1,1,22,44,0.0359071,"\int \frac{2}{3-\cos (4+6 x)} \, dx","Integrate[2/(3 - Cos[4 + 6*x]),x]","\frac{\tan ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\sin (6 x+4)}{-\cos (6 x+4)+2 \sqrt{2}+3}\right)}{3 \sqrt{2}}",1,"ArcTan[Sqrt[2]*Tan[2 + 3*x]]/(3*Sqrt[2])","A",1
2,1,22,44,0.0268198,"\int \frac{2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx","Integrate[(2*Csc[4 + 6*x])/(-Cot[4 + 6*x] + 3*Csc[4 + 6*x]),x]","\frac{\tan ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\sin (6 x+4)}{-\cos (6 x+4)+2 \sqrt{2}+3}\right)}{3 \sqrt{2}}",1,"ArcTan[Sqrt[2]*Tan[2 + 3*x]]/(3*Sqrt[2])","A",1
3,1,22,48,0.0464998,"\int \frac{1}{1+\sin ^2(2+3 x)} \, dx","Integrate[(1 + Sin[2 + 3*x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\sin ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Sqrt[2]*Tan[2 + 3*x]]/(3*Sqrt[2])","A",1
4,1,22,48,0.0572583,"\int \frac{1}{2-\cos ^2(2+3 x)} \, dx","Integrate[(2 - Cos[2 + 3*x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\sin ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Sqrt[2]*Tan[2 + 3*x]]/(3*Sqrt[2])","A",1
5,1,22,48,0.024534,"\int \frac{1}{\cos ^2(2+3 x)+2 \sin ^2(2+3 x)} \, dx","Integrate[(Cos[2 + 3*x]^2 + 2*Sin[2 + 3*x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\sin ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Sqrt[2]*Tan[2 + 3*x]]/(3*Sqrt[2])","A",1
6,1,22,48,0.0182425,"\int \frac{\sec ^2(2+3 x)}{1+2 \tan ^2(2+3 x)} \, dx","Integrate[Sec[2 + 3*x]^2/(1 + 2*Tan[2 + 3*x]^2),x]","\frac{\tan ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\sin ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Sqrt[2]*Tan[2 + 3*x]]/(3*Sqrt[2])","A",1
7,1,22,48,0.0186744,"\int \frac{\csc ^2(2+3 x)}{2+\cot ^2(2+3 x)} \, dx","Integrate[Csc[2 + 3*x]^2/(2 + Cot[2 + 3*x]^2),x]","\frac{\tan ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\sin ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Sqrt[2]*Tan[2 + 3*x]]/(3*Sqrt[2])","A",1
8,1,22,60,0.0372546,"\int \frac{2}{1-3 \cos (4+6 x)} \, dx","Integrate[2/(1 - 3*Cos[4 + 6*x]),x]","-\frac{\tanh ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{\log \left(\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Sqrt[2]*Tan[2 + 3*x]]/Sqrt[2]","A",1
9,1,22,60,0.0387092,"\int \frac{2 \csc (4+6 x)}{-3 \cot (4+6 x)+\csc (4+6 x)} \, dx","Integrate[(2*Csc[4 + 6*x])/(-3*Cot[4 + 6*x] + Csc[4 + 6*x]),x]","-\frac{\tanh ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{\log \left(\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Sqrt[2]*Tan[2 + 3*x]]/Sqrt[2]","A",1
10,1,22,60,0.057678,"\int \frac{1}{-1+3 \sin ^2(2+3 x)} \, dx","Integrate[(-1 + 3*Sin[2 + 3*x]^2)^(-1),x]","-\frac{\tanh ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{\log \left(\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Sqrt[2]*Tan[2 + 3*x]]/Sqrt[2]","A",1
11,1,22,60,0.0688653,"\int \frac{1}{2-3 \cos ^2(2+3 x)} \, dx","Integrate[(2 - 3*Cos[2 + 3*x]^2)^(-1),x]","-\frac{\tanh ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{\log \left(\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Sqrt[2]*Tan[2 + 3*x]]/Sqrt[2]","A",1
12,1,22,60,0.0305578,"\int \frac{1}{-\cos ^2(2+3 x)+2 \sin ^2(2+3 x)} \, dx","Integrate[(-Cos[2 + 3*x]^2 + 2*Sin[2 + 3*x]^2)^(-1),x]","-\frac{\tanh ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{\log \left(\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Sqrt[2]*Tan[2 + 3*x]]/Sqrt[2]","A",1
13,1,22,60,0.0221604,"\int \frac{\sec ^2(2+3 x)}{-1+2 \tan ^2(2+3 x)} \, dx","Integrate[Sec[2 + 3*x]^2/(-1 + 2*Tan[2 + 3*x]^2),x]","-\frac{\tanh ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{\log \left(\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Sqrt[2]*Tan[2 + 3*x]]/Sqrt[2]","A",1
14,1,22,60,0.0311596,"\int \frac{\csc ^2(2+3 x)}{2-\cot ^2(2+3 x)} \, dx","Integrate[Csc[2 + 3*x]^2/(2 - Cot[2 + 3*x]^2),x]","-\frac{\tanh ^{-1}\left(\sqrt{2} \tan (3 x+2)\right)}{3 \sqrt{2}}","\frac{\log \left(\cos (3 x+2)-\sqrt{2} \sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sqrt{2} \sin (3 x+2)+\cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Sqrt[2]*Tan[2 + 3*x]]/Sqrt[2]","A",1
15,1,22,42,0.0294155,"\int \frac{2}{3+\cos (4+6 x)} \, dx","Integrate[2/(3 + Cos[4 + 6*x]),x]","\frac{\tan ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sin (6 x+4)}{\cos (6 x+4)+2 \sqrt{2}+3}\right)}{3 \sqrt{2}}",1,"ArcTan[Tan[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])","A",1
16,1,22,42,0.0214853,"\int \frac{2 \csc (4+6 x)}{\cot (4+6 x)+3 \csc (4+6 x)} \, dx","Integrate[(2*Csc[4 + 6*x])/(Cot[4 + 6*x] + 3*Csc[4 + 6*x]),x]","\frac{\tan ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sin (6 x+4)}{\cos (6 x+4)+2 \sqrt{2}+3}\right)}{3 \sqrt{2}}",1,"ArcTan[Tan[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])","A",1
17,1,22,48,0.0204748,"\int \frac{1}{2-\sin ^2(2+3 x)} \, dx","Integrate[(2 - Sin[2 + 3*x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\cos ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Tan[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])","A",1
18,1,22,48,0.0395005,"\int \frac{1}{1+\cos ^2(2+3 x)} \, dx","Integrate[(1 + Cos[2 + 3*x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\cos ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Tan[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])","A",1
19,1,22,48,0.0196051,"\int \frac{1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx","Integrate[(2*Cos[2 + 3*x]^2 + Sin[2 + 3*x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\cos ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Tan[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])","A",1
20,1,22,48,0.0177234,"\int \frac{\sec ^2(2+3 x)}{2+\tan ^2(2+3 x)} \, dx","Integrate[Sec[2 + 3*x]^2/(2 + Tan[2 + 3*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\cos ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Tan[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])","A",1
21,1,22,48,0.0182254,"\int \frac{\csc ^2(2+3 x)}{1+2 \cot ^2(2+3 x)} \, dx","Integrate[Csc[2 + 3*x]^2/(1 + 2*Cot[2 + 3*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sin (3 x+2) \cos (3 x+2)}{\cos ^2(3 x+2)+\sqrt{2}+1}\right)}{3 \sqrt{2}}",1,"ArcTan[Tan[2 + 3*x]/Sqrt[2]]/(3*Sqrt[2])","A",1
22,1,22,61,0.0263558,"\int -\frac{2}{1+3 \cos (4+6 x)} \, dx","Integrate[-2/(1 + 3*Cos[4 + 6*x]),x]","-\frac{\tanh ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{\log \left(\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Tan[2 + 3*x]/Sqrt[2]]/Sqrt[2]","A",1
23,1,22,61,0.0340648,"\int -\frac{2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx","Integrate[(-2*Csc[4 + 6*x])/(3*Cot[4 + 6*x] + Csc[4 + 6*x]),x]","-\frac{\tanh ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{\log \left(\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Tan[2 + 3*x]/Sqrt[2]]/Sqrt[2]","A",1
24,1,22,61,0.0640353,"\int \frac{1}{-2+3 \sin ^2(2+3 x)} \, dx","Integrate[(-2 + 3*Sin[2 + 3*x]^2)^(-1),x]","-\frac{\tanh ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{\log \left(\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Tan[2 + 3*x]/Sqrt[2]]/Sqrt[2]","A",1
25,1,22,61,0.0574232,"\int \frac{1}{1-3 \cos ^2(2+3 x)} \, dx","Integrate[(1 - 3*Cos[2 + 3*x]^2)^(-1),x]","-\frac{\tanh ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{\log \left(\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Tan[2 + 3*x]/Sqrt[2]]/Sqrt[2]","A",1
26,1,22,61,0.0290687,"\int \frac{1}{-2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx","Integrate[(-2*Cos[2 + 3*x]^2 + Sin[2 + 3*x]^2)^(-1),x]","-\frac{\tanh ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{\log \left(\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Tan[2 + 3*x]/Sqrt[2]]/Sqrt[2]","A",1
27,1,22,61,0.0206494,"\int \frac{\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx","Integrate[Sec[2 + 3*x]^2/(-2 + Tan[2 + 3*x]^2),x]","-\frac{\tanh ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{\log \left(\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Tan[2 + 3*x]/Sqrt[2]]/Sqrt[2]","A",1
28,1,22,61,0.0343977,"\int \frac{\csc ^2(2+3 x)}{1-2 \cot ^2(2+3 x)} \, dx","Integrate[Csc[2 + 3*x]^2/(1 - 2*Cot[2 + 3*x]^2),x]","-\frac{\tanh ^{-1}\left(\frac{\tan (3 x+2)}{\sqrt{2}}\right)}{3 \sqrt{2}}","\frac{\log \left(\sqrt{2} \cos (3 x+2)-\sin (3 x+2)\right)}{6 \sqrt{2}}-\frac{\log \left(\sin (3 x+2)+\sqrt{2} \cos (3 x+2)\right)}{6 \sqrt{2}}",1,"-1/3*ArcTanh[Tan[2 + 3*x]/Sqrt[2]]/Sqrt[2]","A",1
29,1,30,30,0.0614692,"\int (x+\sin (x))^2 \, dx","Integrate[(x + Sin[x])^2,x]","\frac{1}{6} x \left(2 x^2+3\right)+2 \sin (x)-\frac{1}{4} \sin (2 x)-2 x \cos (x)","\frac{x^3}{3}+\frac{x}{2}+2 \sin (x)-2 x \cos (x)-\frac{1}{2} \sin (x) \cos (x)",1,"(x*(3 + 2*x^2))/6 - 2*x*Cos[x] + 2*Sin[x] - Sin[2*x]/4","A",1
30,1,48,56,0.0904187,"\int (x+\sin (x))^3 \, dx","Integrate[(x + Sin[x])^3,x]","\frac{1}{24} \left(6 x \left(x^3+3 x+24 \sin (x)-3 \sin (2 x)\right)-18 \left(4 x^2-7\right) \cos (x)-9 \cos (2 x)+2 \cos (3 x)\right)","\frac{x^4}{4}+\frac{3 x^2}{4}-3 x^2 \cos (x)+\frac{3 \sin ^2(x)}{4}+6 x \sin (x)+\frac{\cos ^3(x)}{3}+5 \cos (x)-\frac{3}{2} x \sin (x) \cos (x)",1,"(-18*(-7 + 4*x^2)*Cos[x] - 9*Cos[2*x] + 2*Cos[3*x] + 6*x*(3*x + x^3 + 24*Sin[x] - 3*Sin[2*x]))/24","A",1
31,1,172,213,0.3224088,"\int \frac{\sin (a+b x)}{c+d x^2} \, dx","Integrate[Sin[a + b*x]/(c + d*x^2),x]","\frac{i \left(\sin \left(a-\frac{i b \sqrt{c}}{\sqrt{d}}\right) \text{Ci}\left(b \left(x+\frac{i \sqrt{c}}{\sqrt{d}}\right)\right)-\sin \left(a+\frac{i b \sqrt{c}}{\sqrt{d}}\right) \text{Ci}\left(b \left(x-\frac{i \sqrt{c}}{\sqrt{d}}\right)\right)+\cos \left(a-\frac{i b \sqrt{c}}{\sqrt{d}}\right) \text{Si}\left(b \left(x+\frac{i \sqrt{c}}{\sqrt{d}}\right)\right)+\cos \left(a+\frac{i b \sqrt{c}}{\sqrt{d}}\right) \text{Si}\left(\frac{i b \sqrt{c}}{\sqrt{d}}-b x\right)\right)}{2 \sqrt{c} \sqrt{d}}","-\frac{\sin \left(a-\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{Ci}\left(x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{\sin \left(a+\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{Ci}\left(\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos \left(a+\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{Si}\left(\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos \left(a-\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{Si}\left(x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right)}{2 \sqrt{-c} \sqrt{d}}",1,"((I/2)*(CosIntegral[b*((I*Sqrt[c])/Sqrt[d] + x)]*Sin[a - (I*b*Sqrt[c])/Sqrt[d]] - CosIntegral[b*(((-I)*Sqrt[c])/Sqrt[d] + x)]*Sin[a + (I*b*Sqrt[c])/Sqrt[d]] + Cos[a - (I*b*Sqrt[c])/Sqrt[d]]*SinIntegral[b*((I*Sqrt[c])/Sqrt[d] + x)] + Cos[a + (I*b*Sqrt[c])/Sqrt[d]]*SinIntegral[(I*b*Sqrt[c])/Sqrt[d] - b*x]))/(Sqrt[c]*Sqrt[d])","C",0
