1,1,77,22,0.0692074,"\int \frac{1}{\sqrt{2}+\cos (z)+\sin (z)} \, dz","Integrate[(Sqrt[2] + Cos[z] + Sin[z])^(-1),z]","\frac{\left((1+i)-i \sqrt{2}\right) \sin \left(\frac{z}{2}\right)-\left(\sqrt{2}+(1+3 i)\right) \cos \left(\frac{z}{2}\right)}{i \left(\sqrt{2}+(-1-i)\right) \sin \left(\frac{z}{2}\right)+\left(\sqrt{2}+(1+i)\right) \cos \left(\frac{z}{2}\right)}","-\frac{1-\sqrt{2} \sin (z)}{\cos (z)-\sin (z)}",1,"(-(((1 + 3*I) + Sqrt[2])*Cos[z/2]) + ((1 + I) - I*Sqrt[2])*Sin[z/2])/(((1 + I) + Sqrt[2])*Cos[z/2] + I*((-1 - I) + Sqrt[2])*Sin[z/2])","C",1
2,1,24,32,0.0243696,"\int \frac{1}{\left(\sqrt{1-x}+\sqrt{1+x}\right)^2} \, dx","Integrate[(Sqrt[1 - x] + Sqrt[1 + x])^(-2),x]","\frac{\sqrt{1-x^2}+x \sin ^{-1}(x)-1}{2 x}","\frac{\sqrt{1-x^2}}{2 x}-\frac{1}{2 x}+\frac{1}{2} \sin ^{-1}(x)",1,"(-1 + Sqrt[1 - x^2] + x*ArcSin[x])/(2*x)","A",1
3,1,16,25,0.0120239,"\int \frac{1}{(1+\cos (x))^2} \, dx","Integrate[(1 + Cos[x])^(-2),x]","\frac{\sin (x) (\cos (x)+2)}{3 (\cos (x)+1)^2}","\frac{\sin (x)}{3 (\cos (x)+1)}+\frac{\sin (x)}{3 (\cos (x)+1)^2}",1,"((2 + Cos[x])*Sin[x])/(3*(1 + Cos[x])^2)","A",1
4,1,68,58,0.0220327,"\int \frac{\sin (x)}{\sqrt{1+x}} \, dx","Integrate[Sin[x]/Sqrt[1 + x],x]","-\frac{e^{-i} \left(\sqrt{-i (x+1)} \operatorname{Gamma}\left(\frac{1}{2},-i (x+1)\right)+e^{2 i} \sqrt{i (x+1)} \operatorname{Gamma}\left(\frac{1}{2},i (x+1)\right)\right)}{2 \sqrt{x+1}}","\sqrt{2 \pi } \cos (1) S\left(\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right)-\sqrt{2 \pi } \sin (1) \operatorname{FresnelC}\left(\sqrt{\frac{2}{\pi }} \sqrt{x+1}\right)",1,"-1/2*(Sqrt[(-I)*(1 + x)]*Gamma[1/2, (-I)*(1 + x)] + E^(2*I)*Sqrt[I*(1 + x)]*Gamma[1/2, I*(1 + x)])/(E^I*Sqrt[1 + x])","C",1
5,1,26,50,0.0439037,"\int \frac{1}{(\cos (x)+\sin (x))^6} \, dx","Integrate[(Cos[x] + Sin[x])^(-6),x]","-\frac{-10 \sin (x)+\sin (5 x)+5 \cos (3 x)}{30 (\sin (x)+\cos (x))^5}","-\frac{\cos (x)-\sin (x)}{15 (\sin (x)+\cos (x))^3}-\frac{\cos (x)-\sin (x)}{10 (\sin (x)+\cos (x))^5}+\frac{2 \sin (x)}{15 (\sin (x)+\cos (x))}",1,"-1/30*(5*Cos[3*x] - 10*Sin[x] + Sin[5*x])/(Cos[x] + Sin[x])^5","A",1
6,1,30,334,0.0055963,"\int \log \left(\frac{1}{x^4}+x^4\right) \, dx","Integrate[Log[x^(-4) + x^4],x]","8 x \, _2F_1\left(\frac{1}{8},1;\frac{9}{8};-x^8\right)+x \log \left(x^4+\frac{1}{x^4}\right)-4 x","x \log \left(x^4+\frac{1}{x^4}\right)-\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left(x^2-\sqrt{2-\sqrt{2}} x+1\right)+\frac{1}{2} \sqrt{2-\sqrt{2}} \log \left(x^2+\sqrt{2-\sqrt{2}} x+1\right)-\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left(x^2-\sqrt{2+\sqrt{2}} x+1\right)+\frac{1}{2} \sqrt{2+\sqrt{2}} \log \left(x^2+\sqrt{2+\sqrt{2}} x+1\right)-4 x-\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right)-\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{\sqrt{2+\sqrt{2}}-2 x}{\sqrt{2-\sqrt{2}}}\right)+\sqrt{2+\sqrt{2}} \tan ^{-1}\left(\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right)+\sqrt{2-\sqrt{2}} \tan ^{-1}\left(\frac{2 x+\sqrt{2+\sqrt{2}}}{\sqrt{2-\sqrt{2}}}\right)",1,"-4*x + 8*x*Hypergeometric2F1[1/8, 1, 9/8, -x^8] + x*Log[x^(-4) + x^4]","C",1
7,1,430,291,1.1101348,"\int \frac{\log (1+x)}{x \sqrt{1+\sqrt{1+x}}} \, dx","Integrate[Log[1 + x]/(x*Sqrt[1 + Sqrt[1 + x]]),x]","-\sqrt{2} \left(2 \operatorname{PolyLog}\left(2,\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}\right)-\operatorname{PolyLog}\left(2,-\left(\left(\sqrt{2}-2\right) \left(\frac{1}{\sqrt{\sqrt{x+1}+1}}+1\right)\right)\right)+\operatorname{PolyLog}\left(2,\left(1+\sqrt{2}\right) \left(\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}-1\right)\right)+\log \left(1+\sqrt{2}\right) \log \left(\frac{1}{\sqrt{\sqrt{x+1}+1}}+1\right)+\log \left(-\left(\left(2+\sqrt{2}\right) \left(\frac{1}{\sqrt{\sqrt{x+1}+1}}-1\right)\right)\right) \log \left(1-\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}\right)\right)+\sqrt{2} \left(2 \operatorname{PolyLog}\left(2,-\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}\right)-\operatorname{PolyLog}\left(2,\left(\sqrt{2}-2\right) \left(\frac{1}{\sqrt{\sqrt{x+1}+1}}-1\right)\right)+\operatorname{PolyLog}\left(2,-\left(\left(1+\sqrt{2}\right) \left(\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}+1\right)\right)\right)+\log \left(1+\sqrt{2}\right) \log \left(1-\frac{1}{\sqrt{\sqrt{x+1}+1}}\right)+\log \left(\left(2+\sqrt{2}\right) \left(\frac{1}{\sqrt{\sqrt{x+1}+1}}+1\right)\right) \log \left(\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}+1\right)\right)-\frac{2 \log (x+1)}{\sqrt{\sqrt{x+1}+1}}+\frac{\log (x+1) \log \left(1-\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}\right)}{\sqrt{2}}-\frac{\log (x+1) \log \left(\frac{\sqrt{2}}{\sqrt{\sqrt{x+1}+1}}+1\right)}{\sqrt{2}}-8 \tanh ^{-1}\left(\frac{1}{\sqrt{\sqrt{x+1}+1}}\right)","\sqrt{2} \operatorname{PolyLog}\left(2,-\frac{\sqrt{2} \left(1-\sqrt{\sqrt{x+1}+1}\right)}{2-\sqrt{2}}\right)-\sqrt{2} \operatorname{PolyLog}\left(2,\frac{\sqrt{2} \left(1-\sqrt{\sqrt{x+1}+1}\right)}{2+\sqrt{2}}\right)-\sqrt{2} \operatorname{PolyLog}\left(2,-\frac{\sqrt{2} \left(\sqrt{\sqrt{x+1}+1}+1\right)}{2-\sqrt{2}}\right)+\sqrt{2} \operatorname{PolyLog}\left(2,\frac{\sqrt{2} \left(\sqrt{\sqrt{x+1}+1}+1\right)}{2+\sqrt{2}}\right)-\frac{2 \log (x+1)}{\sqrt{\sqrt{x+1}+1}}-8 \tanh ^{-1}\left(\sqrt{\sqrt{x+1}+1}\right)-\sqrt{2} \log (x+1) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{x+1}+1}}{\sqrt{2}}\right)+2 \sqrt{2} \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right) \log \left(1-\sqrt{\sqrt{x+1}+1}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right) \log \left(\sqrt{\sqrt{x+1}+1}+1\right)",1,"-8*ArcTanh[1/Sqrt[1 + Sqrt[1 + x]]] - (2*Log[1 + x])/Sqrt[1 + Sqrt[1 + x]] + (Log[1 + x]*Log[1 - Sqrt[2]/Sqrt[1 + Sqrt[1 + x]]])/Sqrt[2] - (Log[1 + x]*Log[1 + Sqrt[2]/Sqrt[1 + Sqrt[1 + x]]])/Sqrt[2] - Sqrt[2]*(Log[1 + Sqrt[2]]*Log[1 + 1/Sqrt[1 + Sqrt[1 + x]]] + Log[-((2 + Sqrt[2])*(-1 + 1/Sqrt[1 + Sqrt[1 + x]]))]*Log[1 - Sqrt[2]/Sqrt[1 + Sqrt[1 + x]]] + 2*PolyLog[2, Sqrt[2]/Sqrt[1 + Sqrt[1 + x]]] - PolyLog[2, -((-2 + Sqrt[2])*(1 + 1/Sqrt[1 + Sqrt[1 + x]]))] + PolyLog[2, (1 + Sqrt[2])*(-1 + Sqrt[2]/Sqrt[1 + Sqrt[1 + x]])]) + Sqrt[2]*(Log[1 + Sqrt[2]]*Log[1 - 1/Sqrt[1 + Sqrt[1 + x]]] + Log[(2 + Sqrt[2])*(1 + 1/Sqrt[1 + Sqrt[1 + x]])]*Log[1 + Sqrt[2]/Sqrt[1 + Sqrt[1 + x]]] + 2*PolyLog[2, -(Sqrt[2]/Sqrt[1 + Sqrt[1 + x]])] - PolyLog[2, (-2 + Sqrt[2])*(-1 + 1/Sqrt[1 + Sqrt[1 + x]])] + PolyLog[2, -((1 + Sqrt[2])*(1 + Sqrt[2]/Sqrt[1 + Sqrt[1 + x]]))])","A",0
