^2 \left(d x^2-2 i\right)}}{d x}-i b x \sin ^{-1}\left(1+i d x^2\right)","a x-\frac{2 b \sqrt{d^2 x^4-2 i d x^2}}{d x}-i b x \sin ^{-1}\left(1+i d x^2\right)",1,"a*x - (2*b*Sqrt[d*x^2*(-2*I + d*x^2)])/(d*x) - I*b*x*ArcSin[1 + I*d*x^2]","A",1
325,1,146,191,0.6913648,"\int \frac{1}{a-i b \sin ^{-1}\left(1+i d x^2\right)} \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^(-1),x]","\frac{x \left(\left(-\sinh \left(\frac{a}{2 b}\right)-i \cosh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\frac{i a}{b}+\sin ^{-1}\left(i d x^2+1\right)\right)\right)+\left(\sinh \left(\frac{a}{2 b}\right)-i \cosh \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{1}{2} \left(\frac{i a}{b}+\sin ^{-1}\left(i d x^2+1\right)\right)\right)\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}","\frac{x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) \text{Shi}\left(\frac{a-i b \sin ^{-1}\left(i d x^2+1\right)}{2 b}\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{i \left(a-i b \sin ^{-1}\left(i d x^2+1\right)\right)}{2 b}\right)}{2 b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}",1,"(x*(CosIntegral[((I*a)/b + ArcSin[1 + I*d*x^2])/2]*((-I)*Cosh[a/(2*b)] - Sinh[a/(2*b)]) + ((-I)*Cosh[a/(2*b)] + Sinh[a/(2*b)])*SinIntegral[((I*a)/b + ArcSin[1 + I*d*x^2])/2]))/(2*b*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))","A",0
326,1,196,244,1.4910843,"\int \frac{1}{\left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^2} \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^(-2),x]","\frac{\frac{x^2 \left(\left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\frac{i a}{b}+\sin ^{-1}\left(i d x^2+1\right)\right)\right)-\left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{1}{2} \left(\frac{i a}{b}+\sin ^{-1}\left(i d x^2+1\right)\right)\right)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)}-\frac{2 b \sqrt{d x^2 \left(d x^2-2 i\right)}}{d \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)}}{4 b^2 x}","\frac{x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{i \left(a-i b \sin ^{-1}\left(i d x^2+1\right)\right)}{2 b}\right)}{4 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) \text{Shi}\left(\frac{a-i b \sin ^{-1}\left(i d x^2+1\right)}{2 b}\right)}{4 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{\sqrt{d^2 x^4-2 i d x^2}}{2 b d x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)}",1,"((-2*b*Sqrt[d*x^2*(-2*I + d*x^2)])/(d*(a - I*b*ArcSin[1 + I*d*x^2])) + (x^2*(CosIntegral[((I*a)/b + ArcSin[1 + I*d*x^2])/2]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]) - (Cosh[a/(2*b)] - I*Sinh[a/(2*b)])*SinIntegral[((I*a)/b + ArcSin[1 + I*d*x^2])/2]))/(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))/(4*b^2*x)","A",0
327,1,227,272,0.7708262,"\int \frac{1}{\left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^3} \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^(-3),x]","\frac{\frac{8 b^2 \sqrt{d x^2 \left(d x^2-2 i\right)}}{d \left(i a+b \sin ^{-1}\left(1+i d x^2\right)\right)^2}-\frac{2 i x^2 \left(\left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{1}{2} \left(\frac{i a}{b}+\sin ^{-1}\left(i d x^2+1\right)\right)\right)+\left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) \text{Si}\left(\frac{1}{2} \left(\frac{i a}{b}+\sin ^{-1}\left(i d x^2+1\right)\right)\right)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)}-\frac{4 b x^2}{a-i b \sin ^{-1}\left(1+i d x^2\right)}}{32 b^3 x}","-\frac{x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) \text{Ci}\left(\frac{i \left(a-i b \sin ^{-1}\left(i d x^2+1\right)\right)}{2 b}\right)}{16 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+\frac{x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) \text{Shi}\left(\frac{a-i b \sin ^{-1}\left(i d x^2+1\right)}{2 b}\right)}{16 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{x}{8 b^2 \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)}-\frac{\sqrt{d^2 x^4-2 i d x^2}}{4 b d x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^2}",1,"((-4*b*x^2)/(a - I*b*ArcSin[1 + I*d*x^2]) + (8*b^2*Sqrt[d*x^2*(-2*I + d*x^2)])/(d*(I*a + b*ArcSin[1 + I*d*x^2])^2) - ((2*I)*x^2*(CosIntegral[((I*a)/b + ArcSin[1 + I*d*x^2])/2]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]) + (Cosh[a/(2*b)] + I*Sinh[a/(2*b)])*SinIntegral[((I*a)/b + ArcSin[1 + I*d*x^2])/2]))/(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))/(32*b^3*x)","A",0
328,1,337,348,0.2877406,"\int \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{5/2} \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(5/2),x]","\frac{15 b^2 x \left(-\sqrt{\pi } \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)+\sqrt{-\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right) \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}\right)}{\sqrt{-\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{5/2}-\frac{5 b \sqrt{d x^2 \left(d x^2+2 i\right)} \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{3/2}}{d x}","-\frac{15 \sqrt{\pi } \sqrt{-\frac{i}{b}} b^3 x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}+\frac{15 \sqrt{\pi } b^2 x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\sqrt{-\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}+15 b^2 x \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}-\frac{5 b \sqrt{d^2 x^4+2 i d x^2} \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{3/2}}{d x}+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{5/2}",1,"(-5*b*Sqrt[d*x^2*(2*I + d*x^2)]*(a + I*b*ArcSin[1 - I*d*x^2])^(3/2))/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^(5/2) + (15*b^2*x*(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]) - Sqrt[Pi]*FresnelC[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]) + Sqrt[Pi]*FresnelS[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])))/(Sqrt[(-I)/b]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))","A",1