32,1,238,271,0.5640948,"\int \frac{\sin (a+b x)}{c+d x+e x^2} \, dx","Integrate[Sin[a + b*x]/(c + d*x + e*x^2),x]","\frac{\sin \left(a+\frac{b \left(\sqrt{d^2-4 c e}-d\right)}{2 e}\right) \text{Ci}\left(\frac{b \left(d+2 e x-\sqrt{d^2-4 c e}\right)}{2 e}\right)-\sin \left(a-\frac{b \left(\sqrt{d^2-4 c e}+d\right)}{2 e}\right) \text{Ci}\left(\frac{b \left(d+2 e x+\sqrt{d^2-4 c e}\right)}{2 e}\right)-\cos \left(a+\frac{b \left(\sqrt{d^2-4 c e}-d\right)}{2 e}\right) \text{Si}\left(\frac{b \left(\sqrt{d^2-4 c e}-d\right)}{2 e}-b x\right)-\cos \left(a-\frac{b \left(\sqrt{d^2-4 c e}+d\right)}{2 e}\right) \text{Si}\left(\frac{b \left(d+2 e x+\sqrt{d^2-4 c e}\right)}{2 e}\right)}{\sqrt{d^2-4 c e}}","\frac{\sin \left(a-\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}\right) \text{Ci}\left(\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}+b x\right)}{\sqrt{d^2-4 c e}}-\frac{\sin \left(a-\frac{b \left(\sqrt{d^2-4 c e}+d\right)}{2 e}\right) \text{Ci}\left(\frac{b \left(d+\sqrt{d^2-4 c e}\right)}{2 e}+b x\right)}{\sqrt{d^2-4 c e}}+\frac{\cos \left(a-\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}\right) \text{Si}\left(\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}+b x\right)}{\sqrt{d^2-4 c e}}-\frac{\cos \left(a-\frac{b \left(\sqrt{d^2-4 c e}+d\right)}{2 e}\right) \text{Si}\left(\frac{b \left(d+\sqrt{d^2-4 c e}\right)}{2 e}+b x\right)}{\sqrt{d^2-4 c e}}",1,"(CosIntegral[(b*(d - Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e)]*Sin[a + (b*(-d + Sqrt[d^2 - 4*c*e]))/(2*e)] - CosIntegral[(b*(d + Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e)]*Sin[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)] - Cos[a + (b*(-d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(-d + Sqrt[d^2 - 4*c*e]))/(2*e) - b*x] - Cos[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(d + Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e)])/Sqrt[d^2 - 4*c*e]","A",0
33,1,10,10,0.022762,"\int \frac{\sin \left(\sqrt{-7+x}\right)}{\sqrt{-7+x}} \, dx","Integrate[Sin[Sqrt[-7 + x]]/Sqrt[-7 + x],x]","-2 \cos \left(\sqrt{x-7}\right)","-2 \cos \left(\sqrt{x-7}\right)",1,"-2*Cos[Sqrt[-7 + x]]","A",1
34,1,46,28,0.7001706,"\int \frac{\sqrt{b-\frac{a}{x^2}} \sin (x)}{\sqrt{a-b x^2}} \, dx","Integrate[(Sqrt[b - a/x^2]*Sin[x])/Sqrt[a - b*x^2],x]","\frac{i x (\text{Ei}(-i x)-\text{Ei}(i x)) \sqrt{b-\frac{a}{x^2}}}{2 \sqrt{a-b x^2}}","\frac{x \text{Si}(x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}}",1,"((I/2)*Sqrt[b - a/x^2]*x*(ExpIntegralEi[(-I)*x] - ExpIntegralEi[I*x]))/Sqrt[a - b*x^2]","C",1
35,1,26,12,0.020005,"\int \frac{1}{x (1+\sin (\log (x)))} \, dx","Integrate[1/(x*(1 + Sin[Log[x]])),x]","\frac{2 \sin \left(\frac{\log (x)}{2}\right)}{\sin \left(\frac{\log (x)}{2}\right)+\cos \left(\frac{\log (x)}{2}\right)}","-\frac{\cos (\log (x))}{\sin (\log (x))+1}",1,"(2*Sin[Log[x]/2])/(Cos[Log[x]/2] + Sin[Log[x]/2])","B",1
36,1,272,100,5.3533284,"\int \sin \left(\frac{a+b x}{c+d x}\right) \, dx","Integrate[Sin[(a + b*x)/(c + d*x)],x]","\frac{2 \cos \left(\frac{b}{d}\right) (b c-a d) \text{Ci}\left(\frac{a d-b c}{d (c+d x)}\right)+d \exp \left(-\frac{i (a d+2 b c+b d x)}{d (c+d x)}\right) \left(i c \left(e^{\frac{2 i b c}{d (c+d x)}}-e^{2 i \left(\frac{a}{c+d x}+\frac{b}{d}\right)}\right)+d x \sin \left(\frac{b}{d}\right) \left(e^{i \left(\frac{2 a}{c+d x}+\frac{b}{d}\right)}+e^{\frac{i b (3 c+d x)}{d (c+d x)}}\right)+2 d x \cos \left(\frac{b}{d}\right) e^{\frac{i (a d+2 b c+b d x)}{d (c+d x)}} \sin \left(\frac{a d-b c}{d (c+d x)}\right)\right)-2 \sin \left(\frac{b}{d}\right) (b c-a d) \text{Si}\left(\frac{a d-b c}{d (c+d x)}\right)}{2 d^2}","\frac{\cos \left(\frac{b}{d}\right) (b c-a d) \text{Ci}\left(\frac{b c-a d}{d (c+d x)}\right)}{d^2}+\frac{\sin \left(\frac{b}{d}\right) (b c-a d) \text{Si}\left(\frac{b c-a d}{d (c+d x)}\right)}{d^2}+\frac{(c+d x) \sin \left(\frac{a+b x}{c+d x}\right)}{d}",1,"(2*(b*c - a*d)*Cos[b/d]*CosIntegral[(-(b*c) + a*d)/(d*(c + d*x))] + (d*(I*c*(E^(((2*I)*b*c)/(d*(c + d*x))) - E^((2*I)*(b/d + a/(c + d*x)))) + d*(E^((I*b*(3*c + d*x))/(d*(c + d*x))) + E^(I*(b/d + (2*a)/(c + d*x))))*x*Sin[b/d] + 2*d*E^((I*(2*b*c + a*d + b*d*x))/(d*(c + d*x)))*x*Cos[b/d]*Sin[(-(b*c) + a*d)/(d*(c + d*x))]))/E^((I*(2*b*c + a*d + b*d*x))/(d*(c + d*x))) - 2*(b*c - a*d)*Sin[b/d]*SinIntegral[(-(b*c) + a*d)/(d*(c + d*x))])/(2*d^2)","C",0
37,1,330,107,7.2697044,"\int \sin ^2\left(\frac{a+b x}{c+d x}\right) \, dx","Integrate[Sin[(a + b*x)/(c + d*x)]^2,x]","\frac{8 \sin \left(\frac{2 b}{d}\right) (b c-a d) \text{Ci}\left(\frac{2 (a d-b c)}{d (c+d x)}\right)-d \exp \left(-\frac{2 i (a d+2 b c+b d x)}{d (c+d x)}\right) \left(d x \left(-4 \exp \left(\frac{2 i (a d+2 b c+b d x)}{d (c+d x)}\right)+e^{4 i \left(\frac{a}{c+d x}+\frac{b}{d}\right)}+e^{\frac{4 i a}{c+d x}}+e^{\frac{4 i b c}{d (c+d x)}}+e^{\frac{4 i b (2 c+d x)}{d (c+d x)}}\right)-4 d x \sin \left(\frac{2 b}{d}\right) \exp \left(\frac{2 i (a d+2 b c+b d x)}{d (c+d x)}\right) \sin \left(\frac{2 (a d-b c)}{d (c+d x)}\right)+2 c \left(e^{4 i \left(\frac{a}{c+d x}+\frac{b}{d}\right)}+e^{\frac{4 i b c}{d (c+d x)}}\right)\right)+8 \cos \left(\frac{2 b}{d}\right) (b c-a d) \text{Si}\left(\frac{2 (a d-b c)}{d (c+d x)}\right)}{8 d^2}","\frac{\sin \left(\frac{2 b}{d}\right) (b c-a d) \text{Ci}\left(\frac{2 (b c-a d)}{d (c+d x)}\right)}{d^2}-\frac{\cos \left(\frac{2 b}{d}\right) (b c-a d) \text{Si}\left(\frac{2 (b c-a d)}{d (c+d x)}\right)}{d^2}+\frac{(c+d x) \sin ^2\left(\frac{a+b x}{c+d x}\right)}{d}",1,"(8*(b*c - a*d)*CosIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))]*Sin[(2*b)/d] - (d*(2*c*(E^(((4*I)*b*c)/(d*(c + d*x))) + E^((4*I)*(b/d + a/(c + d*x)))) + d*(E^(((4*I)*a)/(c + d*x)) + E^(((4*I)*b*c)/(d*(c + d*x))) + E^(((4*I)*b*(2*c + d*x))/(d*(c + d*x))) - 4*E^(((2*I)*(2*b*c + a*d + b*d*x))/(d*(c + d*x))) + E^((4*I)*(b/d + a/(c + d*x))))*x - 4*d*E^(((2*I)*(2*b*c + a*d + b*d*x))/(d*(c + d*x)))*x*Sin[(2*b)/d]*Sin[(2*(-(b*c) + a*d))/(d*(c + d*x))]))/E^(((2*I)*(2*b*c + a*d + b*d*x))/(d*(c + d*x))) + 8*(b*c - a*d)*Cos[(2*b)/d]*SinIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))])/(8*d^2)","C",0
38,1,657,194,7.7053532,"\int \sin ^3\left(\frac{a+b x}{c+d x}\right) \, dx","Integrate[Sin[(a + b*x)/(c + d*x)]^3,x]","-\frac{3 \left(a c d-b c^2\right) \left(\frac{i \left(1+e^{\frac{2 i b}{d}}\right) \left(e^{\frac{2 i b c}{d (c+d x)}}-e^{\frac{2 i a}{c+d x}}\right) \exp \left(-\frac{i (a d+2 b c+b d x)}{d (c+d x)}\right)}{4 (b c-a d)}-\frac{i \left(-1+e^{\frac{2 i b}{d}}\right) \left(e^{\frac{2 i a}{c+d x}}+e^{\frac{2 i b c}{d (c+d x)}}\right) \exp \left(-\frac{i (a d+2 b c+b d x)}{d (c+d x)}\right)}{4 (b c-a d)}\right)}{4 d}+\frac{3 \left(a c d-b c^2\right) \left(\frac{i \left(1+e^{\frac{6 i b}{d}}\right) \left(e^{\frac{6 i b c}{d (c+d x)}}-e^{\frac{6 i a}{c+d x}}\right) \exp \left(-\frac{3 i (a d+2 b c+b d x)}{d (c+d x)}\right)}{12 (b c-a d)}-\frac{i \left(-1+e^{\frac{6 i b}{d}}\right) \left(e^{\frac{6 i a}{c+d x}}+e^{\frac{6 i b c}{d (c+d x)}}\right) \exp \left(-\frac{3 i (a d+2 b c+b d x)}{d (c+d x)}\right)}{12 (b c-a d)}\right)}{4 d}+\frac{3 (a d-b c) \left(-\cos \left(\frac{b}{d}\right) \text{Ci}\left(\frac{a d-b c}{d (c+d x)}\right)+\cos \left(\frac{3 b}{d}\right) \text{Ci}\left(\frac{3 (a d-b c)}{d (c+d x)}\right)+\sin \left(\frac{b}{d}\right) \text{Si}\left(\frac{a d-b c}{d (c+d x)}\right)-\sin \left(\frac{3 b}{d}\right) \text{Si}\left(\frac{3 (a d-b c)}{d (c+d x)}\right)\right)}{4 d^2}+\frac{3}{4} x \sin \left(\frac{b}{d}\right) \cos \left(\frac{a d-b c}{d (c+d x)}\right)-\frac{1}{4} x \sin \left(\frac{3 b}{d}\right) \cos \left(\frac{3 (a d-b c)}{d (c+d x)}\right)+\frac{3}{4} x \cos \left(\frac{b}{d}\right) \sin \left(\frac{a d-b c}{d (c+d x)}\right)-\frac{1}{4} x \cos \left(\frac{3 b}{d}\right) \sin \left(\frac{3 (a d-b c)}{d (c+d x)}\right)","\frac{3 \cos \left(\frac{b}{d}\right) (b c-a d) \text{Ci}\left(\frac{b c-a d}{d (c+d x)}\right)}{4 d^2}-\frac{3 \cos \left(\frac{3 b}{d}\right) (b c-a d) \text{Ci}\left(\frac{3 (b c-a d)}{d (c+d x)}\right)}{4 d^2}+\frac{3 \sin \left(\frac{b}{d}\right) (b c-a d) \text{Si}\left(\frac{b c-a d}{d (c+d x)}\right)}{4 d^2}-\frac{3 \sin \left(\frac{3 b}{d}\right) (b c-a d) \text{Si}\left(\frac{3 (b c-a d)}{d (c+d x)}\right)}{4 d^2}+\frac{(c+d x) \sin ^3\left(\frac{a+b x}{c+d x}\right)}{d}",1,"(-3*(-(b*c^2) + a*c*d)*(((I/4)*(1 + E^(((2*I)*b)/d))*(-E^(((2*I)*a)/(c + d*x)) + E^(((2*I)*b*c)/(d*(c + d*x)))))/((b*c - a*d)*E^((I*(2*b*c + a*d + b*d*x))/(d*(c + d*x)))) - ((I/4)*(-1 + E^(((2*I)*b)/d))*(E^(((2*I)*a)/(c + d*x)) + E^(((2*I)*b*c)/(d*(c + d*x)))))/((b*c - a*d)*E^((I*(2*b*c + a*d + b*d*x))/(d*(c + d*x))))))/(4*d) + (3*(-(b*c^2) + a*c*d)*(((I/12)*(1 + E^(((6*I)*b)/d))*(-E^(((6*I)*a)/(c + d*x)) + E^(((6*I)*b*c)/(d*(c + d*x)))))/((b*c - a*d)*E^(((3*I)*(2*b*c + a*d + b*d*x))/(d*(c + d*x)))) - ((I/12)*(-1 + E^(((6*I)*b)/d))*(E^(((6*I)*a)/(c + d*x)) + E^(((6*I)*b*c)/(d*(c + d*x)))))/((b*c - a*d)*E^(((3*I)*(2*b*c + a*d + b*d*x))/(d*(c + d*x))))))/(4*d) + (3*x*Cos[(-(b*c) + a*d)/(d*(c + d*x))]*Sin[b/d])/4 - (x*Cos[(3*(-(b*c) + a*d))/(d*(c + d*x))]*Sin[(3*b)/d])/4 + (3*x*Cos[b/d]*Sin[(-(b*c) + a*d)/(d*(c + d*x))])/4 - (x*Cos[(3*b)/d]*Sin[(3*(-(b*c) + a*d))/(d*(c + d*x))])/4 + (3*(-(b*c) + a*d)*(-(Cos[b/d]*CosIntegral[(-(b*c) + a*d)/(d*(c + d*x))]) + Cos[(3*b)/d]*CosIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))] + Sin[b/d]*SinIntegral[(-(b*c) + a*d)/(d*(c + d*x))] - Sin[(3*b)/d]*SinIntegral[(3*(-(b*c) + a*d))/(d*(c + d*x))]))/(4*d^2)","C",0