8,1,654,308,0.6004188,"\int \frac{\sqrt{1+\sqrt{1+x}} \log (1+x)}{x} \, dx","Integrate[(Sqrt[1 + Sqrt[1 + x]]*Log[1 + x])/x,x]","-2 \sqrt{2} \operatorname{PolyLog}\left(2,-\left(\left(\sqrt{2}-1\right) \left(\sqrt{\sqrt{x+1}+1}-1\right)\right)\right)+2 \sqrt{2} \operatorname{PolyLog}\left(2,\left(1+\sqrt{2}\right) \left(\sqrt{\sqrt{x+1}+1}-1\right)\right)+2 \sqrt{2} \operatorname{PolyLog}\left(2,\left(\sqrt{2}-1\right) \left(\sqrt{\sqrt{x+1}+1}+1\right)\right)-2 \sqrt{2} \operatorname{PolyLog}\left(2,-\left(\left(1+\sqrt{2}\right) \left(\sqrt{\sqrt{x+1}+1}+1\right)\right)\right)-16 \sqrt{\sqrt{x+1}+1}+\sqrt{2} \log \left(\sqrt{2}-\sqrt{\sqrt{x+1}+1}\right) \log (x+1)-\sqrt{2} \log \left(\sqrt{\sqrt{x+1}+1}+\sqrt{2}\right) \log (x+1)+4 \sqrt{\sqrt{x+1}+1} \log (x+1)-2 \sqrt{2} \log \left(\sqrt{2}-\sqrt{\sqrt{x+1}+1}\right) \log \left(\sqrt{\sqrt{x+1}+1}-1\right)-8 \log \left(\sqrt{\sqrt{x+1}+1}-1\right)-2 \sqrt{2} \log \left(\sqrt{2}-\sqrt{\sqrt{x+1}+1}\right) \log \left(\sqrt{\sqrt{x+1}+1}+1\right)+8 \log \left(\sqrt{\sqrt{x+1}+1}+1\right)+2 \sqrt{2} \log \left(\sqrt{\sqrt{x+1}+1}-1\right) \log \left(\sqrt{\sqrt{x+1}+1}+\sqrt{2}\right)+2 \sqrt{2} \log \left(\sqrt{\sqrt{x+1}+1}+1\right) \log \left(\sqrt{\sqrt{x+1}+1}+\sqrt{2}\right)-2 \sqrt{2} \log \left(\sqrt{\sqrt{x+1}+1}-1\right) \log \left(\left(\sqrt{2}-1\right) \left(\sqrt{\sqrt{x+1}+1}+\sqrt{2}\right)\right)-2 \sqrt{2} \log \left(\sqrt{\sqrt{x+1}+1}+1\right) \log \left(\sqrt{2} \sqrt{\sqrt{x+1}+1}+\sqrt{\sqrt{x+1}+1}+\sqrt{2}+2\right)+2 \sqrt{2} \log \left(\sqrt{\sqrt{x+1}+1}-1\right) \log \left(1-\left(1+\sqrt{2}\right) \left(\sqrt{\sqrt{x+1}+1}-1\right)\right)+2 \sqrt{2} \log \left(\sqrt{\sqrt{x+1}+1}+1\right) \log \left(1-\left(\sqrt{2}-1\right) \left(\sqrt{\sqrt{x+1}+1}+1\right)\right)","2 \sqrt{2} \operatorname{PolyLog}\left(2,-\frac{\sqrt{2} \left(1-\sqrt{\sqrt{x+1}+1}\right)}{2-\sqrt{2}}\right)-2 \sqrt{2} \operatorname{PolyLog}\left(2,\frac{\sqrt{2} \left(1-\sqrt{\sqrt{x+1}+1}\right)}{2+\sqrt{2}}\right)-2 \sqrt{2} \operatorname{PolyLog}\left(2,-\frac{\sqrt{2} \left(\sqrt{\sqrt{x+1}+1}+1\right)}{2-\sqrt{2}}\right)+2 \sqrt{2} \operatorname{PolyLog}\left(2,\frac{\sqrt{2} \left(\sqrt{\sqrt{x+1}+1}+1\right)}{2+\sqrt{2}}\right)-16 \sqrt{\sqrt{x+1}+1}+4 \sqrt{\sqrt{x+1}+1} \log (x+1)+16 \tanh ^{-1}\left(\sqrt{\sqrt{x+1}+1}\right)-2 \sqrt{2} \log (x+1) \tanh ^{-1}\left(\frac{\sqrt{\sqrt{x+1}+1}}{\sqrt{2}}\right)+4 \sqrt{2} \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right) \log \left(1-\sqrt{\sqrt{x+1}+1}\right)-4 \sqrt{2} \tanh ^{-1}\left(\frac{1}{\sqrt{2}}\right) \log \left(\sqrt{\sqrt{x+1}+1}+1\right)",1,"-16*Sqrt[1 + Sqrt[1 + x]] + 4*Sqrt[1 + Sqrt[1 + x]]*Log[1 + x] + Sqrt[2]*Log[1 + x]*Log[Sqrt[2] - Sqrt[1 + Sqrt[1 + x]]] - 8*Log[-1 + Sqrt[1 + Sqrt[1 + x]]] - 2*Sqrt[2]*Log[Sqrt[2] - Sqrt[1 + Sqrt[1 + x]]]*Log[-1 + Sqrt[1 + Sqrt[1 + x]]] + 8*Log[1 + Sqrt[1 + Sqrt[1 + x]]] - 2*Sqrt[2]*Log[Sqrt[2] - Sqrt[1 + Sqrt[1 + x]]]*Log[1 + Sqrt[1 + Sqrt[1 + x]]] - Sqrt[2]*Log[1 + x]*Log[Sqrt[2] + Sqrt[1 + Sqrt[1 + x]]] + 2*Sqrt[2]*Log[-1 + Sqrt[1 + Sqrt[1 + x]]]*Log[Sqrt[2] + Sqrt[1 + Sqrt[1 + x]]] + 2*Sqrt[2]*Log[1 + Sqrt[1 + Sqrt[1 + x]]]*Log[Sqrt[2] + Sqrt[1 + Sqrt[1 + x]]] - 2*Sqrt[2]*Log[-1 + Sqrt[1 + Sqrt[1 + x]]]*Log[(-1 + Sqrt[2])*(Sqrt[2] + Sqrt[1 + Sqrt[1 + x]])] - 2*Sqrt[2]*Log[1 + Sqrt[1 + Sqrt[1 + x]]]*Log[2 + Sqrt[2] + Sqrt[1 + Sqrt[1 + x]] + Sqrt[2]*Sqrt[1 + Sqrt[1 + x]]] + 2*Sqrt[2]*Log[-1 + Sqrt[1 + Sqrt[1 + x]]]*Log[1 - (1 + Sqrt[2])*(-1 + Sqrt[1 + Sqrt[1 + x]])] + 2*Sqrt[2]*Log[1 + Sqrt[1 + Sqrt[1 + x]]]*Log[1 - (-1 + Sqrt[2])*(1 + Sqrt[1 + Sqrt[1 + x]])] - 2*Sqrt[2]*PolyLog[2, -((-1 + Sqrt[2])*(-1 + Sqrt[1 + Sqrt[1 + x]]))] + 2*Sqrt[2]*PolyLog[2, (1 + Sqrt[2])*(-1 + Sqrt[1 + Sqrt[1 + x]])] + 2*Sqrt[2]*PolyLog[2, (-1 + Sqrt[2])*(1 + Sqrt[1 + Sqrt[1 + x]])] - 2*Sqrt[2]*PolyLog[2, -((1 + Sqrt[2])*(1 + Sqrt[1 + Sqrt[1 + x]]))]","B",1
9,1,84,84,0.0464773,"\int \frac{1}{1+\sqrt{x+\sqrt{1+x^2}}} \, dx","Integrate[(1 + Sqrt[x + Sqrt[1 + x^2]])^(-1),x]","\sqrt{\sqrt{x^2+1}+x}+\frac{1}{\sqrt{\sqrt{x^2+1}+x}}-\frac{1}{2 \left(\sqrt{x^2+1}+x\right)}+\frac{1}{2} \log \left(\sqrt{x^2+1}+x\right)-2 \log \left(\sqrt{\sqrt{x^2+1}+x}+1\right)","\sqrt{\sqrt{x^2+1}+x}+\frac{1}{\sqrt{\sqrt{x^2+1}+x}}-\frac{1}{2 \left(\sqrt{x^2+1}+x\right)}+\frac{1}{2} \log \left(\sqrt{x^2+1}+x\right)-2 \log \left(\sqrt{\sqrt{x^2+1}+x}+1\right)",1,"-1/2*1/(x + Sqrt[1 + x^2]) + 1/Sqrt[x + Sqrt[1 + x^2]] + Sqrt[x + Sqrt[1 + x^2]] + Log[x + Sqrt[1 + x^2]]/2 - 2*Log[1 + Sqrt[x + Sqrt[1 + x^2]]]","A",1
10,1,41,41,0.0661711,"\int \frac{\sqrt{1+x}}{x+\sqrt{1+\sqrt{1+x}}} \, dx","Integrate[Sqrt[1 + x]/(x + Sqrt[1 + Sqrt[1 + x]]),x]","2 \sqrt{x+1}+\frac{8 \tanh ^{-1}\left(\frac{2 \sqrt{\sqrt{x+1}+1}+1}{\sqrt{5}}\right)}{\sqrt{5}}","2 \sqrt{x+1}+\frac{8 \tanh ^{-1}\left(\frac{2 \sqrt{\sqrt{x+1}+1}+1}{\sqrt{5}}\right)}{\sqrt{5}}",1,"2*Sqrt[1 + x] + (8*ArcTanh[(1 + 2*Sqrt[1 + Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]","A",1
11,1,71,73,0.068711,"\int \frac{1}{x-\sqrt{1+\sqrt{1+x}}} \, dx","Integrate[(x - Sqrt[1 + Sqrt[1 + x]])^(-1),x]","\frac{1}{5} \left(2 \left(5+\sqrt{5}\right) \log \left(-2 \sqrt{\sqrt{x+1}+1}-\sqrt{5}+1\right)-2 \left(\sqrt{5}-5\right) \log \left(-2 \sqrt{\sqrt{x+1}+1}+\sqrt{5}+1\right)\right)","\frac{2}{5} \left(5+\sqrt{5}\right) \log \left(-2 \sqrt{\sqrt{x+1}+1}-\sqrt{5}+1\right)+\frac{2}{5} \left(5-\sqrt{5}\right) \log \left(-2 \sqrt{\sqrt{x+1}+1}+\sqrt{5}+1\right)",1,"(2*(5 + Sqrt[5])*Log[1 - Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]] - 2*(-5 + Sqrt[5])*Log[1 + Sqrt[5] - 2*Sqrt[1 + Sqrt[1 + x]]])/5","A",1
12,1,52,73,0.0831085,"\int \frac{x}{x+\sqrt{1-\sqrt{1+x}}} \, dx","Integrate[x/(x + Sqrt[1 - Sqrt[1 + x]]),x]","x-4 \sqrt{1-\sqrt{x+1}}+\frac{8 \tanh ^{-1}\left(\frac{2 \sqrt{1-\sqrt{x+1}}+1}{\sqrt{5}}\right)}{\sqrt{5}}","\left(1-\sqrt{x+1}\right)^2-4 \sqrt{1-\sqrt{x+1}}+2 \sqrt{x+1}+\frac{8 \tanh ^{-1}\left(\frac{2 \sqrt{1-\sqrt{x+1}}+1}{\sqrt{5}}\right)}{\sqrt{5}}",1,"x - 4*Sqrt[1 - Sqrt[1 + x]] + (8*ArcTanh[(1 + 2*Sqrt[1 - Sqrt[1 + x]])/Sqrt[5]])/Sqrt[5]","A",1
13,1,357,365,0.4927479,"\int \frac{\sqrt{x+\sqrt{1+x}}}{\sqrt{1+x} \left(1+x^2\right)} \, dx","Integrate[Sqrt[x + Sqrt[1 + x]]/(Sqrt[1 + x]*(1 + x^2)),x]","\frac{1}{4} (1-i)^{3/2} \sqrt{i+\sqrt{1-i}} \tan ^{-1}\left(\frac{-\left(\left(1-2 \sqrt{1-i}\right) \sqrt{x+1}\right)+\sqrt{1-i}+2}{2 \sqrt{i+\sqrt{1-i}} \sqrt{x+\sqrt{x+1}}}\right)+\frac{1}{4} (1+i)^{3/2} \sqrt{\sqrt{1+i}-i} \tan ^{-1}\left(\frac{-\left(\left(1-2 \sqrt{1+i}\right) \sqrt{x+1}\right)+\sqrt{1+i}+2}{2 \sqrt{\sqrt{1+i}-i} \sqrt{x+\sqrt{x+1}}}\right)-\frac{1}{4} (1-i)^{3/2} \sqrt{\sqrt{1-i}-i} \tanh ^{-1}\left(\frac{-\left(\left(1+2 \sqrt{1-i}\right) \sqrt{x+1}\right)-\sqrt{1-i}+2}{2 \sqrt{\sqrt{1-i}-i} \sqrt{x+\sqrt{x+1}}}\right)-\frac{1}{4} (1+i)^{3/2} \sqrt{i+\sqrt{1+i}} \tanh ^{-1}\left(\frac{-\left(\left(1+2 \sqrt{1+i}\right) \sqrt{x+1}\right)-\sqrt{1+i}+2}{2 \sqrt{i+\sqrt{1+i}} \sqrt{x+\sqrt{x+1}}}\right)","-\frac{i \tan ^{-1}\left(\frac{-\left(\left(1-2 \sqrt{1-i}\right) \sqrt{x+1}\right)+\sqrt{1-i}+2}{2 \sqrt{i+\sqrt{1-i}} \sqrt{x+\sqrt{x+1}}}\right)}{2 \sqrt{\frac{1-i}{i+\sqrt{1-i}}}}+\frac{i \tan ^{-1}\left(\frac{-\left(\left(1-2 \sqrt{1+i}\right) \sqrt{x+1}\right)+\sqrt{1+i}+2}{2 \sqrt{\sqrt{1+i}-i} \sqrt{x+\sqrt{x+1}}}\right)}{2 \sqrt{-\frac{1+i}{i-\sqrt{1+i}}}}+\frac{i \tanh ^{-1}\left(\frac{-\left(\left(1+2 \sqrt{1-i}\right) \sqrt{x+1}\right)-\sqrt{1-i}+2}{2 \sqrt{\sqrt{1-i}-i} \sqrt{x+\sqrt{x+1}}}\right)}{2 \sqrt{-\frac{1-i}{i-\sqrt{1-i}}}}-\frac{i \tanh ^{-1}\left(\frac{-\left(\left(1+2 \sqrt{1+i}\right) \sqrt{x+1}\right)-\sqrt{1+i}+2}{2 \sqrt{i+\sqrt{1+i}} \sqrt{x+\sqrt{x+1}}}\right)}{2 \sqrt{\frac{1+i}{i+\sqrt{1+i}}}}",1,"((1 - I)^(3/2)*Sqrt[I + Sqrt[1 - I]]*ArcTan[(2 + Sqrt[1 - I] - (1 - 2*Sqrt[1 - I])*Sqrt[1 + x])/(2*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]])])/4 + ((1 + I)^(3/2)*Sqrt[-I + Sqrt[1 + I]]*ArcTan[(2 + Sqrt[1 + I] - (1 - 2*Sqrt[1 + I])*Sqrt[1 + x])/(2*Sqrt[-I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]])])/4 - ((1 - I)^(3/2)*Sqrt[-I + Sqrt[1 - I]]*ArcTanh[(2 - Sqrt[1 - I] - (1 + 2*Sqrt[1 - I])*Sqrt[1 + x])/(2*Sqrt[-I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]])])/4 - ((1 + I)^(3/2)*Sqrt[I + Sqrt[1 + I]]*ArcTanh[(2 - Sqrt[1 + I] - (1 + 2*Sqrt[1 + I])*Sqrt[1 + x])/(2*Sqrt[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]])])/4","A",1
