1,1,2,2,0.0010609,"\int f'(x) \, dx","Integrate[Derivative[1][f][x],x]","f(x)","f(x)",1,"f[x]","A",1
2,1,4,4,0.0003166,"\int f''(x) \, dx","Integrate[Derivative[2][f][x],x]","f'(x)","f'(x)",1,"Derivative[1][f][x]","A",1
3,1,4,4,0.000274,"\int f^{(3)}(x) \, dx","Integrate[Derivative[3][f][x],x]","f''(x)","f''(x)",1,"Derivative[2][f][x]","A",1
4,0,0,6,0.0006439,"\int f^{(n)}(x) \, dx","Integrate[Derivative[n][f][x],x]","\int f^{(n)}(x) \, dx","f^{(n-1)}(x)",1,"Integrate[Derivative[n][f][x], x]","A",-1
5,1,10,10,0.0401237,"\int u'(x) u''(x) \, dx","Integrate[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",1
6,1,3,3,0.0050707,"\int \frac{f'(x)}{f(x)} \, dx","Integrate[Derivative[1][f][x]/f[x],x]","\log (f(x))","\log (f(x))",1,"Log[f[x]]","A",1
7,1,11,11,0.0142474,"\int \frac{f'(x)}{a+b f(x)} \, dx","Integrate[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",1
8,1,8,8,0.0020275,"\int f(x) f'(x) \, dx","Integrate[f[x]*Derivative[1][f][x],x]","\frac{f(x)^2}{2}","\frac{f(x)^2}{2}",1,"f[x]^2/2","A",1
9,1,14,14,0.0013614,"\int (a+b f(x)) f'(x) \, dx","Integrate[(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",1
10,1,8,8,0.0017069,"\int \frac{f'(x)}{\sqrt{f(x)}} \, dx","Integrate[Derivative[1][f][x]/Sqrt[f[x]],x]","2 \sqrt{f(x)}","2 \sqrt{f(x)}",1,"2*Sqrt[f[x]]","A",1
11,1,15,15,0.0049968,"\int \frac{f'(x)}{\sqrt{a+b f(x)}} \, dx","Integrate[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",1
12,1,12,12,0.0025747,"\int f(x)^n f'(x) \, dx","Integrate[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",1
13,1,19,19,0.0058187,"\int (a+b f(x))^n f'(x) \, dx","Integrate[(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",1
14,1,5,5,0.0042123,"\int \frac{f''(x)}{f'(x)} \, dx","Integrate[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",1
15,1,13,13,0.0098025,"\int \frac{f''(x)}{a+b f'(x)} \, dx","Integrate[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",1
16,1,10,10,0.0010554,"\int f'(x) f''(x) \, dx","Integrate[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",1
17,1,18,18,0.0015426,"\int \left(a+b f'(x)\right) f''(x) \, dx","Integrate[(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",1
18,1,10,10,0.0017548,"\int \frac{f''(x)}{\sqrt{f'(x)}} \, dx","Integrate[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",1
19,1,17,17,0.005648,"\int \frac{f''(x)}{\sqrt{a+b f'(x)}} \, dx","Integrate[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",1
20,1,14,14,0.0023444,"\int f'(x)^n f''(x) \, dx","Integrate[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",1
21,1,21,21,0.0058856,"\int \left(a+b f'(x)\right)^n f''(x) \, dx","Integrate[(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",1
22,0,0,11,0.0267472,"\int f(g(x)) g'(x) \, dx","Integrate[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,"Integrate[f[g[x]]*Derivative[1][g][x], x]","F",-1
23,0,0,13,0.009843,"\int f\left(g'(x)\right) g''(x) \, dx","Integrate[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,"Integrate[f[Derivative[1][g][x]]*Derivative[2][g][x], x]","F",-1
24,1,5,5,0.0045688,"\int \left(g(x) f'(x)+f(x) g'(x)\right) \, dx","Integrate[g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x],x]","f(x) g(x)","f(x) g(x)",1,"f[x]*g[x]","A",1
25,1,7,7,0.0256964,"\int \frac{g(x) f'(x)-f(x) g'(x)}{g(x)^2} \, dx","Integrate[(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
26,1,9,8,0.0116197,"\int \frac{g(x) f'(x)-f(x) g'(x)}{f(x) g(x)} \, dx","Integrate[(g[x]*Derivative[1][f][x] - f[x]*Derivative[1][g][x])/(f[x]*g[x]),x]","\log (f(x))-\log (g(x))","\log \left(\frac{f(x)}{g(x)}\right)",1,"Log[f[x]] - Log[g[x]]","A",1
27,1,6,6,0.0382667,"\int \frac{g(x) f'(x)+f(x) g'(x)}{1+f(x)^2 g(x)^2} \, dx","Integrate[(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",1
28,1,10,8,0.0382645,"\int \frac{g(x) f'(x)-f(x) g'(x)}{f(x)^2+g(x)^2} \, dx","Integrate[(g[x]*Derivative[1][f][x] - f[x]*Derivative[1][g][x])/(f[x]^2 + g[x]^2),x]","-\tan ^{-1}\left(\frac{g(x)}{f(x)}\right)","\tan ^{-1}\left(\frac{f(x)}{g(x)}\right)",1,"-ArcTan[g[x]/f[x]]","A",1
29,1,8,8,0.0279472,"\int -\frac{g(x) f'(x)+f(x) g'(x)}{1+f(x)^2 g(x)^2} \, dx","Integrate[-((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",1