329,1,258,312,0.2275385,"\int \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{3/2} \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(3/2),x]","\frac{3 \sqrt{\pi } b^2 x \left(\left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) \left(-S\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)\right)-\left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)\right)}{\sqrt{i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{3 b \sqrt{d x^2 \left(d x^2+2 i\right)} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{d x}+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{3/2}","-\frac{3 \sqrt{\pi } b^2 x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)}{\sqrt{i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{3 b \sqrt{d^2 x^4+2 i d x^2} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{d x}+\frac{3 \sqrt{\pi } \sqrt{i b} b x \left(-\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}+x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{3/2}",1,"(-3*b*Sqrt[d*x^2*(2*I + d*x^2)]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/(d*x) + x*(a + I*b*ArcSin[1 - I*d*x^2])^(3/2) + (3*b^2*Sqrt[Pi]*x*(-(FresnelS[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)])) - FresnelC[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])))/(Sqrt[I*b]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))","A",1
330,1,259,263,0.0552475,"\int \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)} \, dx","Integrate[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]],x]","\frac{x \left(-\sqrt{\pi } \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)+\sqrt{-\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right) \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}\right)}{\sqrt{-\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}","-\frac{\sqrt{\pi } \sqrt{-\frac{i}{b}} b x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}+\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\sqrt{-\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}+x \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}",1,"(x*(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]) - Sqrt[Pi]*FresnelC[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]) + Sqrt[Pi]*FresnelS[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])))/(Sqrt[(-I)/b]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))","A",1
331,1,180,231,0.0045225,"\int \frac{1}{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}} \, dx","Integrate[1/Sqrt[a + I*b*ArcSin[1 - I*d*x^2]],x]","\frac{\sqrt{\pi } x \left(\left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) \left(-S\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)\right)-\left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)\right)}{\sqrt{i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}","-\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)}{\sqrt{i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)}{\sqrt{i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}",1,"(Sqrt[Pi]*x*(-(FresnelS[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)])) - FresnelC[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])))/(Sqrt[I*b]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))","A",1
332,1,291,291,0.3876485,"\int \frac{1}{\left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{3/2}} \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(-3/2),x]","-\frac{\sqrt{d^2 x^4+2 i d x^2}}{b d x \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}-\frac{\sqrt{\pi } \left(-\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}+\frac{\sqrt{\pi } \left(-\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}","-\frac{\sqrt{d^2 x^4+2 i d x^2}}{b d x \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}-\frac{\sqrt{\pi } \left(-\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}+\frac{\sqrt{\pi } \left(-\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}",1,"-(Sqrt[(2*I)*d*x^2 + d^2*x^4]/(b*d*x*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])) - (((-I)/b)^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]) + (((-I)/b)^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2])","A",1
333,1,308,326,0.8322547,"\int \frac{1}{\left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{5/2}} \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(-5/2),x]","-\frac{\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)}{\sqrt{i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}+\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)}{\sqrt{i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}+\frac{b \sqrt{d x^2 \left(d x^2+2 i\right)}}{d x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{3/2}}+\frac{x}{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}}{3 b^2}","-\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)}{3 \sqrt{i b} b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{i b} \sqrt{\pi }}\right)}{3 \sqrt{i b} b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{x}{3 b^2 \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}-\frac{\sqrt{d^2 x^4+2 i d x^2}}{3 b d x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{3/2}}",1,"-1/3*((b*Sqrt[d*x^2*(2*I + d*x^2)])/(d*x*(a + I*b*ArcSin[1 - I*d*x^2])^(3/2)) + x/Sqrt[a + I*b*ArcSin[1 - I*d*x^2]] + (Sqrt[Pi]*x*FresnelS[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Sqrt[I*b]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2])) + (Sqrt[Pi]*x*FresnelC[Sqrt[a + I*b*ArcSin[1 - I*d*x^2]]/(Sqrt[I*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[I*b]*(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2])))/b^2","A",1
