1,1,2,0,0.2593001,"\int f'(x) \, dx","Int[Derivative[1][f][x],x]","f(x)","f(x)",1,"f[x]","A",1,1,4,0.2500,1,"{6656}"
2,1,4,0,0.0020196,"\int f''(x) \, dx","Int[Derivative[2][f][x],x]","f'(x)","f'(x)",1,"Derivative[1][f][x]","A",1,1,4,0.2500,1,"{6656}"
3,1,4,0,0.0020183,"\int f^{(3)}(x) \, dx","Int[Derivative[3][f][x],x]","f''(x)","f''(x)",1,"Derivative[2][f][x]","A",1,1,4,0.2500,1,"{6656}"
4,1,6,0,0.0029372,"\int f^{(n)}(x) \, dx","Int[Derivative[n][f][x],x]","f^{(n-1)}(x)","f^{(n-1)}(x)",1,"Derivative[-1 + n][f][x]","A",1,1,4,0.2500,1,"{6656}"
5,1,10,0,0.0086893,"\int u'(x) u''(x) \, dx","Int[Derivative[1][u][x]*Derivative[2][u][x],x]","\frac{1}{2} u'(x)^2","\frac{1}{2} u'(x)^2",1,"Derivative[1][u][x]^2/2","A",2,2,9,0.2222,1,"{6663, 30}"
6,1,3,0,0.0095437,"\int \frac{f'(x)}{f(x)} \, dx","Int[Derivative[1][f][x]/f[x],x]","\log (f(x))","\log (f(x))",1,"Log[f[x]]","A",2,2,9,0.2222,1,"{6663, 29}"
7,1,11,0,0.0175042,"\int \frac{f'(x)}{a+b f(x)} \, dx","Int[Derivative[1][f][x]/(a + b*f[x]),x]","\frac{\log (a+b f(x))}{b}","\frac{\log (a+b f(x))}{b}",1,"Log[a + b*f[x]]/b","A",2,2,13,0.1538,1,"{6663, 31}"
8,1,8,0,0.0060985,"\int f(x) f'(x) \, dx","Int[f[x]*Derivative[1][f][x],x]","\frac{f(x)^2}{2}","\frac{f(x)^2}{2}",1,"f[x]^2/2","A",2,2,7,0.2857,1,"{6663, 30}"
9,1,14,0,0.0117097,"\int (a+b f(x)) f'(x) \, dx","Int[(a + b*f[x])*Derivative[1][f][x],x]","a f(x)+\frac{1}{2} b f(x)^2","a f(x)+\frac{1}{2} b f(x)^2",1,"a*f[x] + (b*f[x]^2)/2","A",2,1,11,0.09091,1,"{6663}"
10,1,8,0,0.0097876,"\int \frac{f'(x)}{\sqrt{f(x)}} \, dx","Int[Derivative[1][f][x]/Sqrt[f[x]],x]","2 \sqrt{f(x)}","2 \sqrt{f(x)}",1,"2*Sqrt[f[x]]","A",2,2,11,0.1818,1,"{6663, 30}"
11,1,15,0,0.0266526,"\int \frac{f'(x)}{\sqrt{a+b f(x)}} \, dx","Int[Derivative[1][f][x]/Sqrt[a + b*f[x]],x]","\frac{2 \sqrt{a+b f(x)}}{b}","\frac{2 \sqrt{a+b f(x)}}{b}",1,"(2*Sqrt[a + b*f[x]])/b","A",2,2,15,0.1333,1,"{6663, 32}"
12,1,12,0,0.0123004,"\int f(x)^n f'(x) \, dx","Int[f[x]^n*Derivative[1][f][x],x]","\frac{f(x)^{n+1}}{n+1}","\frac{f(x)^{n+1}}{n+1}",1,"f[x]^(1 + n)/(1 + n)","A",2,2,9,0.2222,1,"{6663, 30}"
13,1,19,0,0.0190744,"\int (a+b f(x))^n f'(x) \, dx","Int[(a + b*f[x])^n*Derivative[1][f][x],x]","\frac{(a+b f(x))^{n+1}}{b (n+1)}","\frac{(a+b f(x))^{n+1}}{b (n+1)}",1,"(a + b*f[x])^(1 + n)/(b*(1 + n))","A",2,2,13,0.1538,1,"{6663, 32}"
14,1,5,0,0.0098675,"\int \frac{f''(x)}{f'(x)} \, dx","Int[Derivative[2][f][x]/Derivative[1][f][x],x]","\log \left(f'(x)\right)","\log \left(f'(x)\right)",1,"Log[Derivative[1][f][x]]","A",2,2,11,0.1818,1,"{6663, 29}"
15,1,13,0,0.0173317,"\int \frac{f''(x)}{a+b f'(x)} \, dx","Int[Derivative[2][f][x]/(a + b*Derivative[1][f][x]),x]","\frac{\log \left(a+b f'(x)\right)}{b}","\frac{\log \left(a+b f'(x)\right)}{b}",1,"Log[a + b*Derivative[1][f][x]]/b","A",2,2,15,0.1333,1,"{6663, 31}"
16,1,10,0,0.007515,"\int f'(x) f''(x) \, dx","Int[Derivative[1][f][x]*Derivative[2][f][x],x]","\frac{1}{2} f'(x)^2","\frac{1}{2} f'(x)^2",1,"Derivative[1][f][x]^2/2","A",2,2,9,0.2222,1,"{6663, 30}"
17,1,18,0,0.0134732,"\int \left(a+b f'(x)\right) f''(x) \, dx","Int[(a + b*Derivative[1][f][x])*Derivative[2][f][x],x]","a f'(x)+\frac{1}{2} b f'(x)^2","a f'(x)+\frac{1}{2} b f'(x)^2",1,"a*Derivative[1][f][x] + (b*Derivative[1][f][x]^2)/2","A",2,1,13,0.07692,1,"{6663}"
18,1,10,0,0.0095441,"\int \frac{f''(x)}{\sqrt{f'(x)}} \, dx","Int[Derivative[2][f][x]/Sqrt[Derivative[1][f][x]],x]","2 \sqrt{f'(x)}","2 \sqrt{f'(x)}",1,"2*Sqrt[Derivative[1][f][x]]","A",2,2,13,0.1538,1,"{6663, 30}"
19,1,17,0,0.023634,"\int \frac{f''(x)}{\sqrt{a+b f'(x)}} \, dx","Int[Derivative[2][f][x]/Sqrt[a + b*Derivative[1][f][x]],x]","\frac{2 \sqrt{a+b f'(x)}}{b}","\frac{2 \sqrt{a+b f'(x)}}{b}",1,"(2*Sqrt[a + b*Derivative[1][f][x]])/b","A",2,2,17,0.1176,1,"{6663, 32}"
20,1,14,0,0.0120575,"\int f'(x)^n f''(x) \, dx","Int[Derivative[1][f][x]^n*Derivative[2][f][x],x]","\frac{f'(x)^{n+1}}{n+1}","\frac{f'(x)^{n+1}}{n+1}",1,"Derivative[1][f][x]^(1 + n)/(1 + n)","A",2,2,11,0.1818,1,"{6663, 30}"
21,1,21,0,0.0175918,"\int \left(a+b f'(x)\right)^n f''(x) \, dx","Int[(a + b*Derivative[1][f][x])^n*Derivative[2][f][x],x]","\frac{\left(a+b f'(x)\right)^{n+1}}{b (n+1)}","\frac{\left(a+b f'(x)\right)^{n+1}}{b (n+1)}",1,"(a + b*Derivative[1][f][x])^(1 + n)/(b*(1 + n))","A",2,2,15,0.1333,1,"{6663, 32}"
22,0,0,0,0.0136354,"\int f(g(x)) g'(x) \, dx","Int[f[g[x]]*Derivative[1][g][x],x]","\int f(g(x)) g'(x) \, dx","\text{Int}\left(f(g(x)) g'(x),x\right)",0,"Defer[Subst][Defer[Int][f[x], x], x, g[x]]","N/A",0,0,0,0,-1,"{}"
23,0,0,0,0.0106663,"\int f\left(g'(x)\right) g''(x) \, dx","Int[f[Derivative[1][g][x]]*Derivative[2][g][x],x]","\int f\left(g'(x)\right) g''(x) \, dx","\text{Int}\left(g''(x) f\left(g'(x)\right),x\right)",0,"Defer[Subst][Defer[Int][f[x], x], x, Derivative[1][g][x]]","N/A",0,0,0,0,-1,"{}"