39,1,53,58,0.1060493,"\int \frac{\sin ^3\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^3/(1 - a^2*x^2),x]","\frac{\text{Si}\left(\frac{3 \sqrt{1-a x}}{\sqrt{a x+1}}\right)-3 \text{Si}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}","\frac{\text{Si}\left(\frac{3 \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}-\frac{3 \text{Si}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}",1,"(-3*SinIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + SinIntegral[(3*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/(4*a)","A",1
40,1,57,58,0.0794845,"\int \frac{\sin ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2),x]","\frac{\text{Ci}\left(\frac{2 \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}-\frac{\log (1-a x)}{4 a}+\frac{\log (a x+1)}{4 a}","\frac{\text{Ci}\left(\frac{2 \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}-\frac{\log \left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}",1,"CosIntegral[(2*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(2*a) - Log[1 - a*x]/(4*a) + Log[1 + a*x]/(4*a)","A",1
41,1,26,26,0.0453471,"\int \frac{\sin \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(1 - a^2*x^2),x]","-\frac{\text{Si}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}","-\frac{\text{Si}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}",1,"-(SinIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)","A",1
42,0,0,40,5.9077428,"\int \frac{\csc \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Csc[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(1 - a^2*x^2),x]","\int \frac{\csc \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","\text{Int}\left(\frac{\csc \left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{(1-a x) (a x+1)},x\right)",0,"Integrate[Csc[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(1 - a^2*x^2), x]","A",-1
43,0,0,42,19.8948665,"\int \frac{\csc ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Csc[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2),x]","\int \frac{\csc ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","\text{Int}\left(\frac{\csc ^2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{(1-a x) (a x+1)},x\right)",0,"Integrate[Csc[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2), x]","A",-1
44,1,26,30,0.0755991,"\int (x+\cos (x))^2 \, dx","Integrate[(x + Cos[x])^2,x]","\frac{1}{6} \left(x \left(2 x^2+12 \sin (x)+3\right)+3 (\sin (x)+4) \cos (x)\right)","\frac{x^3}{3}+\frac{x}{2}+2 x \sin (x)+2 \cos (x)+\frac{1}{2} \sin (x) \cos (x)",1,"(3*Cos[x]*(4 + Sin[x]) + x*(3 + 2*x^2 + 12*Sin[x]))/6","A",1
45,1,51,56,0.1165595,"\int (x+\cos (x))^3 \, dx","Integrate[(x + Cos[x])^3,x]","\frac{1}{12} \left(3 x^4+9 x^2+9 \left(4 x^2-7\right) \sin (x)+9 x \sin (2 x)+\sin (3 x)\right)+6 x \cos (x)+\frac{3}{8} \cos (2 x)","\frac{x^4}{4}+\frac{3 x^2}{4}+3 x^2 \sin (x)-\frac{\sin ^3(x)}{3}-5 \sin (x)+\frac{3 \cos ^2(x)}{4}+6 x \cos (x)+\frac{3}{2} x \sin (x) \cos (x)",1,"6*x*Cos[x] + (3*Cos[2*x])/8 + (9*x^2 + 3*x^4 + 9*(-7 + 4*x^2)*Sin[x] + 9*x*Sin[2*x] + Sin[3*x])/12","A",1
46,1,172,213,0.3358748,"\int \frac{\cos (a+b x)}{c+d x^2} \, dx","Integrate[Cos[a + b*x]/(c + d*x^2),x]","-\frac{i \left(\cos \left(a+\frac{i b \sqrt{c}}{\sqrt{d}}\right) \text{Ci}\left(b \left(x-\frac{i \sqrt{c}}{\sqrt{d}}\right)\right)-\cos \left(a-\frac{i b \sqrt{c}}{\sqrt{d}}\right) \text{Ci}\left(b \left(x+\frac{i \sqrt{c}}{\sqrt{d}}\right)\right)+\sin \left(a-\frac{i b \sqrt{c}}{\sqrt{d}}\right) \text{Si}\left(b \left(x+\frac{i \sqrt{c}}{\sqrt{d}}\right)\right)+\sin \left(a+\frac{i b \sqrt{c}}{\sqrt{d}}\right) \text{Si}\left(\frac{i b \sqrt{c}}{\sqrt{d}}-b x\right)\right)}{2 \sqrt{c} \sqrt{d}}","\frac{\cos \left(a+\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{Ci}\left(\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right)}{2 \sqrt{-c} \sqrt{d}}-\frac{\cos \left(a-\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{Ci}\left(x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{\sin \left(a+\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{Si}\left(\frac{b \sqrt{-c}}{\sqrt{d}}-b x\right)}{2 \sqrt{-c} \sqrt{d}}+\frac{\sin \left(a-\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{Si}\left(x b+\frac{\sqrt{-c} b}{\sqrt{d}}\right)}{2 \sqrt{-c} \sqrt{d}}",1,"((-1/2*I)*(Cos[a + (I*b*Sqrt[c])/Sqrt[d]]*CosIntegral[b*(((-I)*Sqrt[c])/Sqrt[d] + x)] - Cos[a - (I*b*Sqrt[c])/Sqrt[d]]*CosIntegral[b*((I*Sqrt[c])/Sqrt[d] + x)] + Sin[a - (I*b*Sqrt[c])/Sqrt[d]]*SinIntegral[b*((I*Sqrt[c])/Sqrt[d] + x)] + Sin[a + (I*b*Sqrt[c])/Sqrt[d]]*SinIntegral[(I*b*Sqrt[c])/Sqrt[d] - b*x]))/(Sqrt[c]*Sqrt[d])","C",0
47,1,236,271,0.5815836,"\int \frac{\cos (a+b x)}{c+d x+e x^2} \, dx","Integrate[Cos[a + b*x]/(c + d*x + e*x^2),x]","\frac{\cos \left(a+\frac{b \left(\sqrt{d^2-4 c e}-d\right)}{2 e}\right) \text{Ci}\left(\frac{b \left(d+2 e x-\sqrt{d^2-4 c e}\right)}{2 e}\right)-\cos \left(a-\frac{b \left(\sqrt{d^2-4 c e}+d\right)}{2 e}\right) \text{Ci}\left(\frac{b \left(d+2 e x+\sqrt{d^2-4 c e}\right)}{2 e}\right)+\sin \left(a+\frac{b \left(\sqrt{d^2-4 c e}-d\right)}{2 e}\right) \text{Si}\left(\frac{b \left(\sqrt{d^2-4 c e}-d\right)}{2 e}-b x\right)+\sin \left(a-\frac{b \left(\sqrt{d^2-4 c e}+d\right)}{2 e}\right) \text{Si}\left(\frac{b \left(d+2 e x+\sqrt{d^2-4 c e}\right)}{2 e}\right)}{\sqrt{d^2-4 c e}}","\frac{\cos \left(a-\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}\right) \text{Ci}\left(\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}+b x\right)}{\sqrt{d^2-4 c e}}-\frac{\cos \left(a-\frac{b \left(\sqrt{d^2-4 c e}+d\right)}{2 e}\right) \text{Ci}\left(\frac{b \left(d+\sqrt{d^2-4 c e}\right)}{2 e}+b x\right)}{\sqrt{d^2-4 c e}}-\frac{\sin \left(a-\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}\right) \text{Si}\left(\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}+b x\right)}{\sqrt{d^2-4 c e}}+\frac{\sin \left(a-\frac{b \left(\sqrt{d^2-4 c e}+d\right)}{2 e}\right) \text{Si}\left(\frac{b \left(d+\sqrt{d^2-4 c e}\right)}{2 e}+b x\right)}{\sqrt{d^2-4 c e}}",1,"(Cos[a + (b*(-d + Sqrt[d^2 - 4*c*e]))/(2*e)]*CosIntegral[(b*(d - Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e)] - Cos[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*CosIntegral[(b*(d + Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e)] + Sin[a + (b*(-d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(-d + Sqrt[d^2 - 4*c*e]))/(2*e) - b*x] + Sin[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*SinIntegral[(b*(d + Sqrt[d^2 - 4*c*e] + 2*e*x))/(2*e)])/Sqrt[d^2 - 4*c*e]","A",0
48,1,10,10,0.0369539,"\int \frac{x \cos \left(\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \, dx","Integrate[(x*Cos[Sqrt[1 + x^2]])/Sqrt[1 + x^2],x]","\sin \left(\sqrt{x^2+1}\right)","\sin \left(\sqrt{x^2+1}\right)",1,"Sin[Sqrt[1 + x^2]]","A",1
49,1,22,22,0.0553052,"\int \frac{x \cos \left(\sqrt{3} \sqrt{2+x^2}\right)}{\sqrt{2+x^2}} \, dx","Integrate[(x*Cos[Sqrt[3]*Sqrt[2 + x^2]])/Sqrt[2 + x^2],x]","\frac{\sin \left(\sqrt{3} \sqrt{x^2+2}\right)}{\sqrt{3}}","\frac{\sin \left(\sqrt{3} \sqrt{x^2+2}\right)}{\sqrt{3}}",1,"Sin[Sqrt[3]*Sqrt[2 + x^2]]/Sqrt[3]","A",1
50,1,20,24,0.1727181,"\int \frac{(-1+2 x) \cos \left(\sqrt{6+3 (-1+2 x)^2}\right)}{\sqrt{6+3 (-1+2 x)^2}} \, dx","Integrate[((-1 + 2*x)*Cos[Sqrt[6 + 3*(-1 + 2*x)^2]])/Sqrt[6 + 3*(-1 + 2*x)^2],x]","\frac{1}{6} \sin \left(\sqrt{3 (1-2 x)^2+6}\right)","\frac{1}{6} \sin \left(\sqrt{3} \sqrt{(2 x-1)^2+2}\right)",1,"Sin[Sqrt[6 + 3*(1 - 2*x)^2]]/6","A",1
51,1,260,101,5.5101335,"\int \cos \left(\frac{a+b x}{c+d x}\right) \, dx","Integrate[Cos[(a + b*x)/(c + d*x)],x]","\frac{-4 \sin \left(\frac{b}{d}\right) (b c-a d) \text{Ci}\left(\frac{a d-b c}{d (c+d x)}\right)+d \exp \left(-\frac{i (a d+2 b c+b d x)}{d (c+d x)}\right) \left(2 c \left(e^{2 i \left(\frac{a}{c+d x}+\frac{b}{d}\right)}+e^{\frac{2 i b c}{d (c+d x)}}\right)+d x \left(1+e^{\frac{2 i b}{d}}\right) \left(e^{\frac{2 i a}{c+d x}}+e^{\frac{2 i b c}{d (c+d x)}}\right)-4 d x \sin \left(\frac{b}{d}\right) e^{\frac{i (a d+2 b c+b d x)}{d (c+d x)}} \sin \left(\frac{a d-b c}{d (c+d x)}\right)\right)-4 \cos \left(\frac{b}{d}\right) (b c-a d) \text{Si}\left(\frac{a d-b c}{d (c+d x)}\right)}{4 d^2}","-\frac{\sin \left(\frac{b}{d}\right) (b c-a d) \text{Ci}\left(\frac{b c-a d}{d (c+d x)}\right)}{d^2}+\frac{\cos \left(\frac{b}{d}\right) (b c-a d) \text{Si}\left(\frac{b c-a d}{d (c+d x)}\right)}{d^2}+\frac{(c+d x) \cos \left(\frac{a+b x}{c+d x}\right)}{d}",1,"(-4*(b*c - a*d)*CosIntegral[(-(b*c) + a*d)/(d*(c + d*x))]*Sin[b/d] + (d*(2*c*(E^(((2*I)*b*c)/(d*(c + d*x))) + E^((2*I)*(b/d + a/(c + d*x)))) + d*(1 + E^(((2*I)*b)/d))*(E^(((2*I)*a)/(c + d*x)) + E^(((2*I)*b*c)/(d*(c + d*x))))*x - 4*d*E^((I*(2*b*c + a*d + b*d*x))/(d*(c + d*x)))*x*Sin[b/d]*Sin[(-(b*c) + a*d)/(d*(c + d*x))]))/E^((I*(2*b*c + a*d + b*d*x))/(d*(c + d*x))) - 4*(b*c - a*d)*Cos[b/d]*SinIntegral[(-(b*c) + a*d)/(d*(c + d*x))])/(4*d^2)","C",0