14,1,2581,337,6.375628,"\int \frac{\sqrt{x+\sqrt{1+x}}}{1+x^2} \, dx","Integrate[Sqrt[x + Sqrt[1 + x]]/(1 + x^2),x]","\text{Result too large to show}","\frac{1}{2} i \sqrt{i+\sqrt{1-i}} \tan ^{-1}\left(\frac{-\left(\left(1-2 \sqrt{1-i}\right) \sqrt{x+1}\right)+\sqrt{1-i}+2}{2 \sqrt{i+\sqrt{1-i}} \sqrt{x+\sqrt{x+1}}}\right)-\frac{1}{2} i \sqrt{\sqrt{1+i}-i} \tan ^{-1}\left(\frac{-\left(\left(1-2 \sqrt{1+i}\right) \sqrt{x+1}\right)+\sqrt{1+i}+2}{2 \sqrt{\sqrt{1+i}-i} \sqrt{x+\sqrt{x+1}}}\right)+\frac{1}{2} i \sqrt{\sqrt{1-i}-i} \tanh ^{-1}\left(\frac{-\left(\left(1+2 \sqrt{1-i}\right) \sqrt{x+1}\right)-\sqrt{1-i}+2}{2 \sqrt{\sqrt{1-i}-i} \sqrt{x+\sqrt{x+1}}}\right)-\frac{1}{2} i \sqrt{i+\sqrt{1+i}} \tanh ^{-1}\left(\frac{-\left(\left(1+2 \sqrt{1+i}\right) \sqrt{x+1}\right)-\sqrt{1+i}+2}{2 \sqrt{i+\sqrt{1+i}} \sqrt{x+\sqrt{x+1}}}\right)",1,"(((1 + I) + Sqrt[1 - I])*ArcTan[((2 - 3*I) + (3 - I)*Sqrt[1 - I] - 8*Sqrt[1 + x] - 5*Sqrt[1 - I]*Sqrt[1 + x] + (2 + 5*I)*(1 + x) + (5*I)*Sqrt[1 - I]*(1 + x) + 4*Sqrt[I - Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] + 2*Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] - (6 + 2*I)*Sqrt[I - Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] - (8*Sqrt[I - Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/Sqrt[1 - I])/((-4 + 7*I) - (6 - 2*I)*Sqrt[1 - I] + (4 - 2*I)*Sqrt[1 + x] + (6 - 2*I)*Sqrt[1 - I]*Sqrt[1 + x] + (10 + I)*(1 + x) + (8 + 4*I)*Sqrt[1 - I]*(1 + x))])/(2*Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]) + (((-1 - I) + Sqrt[1 - I])*ArcTan[((-2 + 3*I) + (3 - I)*Sqrt[1 - I] + 8*Sqrt[1 + x] - 5*Sqrt[1 - I]*Sqrt[1 + x] - (2 + 5*I)*(1 + x) + (5*I)*Sqrt[1 - I]*(1 + x) - 4*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] + 2*Sqrt[1 - I]*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] + (6 + 2*I)*Sqrt[I + Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] - (8*Sqrt[I + Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/Sqrt[1 - I])/((4 - 7*I) - (6 - 2*I)*Sqrt[1 - I] - (4 - 2*I)*Sqrt[1 + x] + (6 - 2*I)*Sqrt[1 - I]*Sqrt[1 + x] - (10 + I)*(1 + x) + (8 + 4*I)*Sqrt[1 - I]*(1 + x))])/(2*Sqrt[1 - I]*Sqrt[I + Sqrt[1 - I]]) - ((I/2)*((-1 + I) + Sqrt[1 + I])*ArcTan[((1 + 8*I) - 5*(1 + I)^(3/2) - (16 + 8*I)*Sqrt[1 + x] + (10 + 5*I)*Sqrt[1 + I]*Sqrt[1 + x] + (9 - 8*I)*(1 + x) - (5 - 10*I)*Sqrt[1 + I]*(1 + x) - 4*Sqrt[I - Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] + (4 - 2*I)*Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] - 8*Sqrt[I - Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] + (8 - 4*I)*Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/((9 + 20*I) - 12*(1 + I)^(3/2) - (14 + 20*I)*Sqrt[1 + x] + (22 + 12*I)*Sqrt[1 + I]*Sqrt[1 + x] + (6 - 15*I)*(1 + x) + (2 + 12*I)*Sqrt[1 + I]*(1 + x))])/(Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]) - ((I/2)*((1 - I) + Sqrt[1 + I])*ArcTan[((-1 - 8*I) - 5*(1 + I)^(3/2) + (16 + 8*I)*Sqrt[1 + x] + (10 + 5*I)*Sqrt[1 + I]*Sqrt[1 + x] - (9 - 8*I)*(1 + x) - (5 - 10*I)*Sqrt[1 + I]*(1 + x) + 4*Sqrt[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] + (4 - 2*I)*Sqrt[1 + I]*Sqrt[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] + 8*Sqrt[I + Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] + (8 - 4*I)*Sqrt[1 + I]*Sqrt[I + Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]])/((-9 - 20*I) - 12*(1 + I)^(3/2) + (14 + 20*I)*Sqrt[1 + x] + (22 + 12*I)*Sqrt[1 + I]*Sqrt[1 + x] - (6 - 15*I)*(1 + x) + (2 + 12*I)*Sqrt[1 + I]*(1 + x))])/(Sqrt[1 + I]*Sqrt[I + Sqrt[1 + I]]) + ((I/4)*((1 + I) + Sqrt[1 - I])*Log[(Sqrt[1 - I] - Sqrt[1 + x])^2])/(Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]) + (((1 - I) + Sqrt[1 + I])*Log[(Sqrt[1 + I] - Sqrt[1 + x])^2])/(4*Sqrt[1 + I]*Sqrt[I + Sqrt[1 + I]]) + ((I/4)*((-1 - I) + Sqrt[1 - I])*Log[(Sqrt[1 - I] + Sqrt[1 + x])^2])/(Sqrt[1 - I]*Sqrt[I + Sqrt[1 - I]]) + (((-1 + I) + Sqrt[1 + I])*Log[(Sqrt[1 + I] + Sqrt[1 + x])^2])/(4*Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]) - ((I/4)*((1 + I) + Sqrt[1 - I])*Log[(5 + 17*I) + (14*I)*Sqrt[1 - I] - (10 + 22*I)*Sqrt[1 + x] + (5 - 19*I)*Sqrt[1 - I]*Sqrt[1 + x] - (25 + 2*I)*(1 + x) - (15 + 9*I)*Sqrt[1 - I]*(1 + x) - (4 - 4*I)*Sqrt[I - Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] - (6 - 2*I)*Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] - (8 - 8*I)*Sqrt[I - Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] - (12 - 4*I)*Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]]])/(Sqrt[1 - I]*Sqrt[I - Sqrt[1 - I]]) - ((I/4)*((-1 - I) + Sqrt[1 - I])*Log[(-5 - 17*I) + (14*I)*Sqrt[1 - I] + (10 + 22*I)*Sqrt[1 + x] + (5 - 19*I)*Sqrt[1 - I]*Sqrt[1 + x] + (25 + 2*I)*(1 + x) - (15 + 9*I)*Sqrt[1 - I]*(1 + x) + (4 - 4*I)*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] - (6 - 2*I)*Sqrt[1 - I]*Sqrt[I + Sqrt[1 - I]]*Sqrt[x + Sqrt[1 + x]] + (8 - 8*I)*Sqrt[I + Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] - (12 - 4*I)*Sqrt[1 - I]*Sqrt[I + Sqrt[1 - I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]]])/(Sqrt[1 - I]*Sqrt[I + Sqrt[1 - I]]) - (((-1 + I) + Sqrt[1 + I])*Log[(-3 + 5*I) - (2 + 4*I)*Sqrt[1 + I] + (2 - 2*I)*Sqrt[1 + x] - (1 - 3*I)*Sqrt[1 + I]*Sqrt[1 + x] - (8 + 7*I)*(1 + x) + (9 + 3*I)*Sqrt[1 + I]*(1 + x) + (4 + 4*I)*Sqrt[I - Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] - 2*(1 + I)^(3/2)*Sqrt[I - Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] - (8 + 4*I)*Sqrt[I - Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] + 8*Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]]])/(4*Sqrt[1 + I]*Sqrt[I - Sqrt[1 + I]]) - (((1 - I) + Sqrt[1 + I])*Log[(3 - 5*I) - (2 + 4*I)*Sqrt[1 + I] - (2 - 2*I)*Sqrt[1 + x] - (1 - 3*I)*Sqrt[1 + I]*Sqrt[1 + x] + (8 + 7*I)*(1 + x) + (9 + 3*I)*Sqrt[1 + I]*(1 + x) - (4 + 4*I)*Sqrt[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] - 2*(1 + I)^(3/2)*Sqrt[I + Sqrt[1 + I]]*Sqrt[x + Sqrt[1 + x]] + (8 + 4*I)*Sqrt[I + Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]] + 8*Sqrt[1 + I]*Sqrt[I + Sqrt[1 + I]]*Sqrt[1 + x]*Sqrt[x + Sqrt[1 + x]]])/(4*Sqrt[1 + I]*Sqrt[I + Sqrt[1 + I]])","B",0
15,1,74,77,0.0422804,"\int \sqrt{1+\sqrt{x}+\sqrt{1+2 \sqrt{x}+2 x}} \, dx","Integrate[Sqrt[1 + Sqrt[x] + Sqrt[1 + 2*Sqrt[x] + 2*x]],x]","\frac{2 \sqrt{\sqrt{x}+\sqrt{2 x+2 \sqrt{x}+1}+1} \left(6 x^{3/2}+\sqrt{x}+\left(\sqrt{x}-2\right) \sqrt{2 x+2 \sqrt{x}+1}+2\right)}{15 \sqrt{x}}","\frac{2 \sqrt{\sqrt{x}+\sqrt{2 x+2 \sqrt{x}+1}+1} \left(6 x^{3/2}+\sqrt{x}-\left(2-\sqrt{x}\right) \sqrt{2 x+2 \sqrt{x}+1}+2\right)}{15 \sqrt{x}}",1,"(2*Sqrt[1 + Sqrt[x] + Sqrt[1 + 2*Sqrt[x] + 2*x]]*(2 + Sqrt[x] + 6*x^(3/2) + (-2 + Sqrt[x])*Sqrt[1 + 2*Sqrt[x] + 2*x]))/(15*Sqrt[x])","A",1
16,1,112,118,0.0866035,"\int \sqrt{\sqrt{2}+\sqrt{x}+\sqrt{2+2 \sqrt{2} \sqrt{x}+2 x}} \, dx","Integrate[Sqrt[Sqrt[2] + Sqrt[x] + Sqrt[2 + 2*Sqrt[2]*Sqrt[x] + 2*x]],x]","\frac{2 \sqrt{2} \left(3 \sqrt{2} x^{3/2}+\sqrt{2} \sqrt{x}+\sqrt{2} \left(\sqrt{x}-2 \sqrt{2}\right) \sqrt{x+\sqrt{2} \sqrt{x}+1}+4\right) \sqrt{\sqrt{2} \left(\sqrt{x+\sqrt{2} \sqrt{x}+1}+1\right)+\sqrt{x}}}{15 \sqrt{x}}","\frac{2 \sqrt{2} \sqrt{\sqrt{x}+\sqrt{2} \sqrt{x+\sqrt{2} \sqrt{x}+1}+\sqrt{2}} \left(3 \sqrt{2} x^{3/2}+\sqrt{2} \sqrt{x}-\sqrt{2} \left(2 \sqrt{2}-\sqrt{x}\right) \sqrt{x+\sqrt{2} \sqrt{x}+1}+4\right)}{15 \sqrt{x}}",1,"(2*Sqrt[2]*(4 + Sqrt[2]*Sqrt[x] + 3*Sqrt[2]*x^(3/2) + Sqrt[2]*(-2*Sqrt[2] + Sqrt[x])*Sqrt[1 + Sqrt[2]*Sqrt[x] + x])*Sqrt[Sqrt[x] + Sqrt[2]*(1 + Sqrt[1 + Sqrt[2]*Sqrt[x] + x])])/(15*Sqrt[x])","A",1