24,0,0,0,0.0201989,"\int \left(g(x) f'(x)+f(x) g'(x)\right) \, dx","Int[g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x],x]","\int \left(g(x) f'(x)+f(x) g'(x)\right) \, dx","f(x) g(x)",1,"Defer[Int][g[x]*Derivative[1][f][x], x] + Defer[Int][f[x]*Derivative[1][g][x], x]","F",0,0,0,0,-1,"{}"
25,1,7,0,0.0138579,"\int \frac{g(x) f'(x)-f(x) g'(x)}{g(x)^2} \, dx","Int[(g[x]*Derivative[1][f][x] - f[x]*Derivative[1][g][x])/g[x]^2,x]","\frac{f(x)}{g(x)}","\frac{f(x)}{g(x)}",1,"f[x]/g[x]","A",1,1,21,0.04762,1,"{6675}"
26,1,8,0,0.0336994,"\int \frac{g(x) f'(x)-f(x) g'(x)}{f(x) g(x)} \, dx","Int[(g[x]*Derivative[1][f][x] - f[x]*Derivative[1][g][x])/(f[x]*g[x]),x]","\log \left(\frac{f(x)}{g(x)}\right)","\log \left(\frac{f(x)}{g(x)}\right)",1,"Log[f[x]/g[x]]","A",1,1,25,0.04000,1,"{6676}"
27,1,6,0,0.0217194,"\int \frac{g(x) f'(x)+f(x) g'(x)}{1+f(x)^2 g(x)^2} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(1 + f[x]^2*g[x]^2),x]","\tan ^{-1}(f(x) g(x))","\tan ^{-1}(f(x) g(x))",1,"ArcTan[f[x]*g[x]]","A",2,2,29,0.06897,1,"{6664, 203}"
28,1,8,0,0.0274821,"\int \frac{g(x) f'(x)-f(x) g'(x)}{f(x)^2+g(x)^2} \, dx","Int[(g[x]*Derivative[1][f][x] - f[x]*Derivative[1][g][x])/(f[x]^2 + g[x]^2),x]","\tan ^{-1}\left(\frac{f(x)}{g(x)}\right)","\tan ^{-1}\left(\frac{f(x)}{g(x)}\right)",1,"ArcTan[f[x]/g[x]]","A",2,2,28,0.07143,1,"{6711, 203}"
29,1,8,0,0.021762,"\int -\frac{g(x) f'(x)+f(x) g'(x)}{1+f(x)^2 g(x)^2} \, dx","Int[-((g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(1 + f[x]^2*g[x]^2)),x]","-\tan ^{-1}(f(x) g(x))","-\tan ^{-1}(f(x) g(x))",1,"-ArcTan[f[x]*g[x]]","A",2,2,30,0.06667,1,"{6664, 203}"
30,1,6,0,0.0226961,"\int \frac{g(x) f'(x)+f(x) g'(x)}{1-f(x)^2 g(x)^2} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(1 - f[x]^2*g[x]^2),x]","\tanh ^{-1}(f(x) g(x))","\tanh ^{-1}(f(x) g(x))",1,"ArcTanh[f[x]*g[x]]","A",2,2,30,0.06667,1,"{6664, 206}"
31,1,8,0,0.0304914,"\int \frac{-g(x) f'(x)+f(x) g'(x)}{f(x)^2-g(x)^2} \, dx","Int[(-(g[x]*Derivative[1][f][x]) + f[x]*Derivative[1][g][x])/(f[x]^2 - g[x]^2),x]","\tanh ^{-1}\left(\frac{f(x)}{g(x)}\right)","\tanh ^{-1}\left(\frac{f(x)}{g(x)}\right)",1,"ArcTanh[f[x]/g[x]]","A",2,2,30,0.06667,1,"{6711, 207}"
32,1,10,0,0.1784273,"\int \frac{f(x)^{-1+m} g(x)^{-1+n} \left(m g(x) f'(x)+n f(x) g'(x)\right)}{1-f(x)^{2 m} g(x)^{2 n}} \, dx","Int[(f[x]^(-1 + m)*g[x]^(-1 + n)*(m*g[x]*Derivative[1][f][x] + n*f[x]*Derivative[1][g][x]))/(1 - f[x]^(2*m)*g[x]^(2*n)),x]","\tanh ^{-1}\left(f(x)^m g(x)^n\right)","\tanh ^{-1}\left(f(x)^m g(x)^n\right)",1,"ArcTanh[f[x]^m*g[x]^n]","A",2,2,48,0.04167,1,"{6671, 206}"
33,1,12,0,0.4357591,"\int \frac{f(x)^{-1+m} g(x)^{-1+n} \left(-m g(x) f'(x)+n f(x) g'(x)\right)}{f(x)^{2 m}-g(x)^{2 n}} \, dx","Int[(f[x]^(-1 + m)*g[x]^(-1 + n)*(-(m*g[x]*Derivative[1][f][x]) + n*f[x]*Derivative[1][g][x]))/(f[x]^(2*m) - g[x]^(2*n)),x]","\tanh ^{-1}\left(f(x)^{-m} g(x)^n\right)","\tanh ^{-1}\left(f(x)^{-m} g(x)^n\right)",1,"ArcTanh[g[x]^n/f[x]^m]","A",3,3,48,0.06250,1,"{6691, 6671, 206}"
34,1,14,0,0.5515938,"\int \frac{f(x)^{-1+m} g(x)^{-1-n} \left(-m g(x) f'(x)-n f(x) g'(x)\right)}{f(x)^{2 m}-g(x)^{-2 n}} \, dx","Int[(f[x]^(-1 + m)*g[x]^(-1 - n)*(-(m*g[x]*Derivative[1][f][x]) - n*f[x]*Derivative[1][g][x]))/(f[x]^(2*m) - g[x]^(-2*n)),x]","\tanh ^{-1}\left(f(x)^{-m} g(x)^{-n}\right)","\tanh ^{-1}\left(f(x)^{-m} g(x)^{-n}\right)",1,"ArcTanh[1/(f[x]^m*g[x]^n)]","A",3,3,51,0.05882,1,"{6691, 6671, 206}"
35,1,13,0,0.0244243,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x) g(x)} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]*g[x]),x]","\frac{\log (a+b f(x) g(x))}{b}","\frac{\log (a+b f(x) g(x))}{b}",1,"Log[a + b*f[x]*g[x]]/b","A",2,2,26,0.07692,1,"{6664, 31}"
36,1,27,0,0.0393925,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^2 g(x)^2} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^2*g[x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x) g(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x) g(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*f[x]*g[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,30,0.06667,1,"{6664, 205}"
37,1,129,0,0.1096118,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^3 g(x)^3} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^3*g[x]^3),x]","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} f(x) g(x)+b^{2/3} f(x)^2 g(x)^2\right)}{6 a^{2/3} \sqrt[3]{b}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} f(x) g(x)\right)}{3 a^{2/3} \sqrt[3]{b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} f(x) g(x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} \sqrt[3]{b}}","-\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} f(x) g(x)+b^{2/3} f(x)^2 g(x)^2\right)}{6 a^{2/3} \sqrt[3]{b}}+\frac{\log \left(\sqrt[3]{a}+\sqrt[3]{b} f(x) g(x)\right)}{3 a^{2/3} \sqrt[3]{b}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} f(x) g(x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} \sqrt[3]{b}}",1,"-(ArcTan[(a^(1/3) - 2*b^(1/3)*f[x]*g[x])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3))) + Log[a^(1/3) + b^(1/3)*f[x]*g[x]]/(3*a^(2/3)*b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*f[x]*g[x] + b^(2/3)*f[x]^2*g[x]^2]/(6*a^(2/3)*b^(1/3))","A",7,7,30,0.2333,1,"{6664, 200, 31, 634, 617, 204, 628}"