334,1,365,389,0.9949795,"\int \frac{1}{\left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{7/2}} \, dx","Integrate[(a + I*b*ArcSin[1 - I*d*x^2])^(-7/2),x]","\frac{-\frac{\sqrt{\pi } \left(-\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}+\frac{\sqrt{\pi } \left(-\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)}+\frac{-\left(x^2 \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)\right)+\frac{\sqrt{d x^2 \left(d x^2+2 i\right)} \left(b \sin ^{-1}\left(1-i d x^2\right)-i a\right)^2}{b d}-\frac{3 b \sqrt{d x^2 \left(d x^2+2 i\right)}}{d}}{x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{5/2}}}{15 b^2}","-\frac{\sqrt{d^2 x^4+2 i d x^2}}{15 b^3 d x \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}-\frac{\sqrt{\pi } \left(-\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{15 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}+\frac{\sqrt{\pi } \left(-\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{-\frac{i}{b}} \sqrt{a+i b \sin ^{-1}\left(1-i d x^2\right)}}{\sqrt{\pi }}\right)}{15 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1-i d x^2\right)\right)\right)}-\frac{x}{15 b^2 \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{3/2}}-\frac{\sqrt{d^2 x^4+2 i d x^2}}{5 b d x \left(a+i b \sin ^{-1}\left(1-i d x^2\right)\right)^{5/2}}",1,"(((-3*b*Sqrt[d*x^2*(2*I + d*x^2)])/d - x^2*(a + I*b*ArcSin[1 - I*d*x^2]) + (Sqrt[d*x^2*(2*I + d*x^2)]*((-I)*a + b*ArcSin[1 - I*d*x^2])^2)/(b*d))/(x*(a + I*b*ArcSin[1 - I*d*x^2])^(5/2)) - (((-I)/b)^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]) + (((-I)/b)^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[(-I)/b]*Sqrt[a + I*b*ArcSin[1 - I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Cos[ArcSin[1 - I*d*x^2]/2] - Sin[ArcSin[1 - I*d*x^2]/2]))/(15*b^2)","A",1
335,1,337,348,0.2884307,"\int \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{5/2} \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^(5/2),x]","\frac{15 b^2 x \left(-\sqrt{\pi } \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right) \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}\right)}{\sqrt{\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{5/2}-\frac{5 b \sqrt{d x^2 \left(d x^2-2 i\right)} \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{3/2}}{d x}","-\frac{15 \sqrt{\pi } b^2 x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+\frac{15 \sqrt{\pi } b^2 x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+15 b^2 x \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}-\frac{5 b \sqrt{d^2 x^4-2 i d x^2} \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{3/2}}{d x}+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{5/2}",1,"(-5*b*Sqrt[d*x^2*(-2*I + d*x^2)]*(a - I*b*ArcSin[1 + I*d*x^2])^(3/2))/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^(5/2) + (15*b^2*x*(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]) + Sqrt[Pi]*FresnelS[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]) - Sqrt[Pi]*FresnelC[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])))/(Sqrt[I/b]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))","A",1
336,1,255,310,0.2632357,"\int \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{3/2} \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^(3/2),x]","-\frac{3 \sqrt{\pi } (-i b)^{3/2} x \left(\left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) \left(-C\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)\right)-\left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)}+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{3/2}-\frac{3 b \sqrt{d x^2 \left(d x^2-2 i\right)} \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}}{d x}","-\frac{3 \sqrt{\pi } b^2 x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)}{\sqrt{-i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{3 b \sqrt{d^2 x^4-2 i d x^2} \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}}{d x}-\frac{3 \sqrt{\pi } \sqrt{-i b} b x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)}+x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{3/2}",1,"(-3*b*Sqrt[d*x^2*(-2*I + d*x^2)]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/(d*x) + x*(a - I*b*ArcSin[1 + I*d*x^2])^(3/2) - (3*((-I)*b)^(3/2)*Sqrt[Pi]*x*(-(FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)])) - FresnelS[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])))/(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])","A",1
337,1,259,262,0.0548574,"\int \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)} \, dx","Integrate[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]],x]","\frac{x \left(-\sqrt{\pi } \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\pi } \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)+\sqrt{\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right) \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}\right)}{\sqrt{\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}","-\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{\sqrt{\frac{i}{b}} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+x \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}",1,"(x*(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]) + Sqrt[Pi]*FresnelS[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]) - Sqrt[Pi]*FresnelC[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])))/(Sqrt[I/b]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))","A",1