30,1,26,6,0.0561875,"\int \frac{g(x) f'(x)+f(x) g'(x)}{1-f(x)^2 g(x)^2} \, dx","Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(1 - f[x]^2*g[x]^2),x]","\frac{1}{2} \log (f(x) g(x)+1)-\frac{1}{2} \log (1-f(x) g(x))","\tanh ^{-1}(f(x) g(x))",1,"-Log[1 - f[x]*g[x]]/2 + Log[1 + f[x]*g[x]]/2","B",1
31,1,23,8,0.0557544,"\int \frac{-g(x) f'(x)+f(x) g'(x)}{f(x)^2-g(x)^2} \, dx","Integrate[(-(g[x]*Derivative[1][f][x]) + f[x]*Derivative[1][g][x])/(f[x]^2 - g[x]^2),x]","\frac{1}{2} \log (f(x)+g(x))-\frac{1}{2} \log (f(x)-g(x))","\tanh ^{-1}\left(\frac{f(x)}{g(x)}\right)",1,"-Log[f[x] - g[x]]/2 + Log[f[x] + g[x]]/2","B",1
32,1,34,10,0.3696333,"\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","Integrate[(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]","\frac{1}{2} \log \left(f(x)^m g(x)^n+1\right)-\frac{1}{2} \log \left(1-f(x)^m g(x)^n\right)","\tanh ^{-1}\left(f(x)^m g(x)^n\right)",1,"-Log[1 - f[x]^m*g[x]^n]/2 + Log[1 + f[x]^m*g[x]^n]/2","B",1
33,1,31,12,0.3707601,"\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","Integrate[(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]","\frac{1}{2} \log \left(f(x)^m+g(x)^n\right)-\frac{1}{2} \log \left(f(x)^m-g(x)^n\right)","\tanh ^{-1}\left(f(x)^{-m} g(x)^n\right)",1,"-Log[f[x]^m - g[x]^n]/2 + Log[f[x]^m + g[x]^n]/2","B",1
34,1,34,14,0.1847726,"\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","Integrate[(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]","\frac{1}{2} \log \left(f(x)^m g(x)^n+1\right)-\frac{1}{2} \log \left(1-f(x)^m g(x)^n\right)","\tanh ^{-1}\left(f(x)^{-m} g(x)^{-n}\right)",1,"-Log[1 - f[x]^m*g[x]^n]/2 + Log[1 + f[x]^m*g[x]^n]/2","B",1
35,1,13,13,0.0358276,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x) g(x)} \, dx","Integrate[(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",1
36,0,0,27,0.0490327,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^2 g(x)^2} \, dx","Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^2*g[x]^2),x]","\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^2 g(x)^2} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x) g(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^2*g[x]^2), x]","A",-1
37,0,0,129,0.0517402,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^3 g(x)^3} \, dx","Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^3*g[x]^3),x]","\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^3 g(x)^3} \, dx","-\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,"Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^3*g[x]^3), x]","A",-1
38,0,0,35,0.2584455,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b \sqrt{f(x) g(x)}} \, dx","Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*Sqrt[f[x]*g[x]]),x]","\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b \sqrt{f(x) g(x)}} \, dx","\frac{2 \sqrt{f(x) g(x)}}{b}-\frac{2 a \log \left(a+b \sqrt{f(x) g(x)}\right)}{b^2}",1,"Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*Sqrt[f[x]*g[x]]), x]","A",-1
39,0,0,140,0.217838,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b (f(x) g(x))^{3/2}} \, dx","Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*(f[x]*g[x])^(3/2)),x]","\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b (f(x) g(x))^{3/2}} \, dx","\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,"Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*(f[x]*g[x])^(3/2)), x]","A",-1
40,0,0,380,0.2439873,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b (f(x) g(x))^{5/2}} \, dx","Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*(f[x]*g[x])^(5/2)),x]","\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b (f(x) g(x))^{5/2}} \, dx","-\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,"Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*(f[x]*g[x])^(5/2)), x]","A",-1
41,0,0,31,0.264997,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b (f(x) g(x))^n} \, dx","Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*(f[x]*g[x])^n),x]","\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b (f(x) g(x))^n} \, dx","\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,"Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*(f[x]*g[x])^n), x]","A",-1