38,1,35,0,0.0463392,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b \sqrt{f(x) g(x)}} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*Sqrt[f[x]*g[x]]),x]","\frac{2 \sqrt{f(x) g(x)}}{b}-\frac{2 a \log \left(a+b \sqrt{f(x) g(x)}\right)}{b^2}","\frac{2 \sqrt{f(x) g(x)}}{b}-\frac{2 a \log \left(a+b \sqrt{f(x) g(x)}\right)}{b^2}",1,"(2*Sqrt[f[x]*g[x]])/b - (2*a*Log[a + b*Sqrt[f[x]*g[x]]])/b^2","A",4,3,31,0.09677,1,"{6664, 190, 43}"
39,1,140,0,0.1450835,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b (f(x) g(x))^{3/2}} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*(f[x]*g[x])^(3/2)),x]","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sqrt{f(x) g(x)}+b^{2/3} f(x) g(x)\right)}{3 \sqrt[3]{a} b^{2/3}}-\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sqrt{f(x) g(x)}\right)}{3 \sqrt[3]{a} b^{2/3}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sqrt{f(x) g(x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} b^{2/3}}","\frac{\log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \sqrt{f(x) g(x)}+b^{2/3} f(x) g(x)\right)}{3 \sqrt[3]{a} b^{2/3}}-\frac{2 \log \left(\sqrt[3]{a}+\sqrt[3]{b} \sqrt{f(x) g(x)}\right)}{3 \sqrt[3]{a} b^{2/3}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \sqrt{f(x) g(x)}}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} b^{2/3}}",1,"(-2*ArcTan[(a^(1/3) - 2*b^(1/3)*Sqrt[f[x]*g[x]])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*b^(2/3)) - (2*Log[a^(1/3) + b^(1/3)*Sqrt[f[x]*g[x]]])/(3*a^(1/3)*b^(2/3)) + Log[a^(2/3) + b^(2/3)*f[x]*g[x] - a^(1/3)*b^(1/3)*Sqrt[f[x]*g[x]]]/(3*a^(1/3)*b^(2/3))","A",8,8,31,0.2581,1,"{6664, 243, 292, 31, 634, 617, 204, 628}"
40,1,380,0,0.9423153,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b (f(x) g(x))^{5/2}} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*(f[x]*g[x])^(5/2)),x]","-\frac{2 \log \left(\sqrt[5]{a}+\sqrt[5]{b} \sqrt{f(x) g(x)}\right)}{5 a^{3/5} b^{2/5}}+\frac{\left(1-\sqrt{5}\right) \log \left(2 a^{2/5}-\sqrt{5} \sqrt[5]{a} \sqrt[5]{b} \sqrt{f(x) g(x)}-\sqrt[5]{a} \sqrt[5]{b} \sqrt{f(x) g(x)}+2 b^{2/5} f(x) g(x)\right)}{10 a^{3/5} b^{2/5}}+\frac{\left(1+\sqrt{5}\right) \log \left(2 a^{2/5}+\sqrt{5} \sqrt[5]{a} \sqrt[5]{b} \sqrt{f(x) g(x)}-\sqrt[5]{a} \sqrt[5]{b} \sqrt{f(x) g(x)}+2 b^{2/5} f(x) g(x)\right)}{10 a^{3/5} b^{2/5}}-\frac{\sqrt{2 \left(5-\sqrt{5}\right)} \tan ^{-1}\left(\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt[5]{b} \sqrt{f(x) g(x)}}{\sqrt[5]{a}}+\sqrt{\frac{1}{5} \left(5-2 \sqrt{5}\right)}\right)}{5 a^{3/5} b^{2/5}}-\frac{\sqrt{2 \left(5+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{1}{5} \left(5+2 \sqrt{5}\right)}-\frac{\sqrt{\frac{2}{5} \left(5+\sqrt{5}\right)} \sqrt[5]{b} \sqrt{f(x) g(x)}}{\sqrt[5]{a}}\right)}{5 a^{3/5} b^{2/5}}","-\frac{2 \log \left(\sqrt[5]{a}+\sqrt[5]{b} \sqrt{f(x) g(x)}\right)}{5 a^{3/5} b^{2/5}}+\frac{\left(1-\sqrt{5}\right) \log \left(2 a^{2/5}-\sqrt{5} \sqrt[5]{a} \sqrt[5]{b} \sqrt{f(x) g(x)}-\sqrt[5]{a} \sqrt[5]{b} \sqrt{f(x) g(x)}+2 b^{2/5} f(x) g(x)\right)}{10 a^{3/5} b^{2/5}}+\frac{\left(1+\sqrt{5}\right) \log \left(2 a^{2/5}+\sqrt{5} \sqrt[5]{a} \sqrt[5]{b} \sqrt{f(x) g(x)}-\sqrt[5]{a} \sqrt[5]{b} \sqrt{f(x) g(x)}+2 b^{2/5} f(x) g(x)\right)}{10 a^{3/5} b^{2/5}}-\frac{\sqrt{2 \left(5-\sqrt{5}\right)} \tan ^{-1}\left(\frac{2 \sqrt{\frac{2}{5+\sqrt{5}}} \sqrt[5]{b} \sqrt{f(x) g(x)}}{\sqrt[5]{a}}+\sqrt{\frac{1}{5} \left(5-2 \sqrt{5}\right)}\right)}{5 a^{3/5} b^{2/5}}-\frac{\sqrt{2 \left(5+\sqrt{5}\right)} \tan ^{-1}\left(\sqrt{\frac{1}{5} \left(5+2 \sqrt{5}\right)}-\frac{\sqrt{\frac{2}{5} \left(5+\sqrt{5}\right)} \sqrt[5]{b} \sqrt{f(x) g(x)}}{\sqrt[5]{a}}\right)}{5 a^{3/5} b^{2/5}}",1,"-(Sqrt[2*(5 - Sqrt[5])]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + (2*Sqrt[2/(5 + Sqrt[5])]*b^(1/5)*Sqrt[f[x]*g[x]])/a^(1/5)])/(5*a^(3/5)*b^(2/5)) - (Sqrt[2*(5 + Sqrt[5])]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - (Sqrt[(2*(5 + Sqrt[5]))/5]*b^(1/5)*Sqrt[f[x]*g[x]])/a^(1/5)])/(5*a^(3/5)*b^(2/5)) - (2*Log[a^(1/5) + b^(1/5)*Sqrt[f[x]*g[x]]])/(5*a^(3/5)*b^(2/5)) + ((1 - Sqrt[5])*Log[2*a^(2/5) + 2*b^(2/5)*f[x]*g[x] - a^(1/5)*b^(1/5)*Sqrt[f[x]*g[x]] - Sqrt[5]*a^(1/5)*b^(1/5)*Sqrt[f[x]*g[x]]])/(10*a^(3/5)*b^(2/5)) + ((1 + Sqrt[5])*Log[2*a^(2/5) + 2*b^(2/5)*f[x]*g[x] - a^(1/5)*b^(1/5)*Sqrt[f[x]*g[x]] + Sqrt[5]*a^(1/5)*b^(1/5)*Sqrt[f[x]*g[x]]])/(10*a^(3/5)*b^(2/5))","A",8,8,31,0.2581,1,"{6664, 243, 293, 634, 618, 204, 628, 31}"