338,1,180,231,0.0041362,"\int \frac{1}{\sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}} \, dx","Integrate[1/Sqrt[a - I*b*ArcSin[1 + I*d*x^2]],x]","\frac{\sqrt{\pi } x \left(\left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) \left(-C\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)\right)-\left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)\right)}{\sqrt{-i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}","-\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)}{\sqrt{-i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)}{\sqrt{-i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}",1,"(Sqrt[Pi]*x*(-(FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)])) - FresnelS[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)])))/(Sqrt[(-I)*b]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]))","A",1
339,1,291,291,0.3745237,"\int \frac{1}{\left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{3/2}} \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^(-3/2),x]","-\frac{\sqrt{d^2 x^4-2 i d x^2}}{b d x \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}}-\frac{\sqrt{\pi } \left(\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)}+\frac{\sqrt{\pi } \left(\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)}","-\frac{\sqrt{d^2 x^4-2 i d x^2}}{b d x \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}}-\frac{\sqrt{\pi } \left(\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)}+\frac{\sqrt{\pi } \left(\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)}",1,"-(Sqrt[(-2*I)*d*x^2 + d^2*x^4]/(b*d*x*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])) + ((I/b)^(3/2)*Sqrt[Pi]*x*FresnelS[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2]) - ((I/b)^(3/2)*Sqrt[Pi]*x*FresnelC[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])","A",1
340,1,308,326,0.849478,"\int \frac{1}{\left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{5/2}} \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^(-5/2),x]","-\frac{\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)-i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)}{\sqrt{-i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)}{\sqrt{-i b} \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+\frac{b \sqrt{d x^2 \left(d x^2-2 i\right)}}{d x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{3/2}}+\frac{x}{\sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}}}{3 b^2}","-\frac{\sqrt{\pi } \sqrt{-i b} x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)}{3 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{\sqrt{\pi } x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{-i b} \sqrt{\pi }}\right)}{3 \sqrt{-i b} b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{x}{3 b^2 \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}}-\frac{\sqrt{d^2 x^4-2 i d x^2}}{3 b d x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{3/2}}",1,"-1/3*((b*Sqrt[d*x^2*(-2*I + d*x^2)])/(d*x*(a - I*b*ArcSin[1 + I*d*x^2])^(3/2)) + x/Sqrt[a - I*b*ArcSin[1 + I*d*x^2]] + (Sqrt[Pi]*x*FresnelC[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] - I*Sinh[a/(2*b)]))/(Sqrt[(-I)*b]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])) + (Sqrt[Pi]*x*FresnelS[Sqrt[a - I*b*ArcSin[1 + I*d*x^2]]/(Sqrt[(-I)*b]*Sqrt[Pi])]*(Cosh[a/(2*b)] + I*Sinh[a/(2*b)]))/(Sqrt[(-I)*b]*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])))/b^2","A",1
341,1,370,389,0.9811963,"\int \frac{1}{\left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{7/2}} \, dx","Integrate[(a - I*b*ArcSin[1 + I*d*x^2])^(-7/2),x]","\frac{\frac{\sqrt{\pi } \sqrt{\frac{i}{b}} x \left(\sinh \left(\frac{a}{2 b}\right)-i \cosh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+\frac{\sqrt{\pi } \sqrt{\frac{i}{b}} x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{b \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}+\frac{-\left(x^2 \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)\right)+\frac{\sqrt{d x^2 \left(d x^2-2 i\right)} \left(i a+b \sin ^{-1}\left(1+i d x^2\right)\right)^2}{b d}-\frac{3 b \sqrt{d x^2 \left(d x^2-2 i\right)}}{d}}{x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{5/2}}}{15 b^2}","-\frac{\sqrt{d^2 x^4-2 i d x^2}}{15 b^3 d x \sqrt{a-i b \sin ^{-1}\left(1+i d x^2\right)}}+\frac{\sqrt{\pi } \sqrt{\frac{i}{b}} x \left(\sinh \left(\frac{a}{2 b}\right)+i \cosh \left(\frac{a}{2 b}\right)\right) S\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{15 b^3 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{\sqrt{\pi } \left(\frac{i}{b}\right)^{3/2} x \left(\cosh \left(\frac{a}{2 b}\right)+i \sinh \left(\frac{a}{2 b}\right)\right) C\left(\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left(i d x^2+1\right)}}{\sqrt{\pi }}\right)}{15 b^2 \left(\cos \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)-\sin \left(\frac{1}{2} \sin ^{-1}\left(1+i d x^2\right)\right)\right)}-\frac{x}{15 b^2 \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{3/2}}-\frac{\sqrt{d^2 x^4-2 i d x^2}}{5 b d x \left(a-i b \sin ^{-1}\left(1+i d x^2\right)\right)^{5/2}}",1,"(((-3*b*Sqrt[d*x^2*(-2*I + d*x^2)])/d - x^2*(a - I*b*ArcSin[1 + I*d*x^2]) + (Sqrt[d*x^2*(-2*I + d*x^2)]*(I*a + b*ArcSin[1 + I*d*x^2])^2)/(b*d))/(x*(a - I*b*ArcSin[1 + I*d*x^2])^(5/2)) + (Sqrt[I/b]*Sqrt[Pi]*x*FresnelC[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*((-I)*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(b*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])) + (Sqrt[I/b]*Sqrt[Pi]*x*FresnelS[(Sqrt[I/b]*Sqrt[a - I*b*ArcSin[1 + I*d*x^2]])/Sqrt[Pi]]*(I*Cosh[a/(2*b)] + Sinh[a/(2*b)]))/(b*(Cos[ArcSin[1 + I*d*x^2]/2] - Sin[ArcSin[1 + I*d*x^2]/2])))/(15*b^2)","A",1