42,0,0,49,0.2207704,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^n g(x)^n} \, dx","Integrate[(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,"Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^n*g[x]^n), x]","F",-1
43,1,8,8,0.0620076,"\int \left(\cos (x) g\left(e^x\right) f'(\sin (x))+e^x f(\sin (x)) g'\left(e^x\right)\right) \, dx","Integrate[Cos[x]*g[E^x]*Derivative[1][f][Sin[x]] + E^x*f[Sin[x]]*Derivative[1][g][E^x],x]","g\left(e^x\right) f(\sin (x))","g\left(e^x\right) f(\sin (x))",1,"f[Sin[x]]*g[E^x]","A",1
44,0,0,68,0.0481855,"\int F^{a+b x} f^{(3)}(x) \, dx","Integrate[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,"Integrate[F^(a + b*x)*Derivative[3][f][x], x]","F",-1
45,0,0,48,0.0302201,"\int F^{a+b x} f''(x) \, dx","Integrate[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,"Integrate[F^(a + b*x)*Derivative[2][f][x], x]","F",-1
46,0,0,29,0.0202029,"\int F^{a+b x} f'(x) \, dx","Integrate[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,"Integrate[F^(a + b*x)*Derivative[1][f][x], x]","F",-1
47,0,0,13,0.0065524,"\int F^{a+b x} f(x) \, dx","Integrate[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,"Integrate[F^(a + b*x)*f[x], x]","F",-1
48,0,0,42,0.0157566,"\int F^{a+b x} f^{(-1)}(x) \, dx","Integrate[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,"Integrate[F^(a + b*x)*Derivative[-1][f][x], x]","F",-1
49,0,0,61,0.0157165,"\int F^{a+b x} f^{(-2)}(x) \, dx","Integrate[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,"Integrate[F^(a + b*x)*Derivative[-2][f][x], x]","F",-1
50,0,0,81,0.0148186,"\int F^{a+b x} f^{(-3)}(x) \, dx","Integrate[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,"Integrate[F^(a + b*x)*Derivative[-3][f][x], x]","F",-1
51,0,0,46,0.0821738,"\int \left(b^3 F^{a+b x} f(x) \log ^3(F)+F^{a+b x} f^{(3)}(x)\right) \, dx","Integrate[b^3*F^(a + b*x)*f[x]*Log[F]^3 + F^(a + b*x)*Derivative[3][f][x],x]","\int \left(b^3 F^{a+b x} f(x) \log ^3(F)+F^{a+b x} f^{(3)}(x)\right) \, dx","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,"Integrate[b^3*F^(a + b*x)*f[x]*Log[F]^3 + F^(a + b*x)*Derivative[3][f][x], x]","A",-1
52,0,0,54,0.0669415,"\int \sin (a+b x) f^{(3)}(x) \, dx","Integrate[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,"Integrate[Sin[a + b*x]*Derivative[3][f][x], x]","F",-1
53,0,0,40,0.0593751,"\int \sin (a+b x) f''(x) \, dx","Integrate[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,"Integrate[Sin[a + b*x]*Derivative[2][f][x], x]","F",-1
54,0,0,25,0.0469532,"\int \sin (a+b x) f'(x) \, dx","Integrate[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,"Integrate[Sin[a + b*x]*Derivative[1][f][x], x]","F",-1
55,0,0,12,0.0199296,"\int f(x) \sin (a+b x) \, dx","Integrate[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,"Integrate[f[x]*Sin[a + b*x], x]","F",-1
56,0,0,32,0.0238561,"\int \sin (a+b x) f^{(-1)}(x) \, dx","Integrate[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,"Integrate[Sin[a + b*x]*Derivative[-1][f][x], x]","F",-1
57,0,0,47,0.0228964,"\int \sin (a+b x) f^{(-2)}(x) \, dx","Integrate[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,"Integrate[Sin[a + b*x]*Derivative[-2][f][x], x]","F",-1
58,0,0,61,0.023155,"\int \sin (a+b x) f^{(-3)}(x) \, dx","Integrate[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,"Integrate[Sin[a + b*x]*Derivative[-3][f][x], x]","F",-1
59,0,0,38,0.1256584,"\int \left(-b^3 \cos (a+b x) f(x)+\sin (a+b x) f^{(3)}(x)\right) \, dx","Integrate[-(b^3*Cos[a + b*x]*f[x]) + Sin[a + b*x]*Derivative[3][f][x],x]","\int \left(-b^3 \cos (a+b x) f(x)+\sin (a+b x) f^{(3)}(x)\right) \, dx","b^2 (-f(x)) \sin (a+b x)+f''(x) \sin (a+b x)-b f'(x) \cos (a+b x)",1,"Integrate[-(b^3*Cos[a + b*x]*f[x]) + Sin[a + b*x]*Derivative[3][f][x], x]","A",-1
60,0,0,54,0.0457237,"\int \cos (a+b x) f^{(3)}(x) \, dx","Integrate[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,"Integrate[Cos[a + b*x]*Derivative[3][f][x], x]","F",-1
61,0,0,39,0.0279539,"\int \cos (a+b x) f''(x) \, dx","Integrate[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,"Integrate[Cos[a + b*x]*Derivative[2][f][x], x]","F",-1