41,1,31,0,0.0394983,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b (f(x) g(x))^n} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*(f[x]*g[x])^n),x]","\frac{f(x) g(x) \text{Hypergeometric2F1}\left(1,\frac{1}{n},\frac{1}{n}+1,-\frac{b (f(x) g(x))^n}{a}\right)}{a}","\frac{f(x) g(x) \text{Hypergeometric2F1}\left(1,\frac{1}{n},\frac{1}{n}+1,-\frac{b (f(x) g(x))^n}{a}\right)}{a}",1,"(f[x]*g[x]*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*(f[x]*g[x])^n)/a)])/a","A",2,2,29,0.06897,1,"{6664, 245}"
42,0,0,0,0.252081,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^n g(x)^n} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^n*g[x]^n),x]","\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^n g(x)^n} \, dx","\text{Int}\left(\frac{g(x) f'(x)}{a+b f(x)^n g(x)^n},x\right)+\text{Int}\left(\frac{f(x) g'(x)}{a+b f(x)^n g(x)^n},x\right)",0,"Defer[Int][(g[x]*Derivative[1][f][x])/(a + b*f[x]^n*g[x]^n), x] + Defer[Int][(f[x]*Derivative[1][g][x])/(a + b*f[x]^n*g[x]^n), x]","N/A",0,0,0,0,-1,"{}"
43,0,0,0,0.06742,"\int \left(\cos (x) g\left(e^x\right) f'(\sin (x))+e^x f(\sin (x)) g'\left(e^x\right)\right) \, dx","Int[Cos[x]*g[E^x]*Derivative[1][f][Sin[x]] + E^x*f[Sin[x]]*Derivative[1][g][E^x],x]","\int \left(\cos (x) g\left(e^x\right) f'(\sin (x))+e^x f(\sin (x)) g'\left(e^x\right)\right) \, dx","g\left(e^x\right) f(\sin (x))",1,"Defer[Int][Cos[x]*g[E^x]*Derivative[1][f][Sin[x]], x] + Defer[Int][E^x*f[Sin[x]]*Derivative[1][g][E^x], x]","F",0,0,0,0,-1,"{}"
44,0,0,0,0.0682366,"\int F^{a+b x} f^{(3)}(x) \, dx","Int[F^(a + b*x)*Derivative[3][f][x],x]","\int F^{a+b x} f^{(3)}(x) \, dx","-b^3 \log ^3(F) \text{Int}\left(f(x) F^{a+b x},x\right)+b^2 f(x) \log ^2(F) F^{a+b x}+f''(x) F^{a+b x}-b \log (F) f'(x) F^{a+b x}",0,"b^2*F^(a + b*x)*f[x]*Log[F]^2 - b^3*Log[F]^3*Defer[Int][F^(a + b*x)*f[x], x] - b*F^(a + b*x)*Log[F]*Derivative[1][f][x] + F^(a + b*x)*Derivative[2][f][x]","N/A",0,0,0,0,-1,"{}"
45,0,0,0,0.0450604,"\int F^{a+b x} f''(x) \, dx","Int[F^(a + b*x)*Derivative[2][f][x],x]","\int F^{a+b x} f''(x) \, dx","b^2 \log ^2(F) \text{Int}\left(f(x) F^{a+b x},x\right)+f'(x) F^{a+b x}-b f(x) \log (F) F^{a+b x}",0,"-(b*F^(a + b*x)*f[x]*Log[F]) + b^2*Log[F]^2*Defer[Int][F^(a + b*x)*f[x], x] + F^(a + b*x)*Derivative[1][f][x]","N/A",0,0,0,0,-1,"{}"
46,0,0,0,0.0324435,"\int F^{a+b x} f'(x) \, dx","Int[F^(a + b*x)*Derivative[1][f][x],x]","\int F^{a+b x} f'(x) \, dx","f(x) F^{a+b x}-b \log (F) \text{Int}\left(f(x) F^{a+b x},x\right)",0,"F^(a + b*x)*f[x] - b*Log[F]*Defer[Int][F^(a + b*x)*f[x], x]","N/A",0,0,0,0,-1,"{}"
47,0,0,0,0.017011,"\int F^{a+b x} f(x) \, dx","Int[F^(a + b*x)*f[x],x]","\int F^{a+b x} f(x) \, dx","\text{Int}\left(f(x) F^{a+b x},x\right)",0,"Defer[Int][F^(a + b*x)*f[x], x]","N/A",0,0,0,0,-1,"{}"
48,0,0,0,0.0348715,"\int F^{a+b x} f^{(-1)}(x) \, dx","Int[F^(a + b*x)*Derivative[-1][f][x],x]","\int F^{a+b x} f^{(-1)}(x) \, dx","\frac{f^{(-1)}(x) F^{a+b x}}{b \log (F)}-\frac{\text{Int}\left(f(x) F^{a+b x},x\right)}{b \log (F)}",0,"-(Defer[Int][F^(a + b*x)*f[x], x]/(b*Log[F])) + (F^(a + b*x)*Derivative[-1][f][x])/(b*Log[F])","N/A",0,0,0,0,-1,"{}"
49,0,0,0,0.0538706,"\int F^{a+b x} f^{(-2)}(x) \, dx","Int[F^(a + b*x)*Derivative[-2][f][x],x]","\int F^{a+b x} f^{(-2)}(x) \, dx","\frac{\text{Int}\left(f(x) F^{a+b x},x\right)}{b^2 \log ^2(F)}-\frac{f^{(-1)}(x) F^{a+b x}}{b^2 \log ^2(F)}+\frac{f^{(-2)}(x) F^{a+b x}}{b \log (F)}",0,"Defer[Int][F^(a + b*x)*f[x], x]/(b^2*Log[F]^2) + (F^(a + b*x)*Derivative[-2][f][x])/(b*Log[F]) - (F^(a + b*x)*Derivative[-1][f][x])/(b^2*Log[F]^2)","N/A",0,0,0,0,-1,"{}"
50,0,0,0,0.0755787,"\int F^{a+b x} f^{(-3)}(x) \, dx","Int[F^(a + b*x)*Derivative[-3][f][x],x]","\int F^{a+b x} f^{(-3)}(x) \, dx","-\frac{\text{Int}\left(f(x) F^{a+b x},x\right)}{b^3 \log ^3(F)}+\frac{f^{(-1)}(x) F^{a+b x}}{b^3 \log ^3(F)}-\frac{f^{(-2)}(x) F^{a+b x}}{b^2 \log ^2(F)}+\frac{f^{(-3)}(x) F^{a+b x}}{b \log (F)}",0,"-(Defer[Int][F^(a + b*x)*f[x], x]/(b^3*Log[F]^3)) + (F^(a + b*x)*Derivative[-3][f][x])/(b*Log[F]) - (F^(a + b*x)*Derivative[-2][f][x])/(b^2*Log[F]^2) + (F^(a + b*x)*Derivative[-1][f][x])/(b^3*Log[F]^3)","N/A",0,0,0,0,-1,"{}"
51,1,46,0,0.0975617,"\int \left(b^3 F^{a+b x} f(x) \log ^3(F)+F^{a+b x} f^{(3)}(x)\right) \, dx","Int[b^3*F^(a + b*x)*f[x]*Log[F]^3 + F^(a + b*x)*Derivative[3][f][x],x]","b^2 f(x) \log ^2(F) F^{a+b x}+f''(x) F^{a+b x}-b \log (F) f'(x) F^{a+b x}","b^2 f(x) \log ^2(F) F^{a+b x}+f''(x) F^{a+b x}-b \log (F) f'(x) F^{a+b x}",1,"b^2*F^(a + b*x)*f[x]*Log[F]^2 - b*F^(a + b*x)*Log[F]*Derivative[1][f][x] + F^(a + b*x)*Derivative[2][f][x]","A",4,1,30,0.03333,1,"{6657}"
52,0,0,0,0.054083,"\int \sin (a+b x) f^{(3)}(x) \, dx","Int[Sin[a + b*x]*Derivative[3][f][x],x]","\int \sin (a+b x) f^{(3)}(x) \, dx","b^3 \text{Int}(f(x) \cos (a+b x),x)-b^2 f(x) \sin (a+b x)+f''(x) \sin (a+b x)-b f'(x) \cos (a+b x)",0,"-(b^2*f[x]*Sin[a + b*x]) + b^3*Defer[Int][Cos[a + b*x]*f[x], x] - b*Cos[a + b*x]*Derivative[1][f][x] + Sin[a + b*x]*Derivative[2][f][x]","N/A",0,0,0,0,-1,"{}"