342,0,0,43,0.092677,"\int \frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2),x]","\int \frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^n}{1-c^2 x^2} \, dx","\text{Int}\left(\frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^n}{1-c^2 x^2},x\right)",0,"Integrate[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^n/(1 - c^2*x^2), x]","A",-1
343,1,244,261,0.0640709,"\int \frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^3}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]","\frac{6 b^2 \text{Li}_3\left(e^{2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)-6 b \text{Li}_2\left(e^{2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2+\frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^4}{b}-4 \log \left(1-e^{2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3-3 b^3 \text{Li}_4\left(e^{2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{4 c}","\frac{3 b^2 \text{Li}_3\left(e^{-2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{2 c}+\frac{3 b \text{Li}_2\left(e^{-2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 c}-\frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^4}{4 b c}-\frac{\log \left(1-e^{-2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{c}+\frac{3 b^3 \text{Li}_4\left(e^{-2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{4 c}",1,"((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4/b - 4*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3*Log[1 - E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])] - 6*b*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*PolyLog[2, E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])] + 6*b^2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[3, E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])] - 3*b^3*PolyLog[4, E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(4*c)","A",0
344,1,187,194,0.064455,"\int \frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]","\frac{-6 b^2 \text{Li}_2\left(e^{2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)+2 \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2 \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)-3 b \log \left(1-e^{2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)\right)+3 b^3 \text{Li}_3\left(e^{2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{6 b c}","\frac{b \text{Li}_2\left(e^{-2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}-\frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^3}{3 b c}-\frac{\log \left(1-e^{-2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{c}+\frac{b^2 \text{Li}_3\left(e^{-2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{2 c}",1,"(2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]] - 3*b*Log[1 - E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])]) - 6*b^2*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*PolyLog[2, E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])] + 3*b^3*PolyLog[3, E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(6*b*c)","A",0
345,1,127,133,0.0302381,"\int \frac{a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)}{1-c^2 x^2} \, dx","Integrate[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]","\frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)-2 b \log \left(1-e^{2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)\right)-b^2 \text{Li}_2\left(e^{2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{2 b c}","-\frac{\left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2}{2 b c}-\frac{\log \left(1-e^{-2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)}{c}+\frac{b \text{Li}_2\left(e^{-2 \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)}\right)}{2 c}",1,"((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]] - 2*b*Log[1 - E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])]) - b^2*PolyLog[2, E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*b*c)","A",0
346,0,0,43,0.0997195,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)},x\right)",0,"Integrate[1/((1 - c^2*x^2)*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])), x]","A",-1
347,0,0,43,0.847743,"\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","Integrate[1/((1 - c^2*x^2)*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2),x]","\int \frac{1}{\left(1-c^2 x^2\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{\left(1-c^2 x^2\right) \left(a+b \sinh ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right)\right)^2},x\right)",0,"Integrate[1/((1 - c^2*x^2)*(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2), x]","A",-1
348,1,76,76,0.5279085,"\int \sinh ^{-1}\left(c e^{a+b x}\right) \, dx","Integrate[ArcSinh[c*E^(a + b*x)],x]","\frac{\text{Li}_2\left(e^{2 \sinh ^{-1}\left(c e^{a+b x}\right)}\right)}{2 b}-\frac{\sinh ^{-1}\left(c e^{a+b x}\right)^2}{2 b}+\frac{\sinh ^{-1}\left(c e^{a+b x}\right) \log \left(1-e^{2 \sinh ^{-1}\left(c e^{a+b x}\right)}\right)}{b}","\frac{\text{Li}_2\left(e^{2 \sinh ^{-1}\left(c e^{a+b x}\right)}\right)}{2 b}-\frac{\sinh ^{-1}\left(c e^{a+b x}\right)^2}{2 b}+\frac{\sinh ^{-1}\left(c e^{a+b x}\right) \log \left(1-e^{2 \sinh ^{-1}\left(c e^{a+b x}\right)}\right)}{b}",1,"-1/2*ArcSinh[c*E^(a + b*x)]^2/b + (ArcSinh[c*E^(a + b*x)]*Log[1 - E^(2*ArcSinh[c*E^(a + b*x)])])/b + PolyLog[2, E^(2*ArcSinh[c*E^(a + b*x)])]/(2*b)","A",1