52,1,400,107,6.1351705,"\int \cos ^2\left(\frac{a+b x}{c+d x}\right) \, dx","Integrate[Cos[(a + b*x)/(c + d*x)]^2,x]","\frac{\left(a c d-b c^2\right) \left(\frac{\left(-1+e^{\frac{4 i b}{d}}\right) \left(e^{\frac{4 i b c}{d (c+d x)}}-e^{\frac{4 i a}{c+d x}}\right) \exp \left(-\frac{2 i (a d+2 b c+b d x)}{d (c+d x)}\right)}{8 (b c-a d)}-\frac{\left(1+e^{\frac{4 i b}{d}}\right) \left(e^{\frac{4 i a}{c+d x}}+e^{\frac{4 i b c}{d (c+d x)}}\right) \exp \left(-\frac{2 i (a d+2 b c+b d x)}{d (c+d x)}\right)}{8 (b c-a d)}\right)}{d}+\frac{2 a d \sin \left(\frac{2 b}{d}\right) \text{Ci}\left(\frac{2 (a d-b c)}{d (c+d x)}\right)-2 b c \sin \left(\frac{2 b}{d}\right) \text{Ci}\left(\frac{2 (a d-b c)}{d (c+d x)}\right)+2 a d \cos \left(\frac{2 b}{d}\right) \text{Si}\left(\frac{2 (a d-b c)}{d (c+d x)}\right)-2 b c \cos \left(\frac{2 b}{d}\right) \text{Si}\left(\frac{2 (a d-b c)}{d (c+d x)}\right)+d^2 x}{2 d^2}-\frac{1}{2} x \sin \left(\frac{2 b}{d}\right) \sin \left(\frac{2 (a d-b c)}{d (c+d x)}\right)+\frac{1}{2} x \cos \left(\frac{2 b}{d}\right) \cos \left(\frac{2 (a d-b c)}{d (c+d x)}\right)","-\frac{\sin \left(\frac{2 b}{d}\right) (b c-a d) \text{Ci}\left(\frac{2 (b c-a d)}{d (c+d x)}\right)}{d^2}+\frac{\cos \left(\frac{2 b}{d}\right) (b c-a d) \text{Si}\left(\frac{2 (b c-a d)}{d (c+d x)}\right)}{d^2}+\frac{(c+d x) \cos ^2\left(\frac{a+b x}{c+d x}\right)}{d}",1,"((-(b*c^2) + a*c*d)*(((-1 + E^(((4*I)*b)/d))*(-E^(((4*I)*a)/(c + d*x)) + E^(((4*I)*b*c)/(d*(c + d*x)))))/(8*(b*c - a*d)*E^(((2*I)*(2*b*c + a*d + b*d*x))/(d*(c + d*x)))) - ((1 + E^(((4*I)*b)/d))*(E^(((4*I)*a)/(c + d*x)) + E^(((4*I)*b*c)/(d*(c + d*x)))))/(8*(b*c - a*d)*E^(((2*I)*(2*b*c + a*d + b*d*x))/(d*(c + d*x))))))/d + (x*Cos[(2*b)/d]*Cos[(2*(-(b*c) + a*d))/(d*(c + d*x))])/2 - (x*Sin[(2*b)/d]*Sin[(2*(-(b*c) + a*d))/(d*(c + d*x))])/2 + (d^2*x - 2*b*c*CosIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))]*Sin[(2*b)/d] + 2*a*d*CosIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))]*Sin[(2*b)/d] - 2*b*c*Cos[(2*b)/d]*SinIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))] + 2*a*d*Cos[(2*b)/d]*SinIntegral[(2*(-(b*c) + a*d))/(d*(c + d*x))])/(2*d^2)","C",0
53,1,53,58,0.0227666,"\int \frac{\cos ^3\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^3/(1 - a^2*x^2),x]","-\frac{3 \text{Ci}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)+\text{Ci}\left(\frac{3 \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}","-\frac{3 \text{Ci}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}-\frac{\text{Ci}\left(\frac{3 \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}",1,"-1/4*(3*CosIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]] + CosIntegral[(3*Sqrt[1 - a*x])/Sqrt[1 + a*x]])/a","A",1
54,1,51,58,0.0398305,"\int \frac{\cos ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2),x]","-\frac{\text{Ci}\left(\frac{2 \sqrt{1-a x}}{\sqrt{a x+1}}\right)+\log \left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}","-\frac{\text{Ci}\left(\frac{2 \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}-\frac{\log \left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{2 a}",1,"-1/2*(CosIntegral[(2*Sqrt[1 - a*x])/Sqrt[1 + a*x]] + Log[Sqrt[1 - a*x]/Sqrt[1 + a*x]])/a","A",1
55,1,26,26,0.0065232,"\int \frac{\cos \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(1 - a^2*x^2),x]","-\frac{\text{Ci}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}","-\frac{\text{Ci}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}",1,"-(CosIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)","A",1
56,0,0,40,3.0683865,"\int \frac{\sec \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Sec[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(1 - a^2*x^2),x]","\int \frac{\sec \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","\text{Int}\left(\frac{\sec \left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{(1-a x) (a x+1)},x\right)",0,"Integrate[Sec[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(1 - a^2*x^2), x]","A",-1
57,0,0,42,11.8784576,"\int \frac{\sec ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Integrate[Sec[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2),x]","\int \frac{\sec ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","\text{Int}\left(\frac{\sec ^2\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{(1-a x) (a x+1)},x\right)",0,"Integrate[Sec[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2), x]","A",-1
58,1,9,9,0.0130885,"\int \frac{\tan \left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[Tan[Sqrt[x]]/Sqrt[x],x]","-2 \log \left(\cos \left(\sqrt{x}\right)\right)","-2 \log \left(\cos \left(\sqrt{x}\right)\right)",1,"-2*Log[Cos[Sqrt[x]]]","A",1
59,1,18,16,0.0372163,"\int \frac{\tan ^2\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[Tan[Sqrt[x]]^2/Sqrt[x],x]","2 \tan \left(\sqrt{x}\right)-2 \tan ^{-1}\left(\tan \left(\sqrt{x}\right)\right)","2 \tan \left(\sqrt{x}\right)-2 \sqrt{x}",1,"-2*ArcTan[Tan[Sqrt[x]]] + 2*Tan[Sqrt[x]]","A",1
60,1,70,70,0.0229676,"\int \sqrt{x} \tan \left(\sqrt{x}\right) \, dx","Integrate[Sqrt[x]*Tan[Sqrt[x]],x]","2 i \sqrt{x} \text{Li}_2\left(-e^{2 i \sqrt{x}}\right)-\text{Li}_3\left(-e^{2 i \sqrt{x}}\right)+\frac{2}{3} i x^{3/2}-2 x \log \left(1+e^{2 i \sqrt{x}}\right)","2 i \sqrt{x} \text{Li}_2\left(-e^{2 i \sqrt{x}}\right)-\text{Li}_3\left(-e^{2 i \sqrt{x}}\right)+\frac{2}{3} i x^{3/2}-2 x \log \left(1+e^{2 i \sqrt{x}}\right)",1,"((2*I)/3)*x^(3/2) - 2*x*Log[1 + E^((2*I)*Sqrt[x])] + (2*I)*Sqrt[x]*PolyLog[2, -E^((2*I)*Sqrt[x])] - PolyLog[3, -E^((2*I)*Sqrt[x])]","A",1
61,1,18,19,0.7443414,"\int \left(\frac{b \tan \left(a+b x+c x^2\right)}{2 c}+x \tan \left(a+b x+c x^2\right)\right) \, dx","Integrate[(b*Tan[a + b*x + c*x^2])/(2*c) + x*Tan[a + b*x + c*x^2],x]","-\frac{\log (\cos (a+x (b+c x)))}{2 c}","-\frac{\log \left(\cos \left(a+b x+c x^2\right)\right)}{2 c}",1,"-1/2*Log[Cos[a + x*(b + c*x)]]/c","A",1
62,1,26,16,0.0489481,"\int \frac{\cot ^2\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[Cot[Sqrt[x]]^2/Sqrt[x],x]","-2 \cot \left(\sqrt{x}\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2\left(\sqrt{x}\right)\right)","-2 \sqrt{x}-2 \cot \left(\sqrt{x}\right)",1,"-2*Cot[Sqrt[x]]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[Sqrt[x]]^2]","C",1
63,1,85,92,7.2187193,"\int \frac{\sqrt{a+b \sec (c+d x)}}{1+\cos (c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/(1 + Cos[c + d*x]),x]","\frac{\sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{a+b \sec (c+d x)} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}}}","\frac{\sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{a+b \sec (c+d x)} E\left(\sin ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}}}",1,"(EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])","A",1
64,1,35,35,0.0370336,"\int \sec (a+b x) \sec (2 a+2 b x) \, dx","Integrate[Sec[a + b*x]*Sec[2*a + 2*b*x],x]","\frac{\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin (a+b x)\right)}{b}-\frac{\tanh ^{-1}(\sin (a+b x))}{b}","\frac{\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin (a+b x)\right)}{b}-\frac{\tanh ^{-1}(\sin (a+b x))}{b}",1,"-(ArcTanh[Sin[a + b*x]]/b) + (Sqrt[2]*ArcTanh[Sqrt[2]*Sin[a + b*x]])/b","A",1
65,1,35,35,0.0251262,"\int \sec (a+b x) \sec (2 (a+b x)) \, dx","Integrate[Sec[a + b*x]*Sec[2*(a + b*x)],x]","\frac{\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin (a+b x)\right)}{b}-\frac{\tanh ^{-1}(\sin (a+b x))}{b}","\frac{\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin (a+b x)\right)}{b}-\frac{\tanh ^{-1}(\sin (a+b x))}{b}",1,"-(ArcTanh[Sin[a + b*x]]/b) + (Sqrt[2]*ArcTanh[Sqrt[2]*Sin[a + b*x]])/b","A",1
66,1,15,15,0.0062749,"\int \sin (x) \sin (2 x) \, dx","Integrate[Sin[x]*Sin[2*x],x]","\frac{\sin (x)}{2}-\frac{1}{6} \sin (3 x)","\frac{\sin (x)}{2}-\frac{1}{6} \sin (3 x)",1,"Sin[x]/2 - Sin[3*x]/6","A",1
67,1,17,17,0.0074566,"\int \sin (x) \sin (3 x) \, dx","Integrate[Sin[x]*Sin[3*x],x]","\frac{1}{4} \sin (2 x)-\frac{1}{8} \sin (4 x)","\frac{1}{4} \sin (2 x)-\frac{1}{8} \sin (4 x)",1,"Sin[2*x]/4 - Sin[4*x]/8","A",1
68,1,17,17,0.0079571,"\int \sin (x) \sin (4 x) \, dx","Integrate[Sin[x]*Sin[4*x],x]","\frac{1}{6} \sin (3 x)-\frac{1}{10} \sin (5 x)","\frac{1}{6} \sin (3 x)-\frac{1}{10} \sin (5 x)",1,"Sin[3*x]/6 - Sin[5*x]/10","A",1
69,1,25,35,0.047542,"\int \sin (x) \sin (m x) \, dx","Integrate[Sin[x]*Sin[m*x],x]","\frac{\cos (x) \sin (m x)-m \sin (x) \cos (m x)}{m^2-1}","\frac{\sin ((1-m) x)}{2 (1-m)}-\frac{\sin ((m+1) x)}{2 (m+1)}",1,"(-(m*Cos[m*x]*Sin[x]) + Cos[x]*Sin[m*x])/(-1 + m^2)","A",1
70,1,15,15,0.0052721,"\int \cos (2 x) \sin (x) \, dx","Integrate[Cos[2*x]*Sin[x],x]","\frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x)","\frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x)",1,"Cos[x]/2 - Cos[3*x]/6","A",1
71,1,17,17,0.005881,"\int \cos (3 x) \sin (x) \, dx","Integrate[Cos[3*x]*Sin[x],x]","\frac{\cos ^2(x)}{2}-\frac{1}{8} \cos (4 x)","\frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x)",1,"Cos[x]^2/2 - Cos[4*x]/8","A",1
72,1,17,17,0.0072939,"\int \cos (4 x) \sin (x) \, dx","Integrate[Cos[4*x]*Sin[x],x]","\frac{1}{6} \cos (3 x)-\frac{1}{10} \cos (5 x)","\frac{1}{6} \cos (3 x)-\frac{1}{10} \cos (5 x)",1,"Cos[3*x]/6 - Cos[5*x]/10","A",1
73,1,24,35,0.0442148,"\int \cos (m x) \sin (x) \, dx","Integrate[Cos[m*x]*Sin[x],x]","\frac{m \sin (x) \sin (m x)+\cos (x) \cos (m x)}{m^2-1}","-\frac{\cos ((1-m) x)}{2 (1-m)}-\frac{\cos ((m+1) x)}{2 (m+1)}",1,"(Cos[x]*Cos[m*x] + m*Sin[x]*Sin[m*x])/(-1 + m^2)","A",1
74,1,20,20,0.0138907,"\int \sin (x) \tan (2 x) \, dx","Integrate[Sin[x]*Tan[2*x],x]","\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{\sqrt{2}}-\sin (x)","\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{\sqrt{2}}-\sin (x)",1,"ArcTanh[Sqrt[2]*Sin[x]]/Sqrt[2] - Sin[x]","A",1
75,1,21,47,0.0305808,"\int \sin (x) \tan (3 x) \, dx","Integrate[Sin[x]*Tan[3*x],x]","-\sin (x)+\frac{1}{3} \tanh ^{-1}(\sin (x))+\frac{1}{3} \tanh ^{-1}(2 \sin (x))","-\sin (x)-\frac{1}{6} \log (1-2 \sin (x))-\frac{1}{6} \log (1-\sin (x))+\frac{1}{6} \log (\sin (x)+1)+\frac{1}{6} \log (2 \sin (x)+1)",1,"ArcTanh[Sin[x]]/3 + ArcTanh[2*Sin[x]]/3 - Sin[x]","A",1
76,1,69,71,0.0819408,"\int \sin (x) \tan (4 x) \, dx","Integrate[Sin[x]*Tan[4*x],x]","\frac{1}{4} \left(-4 \sin (x)+\sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2-\sqrt{2}}}\right)+\sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2+\sqrt{2}}}\right)\right)","-\sin (x)+\frac{1}{4} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2-\sqrt{2}}}\right)+\frac{1}{4} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2+\sqrt{2}}}\right)",1,"(Sqrt[2 - Sqrt[2]]*ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[2]]] + Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[2]]] - 4*Sin[x])/4","A",1