53,0,0,0,0.0352182,"\int \sin (a+b x) f''(x) \, dx","Int[Sin[a + b*x]*Derivative[2][f][x],x]","\int \sin (a+b x) f''(x) \, dx","b^2 (-\text{Int}(f(x) \sin (a+b x),x))+f'(x) \sin (a+b x)-b f(x) \cos (a+b x)",0,"-(b*Cos[a + b*x]*f[x]) - b^2*Defer[Int][f[x]*Sin[a + b*x], x] + Sin[a + b*x]*Derivative[1][f][x]","N/A",0,0,0,0,-1,"{}"
54,0,0,0,0.0197632,"\int \sin (a+b x) f'(x) \, dx","Int[Sin[a + b*x]*Derivative[1][f][x],x]","\int \sin (a+b x) f'(x) \, dx","f(x) \sin (a+b x)-b \text{Int}(f(x) \cos (a+b x),x)",0,"f[x]*Sin[a + b*x] - b*Defer[Int][Cos[a + b*x]*f[x], x]","N/A",0,0,0,0,-1,"{}"
55,0,0,0,0.0106101,"\int f(x) \sin (a+b x) \, dx","Int[f[x]*Sin[a + b*x],x]","\int f(x) \sin (a+b x) \, dx","\text{Int}(f(x) \sin (a+b x),x)",0,"Defer[Int][f[x]*Sin[a + b*x], x]","N/A",0,0,0,0,-1,"{}"
56,0,0,0,0.0210863,"\int \sin (a+b x) f^{(-1)}(x) \, dx","Int[Sin[a + b*x]*Derivative[-1][f][x],x]","\int \sin (a+b x) f^{(-1)}(x) \, dx","\frac{\text{Int}(f(x) \cos (a+b x),x)}{b}-\frac{f^{(-1)}(x) \cos (a+b x)}{b}",0,"Defer[Int][Cos[a + b*x]*f[x], x]/b - (Cos[a + b*x]*Derivative[-1][f][x])/b","N/A",0,0,0,0,-1,"{}"
57,0,0,0,0.03126,"\int \sin (a+b x) f^{(-2)}(x) \, dx","Int[Sin[a + b*x]*Derivative[-2][f][x],x]","\int \sin (a+b x) f^{(-2)}(x) \, dx","-\frac{\text{Int}(f(x) \sin (a+b x),x)}{b^2}+\frac{f^{(-1)}(x) \sin (a+b x)}{b^2}-\frac{f^{(-2)}(x) \cos (a+b x)}{b}",0,"-(Defer[Int][f[x]*Sin[a + b*x], x]/b^2) - (Cos[a + b*x]*Derivative[-2][f][x])/b + (Sin[a + b*x]*Derivative[-1][f][x])/b^2","N/A",0,0,0,0,-1,"{}"
58,0,0,0,0.043501,"\int \sin (a+b x) f^{(-3)}(x) \, dx","Int[Sin[a + b*x]*Derivative[-3][f][x],x]","\int \sin (a+b x) f^{(-3)}(x) \, dx","-\frac{\text{Int}(f(x) \cos (a+b x),x)}{b^3}+\frac{f^{(-1)}(x) \cos (a+b x)}{b^3}+\frac{f^{(-2)}(x) \sin (a+b x)}{b^2}-\frac{f^{(-3)}(x) \cos (a+b x)}{b}",0,"-(Defer[Int][Cos[a + b*x]*f[x], x]/b^3) - (Cos[a + b*x]*Derivative[-3][f][x])/b + (Sin[a + b*x]*Derivative[-2][f][x])/b^2 + (Cos[a + b*x]*Derivative[-1][f][x])/b^3","N/A",0,0,0,0,-1,"{}"
59,1,38,0,0.0581796,"\int \left(-b^3 \cos (a+b x) f(x)+\sin (a+b x) f^{(3)}(x)\right) \, dx","Int[-(b^3*Cos[a + b*x]*f[x]) + Sin[a + b*x]*Derivative[3][f][x],x]","b^2 (-f(x)) \sin (a+b x)+f''(x) \sin (a+b x)-b f'(x) \cos (a+b x)","b^2 (-f(x)) \sin (a+b x)+f''(x) \sin (a+b x)-b f'(x) \cos (a+b x)",1,"-(b^2*f[x]*Sin[a + b*x]) - b*Cos[a + b*x]*Derivative[1][f][x] + Sin[a + b*x]*Derivative[2][f][x]","A",4,2,25,0.08000,1,"{6659, 6660}"
60,0,0,0,0.0410665,"\int \cos (a+b x) f^{(3)}(x) \, dx","Int[Cos[a + b*x]*Derivative[3][f][x],x]","\int \cos (a+b x) f^{(3)}(x) \, dx","b^3 (-\text{Int}(f(x) \sin (a+b x),x))-b^2 f(x) \cos (a+b x)+f''(x) \cos (a+b x)+b f'(x) \sin (a+b x)",0,"-(b^2*Cos[a + b*x]*f[x]) - b^3*Defer[Int][f[x]*Sin[a + b*x], x] + b*Sin[a + b*x]*Derivative[1][f][x] + Cos[a + b*x]*Derivative[2][f][x]","N/A",0,0,0,0,-1,"{}"
61,0,0,0,0.0316533,"\int \cos (a+b x) f''(x) \, dx","Int[Cos[a + b*x]*Derivative[2][f][x],x]","\int \cos (a+b x) f''(x) \, dx","b^2 (-\text{Int}(f(x) \cos (a+b x),x))+f'(x) \cos (a+b x)+b f(x) \sin (a+b x)",0,"b*f[x]*Sin[a + b*x] - b^2*Defer[Int][Cos[a + b*x]*f[x], x] + Cos[a + b*x]*Derivative[1][f][x]","N/A",0,0,0,0,-1,"{}"
62,0,0,0,0.0187428,"\int \cos (a+b x) f'(x) \, dx","Int[Cos[a + b*x]*Derivative[1][f][x],x]","\int \cos (a+b x) f'(x) \, dx","b \text{Int}(f(x) \sin (a+b x),x)+f(x) \cos (a+b x)",0,"Cos[a + b*x]*f[x] + b*Defer[Int][f[x]*Sin[a + b*x], x]","N/A",0,0,0,0,-1,"{}"
63,0,0,0,0.0116856,"\int \cos (a+b x) f(x) \, dx","Int[Cos[a + b*x]*f[x],x]","\int \cos (a+b x) f(x) \, dx","\text{Int}(f(x) \cos (a+b x),x)",0,"Defer[Int][Cos[a + b*x]*f[x], x]","N/A",0,0,0,0,-1,"{}"
64,0,0,0,0.0202369,"\int \cos (a+b x) f^{(-1)}(x) \, dx","Int[Cos[a + b*x]*Derivative[-1][f][x],x]","\int \cos (a+b x) f^{(-1)}(x) \, dx","\frac{f^{(-1)}(x) \sin (a+b x)}{b}-\frac{\text{Int}(f(x) \sin (a+b x),x)}{b}",0,"-(Defer[Int][f[x]*Sin[a + b*x], x]/b) + (Sin[a + b*x]*Derivative[-1][f][x])/b","N/A",0,0,0,0,-1,"{}"
65,0,0,0,0.0314118,"\int \cos (a+b x) f^{(-2)}(x) \, dx","Int[Cos[a + b*x]*Derivative[-2][f][x],x]","\int \cos (a+b x) f^{(-2)}(x) \, dx","-\frac{\text{Int}(f(x) \cos (a+b x),x)}{b^2}+\frac{f^{(-1)}(x) \cos (a+b x)}{b^2}+\frac{f^{(-2)}(x) \sin (a+b x)}{b}",0,"-(Defer[Int][Cos[a + b*x]*f[x], x]/b^2) + (Sin[a + b*x]*Derivative[-2][f][x])/b + (Cos[a + b*x]*Derivative[-1][f][x])/b^2","N/A",0,0,0,0,-1,"{}"
66,0,0,0,0.0453555,"\int \cos (a+b x) f^{(-3)}(x) \, dx","Int[Cos[a + b*x]*Derivative[-3][f][x],x]","\int \cos (a+b x) f^{(-3)}(x) \, dx","\frac{\text{Int}(f(x) \sin (a+b x),x)}{b^3}-\frac{f^{(-1)}(x) \sin (a+b x)}{b^3}+\frac{f^{(-2)}(x) \cos (a+b x)}{b^2}+\frac{f^{(-3)}(x) \sin (a+b x)}{b}",0,"Defer[Int][f[x]*Sin[a + b*x], x]/b^3 + (Sin[a + b*x]*Derivative[-3][f][x])/b + (Cos[a + b*x]*Derivative[-2][f][x])/b^2 - (Sin[a + b*x]*Derivative[-1][f][x])/b^3","N/A",0,0,0,0,-1,"{}"