349,1,119,165,0.0891745,"\int e^{\sinh ^{-1}(a+b x)} x^3 \, dx","Integrate[E^ArcSinh[a + b*x]*x^3,x]","\frac{15 a \left(3-4 a^2\right) \sinh ^{-1}(a+b x)-\sqrt{a^2+2 a b x+b^2 x^2+1} \left(6 a^4+2 \left(3 a^2-4\right) b^2 x^2+\left(29-6 a^2\right) a b x-83 a^2-6 a b^3 x^3-24 b^4 x^4+16\right)+30 a b^4 x^4+24 b^5 x^5}{120 b^4}","\frac{\left(3-4 a^2\right) a e^{2 \sinh ^{-1}(a+b x)}}{16 b^4}+\frac{\left(3-4 a^2\right) a \sinh ^{-1}(a+b x)}{8 b^4}-\frac{\left(1-6 a^2\right) e^{-\sinh ^{-1}(a+b x)}}{8 b^4}-\frac{\left(1-6 a^2\right) e^{3 \sinh ^{-1}(a+b x)}}{24 b^4}+\frac{3 a e^{-2 \sinh ^{-1}(a+b x)}}{16 b^4}-\frac{3 a e^{4 \sinh ^{-1}(a+b x)}}{32 b^4}+\frac{e^{-3 \sinh ^{-1}(a+b x)}}{48 b^4}+\frac{e^{5 \sinh ^{-1}(a+b x)}}{80 b^4}",1,"(30*a*b^4*x^4 + 24*b^5*x^5 - Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(16 - 83*a^2 + 6*a^4 + a*(29 - 6*a^2)*b*x + 2*(-4 + 3*a^2)*b^2*x^2 - 6*a*b^3*x^3 - 24*b^4*x^4) + 15*a*(3 - 4*a^2)*ArcSinh[a + b*x])/(120*b^4)","A",0
350,1,102,115,0.1179992,"\int e^{\sinh ^{-1}(a+b x)} x^2 \, dx","Integrate[E^ArcSinh[a + b*x]*x^2,x]","\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left(2 a^3-2 a^2 b x+a \left(2 b^2 x^2-13\right)+6 b^3 x^3+3 b x\right)+8 a b^3 x^3+3 (2 a-1) (2 a+1) \sinh ^{-1}(a+b x)+6 b^4 x^4}{24 b^3}","-\frac{\left(1-4 a^2\right) e^{2 \sinh ^{-1}(a+b x)}}{16 b^3}-\frac{\left(1-4 a^2\right) \sinh ^{-1}(a+b x)}{8 b^3}-\frac{a e^{-\sinh ^{-1}(a+b x)}}{2 b^3}-\frac{a e^{3 \sinh ^{-1}(a+b x)}}{6 b^3}-\frac{e^{-2 \sinh ^{-1}(a+b x)}}{16 b^3}+\frac{e^{4 \sinh ^{-1}(a+b x)}}{32 b^3}",1,"(8*a*b^3*x^3 + 6*b^4*x^4 + Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(2*a^3 + 3*b*x - 2*a^2*b*x + 6*b^3*x^3 + a*(-13 + 2*b^2*x^2)) + 3*(-1 + 2*a)*(1 + 2*a)*ArcSinh[a + b*x])/(24*b^3)","A",0
351,1,73,67,0.1080161,"\int e^{\sinh ^{-1}(a+b x)} x \, dx","Integrate[E^ArcSinh[a + b*x]*x,x]","\frac{1}{6} \left(\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left(-a^2+a b x+2 b^2 x^2+2\right)}{b^2}-\frac{3 a \sinh ^{-1}(a+b x)}{b^2}+3 a x^2+2 b x^3\right)","-\frac{a e^{2 \sinh ^{-1}(a+b x)}}{4 b^2}-\frac{a \sinh ^{-1}(a+b x)}{2 b^2}+\frac{e^{-\sinh ^{-1}(a+b x)}}{4 b^2}+\frac{e^{3 \sinh ^{-1}(a+b x)}}{12 b^2}",1,"(3*a*x^2 + 2*b*x^3 + (Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(2 - a^2 + a*b*x + 2*b^2*x^2))/b^2 - (3*a*ArcSinh[a + b*x])/b^2)/6","A",0
352,1,46,31,0.0360613,"\int e^{\sinh ^{-1}(a+b x)} \, dx","Integrate[E^ArcSinh[a + b*x],x]","\frac{(a+b x) \left(\sqrt{a^2+2 a b x+b^2 x^2+1}+a+b x\right)+\sinh ^{-1}(a+b x)}{2 b}","\frac{\sinh ^{-1}(a+b x)}{2 b}+\frac{e^{2 \sinh ^{-1}(a+b x)}}{4 b}",1,"((a + b*x)*(a + b*x + Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) + ArcSinh[a + b*x])/(2*b)","A",0
353,1,99,89,0.0790953,"\int \frac{e^{\sinh ^{-1}(a+b x)}}{x} \, dx","Integrate[E^ArcSinh[a + b*x]/x,x]","\sqrt{a^2+2 a b x+b^2 x^2+1}-\sqrt{a^2+1} \log \left(\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right)+\left(\sqrt{a^2+1}+a\right) \log (x)+a \sinh ^{-1}(a+b x)+b x","\sqrt{a^2+2 a b x+b^2 x^2+1}-\sqrt{a^2+1} \tanh ^{-1}\left(\frac{a^2+a b x+1}{\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}}\right)+a \sinh ^{-1}(a+b x)+a \log (x)+b x",1,"b*x + Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + a*ArcSinh[a + b*x] + (a + Sqrt[1 + a^2])*Log[x] - Sqrt[1 + a^2]*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]]","A",0
354,1,110,99,0.1710518,"\int \frac{e^{\sinh ^{-1}(a+b x)}}{x^2} \, dx","Integrate[E^ArcSinh[a + b*x]/x^2,x]","b \sinh ^{-1}(a+b x)-\frac{\sqrt{a^2+2 a b x+b^2 x^2+1}+\frac{a b x \log \left(\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right)}{\sqrt{a^2+1}}+\left(-\frac{a}{\sqrt{a^2+1}}-1\right) b x \log (x)+a}{x}","-\frac{\sqrt{a^2+2 a b x+b^2 x^2+1}}{x}-\frac{a b \tanh ^{-1}\left(\frac{a^2+a b x+1}{\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}}\right)}{\sqrt{a^2+1}}+b \sinh ^{-1}(a+b x)-\frac{a}{x}+b \log (x)",1,"b*ArcSinh[a + b*x] - (a + Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + (-1 - a/Sqrt[1 + a^2])*b*x*Log[x] + (a*b*x*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]])/Sqrt[1 + a^2])/x","A",0
355,1,129,116,0.2166798,"\int \frac{e^{\sinh ^{-1}(a+b x)}}{x^3} \, dx","Integrate[E^ArcSinh[a + b*x]/x^3,x]","\frac{1}{2} \left(-\frac{\left(a^2+a b x+1\right) \sqrt{a^2+2 a b x+b^2 x^2+1}}{\left(a^2+1\right) x^2}-\frac{b^2 \log \left(\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right)}{\left(a^2+1\right)^{3/2}}+\frac{b^2 \log (x)}{\left(a^2+1\right)^{3/2}}-\frac{a}{x^2}-\frac{2 b}{x}\right)","-\frac{\left(a^2+a b x+1\right) \sqrt{a^2+2 a b x+b^2 x^2+1}}{2 \left(a^2+1\right) x^2}-\frac{b^2 \tanh ^{-1}\left(\frac{a^2+a b x+1}{\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}}\right)}{2 \left(a^2+1\right)^{3/2}}-\frac{a}{2 x^2}-\frac{b}{x}",1,"(-(a/x^2) - (2*b)/x - ((1 + a^2 + a*b*x)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2])/((1 + a^2)*x^2) + (b^2*Log[x])/(1 + a^2)^(3/2) - (b^2*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]])/(1 + a^2)^(3/2))/2","A",0
356,1,162,156,0.1220268,"\int \frac{e^{\sinh ^{-1}(a+b x)}}{x^4} \, dx","Integrate[E^ArcSinh[a + b*x]/x^4,x]","\frac{1}{6} \left(-\frac{3 a b^3 \log (x)}{\left(a^2+1\right)^{5/2}}+\frac{3 a b^3 \log \left(\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right)}{\left(a^2+1\right)^{5/2}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left(2 a^4+a^3 b x+a^2 \left(4-b^2 x^2\right)+a b x+2 b^2 x^2+2\right)}{\left(a^2+1\right)^2 x^3}-\frac{2 a}{x^3}-\frac{3 b}{x^2}\right)","\frac{a b \left(a^2+a b x+1\right) \sqrt{a^2+2 a b x+b^2 x^2+1}}{2 \left(a^2+1\right)^2 x^2}-\frac{\left(a^2+2 a b x+b^2 x^2+1\right)^{3/2}}{3 \left(a^2+1\right) x^3}+\frac{a b^3 \tanh ^{-1}\left(\frac{a^2+a b x+1}{\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}}\right)}{2 \left(a^2+1\right)^{5/2}}-\frac{a}{3 x^3}-\frac{b}{2 x^2}",1,"((-2*a)/x^3 - (3*b)/x^2 - (Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(2 + 2*a^4 + a*b*x + a^3*b*x + 2*b^2*x^2 + a^2*(4 - b^2*x^2)))/((1 + a^2)^2*x^3) - (3*a*b^3*Log[x])/(1 + a^2)^(5/2) + (3*a*b^3*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]])/(1 + a^2)^(5/2))/6","A",0