77,1,100,112,0.1756938,"\int \sin (x) \tan (5 x) \, dx","Integrate[Sin[x]*Tan[5*x],x]","\frac{1}{20} \left(-20 \sin (x)+\left(\sqrt{5}-1\right) \log \left(-4 \sin (x)-\sqrt{5}+1\right)-\left(1+\sqrt{5}\right) \log \left(-4 \sin (x)+\sqrt{5}+1\right)-\left(\sqrt{5}-1\right) \log \left(4 \sin (x)-\sqrt{5}+1\right)+\left(1+\sqrt{5}\right) \log \left(4 \sin (x)+\sqrt{5}+1\right)+4 \tanh ^{-1}(\sin (x))\right)","-\sin (x)-\frac{1}{20} \left(1-\sqrt{5}\right) \log \left(-4 \sin (x)-\sqrt{5}+1\right)-\frac{1}{20} \left(1+\sqrt{5}\right) \log \left(-4 \sin (x)+\sqrt{5}+1\right)+\frac{1}{20} \left(1-\sqrt{5}\right) \log \left(4 \sin (x)-\sqrt{5}+1\right)+\frac{1}{20} \left(1+\sqrt{5}\right) \log \left(4 \sin (x)+\sqrt{5}+1\right)+\frac{1}{5} \tanh ^{-1}(\sin (x))",1,"(4*ArcTanh[Sin[x]] + (-1 + Sqrt[5])*Log[1 - Sqrt[5] - 4*Sin[x]] - (1 + Sqrt[5])*Log[1 + Sqrt[5] - 4*Sin[x]] - (-1 + Sqrt[5])*Log[1 - Sqrt[5] + 4*Sin[x]] + (1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Sin[x]] - 20*Sin[x])/20","A",1
78,1,84,89,0.1504992,"\int \sin (x) \tan (6 x) \, dx","Integrate[Sin[x]*Tan[6*x],x]","\frac{1}{6} \left(-6 \sin (x)+\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin (x)\right)+\sqrt{2-\sqrt{3}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2-\sqrt{3}}}\right)+\sqrt{2+\sqrt{3}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2+\sqrt{3}}}\right)\right)","-\sin (x)+\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{3 \sqrt{2}}+\frac{1}{6} \sqrt{2-\sqrt{3}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{6} \sqrt{2+\sqrt{3}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2+\sqrt{3}}}\right)",1,"(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[x]] + Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[3]]] + Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[3]]] - 6*Sin[x])/6","A",1
79,1,200,105,0.177868,"\int \sin (x) \tan (n x) \, dx","Integrate[Sin[x]*Tan[n*x],x]","-\frac{i e^{-2 i x} \left((2 n+1) e^{i (2 n x+x)} \, _2F_1\left(1,1-\frac{1}{2 n};2-\frac{1}{2 n};-e^{2 i n x}\right)+(2 n-1) \left((2 n+1) e^{i x} \left(\, _2F_1\left(1,-\frac{1}{2 n};1-\frac{1}{2 n};-e^{2 i n x}\right)+e^{2 i x} \, _2F_1\left(1,\frac{1}{2 n};1+\frac{1}{2 n};-e^{2 i n x}\right)\right)-e^{i (2 n+3) x} \, _2F_1\left(1,1+\frac{1}{2 n};2+\frac{1}{2 n};-e^{2 i n x}\right)\right)\right)}{2 \left(4 n^2-1\right)}","-i e^{-i x} \, _2F_1\left(1,-\frac{1}{2 n};1-\frac{1}{2 n};-e^{2 i n x}\right)-i e^{i x} \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);-e^{2 i n x}\right)+\frac{1}{2} i e^{-i x}+\frac{1}{2} i e^{i x}",1,"((-1/2*I)*(E^(I*(x + 2*n*x))*(1 + 2*n)*Hypergeometric2F1[1, 1 - 1/(2*n), 2 - 1/(2*n), -E^((2*I)*n*x)] + (-1 + 2*n)*(-(E^(I*(3 + 2*n)*x)*Hypergeometric2F1[1, 1 + 1/(2*n), 2 + 1/(2*n), -E^((2*I)*n*x)]) + E^(I*x)*(1 + 2*n)*(Hypergeometric2F1[1, -1/2*1/n, 1 - 1/(2*n), -E^((2*I)*n*x)] + E^((2*I)*x)*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -E^((2*I)*n*x)]))))/(E^((2*I)*x)*(-1 + 4*n^2))","A",1
80,1,10,10,0.0107675,"\int \cot (2 x) \sin (x) \, dx","Integrate[Cot[2*x]*Sin[x],x]","\sin (x)-\frac{1}{2} \tanh ^{-1}(\sin (x))","\sin (x)-\frac{1}{2} \tanh ^{-1}(\sin (x))",1,"-1/2*ArcTanh[Sin[x]] + Sin[x]","A",1
81,1,20,20,0.0190604,"\int \cot (3 x) \sin (x) \, dx","Integrate[Cot[3*x]*Sin[x],x]","\sin (x)-\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{3}}\right)}{\sqrt{3}}","\sin (x)-\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{3}}\right)}{\sqrt{3}}",1,"-(ArcTanh[(2*Sin[x])/Sqrt[3]]/Sqrt[3]) + Sin[x]","A",1
82,1,28,28,0.0372032,"\int \cot (4 x) \sin (x) \, dx","Integrate[Cot[4*x]*Sin[x],x]","\sin (x)-\frac{1}{4} \tanh ^{-1}(\sin (x))-\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{2 \sqrt{2}}","\sin (x)-\frac{1}{4} \tanh ^{-1}(\sin (x))-\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{2 \sqrt{2}}",1,"-1/4*ArcTanh[Sin[x]] - ArcTanh[Sqrt[2]*Sin[x]]/(2*Sqrt[2]) + Sin[x]","A",1
83,1,76,82,0.2375704,"\int \cot (5 x) \sin (x) \, dx","Integrate[Cot[5*x]*Sin[x],x]","\frac{1}{10} \left(10 \sin (x)-\sqrt{10-2 \sqrt{5}} \tanh ^{-1}\left(\sqrt{2+\frac{2}{\sqrt{5}}} \sin (x)\right)-\sqrt{2 \left(5+\sqrt{5}\right)} \tanh ^{-1}\left(2 \sqrt{\frac{2}{5+\sqrt{5}}} \sin (x)\right)\right)","\sin (x)-\frac{1}{5} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \tanh ^{-1}\left(2 \sqrt{\frac{2}{5+\sqrt{5}}} \sin (x)\right)-\frac{1}{5} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{2}{5} \left(5+\sqrt{5}\right)} \sin (x)\right)",1,"(-(Sqrt[10 - 2*Sqrt[5]]*ArcTanh[Sqrt[2 + 2/Sqrt[5]]*Sin[x]]) - Sqrt[2*(5 + Sqrt[5])]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Sin[x]] + 10*Sin[x])/10","A",1
84,1,38,38,0.0699141,"\int \cot (6 x) \sin (x) \, dx","Integrate[Cot[6*x]*Sin[x],x]","\sin (x)-\frac{1}{6} \tanh ^{-1}(\sin (x))-\frac{1}{6} \tanh ^{-1}(2 \sin (x))-\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{3}}\right)}{2 \sqrt{3}}","\sin (x)-\frac{1}{6} \tanh ^{-1}(\sin (x))-\frac{1}{6} \tanh ^{-1}(2 \sin (x))-\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"-1/6*ArcTanh[Sin[x]] - ArcTanh[2*Sin[x]]/6 - ArcTanh[(2*Sin[x])/Sqrt[3]]/(2*Sqrt[3]) + Sin[x]","A",1
85,1,174,15,0.414906,"\int \sec (2 x) \sin (x) \, dx","Integrate[Sec[2*x]*Sin[x],x]","\frac{4 \tanh ^{-1}\left(\tan \left(\frac{x}{2}\right)+\sqrt{2}\right)-\log \left(-\sqrt{2} \sin (x)-\sqrt{2} \cos (x)+2\right)+\log \left(-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+2\right)+2 i \tan ^{-1}\left(\frac{\cos \left(\frac{x}{2}\right)-\left(\sqrt{2}-1\right) \sin \left(\frac{x}{2}\right)}{\left(1+\sqrt{2}\right) \cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}\right)-2 i \tan ^{-1}\left(\frac{\cos \left(\frac{x}{2}\right)-\left(1+\sqrt{2}\right) \sin \left(\frac{x}{2}\right)}{\left(\sqrt{2}-1\right) \cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}\right)}{4 \sqrt{2}}","\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{\sqrt{2}}",1,"((2*I)*ArcTan[(Cos[x/2] - (-1 + Sqrt[2])*Sin[x/2])/((1 + Sqrt[2])*Cos[x/2] - Sin[x/2])] - (2*I)*ArcTan[(Cos[x/2] - (1 + Sqrt[2])*Sin[x/2])/((-1 + Sqrt[2])*Cos[x/2] - Sin[x/2])] + 4*ArcTanh[Sqrt[2] + Tan[x/2]] - Log[2 - Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]] + Log[2 + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]])/(4*Sqrt[2])","C",1
86,1,17,21,0.0085446,"\int \sec (3 x) \sin (x) \, dx","Integrate[Sec[3*x]*Sin[x],x]","-\frac{1}{3} \tanh ^{-1}\left(\frac{1}{3} \left(8 \sin ^2(x)-5\right)\right)","\frac{1}{3} \log (\cos (x))-\frac{1}{6} \log \left(3-4 \cos ^2(x)\right)",1,"-1/3*ArcTanh[(-5 + 8*Sin[x]^2)/3]","A",1
87,1,4845,71,57.9341333,"\int \sec (4 x) \sin (x) \, dx","Integrate[Sec[4*x]*Sin[x],x]","\text{Result too large to show}","\frac{\tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{2+\sqrt{2}}}\right)}{2 \sqrt{2 \left(2+\sqrt{2}\right)}}-\frac{\tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{2-\sqrt{2}}}\right)}{2 \sqrt{2 \left(2-\sqrt{2}\right)}}",1,"((-2*(-1)^(3/8)*(1 + Sqrt[2])*x - (2*(-1)^(1/4)*(-2 - (1 - I)*(-1)^(5/8) + (-1)^(5/8)*Sqrt[2])*ArcTan[(-Cos[x] + (1 + Sqrt[2])*Sin[x])/(2*(-1)^(3/8) + Cos[x] - Sqrt[2]*Cos[x] + Sin[x])])/((-1 + I) + 2*(-1)^(3/8) + Sqrt[2]) - (2*(1 - I)^(3/2)*2^(1/4)*((-3 - I) + 2*(-1)^(5/8) + (2 + I)*Sqrt[2] - (2 + 2*I)*(-1)^(3/8)*Sqrt[2] + 2*(-1)^(5/8)*Sqrt[2])*ArcTan[((1 + I) + I*Sqrt[2] + ((-1 + I) + 2*(-1)^(3/8) + Sqrt[2])*Tan[x/2])/(Sqrt[1 - I]*2^(3/4))])/((-1 + I) + 2*(-1)^(3/8) + Sqrt[2]) + 2*(-1)^(3/8)*Log[Sec[x/2]^2] + ((-1)^(3/4)*(-2 - (1 - I)*(-1)^(5/8) + (-1)^(5/8)*Sqrt[2])*Log[-(Sec[x/2]^4*(-2 + (1 - I)*Sqrt[2] + 2*(-1)^(3/8)*(-1 + Sqrt[2])*Cos[x] + Sqrt[2]*Cos[2*x] - 2*(-1)^(3/8)*Sin[x] + Sqrt[2]*Sin[2*x]))])/((-1 + I) + 2*(-1)^(3/8) + Sqrt[2]))*((-1/2 - I/2)/(((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])*(-((-1 - I)^(3/2)*(1 - I)^(1/4)*(1 + I)^(1/4)) - (1 + I)*Cos[x] + I*Sqrt[1 - I]*Sqrt[1 + I]*Cos[x] + (1 - I)*Sin[x] + Sqrt[1 - I]*Sqrt[1 + I]*Sin[x])) - Sin[x]/(Sqrt[-1 - I]*(1 - I)^(1/4)*(1 + I)^(1/4)*((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])*(-((-1 - I)^(3/2)*(1 - I)^(1/4)*(1 + I)^(1/4)) - (1 + I)*Cos[x] + I*Sqrt[1 - I]*Sqrt[1 + I]*Cos[x] + (1 - I)*Sin[x] + Sqrt[1 - I]*Sqrt[1 + I]*Sin[x])) - ((I/2)*Sqrt[-1 - I]*(1 - I)^(1/4)*(1 + I)^(1/4)*Sin[x])/(((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])*(-((-1 - I)^(3/2)*(1 - I)^(1/4)*(1 + I)^(1/4)) - (1 + I)*Cos[x] + I*Sqrt[1 - I]*Sqrt[1 + I]*Cos[x] + (1 - I)*Sin[x] + Sqrt[1 - I]*Sqrt[1 + I]*Sin[x]))))/(-2*(-1)^(3/8)*(1 + Sqrt[2]) - (2*(-1)^(1/4)*(-2 - (1 - I)*(-1)^(5/8) + (-1)^(5/8)*Sqrt[2])*(((1 + Sqrt[2])*Cos[x] + Sin[x])/(2*(-1)^(3/8) + Cos[x] - Sqrt[2]*Cos[x] + Sin[x]) - ((Cos[x] - Sin[x] + Sqrt[2]*Sin[x])*(-Cos[x] + (1 + Sqrt[2])*Sin[x]))/(2*(-1)^(3/8) + Cos[x] - Sqrt[2]*Cos[x] + Sin[x])^2))/(((-1 + I) + 2*(-1)^(3/8) + Sqrt[2])*(1 + (-Cos[x] + (1 + Sqrt[2])*Sin[x])^2/(2*(-1)^(3/8) + Cos[x] - Sqrt[2]*Cos[x] + Sin[x])^2)) + 2*(-1)^(3/8)*Tan[x/2] - ((-1)^(3/4)*(-2 - (1 - I)*(-1)^(5/8) + (-1)^(5/8)*Sqrt[2])*Cos[x/2]^4*(-(Sec[x/2]^4*(-2*(-1)^(3/8)*Cos[x] + 2*Sqrt[2]*Cos[2*x] - 2*(-1)^(3/8)*(-1 + Sqrt[2])*Sin[x] - 2*Sqrt[2]*Sin[2*x])) - 2*Sec[x/2]^4*(-2 + (1 - I)*Sqrt[2] + 2*(-1)^(3/8)*(-1 + Sqrt[2])*Cos[x] + Sqrt[2]*Cos[2*x] - 2*(-1)^(3/8)*Sin[x] + Sqrt[2]*Sin[2*x])*Tan[x/2]))/(((-1 + I) + 2*(-1)^(3/8) + Sqrt[2])*(-2 + (1 - I)*Sqrt[2] + 2*(-1)^(3/8)*(-1 + Sqrt[2])*Cos[x] + Sqrt[2]*Cos[2*x] - 2*(-1)^(3/8)*Sin[x] + Sqrt[2]*Sin[2*x])) - ((1 - I)*((-3 - I) + 2*(-1)^(5/8) + (2 + I)*Sqrt[2] - (2 + 2*I)*(-1)^(3/8)*Sqrt[2] + 2*(-1)^(5/8)*Sqrt[2])*Sec[x/2]^2)/(Sqrt[2]*(1 + ((1/4 + I/4)*((1 + I) + I*Sqrt[2] + ((-1 + I) + 2*(-1)^(3/8) + Sqrt[2])*Tan[x/2])^2)/Sqrt[2]))) + ((-4*Sqrt[-1 - I]*(-1 + Sqrt[2])*ArcTanh[((-I)*((1 + I) + Sqrt[2]) + ((1 + I) + 2*(-1)^(5/8) - Sqrt[2])*Tan[x/2])/(Sqrt[-1 - I]*2^(3/4))] + (-1)^(1/8)*2^(1/4)*(2*ArcTan[(Cos[x] + (1 + Sqrt[2])*Sin[x])/(2*(-1)^(5/8) + (-1 + Sqrt[2])*Cos[x] + Sin[x])] - I*(2*(1 + Sqrt[2])*x + 2*Log[Sec[x/2]^2] - Log[Sec[x/2]^4*(2 - (1 + I)*Sqrt[2] + 2*(-1)^(5/8)*(-1 + Sqrt[2])*Cos[x] - Sqrt[2]*Cos[2*x] + 2*(-1)^(5/8)*Sin[x] + Sqrt[2]*Sin[2*x])])))*(2 + I*Sqrt[-1 + I]*2^(1/4)*((1 + I) + Sqrt[2])*Sin[x]))/(2^(1/4)*(4*Sqrt[-1 + I]*2^(1/4)*((-1 - I) + Sqrt[2]) - 8*(-1 + Sqrt[2])*Cos[x] - 8*Sin[x])*((2*(-1)^(1/8)*(-2 - (1 + I)*Sqrt[2] + (-1)^(1/8)*((1 + I) + I*Sqrt[2])*Cos[x] + (2*I)*(1 + Sqrt[2])*Cos[2*x] + (-1)^(1/8)*Sin[x] - (-1)^(5/8)*Sin[x] + 3*(-1)^(1/8)*Sqrt[2]*Sin[x] - (2*I)*Sin[2*x]))/(2 - (1 + I)*Sqrt[2] + 2*(-1)^(5/8)*(-1 + Sqrt[2])*Cos[x] - Sqrt[2]*Cos[2*x] + 2*(-1)^(5/8)*Sin[x] + Sqrt[2]*Sin[2*x]) - (((1 + I) + 2*(-1)^(5/8) - Sqrt[2])*(-1 + Sqrt[2])*Sec[x/2]^2)/(1 + ((1/4 - I/4)*(I*((1 + I) + Sqrt[2]) + ((-1 - I) - 2*(-1)^(5/8) + Sqrt[2])*Tan[x/2])^2)/Sqrt[2]))) + ((-2*(-1)^(3/8)*Sqrt[2]*(1 + (-1)^(1/4))*x + (2*(-2*I + 2*(-1)^(3/4) + 2*(-1)^(1/8)*Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(7/8)*Sqrt[2])*ArcTan[Cos[x]/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] + (1 + (-1)^(1/4))*Sin[x])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]) - ((4 + 4*I)*(-1)^(5/8)*((3 - 3*I) - (2 - 2*I)*Sqrt[2] + (-1)^(1/8)*Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (1 - I)*(-1)^(5/8)*Sqrt[2] + (1 + I)*(-1)^(7/8)*Sqrt[2])*ArcTanh[(1/2 + I/2)*(-1)^(5/8)*(-1 - (-1)^(1/4) + (-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*Tan[x/2])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]) - 2*(-1)^(7/8)*Sqrt[2]*(-1 + (-1)^(1/4))*Log[Sec[x/2]^2] - ((-1 + (-1)^(1/4))*(2 - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*Log[(1/4 + I/4)*Sec[x/2]^4*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] - (4 - 4*I)*(-1)^(1/8)*Sqrt[2]*Sin[x] - (4 - 4*I)*(-1)^(3/8)*Sqrt[2]*Sin[x] - (2 - 2*I)*Sin[2*x] + (2*I)*Sqrt[2]*Sin[2*x])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]))*(I/(Sqrt[1 - I]*((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(Sqrt[-1 - I]*(1 - I)^(3/4)*(1 + I)^(1/4) + Sqrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] + I*Sqrt[1 - I]*Sin[x] + I*Sqrt[1 + I]*Sin[x])) + 1/(Sqrt[1 + I]*((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(Sqrt[-1 - I]*(1 - I)^(3/4)*(1 + I)^(1/4) + Sqrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] + I*Sqrt[1 - I]*Sin[x] + I*Sqrt[1 + I]*Sin[x])) - (2*Sin[x])/(Sqrt[-1 - I]*(1 - I)^(1/4)*(1 + I)^(3/4)*((-1 + I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(Sqrt[-1 - I]*(1 - I)^(3/4)*(1 + I)^(1/4) + Sqrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] + I*Sqrt[1 - I]*Sin[x] + I*Sqrt[1 + I]*Sin[x]))))/(-2*(-1)^(3/8)*Sqrt[2]*(1 + (-1)^(1/4)) + (2*(-2*I + 2*(-1)^(3/4) + 2*(-1)^(1/8)*Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(7/8)*Sqrt[2])*(-((Cos[x]*((1 + (-1)^(1/4))*Cos[x] - (-1)^(3/4)*Sin[x]))/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] + (1 + (-1)^(1/4))*Sin[x])^2) - Sin[x]/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] + (1 + (-1)^(1/4))*Sin[x])))/((-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*(1 + Cos[x]^2/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] + (1 + (-1)^(1/4))*Sin[x])^2)) - 2*(-1)^(7/8)*Sqrt[2]*(-1 + (-1)^(1/4))*Tan[x/2] - ((2 - 2*I)*(-1 + (-1)^(1/4))*(2 - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*Cos[x/2]^4*((1/4 + I/4)*Sec[x/2]^4*((-4 + 4*I)*(-1)^(1/8)*Sqrt[2]*Cos[x] - (4 - 4*I)*(-1)^(3/8)*Sqrt[2]*Cos[x] - (4 - 4*I)*Cos[2*x] + (4*I)*Sqrt[2]*Cos[2*x] + (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Sin[x] + 4*((1 + I) + Sqrt[2])*Sin[2*x]) + (1/2 + I/2)*Sec[x/2]^4*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] - (4 - 4*I)*(-1)^(1/8)*Sqrt[2]*Sin[x] - (4 - 4*I)*(-1)^(3/8)*Sqrt[2]*Sin[x] - (2 - 2*I)*Sin[2*x] + (2*I)*Sqrt[2]*Sin[2*x])*Tan[x/2]))/((-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] - (4 - 4*I)*(-1)^(1/8)*Sqrt[2]*Sin[x] - (4 - 4*I)*(-1)^(3/8)*Sqrt[2]*Sin[x] - (2 - 2*I)*Sin[2*x] + (2*I)*Sqrt[2]*Sin[2*x])) + (2*(-1)^(3/4)*((3 - 3*I) - (2 - 2*I)*Sqrt[2] + (-1)^(1/8)*Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (1 - I)*(-1)^(5/8)*Sqrt[2] + (1 + I)*(-1)^(7/8)*Sqrt[2])*Sec[x/2]^2)/(1 + ((-1)^(3/4)*(-1 - (-1)^(1/4) + (-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*Tan[x/2])^2)/2)) + ((2*((-1)^(1/8) + (-1)^(3/8))*x - (2*(-1)^(7/8)*(2 - Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*ArcTan[Cos[x]/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] - (1 + (-1)^(1/4))*Sin[x])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]) - ((4 + 4*I)*(-1)^(5/8)*(3*I + (-1)^(1/8) - (-1)^(3/8) - (1 + I)*(-1)^(5/8) - (2*I)*Sqrt[2] + (1 + I)*(-1)^(5/8)*Sqrt[2])*ArcTanh[(1/2 + I/2)*(-1)^(5/8)*(1 + (-1)^(1/4) + (-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*Tan[x/2])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]) + 2*(-1)^(3/8)*(-I + (-1)^(1/4))*Log[Sec[x/2]^2] + ((-1)^(3/8)*(2 - Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*Log[(1/4 + I/4)*Sec[x/2]^4*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] + (4 - 4*I)*(-1)^(1/8)*((1 + I) + Sqrt[2])*Sin[x] + (2 - 2*I)*Sin[2*x] - (2*I)*Sqrt[2]*Sin[2*x])])/(-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2]))*(1/(Sqrt[1 - I]*((-1 - I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(-(Sqrt[-1 + I]*(1 - I)^(1/4)*(1 + I)^(3/4)) + Sqrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] - I*Sqrt[1 - I]*Sin[x] - I*Sqrt[1 + I]*Sin[x])) - I/(Sqrt[1 + I]*((-1 - I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(-(Sqrt[-1 + I]*(1 - I)^(1/4)*(1 + I)^(3/4)) + Sqrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] - I*Sqrt[1 - I]*Sin[x] - I*Sqrt[1 + I]*Sin[x])) + (2*Sin[x])/(Sqrt[-1 + I]*(1 - I)^(3/4)*(1 + I)^(1/4)*((-1 - I) + Sqrt[1 - I]*Sqrt[1 + I])^2*(-(Sqrt[-1 + I]*(1 - I)^(1/4)*(1 + I)^(3/4)) + Sqrt[1 - I]*Cos[x] - Sqrt[1 + I]*Cos[x] - I*Sqrt[1 - I]*Sin[x] - I*Sqrt[1 + I]*Sin[x]))))/(2*((-1)^(1/8) + (-1)^(3/8)) - (2*(-1)^(7/8)*(2 - Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*(-((Cos[x]*(-((1 + (-1)^(1/4))*Cos[x]) - (-1)^(3/4)*Sin[x]))/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] - (1 + (-1)^(1/4))*Sin[x])^2) - Sin[x]/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] - (1 + (-1)^(1/4))*Sin[x])))/((-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*(1 + Cos[x]^2/(-((-1)^(1/8)*Sqrt[2]) + (-1)^(3/4)*Cos[x] - (1 + (-1)^(1/4))*Sin[x])^2)) + 2*(-1)^(3/8)*(-I + (-1)^(1/4))*Tan[x/2] + ((2 - 2*I)*(-1)^(3/8)*(2 - Sqrt[2] - (-1)^(3/8)*Sqrt[2] + (-1)^(5/8)*Sqrt[2])*Cos[x/2]^4*((1/4 + I/4)*Sec[x/2]^4*((4 - 4*I)*(-1)^(1/8)*((1 + I) + Sqrt[2])*Cos[x] + (4 - 4*I)*Cos[2*x] - (4*I)*Sqrt[2]*Cos[2*x] + (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Sin[x] + 4*((1 + I) + Sqrt[2])*Sin[2*x]) + (1/2 + I/2)*Sec[x/2]^4*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] + (4 - 4*I)*(-1)^(1/8)*((1 + I) + Sqrt[2])*Sin[x] + (2 - 2*I)*Sin[2*x] - (2*I)*Sqrt[2]*Sin[2*x])*Tan[x/2]))/((-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*((2 - 2*I) + 6*Sqrt[2] - (4 - 4*I)*(-1)^(7/8)*Sqrt[2]*Cos[x] - 2*((1 + I) + Sqrt[2])*Cos[2*x] + (4 - 4*I)*(-1)^(1/8)*((1 + I) + Sqrt[2])*Sin[x] + (2 - 2*I)*Sin[2*x] - (2*I)*Sqrt[2]*Sin[2*x])) + (2*(-1)^(3/4)*(3*I + (-1)^(1/8) - (-1)^(3/8) - (1 + I)*(-1)^(5/8) - (2*I)*Sqrt[2] + (1 + I)*(-1)^(5/8)*Sqrt[2])*Sec[x/2]^2)/(1 + ((-1)^(3/4)*(1 + (-1)^(1/4) + (-I + (-1)^(3/4) + (-1)^(1/8)*Sqrt[2])*Tan[x/2])^2)/2))","C",0
88,1,57,62,0.0987378,"\int \sec (5 x) \sin (x) \, dx","Integrate[Sec[5*x]*Sin[x],x]","\frac{1}{20} \left(-4 \log (\cos (x))-\left(\sqrt{5}-1\right) \log \left(4 \cos (2 x)-\sqrt{5}-1\right)+\left(1+\sqrt{5}\right) \log \left(4 \cos (2 x)+\sqrt{5}-1\right)\right)","\frac{1}{20} \left(1+\sqrt{5}\right) \log \left(-8 \cos ^2(x)-\sqrt{5}+5\right)+\frac{1}{20} \left(1-\sqrt{5}\right) \log \left(-8 \cos ^2(x)+\sqrt{5}+5\right)-\frac{1}{5} \log (\cos (x))",1,"(-4*Log[Cos[x]] - (-1 + Sqrt[5])*Log[-1 - Sqrt[5] + 4*Cos[2*x]] + (1 + Sqrt[5])*Log[-1 + Sqrt[5] + 4*Cos[2*x]])/20","A",1
89,1,627,85,9.3328205,"\int \sec (6 x) \sin (x) \, dx","Integrate[Sec[6*x]*Sin[x],x]","\frac{1}{24} \left((-4-4 i) (-1)^{3/4} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)-1}{\sqrt{2}}\right)-(4-4 i) \sqrt[4]{-1} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+1}{\sqrt{2}}\right)+\frac{2 \left(1+\sqrt{2}\right) \left(x-\log \left(\sec ^2\left(\frac{x}{2}\right)\right)+2 \sqrt{3} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \tan \left(\frac{x}{2}\right)+2}{\sqrt{6}}\right)+\log \left(-\sec ^2\left(\frac{x}{2}\right) \left(2 \sin (x)-2 \cos (x)+\sqrt{2}\right)\right)\right)}{2+\sqrt{2}}-\sqrt{2} \left(x-\log \left(\sec ^2\left(\frac{x}{2}\right)\right)-2 \sqrt{3} \tanh ^{-1}\left(\frac{\left(\sqrt{2}-1\right) \tan \left(\frac{x}{2}\right)+\sqrt{2}}{\sqrt{3}}\right)+\log \left(\sec ^2\left(\frac{x}{2}\right) \left(-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+1\right)\right)\right)+\frac{2 \left(\sqrt{6} \sin (x)+1\right) \left(\left(2+\sqrt{6}\right) \sin (x)-\left(\left(2+\sqrt{6}\right) \cos (x)\right)+\sqrt{6}+3\right) \left(2 \left(\sqrt{2}+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\left(2+\sqrt{6}\right) \tan \left(\frac{x}{2}\right)+2}{\sqrt{2}}\right)+\left(3+\sqrt{6}\right) \left(x-\log \left(\sec ^2\left(\frac{x}{2}\right)\right)+\log \left(-\sec ^2\left(\frac{x}{2}\right) \left(2 \sin (x)-2 \cos (x)+\sqrt{6}\right)\right)\right)\right)}{-2 \left(4 \left(5+2 \sqrt{6}\right) \sin (x)-6 \sin (2 x)+5 \sqrt{6}+12\right)+\left(12+5 \sqrt{6}\right) \cos (2 x)+2 \left(5 \sqrt{6} \sin (x)+2 \sqrt{6}+5\right) \cos (x)}+\frac{\left(\sqrt{2}-2 \sqrt{3} \sin (x)\right) \left(\left(\sqrt{6}-2\right) \sin (x)-\left(\left(\sqrt{6}-2\right) \cos (x)\right)+\sqrt{6}-3\right) \left(\left(3 \sqrt{2}-2 \sqrt{3}\right) \left(x-\log \left(\sec ^2\left(\frac{x}{2}\right)\right)+\log \left(-\sec ^2\left(\frac{x}{2}\right) \left(-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+\sqrt{3}\right)\right)\right)-2 \left(\sqrt{6}-2\right) \tanh ^{-1}\left(\left(\sqrt{2}-\sqrt{3}\right) \tan \left(\frac{x}{2}\right)+\sqrt{2}\right)\right)}{-2 \left(4 \left(2 \sqrt{6}-5\right) \sin (x)+6 \sin (2 x)+5 \sqrt{6}-12\right)+\left(5 \sqrt{6}-12\right) \cos (2 x)+2 \left(5 \sqrt{6} \sin (x)+2 \sqrt{6}-5\right) \cos (x)}\right)","-\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{3 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{2-\sqrt{3}}}\right)}{6 \sqrt{2-\sqrt{3}}}+\frac{\tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{2+\sqrt{3}}}\right)}{6 \sqrt{2+\sqrt{3}}}",1,"((-4 - 4*I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] - (4 - 4*I)*(-1)^(1/4)*ArcTanh[(1 + Tan[x/2])/Sqrt[2]] + (2*(1 + Sqrt[2])*(x + 2*Sqrt[3]*ArcTanh[(2 + (2 + Sqrt[2])*Tan[x/2])/Sqrt[6]] - Log[Sec[x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[2] - 2*Cos[x] + 2*Sin[x]))]))/(2 + Sqrt[2]) - Sqrt[2]*(x - 2*Sqrt[3]*ArcTanh[(Sqrt[2] + (-1 + Sqrt[2])*Tan[x/2])/Sqrt[3]] - Log[Sec[x/2]^2] + Log[Sec[x/2]^2*(1 + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x])]) + (2*(2*(Sqrt[2] + Sqrt[3])*ArcTanh[(2 + (2 + Sqrt[6])*Tan[x/2])/Sqrt[2]] + (3 + Sqrt[6])*(x - Log[Sec[x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[6] - 2*Cos[x] + 2*Sin[x]))]))*(1 + Sqrt[6]*Sin[x])*(3 + Sqrt[6] - (2 + Sqrt[6])*Cos[x] + (2 + Sqrt[6])*Sin[x]))/((12 + 5*Sqrt[6])*Cos[2*x] + 2*Cos[x]*(5 + 2*Sqrt[6] + 5*Sqrt[6]*Sin[x]) - 2*(12 + 5*Sqrt[6] + 4*(5 + 2*Sqrt[6])*Sin[x] - 6*Sin[2*x])) + ((-2*(-2 + Sqrt[6])*ArcTanh[Sqrt[2] + (Sqrt[2] - Sqrt[3])*Tan[x/2]] + (3*Sqrt[2] - 2*Sqrt[3])*(x - Log[Sec[x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[3] + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]))]))*(Sqrt[2] - 2*Sqrt[3]*Sin[x])*(-3 + Sqrt[6] - (-2 + Sqrt[6])*Cos[x] + (-2 + Sqrt[6])*Sin[x]))/((-12 + 5*Sqrt[6])*Cos[2*x] + 2*Cos[x]*(-5 + 2*Sqrt[6] + 5*Sqrt[6]*Sin[x]) - 2*(-12 + 5*Sqrt[6] + 4*(-5 + 2*Sqrt[6])*Sin[x] + 6*Sin[2*x])))/24","C",0