67,1,29,0,0.0483885,"\int \left(\frac{\cos (a+b x) f(x)}{b^2}+\cos (a+b x) f^{(-2)}(x)\right) \, dx","Int[(Cos[a + b*x]*f[x])/b^2 + Cos[a + b*x]*Derivative[-2][f][x],x]","\frac{f^{(-1)}(x) \cos (a+b x)}{b^2}+\frac{f^{(-2)}(x) \sin (a+b x)}{b}","\frac{f^{(-1)}(x) \cos (a+b x)}{b^2}+\frac{f^{(-2)}(x) \sin (a+b x)}{b}",1,"(Sin[a + b*x]*Derivative[-2][f][x])/b + (Cos[a + b*x]*Derivative[-1][f][x])/b^2","A",3,2,24,0.08333,1,"{6662, 6661}"
68,1,6,0,0.0156386,"\int \cos (f(x) g(x)) \left(g(x) f'(x)+f(x) g'(x)\right) \, dx","Int[Cos[f[x]*g[x]]*(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x]),x]","\sin (f(x) g(x))","\sin (f(x) g(x))",1,"Sin[f[x]*g[x]]","A",2,2,22,0.09091,1,"{6664, 2637}"
69,1,8,0,0.0118512,"\int \cos \left(g(x) f^{(m)}(x)\right) \left(g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)\right) \, dx","Int[Cos[g[x]*Derivative[m][f][x]]*(Derivative[1][g][x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]),x]","\sin \left(g(x) f^{(m)}(x)\right)","\sin \left(g(x) f^{(m)}(x)\right)",1,"Sin[g[x]*Derivative[m][f][x]]","A",2,2,28,0.07143,1,"{6665, 2637}"
70,1,14,0,0.0129398,"\int \cos \left(f^{(-1+m)}(x) g^{(-1+n)}(x)\right) \left(f^{(m)}(x) g^{(-1+n)}(x)+f^{(-1+m)}(x) g^{(n)}(x)\right) \, dx","Int[Cos[Derivative[-1 + m][f][x]*Derivative[-1 + n][g][x]]*(Derivative[m][f][x]*Derivative[-1 + n][g][x] + Derivative[-1 + m][f][x]*Derivative[n][g][x]),x]","\sin \left(f^{(m-1)}(x) g^{(n-1)}(x)\right)","\sin \left(f^{(m-1)}(x) g^{(n-1)}(x)\right)",1,"Sin[Derivative[-1 + m][f][x]*Derivative[-1 + n][g][x]]","A",2,2,38,0.05263,1,"{6666, 2637}"
71,1,27,0,0.0306299,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^2 g(x)^2} \, dx","Int[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^2*g[x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x) g(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x) g(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*f[x]*g[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,30,0.06667,1,"{6664, 205}"
72,1,29,0,0.0286016,"\int \frac{g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)}{a+b g(x)^2 f^{(m)}(x)^2} \, dx","Int[(Derivative[1][g][x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x])/(a + b*g[x]^2*Derivative[m][f][x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x) f^{(m)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x) f^{(m)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*g[x]*Derivative[m][f][x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,36,0.05556,1,"{6665, 205}"
73,1,31,0,0.0295801,"\int \frac{f^{(1+m)}(x) g^{(n)}(x)+f^{(m)}(x) g^{(1+n)}(x)}{a+b f^{(m)}(x)^2 g^{(n)}(x)^2} \, dx","Int[(Derivative[1 + m][f][x]*Derivative[n][g][x] + Derivative[m][f][x]*Derivative[1 + n][g][x])/(a + b*Derivative[m][f][x]^2*Derivative[n][g][x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f^{(m)}(x) g^{(n)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f^{(m)}(x) g^{(n)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*Derivative[m][f][x]*Derivative[n][g][x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,42,0.04762,1,"{6666, 205}"
74,1,6,0,0.0112031,"\int F'(f(x) g(x)) \left(g(x) f'(x)+f(x) g'(x)\right) \, dx","Int[Derivative[1][F][f[x]*g[x]]*(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x]),x]","F(f(x) g(x))","F(f(x) g(x))",1,"F[f[x]*g[x]]","A",2,2,24,0.08333,1,"{6664, 6656}"
75,1,8,0,0.009394,"\int F'\left(g(x) f^{(m)}(x)\right) \left(g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)\right) \, dx","Int[Derivative[1][F][g[x]*Derivative[m][f][x]]*(Derivative[1][g][x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]),x]","F\left(g(x) f^{(m)}(x)\right)","F\left(g(x) f^{(m)}(x)\right)",1,"F[g[x]*Derivative[m][f][x]]","A",2,2,30,0.06667,1,"{6665, 6656}"
76,1,14,0,0.0114575,"\int F'\left(f^{(-1+m)}(x) g^{(-1+n)}(x)\right) \left(f^{(m)}(x) g^{(-1+n)}(x)+f^{(-1+m)}(x) g^{(n)}(x)\right) \, dx","Int[Derivative[1][F][Derivative[-1 + m][f][x]*Derivative[-1 + n][g][x]]*(Derivative[m][f][x]*Derivative[-1 + n][g][x] + Derivative[-1 + m][f][x]*Derivative[n][g][x]),x]","F\left(f^{(m-1)}(x) g^{(n-1)}(x)\right)","F\left(f^{(m-1)}(x) g^{(n-1)}(x)\right)",1,"F[Derivative[-1 + m][f][x]*Derivative[-1 + n][g][x]]","A",2,2,40,0.05000,1,"{6666, 6656}"
77,1,8,0,0.0206322,"\int \cos \left(f(x)^2 g(x)\right) f(x) \left(2 g(x) f'(x)+f(x) g'(x)\right) \, dx","Int[Cos[f[x]^2*g[x]]*f[x]*(2*g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x]),x]","\sin \left(f(x)^2 g(x)\right)","\sin \left(f(x)^2 g(x)\right)",1,"Sin[f[x]^2*g[x]]","A",2,2,27,0.07407,1,"{6667, 2637}"