357,1,192,207,0.808771,"\int \frac{e^{\sinh ^{-1}(a+b x)}}{x^5} \, dx","Integrate[E^ArcSinh[a + b*x]/x^5,x]","\frac{1}{24} \left(\frac{3 (2 a-1) (2 a+1) b^4 \log (x)}{\left(a^2+1\right)^{7/2}}-\frac{3 (2 a-1) (2 a+1) b^4 \log \left(\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}+a^2+a b x+1\right)}{\left(a^2+1\right)^{7/2}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left(\frac{a \left(2 a^2-13\right) b^3 x^3}{\left(a^2+1\right)^3}-\frac{\left(2 a^2-3\right) b^2 x^2}{\left(a^2+1\right)^2}+\frac{2 a b x}{a^2+1}+6\right)}{x^4}-\frac{6 a}{x^4}-\frac{8 b}{x^3}\right)","\frac{\left(1-4 a^2\right) b^2 \left(a^2+a b x+1\right) \sqrt{a^2+2 a b x+b^2 x^2+1}}{8 \left(a^2+1\right)^3 x^2}-\frac{\left(a^2+2 a b x+b^2 x^2+1\right)^{3/2}}{4 \left(a^2+1\right) x^4}+\frac{5 a b \left(a^2+2 a b x+b^2 x^2+1\right)^{3/2}}{12 \left(a^2+1\right)^2 x^3}+\frac{\left(1-4 a^2\right) b^4 \tanh ^{-1}\left(\frac{a^2+a b x+1}{\sqrt{a^2+1} \sqrt{a^2+2 a b x+b^2 x^2+1}}\right)}{8 \left(a^2+1\right)^{7/2}}-\frac{a}{4 x^4}-\frac{b}{3 x^3}",1,"((-6*a)/x^4 - (8*b)/x^3 - (Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(6 + (2*a*b*x)/(1 + a^2) - ((-3 + 2*a^2)*b^2*x^2)/(1 + a^2)^2 + (a*(-13 + 2*a^2)*b^3*x^3)/(1 + a^2)^3))/x^4 + (3*(-1 + 2*a)*(1 + 2*a)*b^4*Log[x])/(1 + a^2)^(7/2) - (3*(-1 + 2*a)*(1 + 2*a)*b^4*Log[1 + a^2 + a*b*x + Sqrt[1 + a^2]*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]])/(1 + a^2)^(7/2))/24","A",0
358,1,198,359,0.3327594,"\int e^{\sinh ^{-1}(a+b x)^2} x^3 \, dx","Integrate[E^ArcSinh[a + b*x]^2*x^3,x]","\frac{\sqrt{\pi } \left(-8 e^{15/4} a^3 \text{erfi}\left(\sinh ^{-1}(a+b x)+\frac{1}{2}\right)+12 e^3 a^2 \text{erfi}\left(\sinh ^{-1}(a+b x)+1\right)+2 e^{15/4} \left(4 a^2-3\right) a \text{erfi}\left(\frac{1}{2}-\sinh ^{-1}(a+b x)\right)+2 e^3 \left(6 a^2-1\right) \text{erfi}\left(1-\sinh ^{-1}(a+b x)\right)+6 e^{7/4} a \text{erfi}\left(\frac{3}{2}-\sinh ^{-1}(a+b x)\right)+6 e^{15/4} a \text{erfi}\left(\sinh ^{-1}(a+b x)+\frac{1}{2}\right)-6 e^{7/4} a \text{erfi}\left(\sinh ^{-1}(a+b x)+\frac{3}{2}\right)+\text{erfi}\left(2-\sinh ^{-1}(a+b x)\right)-2 e^3 \text{erfi}\left(\sinh ^{-1}(a+b x)+1\right)+\text{erfi}\left(\sinh ^{-1}(a+b x)+2\right)\right)}{32 e^4 b^4}","-\frac{\sqrt{\pi } a^3 \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)-1\right)\right)}{4 \sqrt[4]{e} b^4}-\frac{\sqrt{\pi } a^3 \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)+1\right)\right)}{4 \sqrt[4]{e} b^4}+\frac{3 \sqrt{\pi } a^2 \text{erfi}\left(1-\sinh ^{-1}(a+b x)\right)}{8 e b^4}+\frac{3 \sqrt{\pi } a^2 \text{erfi}\left(\sinh ^{-1}(a+b x)+1\right)}{8 e b^4}-\frac{3 \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)-3\right)\right)}{16 e^{9/4} b^4}+\frac{3 \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)-1\right)\right)}{16 \sqrt[4]{e} b^4}+\frac{3 \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)+1\right)\right)}{16 \sqrt[4]{e} b^4}-\frac{3 \sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)+3\right)\right)}{16 e^{9/4} b^4}-\frac{\sqrt{\pi } \text{erfi}\left(1-\sinh ^{-1}(a+b x)\right)}{16 e b^4}+\frac{\sqrt{\pi } \text{erfi}\left(2-\sinh ^{-1}(a+b x)\right)}{32 e^4 b^4}-\frac{\sqrt{\pi } \text{erfi}\left(\sinh ^{-1}(a+b x)+1\right)}{16 e b^4}+\frac{\sqrt{\pi } \text{erfi}\left(\sinh ^{-1}(a+b x)+2\right)}{32 e^4 b^4}",1,"(Sqrt[Pi]*(2*a*(-3 + 4*a^2)*E^(15/4)*Erfi[1/2 - ArcSinh[a + b*x]] + 2*(-1 + 6*a^2)*E^3*Erfi[1 - ArcSinh[a + b*x]] + 6*a*E^(7/4)*Erfi[3/2 - ArcSinh[a + b*x]] + Erfi[2 - ArcSinh[a + b*x]] + 6*a*E^(15/4)*Erfi[1/2 + ArcSinh[a + b*x]] - 8*a^3*E^(15/4)*Erfi[1/2 + ArcSinh[a + b*x]] - 2*E^3*Erfi[1 + ArcSinh[a + b*x]] + 12*a^2*E^3*Erfi[1 + ArcSinh[a + b*x]] - 6*a*E^(7/4)*Erfi[3/2 + ArcSinh[a + b*x]] + Erfi[2 + ArcSinh[a + b*x]]))/(32*b^4*E^4)","A",1
359,1,138,251,0.2050867,"\int e^{\sinh ^{-1}(a+b x)^2} x^2 \, dx","Integrate[E^ArcSinh[a + b*x]^2*x^2,x]","-\frac{\sqrt{\pi } \left(-4 e^2 a^2 \text{erfi}\left(\sinh ^{-1}(a+b x)+\frac{1}{2}\right)+e^2 \left(4 a^2-1\right) \text{erfi}\left(\frac{1}{2}-\sinh ^{-1}(a+b x)\right)+4 e^{5/4} a \text{erfi}\left(1-\sinh ^{-1}(a+b x)\right)+4 e^{5/4} a \text{erfi}\left(\sinh ^{-1}(a+b x)+1\right)+\text{erfi}\left(\frac{3}{2}-\sinh ^{-1}(a+b x)\right)+e^2 \text{erfi}\left(\sinh ^{-1}(a+b x)+\frac{1}{2}\right)-\text{erfi}\left(\sinh ^{-1}(a+b x)+\frac{3}{2}\right)\right)}{16 e^{9/4} b^3}","\frac{\sqrt{\pi } a^2 \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)-1\right)\right)}{4 \sqrt[4]{e} b^3}+\frac{\sqrt{\pi } a^2 \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)+1\right)\right)}{4 \sqrt[4]{e} b^3}-\frac{\sqrt{\pi } a \text{erfi}\left(1-\sinh ^{-1}(a+b x)\right)}{4 e b^3}-\frac{\sqrt{\pi } a \text{erfi}\left(\sinh ^{-1}(a+b x)+1\right)}{4 e b^3}+\frac{\sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)-3\right)\right)}{16 e^{9/4} b^3}-\frac{\sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)-1\right)\right)}{16 \sqrt[4]{e} b^3}-\frac{\sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)+1\right)\right)}{16 \sqrt[4]{e} b^3}+\frac{\sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)+3\right)\right)}{16 e^{9/4} b^3}",1,"-1/16*(Sqrt[Pi]*((-1 + 4*a^2)*E^2*Erfi[1/2 - ArcSinh[a + b*x]] + 4*a*E^(5/4)*Erfi[1 - ArcSinh[a + b*x]] + Erfi[3/2 - ArcSinh[a + b*x]] + E^2*Erfi[1/2 + ArcSinh[a + b*x]] - 4*a^2*E^2*Erfi[1/2 + ArcSinh[a + b*x]] + 4*a*E^(5/4)*Erfi[1 + ArcSinh[a + b*x]] - Erfi[3/2 + ArcSinh[a + b*x]]))/(b^3*E^(9/4))","A",1