90,1,37,7,0.0059842,"\int \csc (2 x) \sin (x) \, dx","Integrate[Csc[2*x]*Sin[x],x]","\frac{1}{2} \left(\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)\right)","\frac{1}{2} \tanh ^{-1}(\sin (x))",1,"(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])/2","B",1
91,1,15,45,0.0186661,"\int \csc (3 x) \sin (x) \, dx","Integrate[Csc[3*x]*Sin[x],x]","\frac{\tanh ^{-1}\left(\frac{\tan (x)}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{\log \left(\sin (x)+\sqrt{3} \cos (x)\right)}{2 \sqrt{3}}-\frac{\log \left(\sqrt{3} \cos (x)-\sin (x)\right)}{2 \sqrt{3}}",1,"ArcTanh[Tan[x]/Sqrt[3]]/Sqrt[3]","A",1
92,1,26,26,0.0206012,"\int \csc (4 x) \sin (x) \, dx","Integrate[Csc[4*x]*Sin[x],x]","\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{2 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\sin (x))","\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{2 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\sin (x))",1,"-1/4*ArcTanh[Sin[x]] + ArcTanh[Sqrt[2]*Sin[x]]/(2*Sqrt[2])","A",1
93,1,84,165,0.1060306,"\int \csc (5 x) \sin (x) \, dx","Integrate[Csc[5*x]*Sin[x],x]","\frac{\sqrt{5+\sqrt{5}} \tanh ^{-1}\left(\frac{\left(\sqrt{5}-3\right) \tan (x)}{\sqrt{10-2 \sqrt{5}}}\right)+\sqrt{5-\sqrt{5}} \tanh ^{-1}\left(\frac{\left(3+\sqrt{5}\right) \tan (x)}{\sqrt{2 \left(5+\sqrt{5}\right)}}\right)}{5 \sqrt{2}}","-\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \log \left(\sqrt{5-2 \sqrt{5}} \cos (x)-\sin (x)\right)+\frac{1}{10} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \log \left(\sqrt{5+2 \sqrt{5}} \cos (x)-\sin (x)\right)+\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \log \left(\sin (x)+\sqrt{5-2 \sqrt{5}} \cos (x)\right)-\frac{1}{10} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \log \left(\sin (x)+\sqrt{5+2 \sqrt{5}} \cos (x)\right)",1,"(Sqrt[5 + Sqrt[5]]*ArcTanh[((-3 + Sqrt[5])*Tan[x])/Sqrt[10 - 2*Sqrt[5]]] + Sqrt[5 - Sqrt[5]]*ArcTanh[((3 + Sqrt[5])*Tan[x])/Sqrt[2*(5 + Sqrt[5])]])/(5*Sqrt[2])","A",0
94,1,30,36,0.0361218,"\int \csc (6 x) \sin (x) \, dx","Integrate[Csc[6*x]*Sin[x],x]","\frac{1}{6} \left(\tanh ^{-1}(\sin (x))+\tanh ^{-1}(2 \sin (x))-\sqrt{3} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{3}}\right)\right)","\frac{1}{6} \tanh ^{-1}(\sin (x))+\frac{1}{6} \tanh ^{-1}(2 \sin (x))-\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(ArcTanh[Sin[x]] + ArcTanh[2*Sin[x]] - Sqrt[3]*ArcTanh[(2*Sin[x])/Sqrt[3]])/6","A",1
95,1,6,8,0.0108946,"\int \csc (x) \sin (3 x) \, dx","Integrate[Csc[x]*Sin[3*x],x]","x+\sin (2 x)","x+2 \sin (x) \cos (x)",1,"x + Sin[2*x]","A",1
96,1,8,8,0.0029957,"\int \csc (3 x) \sin (6 x) \, dx","Integrate[Csc[3*x]*Sin[6*x],x]","\frac{2}{3} \sin (3 x)","\frac{2}{3} \sin (3 x)",1,"(2*Sin[3*x])/3","A",1
97,1,15,15,0.0046083,"\int \cos (x) \sin (2 x) \, dx","Integrate[Cos[x]*Sin[2*x],x]","-\frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x)","-\frac{\cos (x)}{2}-\frac{1}{6} \cos (3 x)",1,"-1/2*Cos[x] - Cos[3*x]/6","A",1
98,1,17,17,0.0058824,"\int \cos (x) \sin (3 x) \, dx","Integrate[Cos[x]*Sin[3*x],x]","-\frac{1}{2} \cos ^2(x)-\frac{1}{8} \cos (4 x)","-\frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x)",1,"-1/2*Cos[x]^2 - Cos[4*x]/8","A",1
99,1,17,17,0.0054646,"\int \cos (x) \sin (4 x) \, dx","Integrate[Cos[x]*Sin[4*x],x]","-\frac{1}{6} \cos (3 x)-\frac{1}{10} \cos (5 x)","-\frac{1}{6} \cos (3 x)-\frac{1}{10} \cos (5 x)",1,"-1/6*Cos[3*x] - Cos[5*x]/10","A",1
100,1,26,35,0.046708,"\int \cos (x) \sin (m x) \, dx","Integrate[Cos[x]*Sin[m*x],x]","\frac{\sin (x) \sin (m x)+m \cos (x) \cos (m x)}{1-m^2}","\frac{\cos ((1-m) x)}{2 (1-m)}-\frac{\cos ((m+1) x)}{2 (m+1)}",1,"(m*Cos[x]*Cos[m*x] + Sin[x]*Sin[m*x])/(1 - m^2)","A",1
101,1,15,15,0.0051903,"\int \cos (x) \cos (2 x) \, dx","Integrate[Cos[x]*Cos[2*x],x]","\frac{\sin (x)}{2}+\frac{1}{6} \sin (3 x)","\frac{\sin (x)}{2}+\frac{1}{6} \sin (3 x)",1,"Sin[x]/2 + Sin[3*x]/6","A",1
102,1,17,17,0.0054398,"\int \cos (x) \cos (3 x) \, dx","Integrate[Cos[x]*Cos[3*x],x]","\frac{1}{4} \sin (2 x)+\frac{1}{8} \sin (4 x)","\frac{1}{4} \sin (2 x)+\frac{1}{8} \sin (4 x)",1,"Sin[2*x]/4 + Sin[4*x]/8","A",1
103,1,17,17,0.0053468,"\int \cos (x) \cos (4 x) \, dx","Integrate[Cos[x]*Cos[4*x],x]","\frac{1}{6} \sin (3 x)+\frac{1}{10} \sin (5 x)","\frac{1}{6} \sin (3 x)+\frac{1}{10} \sin (5 x)",1,"Sin[3*x]/6 + Sin[5*x]/10","A",1
104,1,25,35,0.0380745,"\int \cos (x) \cos (m x) \, dx","Integrate[Cos[x]*Cos[m*x],x]","\frac{m \cos (x) \sin (m x)-\sin (x) \cos (m x)}{m^2-1}","\frac{\sin ((1-m) x)}{2 (1-m)}+\frac{\sin ((m+1) x)}{2 (m+1)}",1,"(-(Cos[m*x]*Sin[x]) + m*Cos[x]*Sin[m*x])/(-1 + m^2)","A",1
105,1,183,20,0.2344652,"\int \cos (x) \tan (2 x) \, dx","Integrate[Cos[x]*Tan[2*x],x]","\frac{-4 \sqrt{2} \cos (x)+4 \tanh ^{-1}\left(\tan \left(\frac{x}{2}\right)+\sqrt{2}\right)-\log \left(-\sqrt{2} \sin (x)-\sqrt{2} \cos (x)+2\right)+\log \left(-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+2\right)+2 i \tan ^{-1}\left(\frac{\cos \left(\frac{x}{2}\right)-\left(\sqrt{2}-1\right) \sin \left(\frac{x}{2}\right)}{\left(1+\sqrt{2}\right) \cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}\right)-2 i \tan ^{-1}\left(\frac{\cos \left(\frac{x}{2}\right)-\left(1+\sqrt{2}\right) \sin \left(\frac{x}{2}\right)}{\left(\sqrt{2}-1\right) \cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}\right)}{4 \sqrt{2}}","\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{\sqrt{2}}-\cos (x)",1,"((2*I)*ArcTan[(Cos[x/2] - (-1 + Sqrt[2])*Sin[x/2])/((1 + Sqrt[2])*Cos[x/2] - Sin[x/2])] - (2*I)*ArcTan[(Cos[x/2] - (1 + Sqrt[2])*Sin[x/2])/((-1 + Sqrt[2])*Cos[x/2] - Sin[x/2])] + 4*ArcTanh[Sqrt[2] + Tan[x/2]] - 4*Sqrt[2]*Cos[x] - Log[2 - Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]] + Log[2 + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]])/(4*Sqrt[2])","C",1
106,1,48,21,0.0533203,"\int \cos (x) \tan (3 x) \, dx","Integrate[Cos[x]*Tan[3*x],x]","-\cos (x)-\frac{\tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)-2}{\sqrt{3}}\right)}{\sqrt{3}}+\frac{\tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+2}{\sqrt{3}}\right)}{\sqrt{3}}","\frac{\tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{3}}\right)}{\sqrt{3}}-\cos (x)",1,"-(ArcTanh[(-2 + Tan[x/2])/Sqrt[3]]/Sqrt[3]) + ArcTanh[(2 + Tan[x/2])/Sqrt[3]]/Sqrt[3] - Cos[x]","B",1
107,1,6196,71,59.5466956,"\int \cos (x) \tan (4 x) \, dx","Integrate[Cos[x]*Tan[4*x],x]","\text{Result too large to show}","-\cos (x)+\frac{1}{4} \sqrt{2-\sqrt{2}} \tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{2-\sqrt{2}}}\right)+\frac{1}{4} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{2+\sqrt{2}}}\right)",1,"Result too large to show","C",0
108,1,215,84,0.5902417,"\int \cos (x) \tan (5 x) \, dx","Integrate[Cos[x]*Tan[5*x],x]","-\cos (x)+\frac{\left(1+\sqrt{5}\right) \tanh ^{-1}\left(\frac{4-\left(\sqrt{5}-1\right) \tan \left(\frac{x}{2}\right)}{\sqrt{2 \left(5+\sqrt{5}\right)}}\right)}{\sqrt{10 \left(5+\sqrt{5}\right)}}+\frac{\left(1+\sqrt{5}\right) \tanh ^{-1}\left(\frac{\left(\sqrt{5}-1\right) \tan \left(\frac{x}{2}\right)+4}{\sqrt{2 \left(5+\sqrt{5}\right)}}\right)}{\sqrt{10 \left(5+\sqrt{5}\right)}}+\frac{\left(\sqrt{5}-1\right) \tanh ^{-1}\left(\frac{4-\left(1+\sqrt{5}\right) \tan \left(\frac{x}{2}\right)}{\sqrt{10-2 \sqrt{5}}}\right)}{\sqrt{50-10 \sqrt{5}}}+\frac{\left(\sqrt{5}-1\right) \tanh ^{-1}\left(\frac{\left(1+\sqrt{5}\right) \tan \left(\frac{x}{2}\right)+4}{\sqrt{10-2 \sqrt{5}}}\right)}{\sqrt{50-10 \sqrt{5}}}","-\cos (x)+\frac{1}{5} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \tanh ^{-1}\left(2 \sqrt{\frac{2}{5+\sqrt{5}}} \cos (x)\right)+\frac{1}{5} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \tanh ^{-1}\left(\sqrt{\frac{2}{5} \left(5+\sqrt{5}\right)} \cos (x)\right)",1,"((1 + Sqrt[5])*ArcTanh[(4 - (-1 + Sqrt[5])*Tan[x/2])/Sqrt[2*(5 + Sqrt[5])]])/Sqrt[10*(5 + Sqrt[5])] + ((1 + Sqrt[5])*ArcTanh[(4 + (-1 + Sqrt[5])*Tan[x/2])/Sqrt[2*(5 + Sqrt[5])]])/Sqrt[10*(5 + Sqrt[5])] + ((-1 + Sqrt[5])*ArcTanh[(4 - (1 + Sqrt[5])*Tan[x/2])/Sqrt[10 - 2*Sqrt[5]]])/Sqrt[50 - 10*Sqrt[5]] + ((-1 + Sqrt[5])*ArcTanh[(4 + (1 + Sqrt[5])*Tan[x/2])/Sqrt[10 - 2*Sqrt[5]]])/Sqrt[50 - 10*Sqrt[5]] - Cos[x]","B",0