17,1,85,83,0.0450511,"\int \frac{\sqrt{x+\sqrt{1+x}}}{x^2} \, dx","Integrate[Sqrt[x + Sqrt[1 + x]]/x^2,x]","-\frac{\sqrt{x+\sqrt{x+1}}}{x}+\frac{1}{4} \tan ^{-1}\left(\frac{-\sqrt{x+1}-3}{2 \sqrt{x+\sqrt{x+1}}}\right)-\frac{3}{4} \tanh ^{-1}\left(\frac{3 \sqrt{x+1}-1}{2 \sqrt{x+\sqrt{x+1}}}\right)","-\frac{\sqrt{x+\sqrt{x+1}}}{x}-\frac{1}{4} \tan ^{-1}\left(\frac{\sqrt{x+1}+3}{2 \sqrt{x+\sqrt{x+1}}}\right)+\frac{3}{4} \tanh ^{-1}\left(\frac{1-3 \sqrt{x+1}}{2 \sqrt{x+\sqrt{x+1}}}\right)",1,"-(Sqrt[x + Sqrt[1 + x]]/x) + ArcTan[(-3 - Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])]/4 - (3*ArcTanh[(-1 + 3*Sqrt[1 + x])/(2*Sqrt[x + Sqrt[1 + x]])])/4","A",1
18,1,98,96,0.2019752,"\int \sqrt{\sqrt{1+\frac{1}{x}}+\frac{1}{x}} \, dx","Integrate[Sqrt[Sqrt[1 + x^(-1)] + x^(-1)],x]","\sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}} x-\frac{1}{4} \tan ^{-1}\left(\frac{-\sqrt{\frac{1}{x}+1}-3}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right)+\frac{3}{4} \tanh ^{-1}\left(\frac{3 \sqrt{\frac{1}{x}+1}-1}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right)","\sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}} x+\frac{1}{4} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{x}+1}+3}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right)-\frac{3}{4} \tanh ^{-1}\left(\frac{1-3 \sqrt{\frac{1}{x}+1}}{2 \sqrt{\sqrt{\frac{1}{x}+1}+\frac{1}{x}}}\right)",1,"Sqrt[Sqrt[1 + x^(-1)] + x^(-1)]*x - ArcTan[(-3 - Sqrt[1 + x^(-1)])/(2*Sqrt[Sqrt[1 + x^(-1)] + x^(-1)])]/4 + (3*ArcTanh[(-1 + 3*Sqrt[1 + x^(-1)])/(2*Sqrt[Sqrt[1 + x^(-1)] + x^(-1)])])/4","A",1
19,1,112,25,0.1299257,"\int \frac{\sqrt{1+e^{-x}}}{-e^{-x}+e^x} \, dx","Integrate[Sqrt[1 + E^(-x)]/(-E^(-x) + E^x),x]","\frac{e^{x/2} \sqrt{e^{-x}+1} \left(\log \left(1-e^{x/2}\right)-\log \left(e^{x/2}+1\right)+\log \left(\sqrt{2} \sqrt{e^x+1}-e^{x/2}+1\right)-\log \left(\sqrt{2} \sqrt{e^x+1}+e^{x/2}+1\right)\right)}{\sqrt{2} \sqrt{e^x+1}}","-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{e^{-x}+1}}{\sqrt{2}}\right)",1,"(E^(x/2)*Sqrt[1 + E^(-x)]*(Log[1 - E^(x/2)] - Log[1 + E^(x/2)] + Log[1 - E^(x/2) + Sqrt[2]*Sqrt[1 + E^x]] - Log[1 + E^(x/2) + Sqrt[2]*Sqrt[1 + E^x]]))/(Sqrt[2]*Sqrt[1 + E^x])","B",1
20,1,126,25,0.1156159,"\int \sqrt{1+e^{-x}} \text{csch}(x) \, dx","Integrate[Sqrt[1 + E^(-x)]*Csch[x],x]","\frac{\sqrt{2} e^{x/2} \sqrt{e^{-x}+1} \left(\log \left(1-e^{-x/2}\right)+\log \left(e^{-x/2}+1\right)-\log \left(e^{-x/2} \left(\sqrt{2} \sqrt{e^x+1}+e^{x/2}-1\right)\right)-\log \left(e^{-x/2} \left(\sqrt{2} \sqrt{e^x+1}+e^{x/2}+1\right)\right)\right)}{\sqrt{e^x+1}}","-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{e^{-x}+1}}{\sqrt{2}}\right)",1,"(Sqrt[2]*E^(x/2)*Sqrt[1 + E^(-x)]*(Log[1 - E^(-1/2*x)] + Log[1 + E^(-1/2*x)] - Log[(-1 + E^(x/2) + Sqrt[2]*Sqrt[1 + E^x])/E^(x/2)] - Log[(1 + E^(x/2) + Sqrt[2]*Sqrt[1 + E^x])/E^(x/2)]))/Sqrt[1 + E^x]","B",1
21,1,478,108,6.3568022,"\int \frac{1}{(\cos (x)+\cos (3 x))^5} \, dx","Integrate[(Cos[x] + Cos[3*x])^(-5),x]","\frac{1483 \log \left(2 \sin (x)+\sqrt{2}\right)}{1024 \sqrt{2}}+\frac{83 \sin (x)}{512 (\cos (x)-\sin (x))^2}+\frac{\sin (x)}{128 (\cos (x)-\sin (x))^4}-\frac{437}{1024 (\cos (x)-\sin (x))}+\frac{437}{1024 (\sin (x)+\cos (x))}-\frac{43}{512 \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^2}+\frac{43}{512 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}+\frac{83 \sin (x)}{512 (\sin (x)+\cos (x))^2}-\frac{17}{768 (\cos (x)-\sin (x))^3}+\frac{17}{768 (\sin (x)+\cos (x))^3}-\frac{1}{512 \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^4}+\frac{1}{512 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4}+\frac{\sin (x)}{128 (\sin (x)+\cos (x))^4}+\frac{523}{256} \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\frac{523}{256} \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\frac{1483 \log \left(-\sqrt{2} \sin (x)-\sqrt{2} \cos (x)+2\right)}{2048 \sqrt{2}}+\frac{\left(\frac{1483}{4096}-\frac{1483 i}{4096}\right) \left(\sqrt{2}+(-1-i)\right) \log \left(-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+2\right)}{\sqrt{2}+(-1+i)}-\frac{1483 i \tan ^{-1}\left(\frac{-\sqrt{2} \sin \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}{-\sin \left(\frac{x}{2}\right)+\sqrt{2} \cos \left(\frac{x}{2}\right)-\cos \left(\frac{x}{2}\right)}\right)}{1024 \sqrt{2}}+\frac{\left(\frac{1483}{2048}+\frac{1483 i}{2048}\right) \left(\sqrt{2}+(-1-i)\right) \tan ^{-1}\left(\frac{-\sqrt{2} \sin \left(\frac{x}{2}\right)+\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}{-\sin \left(\frac{x}{2}\right)+\sqrt{2} \cos \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}\right)}{\sqrt{2}+(-1+i)}","-\frac{437 \sin (x)}{512 \left(1-2 \sin ^2(x)\right)}+\frac{203 \sin (x)}{768 \left(1-2 \sin ^2(x)\right)^2}-\frac{17 \sin (x)}{192 \left(1-2 \sin ^2(x)\right)^3}+\frac{\sin (x)}{32 \left(1-2 \sin ^2(x)\right)^4}-\frac{523}{256} \tanh ^{-1}(\sin (x))+\frac{1483 \tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{512 \sqrt{2}}-\frac{1}{128} \tan (x) \sec ^3(x)-\frac{43}{256} \tan (x) \sec (x)",1,"(((-1483*I)/1024)*ArcTan[(Cos[x/2] - Sin[x/2] - Sqrt[2]*Sin[x/2])/(-Cos[x/2] + Sqrt[2]*Cos[x/2] - Sin[x/2])])/Sqrt[2] + ((1483/2048 + (1483*I)/2048)*((-1 - I) + Sqrt[2])*ArcTan[(Cos[x/2] + Sin[x/2] - Sqrt[2]*Sin[x/2])/(Cos[x/2] + Sqrt[2]*Cos[x/2] - Sin[x/2])])/((-1 + I) + Sqrt[2]) + (523*Log[Cos[x/2] - Sin[x/2]])/256 - (523*Log[Cos[x/2] + Sin[x/2]])/256 + (1483*Log[Sqrt[2] + 2*Sin[x]])/(1024*Sqrt[2]) - (1483*Log[2 - Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]])/(2048*Sqrt[2]) + ((1483/4096 - (1483*I)/4096)*((-1 - I) + Sqrt[2])*Log[2 + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]])/((-1 + I) + Sqrt[2]) - 1/(512*(Cos[x/2] - Sin[x/2])^4) - 43/(512*(Cos[x/2] - Sin[x/2])^2) + 1/(512*(Cos[x/2] + Sin[x/2])^4) + 43/(512*(Cos[x/2] + Sin[x/2])^2) - 17/(768*(Cos[x] - Sin[x])^3) - 437/(1024*(Cos[x] - Sin[x])) + Sin[x]/(128*(Cos[x] - Sin[x])^4) + (83*Sin[x])/(512*(Cos[x] - Sin[x])^2) + Sin[x]/(128*(Cos[x] + Sin[x])^4) + 17/(768*(Cos[x] + Sin[x])^3) + (83*Sin[x])/(512*(Cos[x] + Sin[x])^2) + 437/(1024*(Cos[x] + Sin[x]))","C",0
22,1,56,29,0.0417675,"\int \frac{1}{(1+\cos (x)+\sin (x))^2} \, dx","Integrate[(1 + Cos[x] + Sin[x])^(-2),x]","\frac{1}{2} \tan \left(\frac{x}{2}\right)+\log \left(\cos \left(\frac{x}{2}\right)\right)+\frac{\sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","-\log \left(\tan \left(\frac{x}{2}\right)+1\right)-\frac{\cos (x)-\sin (x)}{\sin (x)+\cos (x)+1}",1,"Log[Cos[x/2]] - Log[Cos[x/2] + Sin[x/2]] + Sin[x/2]/(Cos[x/2] + Sin[x/2]) + Tan[x/2]/2","A",1
23,1,26,26,0.018706,"\int \sqrt{1+\tanh (4 x)} \, dx","Integrate[Sqrt[1 + Tanh[4*x]],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{\tanh (4 x)+1}}{\sqrt{2}}\right)}{2 \sqrt{2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{\tanh (4 x)+1}}{\sqrt{2}}\right)}{2 \sqrt{2}}",1,"ArcTanh[Sqrt[1 + Tanh[4*x]]/Sqrt[2]]/(2*Sqrt[2])","A",1
24,1,121,110,0.1337747,"\int \frac{\tanh (x)}{\sqrt{e^x+e^{2 x}}} \, dx","Integrate[Tanh[x]/Sqrt[E^x + E^(2*x)],x]","\frac{2 e^x-(1-i)^{3/2} e^{x/2} \sqrt{e^x+1} \tanh ^{-1}\left(\frac{\sqrt{1-i} e^{x/2}}{\sqrt{e^x+1}}\right)-(1+i)^{3/2} e^{x/2} \sqrt{e^x+1} \tanh ^{-1}\left(\frac{\sqrt{1+i} e^{x/2}}{\sqrt{e^x+1}}\right)+2}{\sqrt{e^x \left(e^x+1\right)}}","2 e^{-x} \sqrt{e^x+e^{2 x}}-\frac{\tan ^{-1}\left(\frac{i-(1-2 i) e^x}{2 \sqrt{1+i} \sqrt{e^x+e^{2 x}}}\right)}{\sqrt{1+i}}+\frac{\tan ^{-1}\left(\frac{(1+2 i) e^x+i}{2 \sqrt{1-i} \sqrt{e^x+e^{2 x}}}\right)}{\sqrt{1-i}}",1,"(2 + 2*E^x - (1 - I)^(3/2)*E^(x/2)*Sqrt[1 + E^x]*ArcTanh[(Sqrt[1 - I]*E^(x/2))/Sqrt[1 + E^x]] - (1 + I)^(3/2)*E^(x/2)*Sqrt[1 + E^x]*ArcTanh[(Sqrt[1 + I]*E^(x/2))/Sqrt[1 + E^x]])/Sqrt[E^x*(1 + E^x)]","A",1