360,1,76,117,0.1079713,"\int e^{\sinh ^{-1}(a+b x)^2} x \, dx","Integrate[E^ArcSinh[a + b*x]^2*x,x]","\frac{\sqrt{\pi } \left(2 e^{3/4} a \text{erfi}\left(\frac{1}{2}-\sinh ^{-1}(a+b x)\right)+\text{erfi}\left(1-\sinh ^{-1}(a+b x)\right)-2 e^{3/4} a \text{erfi}\left(\sinh ^{-1}(a+b x)+\frac{1}{2}\right)+\text{erfi}\left(\sinh ^{-1}(a+b x)+1\right)\right)}{8 e b^2}","\frac{\sqrt{\pi } \text{erfi}\left(1-\sinh ^{-1}(a+b x)\right)}{8 e b^2}+\frac{\sqrt{\pi } \text{erfi}\left(\sinh ^{-1}(a+b x)+1\right)}{8 e b^2}-\frac{\sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)-1\right)\right)}{4 \sqrt[4]{e} b^2}-\frac{\sqrt{\pi } a \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)+1\right)\right)}{4 \sqrt[4]{e} b^2}",1,"(Sqrt[Pi]*(2*a*E^(3/4)*Erfi[1/2 - ArcSinh[a + b*x]] + Erfi[1 - ArcSinh[a + b*x]] - 2*a*E^(3/4)*Erfi[1/2 + ArcSinh[a + b*x]] + Erfi[1 + ArcSinh[a + b*x]]))/(8*b^2*E)","A",1
361,1,44,65,0.0385151,"\int e^{\sinh ^{-1}(a+b x)^2} \, dx","Integrate[E^ArcSinh[a + b*x]^2,x]","\frac{\sqrt{\pi } \left(\text{erfi}\left(\sinh ^{-1}(a+b x)+\frac{1}{2}\right)+\text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)-1\right)\right)\right)}{4 \sqrt[4]{e} b}","\frac{\sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)-1\right)\right)}{4 \sqrt[4]{e} b}+\frac{\sqrt{\pi } \text{erfi}\left(\frac{1}{2} \left(2 \sinh ^{-1}(a+b x)+1\right)\right)}{4 \sqrt[4]{e} b}",1,"(Sqrt[Pi]*(Erfi[1/2 + ArcSinh[a + b*x]] + Erfi[(-1 + 2*ArcSinh[a + b*x])/2]))/(4*b*E^(1/4))","A",1
362,0,0,17,0.1419417,"\int \frac{e^{\sinh ^{-1}(a+b x)^2}}{x} \, dx","Integrate[E^ArcSinh[a + b*x]^2/x,x]","\int \frac{e^{\sinh ^{-1}(a+b x)^2}}{x} \, dx","\text{Int}\left(\frac{e^{\sinh ^{-1}(a+b x)^2}}{x},x\right)",0,"Integrate[E^ArcSinh[a + b*x]^2/x, x]","A",-1
363,0,0,17,0.4740814,"\int \frac{e^{\sinh ^{-1}(a+b x)^2}}{x^2} \, dx","Integrate[E^ArcSinh[a + b*x]^2/x^2,x]","\int \frac{e^{\sinh ^{-1}(a+b x)^2}}{x^2} \, dx","\text{Int}\left(\frac{e^{\sinh ^{-1}(a+b x)^2}}{x^2},x\right)",0,"Integrate[E^ArcSinh[a + b*x]^2/x^2, x]","A",-1
364,1,52,60,0.018276,"\int \frac{\sinh ^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx","Integrate[ArcSinh[a + b*x]/((a*d)/b + d*x),x]","\frac{\text{Li}_2\left(e^{2 \sinh ^{-1}(a+b x)}\right)-\sinh ^{-1}(a+b x) \left(\sinh ^{-1}(a+b x)-2 \log \left(1-e^{2 \sinh ^{-1}(a+b x)}\right)\right)}{2 d}","\frac{\text{Li}_2\left(e^{2 \sinh ^{-1}(a+b x)}\right)}{2 d}-\frac{\sinh ^{-1}(a+b x)^2}{2 d}+\frac{\sinh ^{-1}(a+b x) \log \left(1-e^{2 \sinh ^{-1}(a+b x)}\right)}{d}",1,"(-(ArcSinh[a + b*x]*(ArcSinh[a + b*x] - 2*Log[1 - E^(2*ArcSinh[a + b*x])])) + PolyLog[2, E^(2*ArcSinh[a + b*x])])/(2*d)","A",1
365,1,3,3,0.0642199,"\int \frac{x}{\sqrt{1+x^2} \sinh ^{-1}(x)} \, dx","Integrate[x/(Sqrt[1 + x^2]*ArcSinh[x]),x]","\text{Shi}\left(\sinh ^{-1}(x)\right)","\text{Shi}\left(\sinh ^{-1}(x)\right)",1,"SinhIntegral[ArcSinh[x]]","A",1
366,1,41,45,0.0268117,"\int x^3 \sinh ^{-1}\left(a+b x^4\right) \, dx","Integrate[x^3*ArcSinh[a + b*x^4],x]","\frac{\left(a+b x^4\right) \sinh ^{-1}\left(a+b x^4\right)-\sqrt{\left(a+b x^4\right)^2+1}}{4 b}","\frac{\left(a+b x^4\right) \sinh ^{-1}\left(a+b x^4\right)}{4 b}-\frac{\sqrt{\left(a+b x^4\right)^2+1}}{4 b}",1,"(-Sqrt[1 + (a + b*x^4)^2] + (a + b*x^4)*ArcSinh[a + b*x^4])/(4*b)","A",1
367,1,41,46,0.0417135,"\int x^{-1+n} \sinh ^{-1}\left(a+b x^n\right) \, dx","Integrate[x^(-1 + n)*ArcSinh[a + b*x^n],x]","\frac{\left(a+b x^n\right) \sinh ^{-1}\left(a+b x^n\right)-\sqrt{\left(a+b x^n\right)^2+1}}{b n}","\frac{\left(a+b x^n\right) \sinh ^{-1}\left(a+b x^n\right)}{b n}-\frac{\sqrt{\left(a+b x^n\right)^2+1}}{b n}",1,"(-Sqrt[1 + (a + b*x^n)^2] + (a + b*x^n)*ArcSinh[a + b*x^n])/(b*n)","A",1
368,1,131,49,0.1282713,"\int \sinh ^{-1}\left(\frac{c}{a+b x}\right) \, dx","Integrate[ArcSinh[c/(a + b*x)],x]","\frac{(a+b x) \sqrt{\frac{a^2+2 a b x+b^2 x^2+c^2}{(a+b x)^2}} \left(c \tanh ^{-1}\left(\frac{a+b x}{\sqrt{a^2+2 a b x+b^2 x^2+c^2}}\right)+a \tanh ^{-1}\left(\frac{\sqrt{(a+b x)^2+c^2}}{c}\right)\right)}{b \sqrt{a^2+2 a b x+b^2 x^2+c^2}}+x \sinh ^{-1}\left(\frac{c}{a+b x}\right)","\frac{c \tanh ^{-1}\left(\sqrt{\frac{1}{\left(\frac{a}{c}+\frac{b x}{c}\right)^2}+1}\right)}{b}+\frac{(a+b x) \text{csch}^{-1}\left(\frac{a}{c}+\frac{b x}{c}\right)}{b}",1,"x*ArcSinh[c/(a + b*x)] + ((a + b*x)*Sqrt[(a^2 + c^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(c*ArcTanh[(a + b*x)/Sqrt[a^2 + c^2 + 2*a*b*x + b^2*x^2]] + a*ArcTanh[Sqrt[c^2 + (a + b*x)^2]/c]))/(b*Sqrt[a^2 + c^2 + 2*a*b*x + b^2*x^2])","B",1
369,1,28,27,0.5584615,"\int \frac{x}{\sinh ^{-1}(\sinh (x))} \, dx","Integrate[x/ArcSinh[Sinh[x]],x]","x \sqrt{\cosh ^2(x)} \text{sech}(x) \log \left(\sinh ^{-1}(\sinh (x))\right)-\sinh ^{-1}(\sinh (x)) \left(\log \left(\sinh ^{-1}(\sinh (x))\right)-1\right)","\sinh ^{-1}(\sinh (x))+\log \left(\sinh ^{-1}(\sinh (x))\right) \left(x \sqrt{\cosh ^2(x)} \text{sech}(x)-\sinh ^{-1}(\sinh (x))\right)",1,"-(ArcSinh[Sinh[x]]*(-1 + Log[ArcSinh[Sinh[x]]])) + x*Sqrt[Cosh[x]^2]*Log[ArcSinh[Sinh[x]]]*Sech[x]","A",1
370,1,37,37,0.0442324,"\int \frac{\sinh ^{-1}\left(\sqrt{-1+b x^2}\right)^n}{\sqrt{-1+b x^2}} \, dx","Integrate[ArcSinh[Sqrt[-1 + b*x^2]]^n/Sqrt[-1 + b*x^2],x]","\frac{\sqrt{b x^2} \sinh ^{-1}\left(\sqrt{b x^2-1}\right)^{n+1}}{b (n+1) x}","\frac{\sqrt{b x^2} \sinh ^{-1}\left(\sqrt{b x^2-1}\right)^{n+1}}{b (n+1) x}",1,"(Sqrt[b*x^2]*ArcSinh[Sqrt[-1 + b*x^2]]^(1 + n))/(b*(1 + n)*x)","A",1
371,1,24,29,0.0237753,"\int \frac{1}{\sqrt{-1+b x^2} \sinh ^{-1}\left(\sqrt{-1+b x^2}\right)} \, dx","Integrate[1/(Sqrt[-1 + b*x^2]*ArcSinh[Sqrt[-1 + b*x^2]]),x]","\frac{x \log \left(\sinh ^{-1}\left(\sqrt{b x^2-1}\right)\right)}{\sqrt{b x^2}}","\frac{\sqrt{b x^2} \log \left(\sinh ^{-1}\left(\sqrt{b x^2-1}\right)\right)}{b x}",1,"(x*Log[ArcSinh[Sqrt[-1 + b*x^2]]])/Sqrt[b*x^2]","A",1