62,0,0,24,0.0182276,"\int \cos (a+b x) f'(x) \, dx","Integrate[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,"Integrate[Cos[a + b*x]*Derivative[1][f][x], x]","F",-1
63,0,0,12,0.0087871,"\int \cos (a+b x) f(x) \, dx","Integrate[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,"Integrate[Cos[a + b*x]*f[x], x]","F",-1
64,0,0,32,0.0234654,"\int \cos (a+b x) f^{(-1)}(x) \, dx","Integrate[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,"Integrate[Cos[a + b*x]*Derivative[-1][f][x], x]","F",-1
65,0,0,46,0.0243373,"\int \cos (a+b x) f^{(-2)}(x) \, dx","Integrate[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,"Integrate[Cos[a + b*x]*Derivative[-2][f][x], x]","F",-1
66,0,0,60,0.024357,"\int \cos (a+b x) f^{(-3)}(x) \, dx","Integrate[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,"Integrate[Cos[a + b*x]*Derivative[-3][f][x], x]","F",-1
67,0,0,29,0.0932639,"\int \left(\frac{\cos (a+b x) f(x)}{b^2}+\cos (a+b x) f^{(-2)}(x)\right) \, dx","Integrate[(Cos[a + b*x]*f[x])/b^2 + Cos[a + b*x]*Derivative[-2][f][x],x]","\int \left(\frac{\cos (a+b x) f(x)}{b^2}+\cos (a+b x) f^{(-2)}(x)\right) \, dx","\frac{f^{(-1)}(x) \cos (a+b x)}{b^2}+\frac{f^{(-2)}(x) \sin (a+b x)}{b}",1,"Integrate[(Cos[a + b*x]*f[x])/b^2 + Cos[a + b*x]*Derivative[-2][f][x], x]","A",-1
68,1,6,6,0.0388816,"\int \cos (f(x) g(x)) \left(g(x) f'(x)+f(x) g'(x)\right) \, dx","Integrate[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",1
69,1,8,8,0.0398692,"\int \cos \left(g(x) f^{(m)}(x)\right) \left(g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)\right) \, dx","Integrate[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",1
70,1,14,14,0.0486969,"\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","Integrate[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",1
71,0,0,27,0.0490526,"\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^2 g(x)^2} \, dx","Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^2*g[x]^2),x]","\int \frac{g(x) f'(x)+f(x) g'(x)}{a+b f(x)^2 g(x)^2} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x) g(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x])/(a + b*f[x]^2*g[x]^2), x]","A",-1
72,0,0,29,0.0541252,"\int \frac{g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)}{a+b g(x)^2 f^{(m)}(x)^2} \, dx","Integrate[(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]","\int \frac{g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)}{a+b g(x)^2 f^{(m)}(x)^2} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x) f^{(m)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(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]","A",-1
73,0,0,31,0.0552083,"\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","Integrate[(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]","\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","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f^{(m)}(x) g^{(n)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(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]","A",-1
74,1,6,6,0.0420344,"\int F'(f(x) g(x)) \left(g(x) f'(x)+f(x) g'(x)\right) \, dx","Integrate[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",1
75,1,8,8,0.0446393,"\int F'\left(g(x) f^{(m)}(x)\right) \left(g'(x) f^{(m)}(x)+g(x) f^{(1+m)}(x)\right) \, dx","Integrate[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",1
76,1,14,14,0.0404031,"\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","Integrate[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",1
77,1,8,8,0.0605168,"\int \cos \left(f(x)^2 g(x)\right) f(x) \left(2 g(x) f'(x)+f(x) g'(x)\right) \, dx","Integrate[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",1
78,1,10,10,0.0756676,"\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","Integrate[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",1
79,1,10,10,0.068497,"\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","Integrate[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",1
80,1,16,16,0.0717842,"\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","Integrate[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",1