25,1,54,40,1.9299428,"\int \sqrt{\text{sech}(x) \sinh (2 x)} \, dx","Integrate[Sqrt[Sech[x]*Sinh[2*x]],x]","-\frac{2}{3} \sqrt{2} \sqrt{\sinh (x)} \tanh \left(\frac{x}{2}\right) \left(\sqrt{\text{sech}^2\left(\frac{x}{2}\right)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};\tanh ^2\left(\frac{x}{2}\right)\right)-3\right)","\frac{2 i \sqrt{2} \sqrt{\sinh (x)} E\left(\left.\frac{\pi }{4}-\frac{i x}{2}\right|2\right)}{\sqrt{i \sinh (x)}}",1,"(-2*Sqrt[2]*(-3 + Hypergeometric2F1[1/2, 3/4, 7/4, Tanh[x/2]^2]*Sqrt[Sech[x/2]^2])*Sqrt[Sinh[x]]*Tanh[x/2])/3","C",1
26,1,910,185,0.5801092,"\int \log \left(x^2+\sqrt{1-x^2}\right) \, dx","Integrate[Log[x^2 + Sqrt[1 - x^2]],x]","\frac{4 \sqrt{5} \log \left(x^2+\sqrt{1-x^2}\right) x-8 \sqrt{5} x-4 \sqrt{5} \sin ^{-1}(x)+\sqrt{10 \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)+5 \sqrt{2 \left(-1+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)-\left(-5+\sqrt{5}\right) \sqrt{2 \left(1+\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{2}{-1+\sqrt{5}}} x\right)+3 \sqrt{5 \left(2+\sqrt{5}\right)} \log \left(x-\sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)}\right)-5 \sqrt{2+\sqrt{5}} \log \left(x-\sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)}\right)-3 \sqrt{5 \left(2+\sqrt{5}\right)} \log \left(x+\sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)}\right)+5 \sqrt{2+\sqrt{5}} \log \left(x+\sqrt{\frac{1}{2} \left(-1+\sqrt{5}\right)}\right)-3 i \sqrt{5 \left(-2+\sqrt{5}\right)} \log \left(x-i \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)}\right)-5 i \sqrt{-2+\sqrt{5}} \log \left(x-i \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)}\right)+3 i \sqrt{5 \left(-2+\sqrt{5}\right)} \log \left(x+i \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)}\right)+5 i \sqrt{-2+\sqrt{5}} \log \left(x+i \sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)}\right)+3 i \sqrt{5 \left(-2+\sqrt{5}\right)} \log \left(-i \sqrt{2 \left(1+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)} \sqrt{1-x^2}+2\right)+5 i \sqrt{-2+\sqrt{5}} \log \left(-i \sqrt{2 \left(1+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)} \sqrt{1-x^2}+2\right)-3 i \sqrt{5 \left(-2+\sqrt{5}\right)} \log \left(i \sqrt{2 \left(1+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)} \sqrt{1-x^2}+2\right)-5 i \sqrt{-2+\sqrt{5}} \log \left(i \sqrt{2 \left(1+\sqrt{5}\right)} x+\sqrt{2 \left(3+\sqrt{5}\right)} \sqrt{1-x^2}+2\right)-3 \sqrt{5 \left(2+\sqrt{5}\right)} \log \left(-\sqrt{2 \left(-1+\sqrt{5}\right)} x+\sqrt{2} \sqrt{\left(-3+\sqrt{5}\right) \left(x^2-1\right)}+2\right)+5 \sqrt{2+\sqrt{5}} \log \left(-\sqrt{2 \left(-1+\sqrt{5}\right)} x+\sqrt{2} \sqrt{\left(-3+\sqrt{5}\right) \left(x^2-1\right)}+2\right)+3 \sqrt{5 \left(2+\sqrt{5}\right)} \log \left(\sqrt{2 \left(-1+\sqrt{5}\right)} x+\sqrt{2} \sqrt{\left(-3+\sqrt{5}\right) \left(x^2-1\right)}+2\right)-5 \sqrt{2+\sqrt{5}} \log \left(\sqrt{2 \left(-1+\sqrt{5}\right)} x+\sqrt{2} \sqrt{\left(-3+\sqrt{5}\right) \left(x^2-1\right)}+2\right)}{4 \sqrt{5}}","x \log \left(x^2+\sqrt{1-x^2}\right)+\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\frac{\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} x}{\sqrt{1-x^2}}\right)-\sqrt{\frac{1}{2} \left(\sqrt{5}-1\right)} \tanh ^{-1}\left(\frac{\sqrt{\frac{1}{2} \left(\sqrt{5}-1\right)} x}{\sqrt{1-x^2}}\right)-2 x-\sin ^{-1}(x)+\sqrt{\frac{1}{2} \left(1+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{2}{1+\sqrt{5}}} x\right)+\sqrt{\frac{1}{2} \left(\sqrt{5}-1\right)} \tanh ^{-1}\left(\sqrt{\frac{2}{\sqrt{5}-1}} x\right)",1,"(-8*Sqrt[5]*x - 4*Sqrt[5]*ArcSin[x] + 5*Sqrt[2*(-1 + Sqrt[5])]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] + Sqrt[10*(-1 + Sqrt[5])]*ArcTan[Sqrt[2/(1 + Sqrt[5])]*x] - (-5 + Sqrt[5])*Sqrt[2*(1 + Sqrt[5])]*ArcTanh[Sqrt[2/(-1 + Sqrt[5])]*x] - 5*Sqrt[2 + Sqrt[5]]*Log[-Sqrt[(-1 + Sqrt[5])/2] + x] + 3*Sqrt[5*(2 + Sqrt[5])]*Log[-Sqrt[(-1 + Sqrt[5])/2] + x] + 5*Sqrt[2 + Sqrt[5]]*Log[Sqrt[(-1 + Sqrt[5])/2] + x] - 3*Sqrt[5*(2 + Sqrt[5])]*Log[Sqrt[(-1 + Sqrt[5])/2] + x] - (5*I)*Sqrt[-2 + Sqrt[5]]*Log[(-I)*Sqrt[(1 + Sqrt[5])/2] + x] - (3*I)*Sqrt[5*(-2 + Sqrt[5])]*Log[(-I)*Sqrt[(1 + Sqrt[5])/2] + x] + (5*I)*Sqrt[-2 + Sqrt[5]]*Log[I*Sqrt[(1 + Sqrt[5])/2] + x] + (3*I)*Sqrt[5*(-2 + Sqrt[5])]*Log[I*Sqrt[(1 + Sqrt[5])/2] + x] + 4*Sqrt[5]*x*Log[x^2 + Sqrt[1 - x^2]] + (5*I)*Sqrt[-2 + Sqrt[5]]*Log[2 - I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] + (3*I)*Sqrt[5*(-2 + Sqrt[5])]*Log[2 - I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] - (5*I)*Sqrt[-2 + Sqrt[5]]*Log[2 + I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] - (3*I)*Sqrt[5*(-2 + Sqrt[5])]*Log[2 + I*Sqrt[2*(1 + Sqrt[5])]*x + Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 - x^2]] + 5*Sqrt[2 + Sqrt[5]]*Log[2 - Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]] - 3*Sqrt[5*(2 + Sqrt[5])]*Log[2 - Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]] - 5*Sqrt[2 + Sqrt[5]]*Log[2 + Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]] + 3*Sqrt[5*(2 + Sqrt[5])]*Log[2 + Sqrt[2*(-1 + Sqrt[5])]*x + Sqrt[2]*Sqrt[(-3 + Sqrt[5])*(-1 + x^2)]])/(4*Sqrt[5])","C",0
27,1,102,102,0.0191493,"\int \frac{\log \left(1+e^x\right)}{1+e^{2 x}} \, dx","Integrate[Log[1 + E^x]/(1 + E^(2*x)),x]","-\operatorname{PolyLog}\left(2,-e^x\right)-\frac{1}{2} \operatorname{PolyLog}\left(2,\left(\frac{1}{2}-\frac{i}{2}\right) \left(e^x+1\right)\right)-\frac{1}{2} \operatorname{PolyLog}\left(2,\left(\frac{1}{2}+\frac{i}{2}\right) \left(e^x+1\right)\right)-\frac{1}{2} \log \left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(-e^x+i\right)\right) \log \left(e^x+1\right)-\frac{1}{2} \log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(e^x+i\right)\right) \log \left(e^x+1\right)","-\operatorname{PolyLog}\left(2,-e^x\right)-\frac{1}{2} \operatorname{PolyLog}\left(2,\left(\frac{1}{2}-\frac{i}{2}\right) \left(e^x+1\right)\right)-\frac{1}{2} \operatorname{PolyLog}\left(2,\left(\frac{1}{2}+\frac{i}{2}\right) \left(e^x+1\right)\right)-\frac{1}{2} \log \left(\left(\frac{1}{2}-\frac{i}{2}\right) \left(-e^x+i\right)\right) \log \left(e^x+1\right)-\frac{1}{2} \log \left(\left(-\frac{1}{2}-\frac{i}{2}\right) \left(e^x+i\right)\right) \log \left(e^x+1\right)",1,"-1/2*(Log[(1/2 - I/2)*(I - E^x)]*Log[1 + E^x]) - (Log[(-1/2 - I/2)*(I + E^x)]*Log[1 + E^x])/2 - PolyLog[2, -E^x] - PolyLog[2, (1/2 - I/2)*(1 + E^x)]/2 - PolyLog[2, (1/2 + I/2)*(1 + E^x)]/2","A",1
28,1,122,159,0.0952251,"\int \cosh (x) \log ^2\left(1+\cosh ^2(x)\right) \, dx","Integrate[Cosh[x]*Log[1 + Cosh[x]^2]^2,x]","4 i \sqrt{2} \operatorname{PolyLog}\left(2,\frac{\sqrt{2} \sinh (x)+2 i}{\sqrt{2} \sinh (x)-2 i}\right)+\sinh (x) \left(\log ^2\left(\sinh ^2(x)+2\right)-4 \log \left(\sinh ^2(x)+2\right)+8\right)+4 \sqrt{2} \tan ^{-1}\left(\frac{\sinh (x)}{\sqrt{2}}\right) \left(\log \left(\sinh ^2(x)+2\right)+2 \log \left(\frac{4 i}{-\sqrt{2} \sinh (x)+2 i}\right)+i \tan ^{-1}\left(\frac{\sinh (x)}{\sqrt{2}}\right)-2\right)","4 i \sqrt{2} \operatorname{PolyLog}\left(2,1-\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right)+8 \sinh (x)+\sinh (x) \log ^2\left(\sinh ^2(x)+2\right)-4 \sinh (x) \log \left(\sinh ^2(x)+2\right)+4 i \sqrt{2} \tan ^{-1}\left(\frac{\sinh (x)}{\sqrt{2}}\right)^2-8 \sqrt{2} \tan ^{-1}\left(\frac{\sinh (x)}{\sqrt{2}}\right)+4 \sqrt{2} \log \left(\sinh ^2(x)+2\right) \tan ^{-1}\left(\frac{\sinh (x)}{\sqrt{2}}\right)+8 \sqrt{2} \log \left(\frac{2 \sqrt{2}}{\sqrt{2}+i \sinh (x)}\right) \tan ^{-1}\left(\frac{\sinh (x)}{\sqrt{2}}\right)",1,"4*Sqrt[2]*ArcTan[Sinh[x]/Sqrt[2]]*(-2 + I*ArcTan[Sinh[x]/Sqrt[2]] + 2*Log[(4*I)/(2*I - Sqrt[2]*Sinh[x])] + Log[2 + Sinh[x]^2]) + (4*I)*Sqrt[2]*PolyLog[2, (2*I + Sqrt[2]*Sinh[x])/(-2*I + Sqrt[2]*Sinh[x])] + (8 - 4*Log[2 + Sinh[x]^2] + Log[2 + Sinh[x]^2]^2)*Sinh[x]","A",0