78,1,10,0,0.0204393,"\int \cos \left(g(x)^2 f^{(m)}(x)\right) g(x) \left(2 g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)\right) \, dx","Int[Cos[g[x]^2*Derivative[m][f][x]]*g[x]*(2*Derivative[1][g][x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]),x]","\sin \left(g(x)^2 f^{(m)}(x)\right)","\sin \left(g(x)^2 f^{(m)}(x)\right)",1,"Sin[g[x]^2*Derivative[m][f][x]]","A",2,2,33,0.06061,1,"{6669, 2637}"
79,1,10,0,0.0198951,"\int \cos \left(g(x) f^{(m)}(x)^2\right) f^{(m)}(x) \left(g'(x) f^{(m)}(x)+2 g(x) f^{(1+m)}(x)\right) \, dx","Int[Cos[g[x]*Derivative[m][f][x]^2]*Derivative[m][f][x]*(Derivative[1][g][x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]),x]","\sin \left(g(x) f^{(m)}(x)^2\right)","\sin \left(g(x) f^{(m)}(x)^2\right)",1,"Sin[g[x]*Derivative[m][f][x]^2]","A",2,2,35,0.05714,1,"{6668, 2637}"
80,1,16,0,0.0224152,"\int \cos \left(f^{(-1+m)}(x)^2 g^{(-1+n)}(x)\right) f^{(-1+m)}(x) \left(2 f^{(m)}(x) g^{(-1+n)}(x)+f^{(-1+m)}(x) g^{(n)}(x)\right) \, dx","Int[Cos[Derivative[-1 + m][f][x]^2*Derivative[-1 + n][g][x]]*Derivative[-1 + m][f][x]*(2*Derivative[m][f][x]*Derivative[-1 + n][g][x] + Derivative[-1 + m][f][x]*Derivative[n][g][x]),x]","\sin \left(f^{(m-1)}(x)^2 g^{(n-1)}(x)\right)","\sin \left(f^{(m-1)}(x)^2 g^{(n-1)}(x)\right)",1,"Sin[Derivative[-1 + m][f][x]^2*Derivative[-1 + n][g][x]]","A",2,2,47,0.04255,1,"{6670, 2637}"
81,1,29,0,0.0391728,"\int \frac{f(x) \left(2 g(x) f'(x)+f(x) g'(x)\right)}{a+b f(x)^4 g(x)^2} \, dx","Int[(f[x]*(2*g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x]))/(a + b*f[x]^4*g[x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x)^2 g(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x)^2 g(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*f[x]^2*g[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,33,0.06061,1,"{6667, 205}"
82,1,31,0,0.039147,"\int \frac{g(x) \left(2 g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)\right)}{a+b g(x)^4 f^{(m)}(x)^2} \, dx","Int[(g[x]*(2*Derivative[1][g][x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]))/(a + b*g[x]^4*Derivative[m][f][x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x)^2 f^{(m)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x)^2 f^{(m)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*g[x]^2*Derivative[m][f][x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,39,0.05128,1,"{6669, 205}"
83,1,31,0,0.0399527,"\int \frac{f^{(m)}(x) \left(g'(x) f^{(m)}(x)+2 g(x) f^{(1+m)}(x)\right)}{a+b g(x)^2 f^{(m)}(x)^4} \, dx","Int[(Derivative[m][f][x]*(Derivative[1][g][x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]))/(a + b*g[x]^2*Derivative[m][f][x]^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x) f^{(m)}(x)^2}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x) f^{(m)}(x)^2}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*g[x]*Derivative[m][f][x]^2)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,41,0.04878,1,"{6668, 205}"
84,1,33,0,0.0415156,"\int \frac{f^{(m)}(x) \left(2 f^{(1+m)}(x) g^{(n)}(x)+f^{(m)}(x) g^{(1+n)}(x)\right)}{a+b f^{(m)}(x)^4 g^{(n)}(x)^2} \, dx","Int[(Derivative[m][f][x]*(2*Derivative[1 + m][f][x]*Derivative[n][g][x] + Derivative[m][f][x]*Derivative[1 + n][g][x]))/(a + b*Derivative[m][f][x]^4*Derivative[n][g][x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f^{(m)}(x)^2 g^{(n)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f^{(m)}(x)^2 g^{(n)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*Derivative[m][f][x]^2*Derivative[n][g][x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,47,0.04255,1,"{6670, 205}"
85,1,8,0,0.0182497,"\int f(x) F'\left(f(x)^2 g(x)\right) \left(2 g(x) f'(x)+f(x) g'(x)\right) \, dx","Int[f[x]*Derivative[1][F][f[x]^2*g[x]]*(2*g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x]),x]","F\left(f(x)^2 g(x)\right)","F\left(f(x)^2 g(x)\right)",1,"F[f[x]^2*g[x]]","A",2,2,29,0.06897,1,"{6667, 6656}"
86,1,10,0,0.0181497,"\int g(x) F'\left(g(x)^2 f^{(m)}(x)\right) \left(2 g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)\right) \, dx","Int[g[x]*Derivative[1][F][g[x]^2*Derivative[m][f][x]]*(2*Derivative[1][g][x]*Derivative[m][f][x] + g[x]*Derivative[1 + m][f][x]),x]","F\left(g(x)^2 f^{(m)}(x)\right)","F\left(g(x)^2 f^{(m)}(x)\right)",1,"F[g[x]^2*Derivative[m][f][x]]","A",2,2,35,0.05714,1,"{6669, 6656}"
87,1,10,0,0.0185325,"\int F'\left(g(x) f^{(m)}(x)^2\right) f^{(m)}(x) \left(g'(x) f^{(m)}(x)+2 g(x) f^{(1+m)}(x)\right) \, dx","Int[Derivative[1][F][g[x]*Derivative[m][f][x]^2]*Derivative[m][f][x]*(Derivative[1][g][x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]),x]","F\left(g(x) f^{(m)}(x)^2\right)","F\left(g(x) f^{(m)}(x)^2\right)",1,"F[g[x]*Derivative[m][f][x]^2]","A",2,2,37,0.05405,1,"{6668, 6656}"