81,0,0,29,0.0741464,"\int \frac{f(x) \left(2 g(x) f'(x)+f(x) g'(x)\right)}{a+b f(x)^4 g(x)^2} \, dx","Integrate[(f[x]*(2*g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x]))/(a + b*f[x]^4*g[x]^2),x]","\int \frac{f(x) \left(2 g(x) f'(x)+f(x) g'(x)\right)}{a+b f(x)^4 g(x)^2} \, dx","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x)^2 g(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(f[x]*(2*g[x]*Derivative[1][f][x] + f[x]*Derivative[1][g][x]))/(a + b*f[x]^4*g[x]^2), x]","A",-1
82,0,0,31,0.0661914,"\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","Integrate[(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]","\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","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x)^2 f^{(m)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(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]","A",-1
83,0,0,31,0.0661706,"\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","Integrate[(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]","\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","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x) f^{(m)}(x)^2}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(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]","A",-1
84,0,0,33,0.0710446,"\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","Integrate[(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]","\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","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f^{(m)}(x)^2 g^{(n)}(x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(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]","A",-1
85,1,8,8,0.0531225,"\int f(x) F'\left(f(x)^2 g(x)\right) \left(2 g(x) f'(x)+f(x) g'(x)\right) \, dx","Integrate[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",1
86,1,10,10,0.0533696,"\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","Integrate[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",1
87,1,10,10,0.0515788,"\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","Integrate[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",1
88,1,16,16,0.0580141,"\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","Integrate[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",1
89,1,10,10,0.085549,"\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","Integrate[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",1
90,1,12,12,0.0891452,"\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","Integrate[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",1
91,1,14,14,0.0976595,"\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","Integrate[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",1
92,0,0,31,0.0901167,"\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","Integrate[(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]","\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","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f(x)^2 g(x)^3}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(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]","A",-1
93,0,0,33,0.0888543,"\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","Integrate[(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]","\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","\frac{\tan ^{-1}\left(\frac{\sqrt{b} g(x)^3 f^{(m)}(x)^2}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(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]","A",-1
94,0,0,35,0.0879958,"\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","Integrate[(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]","\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","\frac{\tan ^{-1}\left(\frac{\sqrt{b} f^{(m)}(x)^2 g^{(n)}(x)^3}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"Integrate[(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]","A",-1
95,1,10,10,0.0582916,"\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","Integrate[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",1
96,1,12,12,0.0647688,"\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","Integrate[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",1
97,1,14,14,0.0632367,"\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","Integrate[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",1