29,1,347,395,0.2476339,"\int \cosh (x) \log ^2\left(\cosh ^2(x)+\sinh (x)\right) \, dx","Integrate[Cosh[x]*Log[Cosh[x]^2 + Sinh[x]]^2,x]","-\frac{1}{2} i \left(\sqrt{3}-i\right) \left(2 \operatorname{PolyLog}\left(2,\frac{-2 i \sinh (x)+\sqrt{3}-i}{2 \sqrt{3}}\right)+\log \left(2 \sinh (x)+i \sqrt{3}+1\right) \left(2 \log \left(\frac{2 i \sinh (x)+\sqrt{3}+i}{2 \sqrt{3}}\right)+\log \left(2 \sinh (x)+i \sqrt{3}+1\right)\right)\right)+\frac{1}{2} i \left(\sqrt{3}+i\right) \left(2 \operatorname{PolyLog}\left(2,\frac{2 i \sinh (x)+\sqrt{3}+i}{2 \sqrt{3}}\right)+\log \left(2 \sinh (x)-i \sqrt{3}+1\right) \left(2 \log \left(\frac{-2 i \sinh (x)+\sqrt{3}-i}{2 \sqrt{3}}\right)+\log \left(2 \sinh (x)-i \sqrt{3}+1\right)\right)\right)+8 \sinh (x)+\sinh (x) \log ^2\left(\sinh ^2(x)+\sinh (x)+1\right)+\left(1-i \sqrt{3}\right) \log \left(2 \sinh (x)-i \sqrt{3}+1\right) \log \left(\sinh ^2(x)+\sinh (x)+1\right)+\left(1+i \sqrt{3}\right) \log \left(2 \sinh (x)+i \sqrt{3}+1\right) \log \left(\sinh ^2(x)+\sinh (x)+1\right)-4 \sinh (x) \log \left(\sinh ^2(x)+\sinh (x)+1\right)-2 \log \left(\sinh ^2(x)+\sinh (x)+1\right)-4 \sqrt{3} \tan ^{-1}\left(\frac{2 \sinh (x)+1}{\sqrt{3}}\right)","-\left(1+i \sqrt{3}\right) \operatorname{PolyLog}\left(2,-\frac{2 i \sinh (x)-\sqrt{3}+i}{2 \sqrt{3}}\right)-\left(1-i \sqrt{3}\right) \operatorname{PolyLog}\left(2,\frac{2 i \sinh (x)+\sqrt{3}+i}{2 \sqrt{3}}\right)+8 \sinh (x)+\sinh (x) \log ^2\left(\sinh ^2(x)+\sinh (x)+1\right)-\frac{1}{2} \left(1-i \sqrt{3}\right) \log ^2\left(2 \sinh (x)-i \sqrt{3}+1\right)-\frac{1}{2} \left(1+i \sqrt{3}\right) \log ^2\left(2 \sinh (x)+i \sqrt{3}+1\right)+\left(1-i \sqrt{3}\right) \log \left(\sinh ^2(x)+\sinh (x)+1\right) \log \left(2 \sinh (x)-i \sqrt{3}+1\right)+\left(1+i \sqrt{3}\right) \log \left(2 \sinh (x)+i \sqrt{3}+1\right) \log \left(\sinh ^2(x)+\sinh (x)+1\right)-2 \log \left(\sinh ^2(x)+\sinh (x)+1\right)-4 \sinh (x) \log \left(\sinh ^2(x)+\sinh (x)+1\right)-\left(1-i \sqrt{3}\right) \log \left(-\frac{i \left(2 \sinh (x)+i \sqrt{3}+1\right)}{2 \sqrt{3}}\right) \log \left(2 \sinh (x)-i \sqrt{3}+1\right)-\left(1+i \sqrt{3}\right) \log \left(\frac{i \left(2 \sinh (x)-i \sqrt{3}+1\right)}{2 \sqrt{3}}\right) \log \left(2 \sinh (x)+i \sqrt{3}+1\right)-4 \sqrt{3} \tan ^{-1}\left(\frac{2 \sinh (x)+1}{\sqrt{3}}\right)",1,"-4*Sqrt[3]*ArcTan[(1 + 2*Sinh[x])/Sqrt[3]] - 2*Log[1 + Sinh[x] + Sinh[x]^2] + (1 - I*Sqrt[3])*Log[1 - I*Sqrt[3] + 2*Sinh[x]]*Log[1 + Sinh[x] + Sinh[x]^2] + (1 + I*Sqrt[3])*Log[1 + I*Sqrt[3] + 2*Sinh[x]]*Log[1 + Sinh[x] + Sinh[x]^2] - (I/2)*(-I + Sqrt[3])*(Log[1 + I*Sqrt[3] + 2*Sinh[x]]*(2*Log[(I + Sqrt[3] + (2*I)*Sinh[x])/(2*Sqrt[3])] + Log[1 + I*Sqrt[3] + 2*Sinh[x]]) + 2*PolyLog[2, (-I + Sqrt[3] - (2*I)*Sinh[x])/(2*Sqrt[3])]) + (I/2)*(I + Sqrt[3])*(Log[1 - I*Sqrt[3] + 2*Sinh[x]]*(2*Log[(-I + Sqrt[3] - (2*I)*Sinh[x])/(2*Sqrt[3])] + Log[1 - I*Sqrt[3] + 2*Sinh[x]]) + 2*PolyLog[2, (I + Sqrt[3] + (2*I)*Sinh[x])/(2*Sqrt[3])]) + 8*Sinh[x] - 4*Log[1 + Sinh[x] + Sinh[x]^2]*Sinh[x] + Log[1 + Sinh[x] + Sinh[x]^2]^2*Sinh[x]","A",1
30,1,868,981,0.5568527,"\int \frac{\log \left(x+\sqrt{1+x}\right)}{1+x^2} \, dx","Integrate[Log[x + Sqrt[1 + x]]/(1 + x^2),x]","\frac{1}{2} i \left(2 i \tan ^{-1}(x) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)+\log \left(\frac{2 \left(\sqrt{1-i}-\sqrt{x+1}\right)}{1+2 \sqrt{1-i}-\sqrt{5}}\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)-\log \left(\frac{2 \left(\sqrt{1+i}-\sqrt{x+1}\right)}{1+2 \sqrt{1+i}-\sqrt{5}}\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)+\log \left(\frac{2 \left(\sqrt{x+1}+\sqrt{1-i}\right)}{-1+2 \sqrt{1-i}+\sqrt{5}}\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)-\log \left(\frac{2 \left(\sqrt{x+1}+\sqrt{1+i}\right)}{-1+2 \sqrt{1+i}+\sqrt{5}}\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)+2 i \tan ^{-1}(x) \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)+\log \left(\frac{2 \left(\sqrt{1-i}-\sqrt{x+1}\right)}{1+2 \sqrt{1-i}+\sqrt{5}}\right) \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)-\log \left(\frac{2 \left(\sqrt{1+i}-\sqrt{x+1}\right)}{1+2 \sqrt{1+i}+\sqrt{5}}\right) \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)+\log \left(\frac{2 \left(\sqrt{x+1}+\sqrt{1-i}\right)}{-1+2 \sqrt{1-i}-\sqrt{5}}\right) \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)-\log \left(\frac{2 \left(\sqrt{x+1}+\sqrt{1+i}\right)}{-1+2 \sqrt{1+i}-\sqrt{5}}\right) \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)-2 i \tan ^{-1}(x) \log \left(x+\sqrt{x+1}\right)+\operatorname{PolyLog}\left(2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{-1+2 \sqrt{1-i}+\sqrt{5}}\right)-\operatorname{PolyLog}\left(2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{-1+2 \sqrt{1+i}+\sqrt{5}}\right)+\operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1-i}-\sqrt{5}}\right)-\operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1+i}-\sqrt{5}}\right)+\operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1-i}+\sqrt{5}}\right)+\operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1-i}+\sqrt{5}}\right)-\operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1+i}+\sqrt{5}}\right)-\operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1+i}+\sqrt{5}}\right)\right)","\frac{1}{2} i \log \left(\sqrt{1-i}-\sqrt{x+1}\right) \log \left(x+\sqrt{x+1}\right)-\frac{1}{2} i \log \left(\sqrt{1+i}-\sqrt{x+1}\right) \log \left(x+\sqrt{x+1}\right)+\frac{1}{2} i \log \left(\sqrt{x+1}+\sqrt{1-i}\right) \log \left(x+\sqrt{x+1}\right)-\frac{1}{2} i \log \left(\sqrt{x+1}+\sqrt{1+i}\right) \log \left(x+\sqrt{x+1}\right)-\frac{1}{2} i \log \left(\sqrt{x+1}+\sqrt{1-i}\right) \log \left(\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1-2 \sqrt{1-i}-\sqrt{5}}\right)-\frac{1}{2} i \log \left(\sqrt{1-i}-\sqrt{x+1}\right) \log \left(\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1-i}-\sqrt{5}}\right)+\frac{1}{2} i \log \left(\sqrt{x+1}+\sqrt{1+i}\right) \log \left(\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1-2 \sqrt{1+i}-\sqrt{5}}\right)+\frac{1}{2} i \log \left(\sqrt{1+i}-\sqrt{x+1}\right) \log \left(\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1+i}-\sqrt{5}}\right)-\frac{1}{2} i \log \left(\sqrt{x+1}+\sqrt{1-i}\right) \log \left(\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1-i}+\sqrt{5}}\right)-\frac{1}{2} i \log \left(\sqrt{1-i}-\sqrt{x+1}\right) \log \left(\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1-i}+\sqrt{5}}\right)+\frac{1}{2} i \log \left(\sqrt{x+1}+\sqrt{1+i}\right) \log \left(\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1+i}+\sqrt{5}}\right)+\frac{1}{2} i \log \left(\sqrt{1+i}-\sqrt{x+1}\right) \log \left(\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1+i}+\sqrt{5}}\right)-\frac{1}{2} i \operatorname{PolyLog}\left(2,\frac{2 \left(\sqrt{1-i}-\sqrt{x+1}\right)}{1+2 \sqrt{1-i}-\sqrt{5}}\right)-\frac{1}{2} i \operatorname{PolyLog}\left(2,\frac{2 \left(\sqrt{1-i}-\sqrt{x+1}\right)}{1+2 \sqrt{1-i}+\sqrt{5}}\right)+\frac{1}{2} i \operatorname{PolyLog}\left(2,\frac{2 \left(\sqrt{1+i}-\sqrt{x+1}\right)}{1+2 \sqrt{1+i}-\sqrt{5}}\right)+\frac{1}{2} i \operatorname{PolyLog}\left(2,\frac{2 \left(\sqrt{1+i}-\sqrt{x+1}\right)}{1+2 \sqrt{1+i}+\sqrt{5}}\right)-\frac{1}{2} i \operatorname{PolyLog}\left(2,-\frac{2 \left(\sqrt{x+1}+\sqrt{1-i}\right)}{1-2 \sqrt{1-i}-\sqrt{5}}\right)-\frac{1}{2} i \operatorname{PolyLog}\left(2,-\frac{2 \left(\sqrt{x+1}+\sqrt{1-i}\right)}{1-2 \sqrt{1-i}+\sqrt{5}}\right)+\frac{1}{2} i \operatorname{PolyLog}\left(2,-\frac{2 \left(\sqrt{x+1}+\sqrt{1+i}\right)}{1-2 \sqrt{1+i}-\sqrt{5}}\right)+\frac{1}{2} i \operatorname{PolyLog}\left(2,-\frac{2 \left(\sqrt{x+1}+\sqrt{1+i}\right)}{1-2 \sqrt{1+i}+\sqrt{5}}\right)",1,"(I/2)*((2*I)*ArcTan[x]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 - I] - Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 + I] - Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] + Sqrt[1 + x]))/(-1 + 2*Sqrt[1 - I] + Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] + Sqrt[1 + x]))/(-1 + 2*Sqrt[1 + I] + Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + (2*I)*ArcTan[x]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 - I] + Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] - Sqrt[1 + x]))/(1 + 2*Sqrt[1 + I] + Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] + Log[(2*(Sqrt[1 - I] + Sqrt[1 + x]))/(-1 + 2*Sqrt[1 - I] - Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - Log[(2*(Sqrt[1 + I] + Sqrt[1 + x]))/(-1 + 2*Sqrt[1 + I] - Sqrt[5])]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - (2*I)*ArcTan[x]*Log[x + Sqrt[1 + x]] + PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(-1 + 2*Sqrt[1 - I] + Sqrt[5])] - PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(-1 + 2*Sqrt[1 + I] + Sqrt[5])] + PolyLog[2, (1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] - Sqrt[5])] - PolyLog[2, (1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I] - Sqrt[5])] + PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 - I] + Sqrt[5])] + PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 - I] + Sqrt[5])] - PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - 2*Sqrt[1 + I] + Sqrt[5])] - PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 + 2*Sqrt[1 + I] + Sqrt[5])])","A",1
31,1,1076,555,1.575637,"\int \frac{\log ^2\left(x+\sqrt{1+x}\right)}{(1+x)^2} \, dx","Integrate[Log[x + Sqrt[1 + x]]^2/(1 + x)^2,x]","\frac{1}{2} \left(\sqrt{5} \log ^2\left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)+3 \log ^2\left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)-12 \log \left(\frac{2 \sqrt{x+1}}{-1+\sqrt{5}}\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)+6 \log (x+1) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)-2 \sqrt{5} \log \left(-2 \sqrt{x+1}+\sqrt{5}-1\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)-6 \log \left(-2 \sqrt{x+1}+\sqrt{5}-1\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)+2 \sqrt{5} \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)-6 \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)-2 \sqrt{5} \log \left(\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)+6 \log \left(\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right) \log \left(\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right)-\sqrt{5} \log ^2\left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)+3 \log ^2\left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)-\frac{2 \log ^2\left(x+\sqrt{x+1}\right)}{x+1}-6 \log \left(\frac{1}{2} \left(1+\sqrt{5}\right)\right) \log (x+1)+2 \log (x+1)+6 \log (x+1) \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)-2 \sqrt{5} \log \left(-2 \sqrt{x+1}+\sqrt{5}-1\right) \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)-6 \log \left(-2 \sqrt{x+1}+\sqrt{5}-1\right) \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right)-6 \log (x+1) \log \left(x+\sqrt{x+1}\right)+2 \sqrt{5} \log \left(-2 \sqrt{x+1}+\sqrt{5}-1\right) \log \left(x+\sqrt{x+1}\right)+6 \log \left(-2 \sqrt{x+1}+\sqrt{5}-1\right) \log \left(x+\sqrt{x+1}\right)+\frac{4 \log \left(x+\sqrt{x+1}\right)}{\sqrt{x+1}}+\sqrt{5} \log (5) \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right)+3 \log (5) \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right)-2 \sqrt{5} \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right)-2 \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right)+2 \sqrt{5} \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right) \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right)-6 \log \left(\sqrt{x+1}+\frac{1}{2} \left(1+\sqrt{5}\right)\right) \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right)-2 \sqrt{5} \log \left(x+\sqrt{x+1}\right) \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right)+6 \log \left(x+\sqrt{x+1}\right) \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right)+2 \sqrt{5} \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right)-2 \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right)+12 \operatorname{PolyLog}\left(2,-\frac{2 \sqrt{x+1}}{1+\sqrt{5}}\right)-4 \sqrt{5} \operatorname{PolyLog}\left(2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{2 \sqrt{5}}\right)-12 \operatorname{PolyLog}\left(2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{-1+\sqrt{5}}\right)\right)","6 \operatorname{PolyLog}\left(2,-\frac{2 \sqrt{x+1}}{1+\sqrt{5}}\right)-\left(3+\sqrt{5}\right) \operatorname{PolyLog}\left(2,-\frac{2 \sqrt{x+1}-\sqrt{5}+1}{2 \sqrt{5}}\right)-\left(3-\sqrt{5}\right) \operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right)-6 \operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}}{1-\sqrt{5}}+1\right)-\frac{\log ^2\left(x+\sqrt{x+1}\right)}{x+1}-\frac{1}{2} \left(3+\sqrt{5}\right) \log ^2\left(2 \sqrt{x+1}-\sqrt{5}+1\right)-\frac{1}{2} \left(3-\sqrt{5}\right) \log ^2\left(2 \sqrt{x+1}+\sqrt{5}+1\right)-6 \log \left(\sqrt{x+1}\right) \log \left(x+\sqrt{x+1}\right)+\left(3+\sqrt{5}\right) \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right) \log \left(x+\sqrt{x+1}\right)+\left(3-\sqrt{5}\right) \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right) \log \left(x+\sqrt{x+1}\right)+\frac{2 \log \left(x+\sqrt{x+1}\right)}{\sqrt{x+1}}+\log (x+1)+6 \log \left(\frac{1}{2} \left(\sqrt{5}-1\right)\right) \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right)-\left(1+\sqrt{5}\right) \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right)-\left(3-\sqrt{5}\right) \log \left(-\frac{2 \sqrt{x+1}-\sqrt{5}+1}{2 \sqrt{5}}\right) \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right)-\left(1-\sqrt{5}\right) \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right)-\left(3+\sqrt{5}\right) \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right) \log \left(\frac{2 \sqrt{x+1}+\sqrt{5}+1}{2 \sqrt{5}}\right)+6 \log \left(\sqrt{x+1}\right) \log \left(\frac{2 \sqrt{x+1}}{1+\sqrt{5}}+1\right)",1,"(2*Log[1 + x] - 6*Log[(1 + Sqrt[5])/2]*Log[1 + x] - 12*Log[(2*Sqrt[1 + x])/(-1 + Sqrt[5])]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + 6*Log[1 + x]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - 6*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] - 2*Sqrt[5]*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]] + 3*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]^2 + Sqrt[5]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]^2 + 6*Log[1 + x]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - 6*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] - 2*Sqrt[5]*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]] + 3*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]^2 - Sqrt[5]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]^2 + (4*Log[x + Sqrt[1 + x]])/Sqrt[1 + x] - 6*Log[1 + x]*Log[x + Sqrt[1 + x]] + 6*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] + 2*Sqrt[5]*Log[-1 + Sqrt[5] - 2*Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] - (2*Log[x + Sqrt[1 + x]]^2)/(1 + x) - 2*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] - 2*Sqrt[5]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] + 3*Log[5]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] + Sqrt[5]*Log[5]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] - 2*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + 2*Sqrt[5]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] - 6*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + 2*Sqrt[5]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] - 6*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + 2*Sqrt[5]*Log[(1 + Sqrt[5])/2 + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + 6*Log[x + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] - 2*Sqrt[5]*Log[x + Sqrt[1 + x]]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] + 6*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])] - 2*Sqrt[5]*Log[1/2 - Sqrt[5]/2 + Sqrt[1 + x]]*Log[(1 + Sqrt[5] + 2*Sqrt[1 + x])/(2*Sqrt[5])] + 12*PolyLog[2, (-2*Sqrt[1 + x])/(1 + Sqrt[5])] - 4*Sqrt[5]*PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(2*Sqrt[5])] - 12*PolyLog[2, (-1 + Sqrt[5] - 2*Sqrt[1 + x])/(-1 + Sqrt[5])])/2","A",0
32,1,303,313,0.1046142,"\int \frac{\log \left(x+\sqrt{1+x}\right)}{x} \, dx","Integrate[Log[x + Sqrt[1 + x]]/x,x]","-\operatorname{PolyLog}\left(2,\frac{2 \left(\sqrt{x+1}+1\right)}{1-\sqrt{5}}\right)+\operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}-\sqrt{5}+1}{3-\sqrt{5}}\right)+\operatorname{PolyLog}\left(2,-\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+\sqrt{5}}\right)+\operatorname{PolyLog}\left(2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{3+\sqrt{5}}\right)+\log \left(1-\sqrt{x+1}\right) \log \left(x+\sqrt{x+1}\right)+\log \left(\sqrt{x+1}+1\right) \log \left(x+\sqrt{x+1}\right)-\log \left(\frac{1}{2} \left(1+\sqrt{5}\right)\right) \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right)-\log \left(\frac{1}{2} \left(3-\sqrt{5}\right)\right) \log \left(2 \sqrt{x+1}-\sqrt{5}+1\right)-\log \left(\frac{1}{2} \left(3+\sqrt{5}\right)\right) \log \left(2 \sqrt{x+1}+\sqrt{5}+1\right)-\log \left(\sqrt{x+1}+1\right) \log \left(-\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-\sqrt{5}}\right)","-\operatorname{PolyLog}\left(2,\frac{2 \left(1-\sqrt{x+1}\right)}{3-\sqrt{5}}\right)-\operatorname{PolyLog}\left(2,\frac{2 \left(1-\sqrt{x+1}\right)}{3+\sqrt{5}}\right)-\operatorname{PolyLog}\left(2,\frac{2 \left(\sqrt{x+1}+1\right)}{1-\sqrt{5}}\right)-\operatorname{PolyLog}\left(2,\frac{2 \left(\sqrt{x+1}+1\right)}{1+\sqrt{5}}\right)+\log \left(\sqrt{x+1}-1\right) \log \left(x+\sqrt{x+1}\right)+\log \left(\sqrt{x+1}+1\right) \log \left(x+\sqrt{x+1}\right)-\log \left(\sqrt{x+1}-1\right) \log \left(\frac{2 \sqrt{x+1}-\sqrt{5}+1}{3-\sqrt{5}}\right)-\log \left(\sqrt{x+1}+1\right) \log \left(-\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+\sqrt{5}}\right)-\log \left(\sqrt{x+1}+1\right) \log \left(-\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-\sqrt{5}}\right)-\log \left(\sqrt{x+1}-1\right) \log \left(\frac{2 \sqrt{x+1}+\sqrt{5}+1}{3+\sqrt{5}}\right)",1,"Log[1 - Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] + Log[1 + Sqrt[1 + x]]*Log[x + Sqrt[1 + x]] - Log[(3 - Sqrt[5])/2]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] - Log[(1 + Sqrt[5])/2]*Log[1 - Sqrt[5] + 2*Sqrt[1 + x]] - Log[(3 + Sqrt[5])/2]*Log[1 + Sqrt[5] + 2*Sqrt[1 + x]] - Log[1 + Sqrt[1 + x]]*Log[-((1 + Sqrt[5] + 2*Sqrt[1 + x])/(1 - Sqrt[5]))] - PolyLog[2, (2*(1 + Sqrt[1 + x]))/(1 - Sqrt[5])] + PolyLog[2, (1 - Sqrt[5] + 2*Sqrt[1 + x])/(3 - Sqrt[5])] + PolyLog[2, -((1 - Sqrt[5] + 2*Sqrt[1 + x])/(1 + Sqrt[5]))] + PolyLog[2, (1 + Sqrt[5] + 2*Sqrt[1 + x])/(3 + Sqrt[5])]","A",1
33,1,262,80,0.2564283,"\int \tan ^{-1}(2 \tan (x)) \, dx","Integrate[ArcTan[2*Tan[x]],x]","x \tan ^{-1}(2 \tan (x))-\frac{1}{4} i \left(i \left(\operatorname{PolyLog}\left(2,\frac{2 \tan (x)-i}{6 \tan (x)+3 i}\right)-\operatorname{PolyLog}\left(2,\frac{6 \tan (x)-3 i}{2 \tan (x)+i}\right)\right)+2 i \cos ^{-1}\left(\frac{5}{3}\right) \tan ^{-1}(2 \tan (x))+4 i x \tan ^{-1}\left(\frac{\cot (x)}{2}\right)-\log \left(\frac{-4 \tan (x)+4 i}{2 \tan (x)+i}\right) \left(\cos ^{-1}\left(\frac{5}{3}\right)-2 \tan ^{-1}(2 \tan (x))\right)-\log \left(\frac{4 (\tan (x)+i)}{6 \tan (x)+3 i}\right) \left(2 \tan ^{-1}(2 \tan (x))+\cos ^{-1}\left(\frac{5}{3}\right)\right)+\log \left(\frac{2 i \sqrt{\frac{2}{3}} e^{-i x}}{\sqrt{3 \cos (2 x)-5}}\right) \left(2 \tan ^{-1}(2 \tan (x))+2 \tan ^{-1}\left(\frac{\cot (x)}{2}\right)+\cos ^{-1}\left(\frac{5}{3}\right)\right)+\log \left(\frac{2 i \sqrt{\frac{2}{3}} e^{i x}}{\sqrt{3 \cos (2 x)-5}}\right) \left(-2 \tan ^{-1}(2 \tan (x))-2 \tan ^{-1}\left(\frac{\cot (x)}{2}\right)+\cos ^{-1}\left(\frac{5}{3}\right)\right)\right)","-\frac{1}{4} \operatorname{PolyLog}\left(2,\frac{1}{3} e^{2 i x}\right)+\frac{1}{4} \operatorname{PolyLog}\left(2,3 e^{2 i x}\right)+\frac{1}{2} i x \log \left(1-3 e^{2 i x}\right)-\frac{1}{2} i x \log \left(1-\frac{1}{3} e^{2 i x}\right)+x \tan ^{-1}(2 \tan (x))",1,"x*ArcTan[2*Tan[x]] - (I/4)*((4*I)*x*ArcTan[Cot[x]/2] + (2*I)*ArcCos[5/3]*ArcTan[2*Tan[x]] + (ArcCos[5/3] + 2*ArcTan[Cot[x]/2] + 2*ArcTan[2*Tan[x]])*Log[((2*I)*Sqrt[2/3])/(E^(I*x)*Sqrt[-5 + 3*Cos[2*x]])] + (ArcCos[5/3] - 2*ArcTan[Cot[x]/2] - 2*ArcTan[2*Tan[x]])*Log[((2*I)*Sqrt[2/3]*E^(I*x))/Sqrt[-5 + 3*Cos[2*x]]] - (ArcCos[5/3] - 2*ArcTan[2*Tan[x]])*Log[(4*I - 4*Tan[x])/(I + 2*Tan[x])] - (ArcCos[5/3] + 2*ArcTan[2*Tan[x]])*Log[(4*(I + Tan[x]))/(3*I + 6*Tan[x])] + I*(-PolyLog[2, (-3*I + 6*Tan[x])/(I + 2*Tan[x])] + PolyLog[2, (-I + 2*Tan[x])/(3*I + 6*Tan[x])]))","B",1
34,1,44,57,0.0588625,"\int \frac{\tan ^{-1}(x) \log (x)}{x} \, dx","Integrate[(ArcTan[x]*Log[x])/x,x]","\frac{1}{2} i (-\operatorname{PolyLog}(3,-i x)+\operatorname{PolyLog}(3,i x)+\log (x) \operatorname{PolyLog}(2,-i x)-\log (x) \operatorname{PolyLog}(2,i x))","-\frac{1}{2} i \operatorname{PolyLog}(3,-i x)+\frac{1}{2} i \operatorname{PolyLog}(3,i x)+\frac{1}{2} i \log (x) \operatorname{PolyLog}(2,-i x)-\frac{1}{2} i \log (x) \operatorname{PolyLog}(2,i x)",1,"(I/2)*(Log[x]*PolyLog[2, (-I)*x] - Log[x]*PolyLog[2, I*x] - PolyLog[3, (-I)*x] + PolyLog[3, I*x])","A",1
35,1,131,121,0.2176844,"\int \sqrt{1+x^2} \tan ^{-1}(x)^2 \, dx","Integrate[Sqrt[1 + x^2]*ArcTan[x]^2,x]","i \tan ^{-1}(x) \operatorname{PolyLog}\left(2,-i e^{i \tan ^{-1}(x)}\right)-i \tan ^{-1}(x) \operatorname{PolyLog}\left(2,i e^{i \tan ^{-1}(x)}\right)-\operatorname{PolyLog}\left(3,-i e^{i \tan ^{-1}(x)}\right)+\operatorname{PolyLog}\left(3,i e^{i \tan ^{-1}(x)}\right)+\frac{1}{2} x \sqrt{x^2+1} \tan ^{-1}(x)^2-\sqrt{x^2+1} \tan ^{-1}(x)+\tanh ^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)-i \tan ^{-1}\left(e^{i \tan ^{-1}(x)}\right) \tan ^{-1}(x)^2","i \tan ^{-1}(x) \operatorname{PolyLog}\left(2,-i e^{i \tan ^{-1}(x)}\right)-i \tan ^{-1}(x) \operatorname{PolyLog}\left(2,i e^{i \tan ^{-1}(x)}\right)-\operatorname{PolyLog}\left(3,-i e^{i \tan ^{-1}(x)}\right)+\operatorname{PolyLog}\left(3,i e^{i \tan ^{-1}(x)}\right)+\frac{1}{2} x \sqrt{x^2+1} \tan ^{-1}(x)^2-\sqrt{x^2+1} \tan ^{-1}(x)-i \tan ^{-1}\left(e^{i \tan ^{-1}(x)}\right) \tan ^{-1}(x)^2+\sinh ^{-1}(x)",1,"-(Sqrt[1 + x^2]*ArcTan[x]) + (x*Sqrt[1 + x^2]*ArcTan[x]^2)/2 - I*ArcTan[E^(I*ArcTan[x])]*ArcTan[x]^2 + ArcTanh[x/Sqrt[1 + x^2]] + I*ArcTan[x]*PolyLog[2, (-I)*E^(I*ArcTan[x])] - I*ArcTan[x]*PolyLog[2, I*E^(I*ArcTan[x])] - PolyLog[3, (-I)*E^(I*ArcTan[x])] + PolyLog[3, I*E^(I*ArcTan[x])]","A",0