88,1,16,0,0.0210389,"\int F'\left(f^{(-1+m)}(x)^2 g^{(-1+n)}(x)\right) f^{(-1+m)}(x) \left(2 f^{(m)}(x) g^{(-1+n)}(x)+f^{(-1+m)}(x) g^{(n)}(x)\right) \, dx","Int[Derivative[1][F][Derivative[-1 + m][f][x]^2*Derivative[-1 + n][g][x]]*Derivative[-1 + m][f][x]*(2*Derivative[m][f][x]*Derivative[-1 + n][g][x] + Derivative[-1 + m][f][x]*Derivative[n][g][x]),x]","F\left(f^{(m-1)}(x)^2 g^{(n-1)}(x)\right)","F\left(f^{(m-1)}(x)^2 g^{(n-1)}(x)\right)",1,"F[Derivative[-1 + m][f][x]^2*Derivative[-1 + n][g][x]]","A",2,2,49,0.04082,1,"{6670, 6656}"
89,1,10,0,0.0321748,"\int \cos \left(f(x)^2 g(x)^3\right) f(x) g(x)^2 \left(2 g(x) f'(x)+3 f(x) g'(x)\right) \, dx","Int[Cos[f[x]^2*g[x]^3]*f[x]*g[x]^2*(2*g[x]*Derivative[1][f][x] + 3*f[x]*Derivative[1][g][x]),x]","\sin \left(f(x)^2 g(x)^3\right)","\sin \left(f(x)^2 g(x)^3\right)",1,"Sin[f[x]^2*g[x]^3]","A",2,2,34,0.05882,1,"{6671, 2637}"
90,1,12,0,0.0347795,"\int \cos \left(g(x)^3 f^{(m)}(x)^2\right) g(x)^2 f^{(m)}(x) \left(3 g'(x) f^{(m)}(x)+2 g(x) f^{(1+m)}(x)\right) \, dx","Int[Cos[g[x]^3*Derivative[m][f][x]^2]*g[x]^2*Derivative[m][f][x]*(3*Derivative[1][g][x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]),x]","\sin \left(g(x)^3 f^{(m)}(x)^2\right)","\sin \left(g(x)^3 f^{(m)}(x)^2\right)",1,"Sin[g[x]^3*Derivative[m][f][x]^2]","A",2,2,42,0.04762,1,"{6672, 2637}"
91,1,14,0,0.0365007,"\int \cos \left(f^{(m)}(x)^2 g^{(n)}(x)^3\right) f^{(m)}(x) g^{(n)}(x)^2 \left(2 f^{(1+m)}(x) g^{(n)}(x)+3 f^{(m)}(x) g^{(1+n)}(x)\right) \, dx","Int[Cos[Derivative[m][f][x]^2*Derivative[n][g][x]^3]*Derivative[m][f][x]*Derivative[n][g][x]^2*(2*Derivative[1 + m][f][x]*Derivative[n][g][x] + 3*Derivative[m][f][x]*Derivative[1 + n][g][x]),x]","\sin \left(f^{(m)}(x)^2 g^{(n)}(x)^3\right)","\sin \left(f^{(m)}(x)^2 g^{(n)}(x)^3\right)",1,"Sin[Derivative[m][f][x]^2*Derivative[n][g][x]^3]","A",2,2,50,0.04000,1,"{6673, 2637}"
92,1,31,0,0.0703551,"\int \frac{f(x) g(x)^2 \left(2 g(x) f'(x)+3 f(x) g'(x)\right)}{a+b f(x)^4 g(x)^6} \, dx","Int[(f[x]*g[x]^2*(2*g[x]*Derivative[1][f][x] + 3*f[x]*Derivative[1][g][x]))/(a + b*f[x]^4*g[x]^6),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x)^2 g(x)^3}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x)^2 g(x)^3}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*f[x]^2*g[x]^3)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,38,0.05263,1,"{6671, 205}"
93,1,33,0,0.0693361,"\int \frac{g(x)^2 f^{(m)}(x) \left(3 g'(x) f^{(m)}(x)+2 g(x) f^{(1+m)}(x)\right)}{a+b g(x)^6 f^{(m)}(x)^4} \, dx","Int[(g[x]^2*Derivative[m][f][x]*(3*Derivative[1][g][x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]))/(a + b*g[x]^6*Derivative[m][f][x]^4),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x)^3 f^{(m)}(x)^2}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x)^3 f^{(m)}(x)^2}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*g[x]^3*Derivative[m][f][x]^2)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,46,0.04348,1,"{6672, 205}"
94,1,35,0,0.071932,"\int \frac{f^{(m)}(x) g^{(n)}(x)^2 \left(2 f^{(1+m)}(x) g^{(n)}(x)+3 f^{(m)}(x) g^{(1+n)}(x)\right)}{a+b f^{(m)}(x)^4 g^{(n)}(x)^6} \, dx","Int[(Derivative[m][f][x]*Derivative[n][g][x]^2*(2*Derivative[1 + m][f][x]*Derivative[n][g][x] + 3*Derivative[m][f][x]*Derivative[1 + n][g][x]))/(a + b*Derivative[m][f][x]^4*Derivative[n][g][x]^6),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f^{(m)}(x)^2 g^{(n)}(x)^3}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f^{(m)}(x)^2 g^{(n)}(x)^3}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"ArcTan[(Sqrt[b]*Derivative[m][f][x]^2*Derivative[n][g][x]^3)/Sqrt[a]]/(Sqrt[a]*Sqrt[b])","A",2,2,54,0.03704,1,"{6673, 205}"
95,1,10,0,0.0365821,"\int f(x) g(x)^2 F'\left(f(x)^2 g(x)^3\right) \left(2 g(x) f'(x)+3 f(x) g'(x)\right) \, dx","Int[f[x]*g[x]^2*Derivative[1][F][f[x]^2*g[x]^3]*(2*g[x]*Derivative[1][f][x] + 3*f[x]*Derivative[1][g][x]),x]","F\left(f(x)^2 g(x)^3\right)","F\left(f(x)^2 g(x)^3\right)",1,"F[f[x]^2*g[x]^3]","A",2,2,36,0.05556,1,"{6671, 6656}"
96,1,12,0,0.0324511,"\int g(x)^2 F'\left(g(x)^3 f^{(m)}(x)^2\right) f^{(m)}(x) \left(3 g'(x) f^{(m)}(x)+2 g(x) f^{(1+m)}(x)\right) \, dx","Int[g[x]^2*Derivative[1][F][g[x]^3*Derivative[m][f][x]^2]*Derivative[m][f][x]*(3*Derivative[1][g][x]*Derivative[m][f][x] + 2*g[x]*Derivative[1 + m][f][x]),x]","F\left(g(x)^3 f^{(m)}(x)^2\right)","F\left(g(x)^3 f^{(m)}(x)^2\right)",1,"F[g[x]^3*Derivative[m][f][x]^2]","A",2,2,44,0.04545,1,"{6672, 6656}"
97,1,14,0,0.0337159,"\int F'\left(f^{(m)}(x)^2 g^{(n)}(x)^3\right) f^{(m)}(x) g^{(n)}(x)^2 \left(2 f^{(1+m)}(x) g^{(n)}(x)+3 f^{(m)}(x) g^{(1+n)}(x)\right) \, dx","Int[Derivative[1][F][Derivative[m][f][x]^2*Derivative[n][g][x]^3]*Derivative[m][f][x]*Derivative[n][g][x]^2*(2*Derivative[1 + m][f][x]*Derivative[n][g][x] + 3*Derivative[m][f][x]*Derivative[1 + n][g][x]),x]","F\left(f^{(m)}(x)^2 g^{(n)}(x)^3\right)","F\left(f^{(m)}(x)^2 g^{(n)}(x)^3\right)",1,"F[Derivative[m][f][x]^2*Derivative[n][g][x]^3]","A",2,2,52,0.03846,1,"{6673, 6656}"