109,1,679,89,9.0421155,"\int \cos (x) \tan (6 x) \, dx","Integrate[Cos[x]*Tan[6*x],x]","-\cos (x)+\left(-\frac{1}{6}-\frac{i}{6}\right) \sqrt[4]{-1} \tan ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) \sqrt[4]{-1} \sec \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)+\left(\frac{1}{6}+\frac{i}{6}\right) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{x}{2}\right) \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)\right)-\frac{\left(1+\sqrt{2}\right) \left(x-\log \left(\sec ^2\left(\frac{x}{2}\right)\right)-2 \sqrt{3} \tanh ^{-1}\left(\frac{\left(2+\sqrt{2}\right) \tan \left(\frac{x}{2}\right)+2}{\sqrt{6}}\right)+\log \left(-\sec ^2\left(\frac{x}{2}\right) \left(2 \sin (x)-2 \cos (x)+\sqrt{2}\right)\right)\right)}{12 \left(2+\sqrt{2}\right)}+\frac{x-\log \left(\sec ^2\left(\frac{x}{2}\right)\right)+2 \sqrt{3} \tanh ^{-1}\left(\frac{\left(\sqrt{2}-1\right) \tan \left(\frac{x}{2}\right)+\sqrt{2}}{\sqrt{3}}\right)+\log \left(\sec ^2\left(\frac{x}{2}\right) \left(-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+1\right)\right)}{12 \sqrt{2}}-\frac{\left(\sqrt{2}-\sqrt{3} \sin (x)\right) \left(\left(\sqrt{6}-2\right) \sin (x)-\left(\left(\sqrt{6}-2\right) \cos (x)\right)+\sqrt{6}-3\right) \left(2 \left(\sqrt{6}-2\right) \tanh ^{-1}\left(\left(\sqrt{2}-\sqrt{3}\right) \tan \left(\frac{x}{2}\right)+\sqrt{2}\right)+\left(3 \sqrt{2}-2 \sqrt{3}\right) \left(x-\log \left(\sec ^2\left(\frac{x}{2}\right)\right)+\log \left(-\sec ^2\left(\frac{x}{2}\right) \left(-\sqrt{2} \sin (x)+\sqrt{2} \cos (x)+\sqrt{3}\right)\right)\right)\right)}{12 \left(20 \sqrt{6} \sin (x)-50 \sin (x)-5 \sqrt{6} \sin (2 x)+12 \sin (2 x)+\left(20-8 \sqrt{6}\right) \cos (x)+\left(12-5 \sqrt{6}\right) \cos (2 x)+15 \sqrt{6}-36\right)}+\frac{\left(\sqrt{6} \sin (x)+2\right) \left(\left(2+\sqrt{6}\right) \sin (x)-\left(\left(2+\sqrt{6}\right) \cos (x)\right)+\sqrt{6}+3\right) \left(\left(3+\sqrt{6}\right) \left(x-\log \left(\sec ^2\left(\frac{x}{2}\right)\right)+\log \left(-\sec ^2\left(\frac{x}{2}\right) \left(2 \sin (x)-2 \cos (x)+\sqrt{6}\right)\right)\right)-2 \left(\sqrt{2}+\sqrt{3}\right) \tanh ^{-1}\left(\frac{\left(2+\sqrt{6}\right) \tan \left(\frac{x}{2}\right)+2}{\sqrt{2}}\right)\right)}{12 \left(-20 \sqrt{6} \sin (x)-50 \sin (x)+5 \sqrt{6} \sin (2 x)+12 \sin (2 x)+4 \left(5+2 \sqrt{6}\right) \cos (x)+\left(12+5 \sqrt{6}\right) \cos (2 x)-15 \sqrt{6}-36\right)}","-\cos (x)+\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{3 \sqrt{2}}+\frac{1}{6} \sqrt{2-\sqrt{3}} \tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{2-\sqrt{3}}}\right)+\frac{1}{6} \sqrt{2+\sqrt{3}} \tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{2+\sqrt{3}}}\right)",1,"(-1/6 - I/6)*(-1)^(1/4)*ArcTan[(1/2 + I/2)*(-1)^(1/4)*Sec[x/2]*(Cos[x/2] + Sin[x/2])] + (1/6 + I/6)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[x/2]*(Cos[x/2] - Sin[x/2])] - Cos[x] - ((1 + Sqrt[2])*(x - 2*Sqrt[3]*ArcTanh[(2 + (2 + Sqrt[2])*Tan[x/2])/Sqrt[6]] - Log[Sec[x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[2] - 2*Cos[x] + 2*Sin[x]))]))/(12*(2 + Sqrt[2])) + (x + 2*Sqrt[3]*ArcTanh[(Sqrt[2] + (-1 + Sqrt[2])*Tan[x/2])/Sqrt[3]] - Log[Sec[x/2]^2] + Log[Sec[x/2]^2*(1 + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x])])/(12*Sqrt[2]) - ((2*(-2 + Sqrt[6])*ArcTanh[Sqrt[2] + (Sqrt[2] - Sqrt[3])*Tan[x/2]] + (3*Sqrt[2] - 2*Sqrt[3])*(x - Log[Sec[x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[3] + Sqrt[2]*Cos[x] - Sqrt[2]*Sin[x]))]))*(Sqrt[2] - Sqrt[3]*Sin[x])*(-3 + Sqrt[6] - (-2 + Sqrt[6])*Cos[x] + (-2 + Sqrt[6])*Sin[x]))/(12*(-36 + 15*Sqrt[6] + (20 - 8*Sqrt[6])*Cos[x] + (12 - 5*Sqrt[6])*Cos[2*x] - 50*Sin[x] + 20*Sqrt[6]*Sin[x] + 12*Sin[2*x] - 5*Sqrt[6]*Sin[2*x])) + ((-2*(Sqrt[2] + Sqrt[3])*ArcTanh[(2 + (2 + Sqrt[6])*Tan[x/2])/Sqrt[2]] + (3 + Sqrt[6])*(x - Log[Sec[x/2]^2] + Log[-(Sec[x/2]^2*(Sqrt[6] - 2*Cos[x] + 2*Sin[x]))]))*(2 + Sqrt[6]*Sin[x])*(3 + Sqrt[6] - (2 + Sqrt[6])*Cos[x] + (2 + Sqrt[6])*Sin[x]))/(12*(-36 - 15*Sqrt[6] + 4*(5 + 2*Sqrt[6])*Cos[x] + (12 + 5*Sqrt[6])*Cos[2*x] - 50*Sin[x] - 20*Sqrt[6]*Sin[x] + 12*Sin[2*x] + 5*Sqrt[6]*Sin[2*x]))","C",0
110,1,25,10,0.0134161,"\int \cos (x) \cot (2 x) \, dx","Integrate[Cos[x]*Cot[2*x],x]","\cos (x)+\frac{1}{2} \log \left(\sin \left(\frac{x}{2}\right)\right)-\frac{1}{2} \log \left(\cos \left(\frac{x}{2}\right)\right)","\cos (x)-\frac{1}{2} \tanh ^{-1}(\cos (x))",1,"Cos[x] - Log[Cos[x/2]]/2 + Log[Sin[x/2]]/2","B",1
111,1,47,45,0.0173342,"\int \cos (x) \cot (3 x) \, dx","Integrate[Cos[x]*Cot[3*x],x]","\cos (x)+\frac{1}{3} \log \left(\sin \left(\frac{x}{2}\right)\right)-\frac{1}{3} \log \left(\cos \left(\frac{x}{2}\right)\right)+\frac{1}{6} \log (1-2 \cos (x))-\frac{1}{6} \log (2 \cos (x)+1)","\cos (x)+\frac{1}{6} \log (1-2 \cos (x))+\frac{1}{6} \log (1-\cos (x))-\frac{1}{6} \log (\cos (x)+1)-\frac{1}{6} \log (2 \cos (x)+1)",1,"Cos[x] - Log[Cos[x/2]]/3 + Log[1 - 2*Cos[x]]/6 - Log[1 + 2*Cos[x]]/6 + Log[Sin[x/2]]/3","A",1
112,1,73,28,0.0662778,"\int \cos (x) \cot (4 x) \, dx","Integrate[Cos[x]*Cot[4*x],x]","\frac{1}{4} \left(4 \cos (x)+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)+(-1-i) (-1)^{3/4} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)-1}{\sqrt{2}}\right)-(1-i) \sqrt[4]{-1} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+1}{\sqrt{2}}\right)\right)","\cos (x)-\frac{1}{4} \tanh ^{-1}(\cos (x))-\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{2 \sqrt{2}}",1,"((-1 - I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] - (1 - I)*(-1)^(1/4)*ArcTanh[(1 + Tan[x/2])/Sqrt[2]] + 4*Cos[x] - Log[Cos[x/2]] + Log[Sin[x/2]])/4","C",1
113,1,133,110,0.1246694,"\int \cos (x) \cot (5 x) \, dx","Integrate[Cos[x]*Cot[5*x],x]","\frac{1}{100} \left(100 \cos (x)+20 \log \left(\sin \left(\frac{x}{2}\right)\right)-20 \log \left(\cos \left(\frac{x}{2}\right)\right)+\sqrt{5} \left(\sqrt{5}-5\right) \log \left(-4 \cos (x)-\sqrt{5}+1\right)+\sqrt{5} \left(5+\sqrt{5}\right) \log \left(-4 \cos (x)+\sqrt{5}+1\right)-\sqrt{5} \left(\sqrt{5}-5\right) \log \left(4 \cos (x)-\sqrt{5}+1\right)-\sqrt{5} \left(5+\sqrt{5}\right) \log \left(4 \cos (x)+\sqrt{5}+1\right)\right)","\cos (x)+\frac{1}{20} \left(1-\sqrt{5}\right) \log \left(-4 \cos (x)-\sqrt{5}+1\right)+\frac{1}{20} \left(1+\sqrt{5}\right) \log \left(-4 \cos (x)+\sqrt{5}+1\right)-\frac{1}{20} \left(1-\sqrt{5}\right) \log \left(4 \cos (x)-\sqrt{5}+1\right)-\frac{1}{20} \left(1+\sqrt{5}\right) \log \left(4 \cos (x)+\sqrt{5}+1\right)-\frac{1}{5} \tanh ^{-1}(\cos (x))",1,"(100*Cos[x] - 20*Log[Cos[x/2]] + Sqrt[5]*(-5 + Sqrt[5])*Log[1 - Sqrt[5] - 4*Cos[x]] + Sqrt[5]*(5 + Sqrt[5])*Log[1 + Sqrt[5] - 4*Cos[x]] - Sqrt[5]*(-5 + Sqrt[5])*Log[1 - Sqrt[5] + 4*Cos[x]] - Sqrt[5]*(5 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cos[x]] + 20*Log[Sin[x/2]])/100","A",1
114,1,87,38,0.0900952,"\int \cos (x) \cot (6 x) \, dx","Integrate[Cos[x]*Cot[6*x],x]","\frac{1}{12} \left(12 \cos (x)+2 \log \left(\sin \left(\frac{x}{2}\right)\right)-2 \log \left(\cos \left(\frac{x}{2}\right)\right)+\log (1-2 \cos (x))-\log (2 \cos (x)+1)+2 \sqrt{3} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)-2}{\sqrt{3}}\right)-2 \sqrt{3} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+2}{\sqrt{3}}\right)\right)","\cos (x)-\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))-\frac{\tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(2*Sqrt[3]*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]] - 2*Sqrt[3]*ArcTanh[(2 + Tan[x/2])/Sqrt[3]] + 12*Cos[x] - 2*Log[Cos[x/2]] + Log[1 - 2*Cos[x]] - Log[1 + 2*Cos[x]] + 2*Log[Sin[x/2]])/12","B",1
115,1,179,92,0.1827504,"\int \cos (x) \cot (n x) \, dx","Integrate[Cos[x]*Cot[n*x],x]","\frac{1}{2} e^{-2 i x} \left(-\frac{e^{i (2 n x+x)} \, _2F_1\left(1,1-\frac{1}{2 n};2-\frac{1}{2 n};e^{2 i n x}\right)}{2 n-1}-\frac{e^{i (2 n+3) x} \, _2F_1\left(1,1+\frac{1}{2 n};2+\frac{1}{2 n};e^{2 i n x}\right)}{2 n+1}+e^{i x} \, _2F_1\left(1,-\frac{1}{2 n};1-\frac{1}{2 n};e^{2 i n x}\right)-e^{3 i x} \, _2F_1\left(1,\frac{1}{2 n};1+\frac{1}{2 n};e^{2 i n x}\right)\right)","e^{-i x} \, _2F_1\left(1,-\frac{1}{2 n};1-\frac{1}{2 n};e^{2 i n x}\right)-e^{i x} \, _2F_1\left(1,\frac{1}{2 n};\frac{1}{2} \left(2+\frac{1}{n}\right);e^{2 i n x}\right)-\frac{e^{-i x}}{2}+\frac{e^{i x}}{2}",1,"(-((E^(I*(x + 2*n*x))*Hypergeometric2F1[1, 1 - 1/(2*n), 2 - 1/(2*n), E^((2*I)*n*x)])/(-1 + 2*n)) - (E^(I*(3 + 2*n)*x)*Hypergeometric2F1[1, 1 + 1/(2*n), 2 + 1/(2*n), E^((2*I)*n*x)])/(1 + 2*n) + E^(I*x)*Hypergeometric2F1[1, -1/2*1/n, 1 - 1/(2*n), E^((2*I)*n*x)] - E^((3*I)*x)*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), E^((2*I)*n*x)])/(2*E^((2*I)*x))","A",1
116,1,15,15,0.0070335,"\int \cos (x) \sec (2 x) \, dx","Integrate[Cos[x]*Sec[2*x],x]","\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{\sqrt{2}}","\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{\sqrt{2}}",1,"ArcTanh[Sqrt[2]*Sin[x]]/Sqrt[2]","A",1
117,1,15,44,0.0154688,"\int \cos (x) \sec (3 x) \, dx","Integrate[Cos[x]*Sec[3*x],x]","\frac{\tanh ^{-1}\left(\sqrt{3} \tan (x)\right)}{\sqrt{3}}","\frac{\log \left(\sqrt{3} \sin (x)+\cos (x)\right)}{2 \sqrt{3}}-\frac{\log \left(\cos (x)-\sqrt{3} \sin (x)\right)}{2 \sqrt{3}}",1,"ArcTanh[Sqrt[3]*Tan[x]]/Sqrt[3]","A",1
118,1,67,71,0.1038289,"\int \cos (x) \sec (4 x) \, dx","Integrate[Cos[x]*Sec[4*x],x]","\frac{1}{4} \sqrt{2+\sqrt{2}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2-\sqrt{2}}}\right)-\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2+\sqrt{2}}}\right)}{2 \sqrt{2 \left(2+\sqrt{2}\right)}}","\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2-\sqrt{2}}}\right)}{2 \sqrt{2 \left(2-\sqrt{2}\right)}}-\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2+\sqrt{2}}}\right)}{2 \sqrt{2 \left(2+\sqrt{2}\right)}}",1,"(Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[2]]])/4 - ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])])","A",1
119,1,84,163,0.0991152,"\int \cos (x) \sec (5 x) \, dx","Integrate[Cos[x]*Sec[5*x],x]","\frac{\sqrt{5+\sqrt{5}} \tanh ^{-1}\left(\frac{\left(5+\sqrt{5}\right) \tan (x)}{\sqrt{10-2 \sqrt{5}}}\right)+\sqrt{5-\sqrt{5}} \tanh ^{-1}\left(\frac{\left(\sqrt{5}-5\right) \tan (x)}{\sqrt{2 \left(5+\sqrt{5}\right)}}\right)}{5 \sqrt{2}}","\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \log \left(\cos (x)-\sqrt{5-2 \sqrt{5}} \sin (x)\right)-\frac{1}{10} \sqrt{\frac{1}{2} \left(5-\sqrt{5}\right)} \log \left(\sqrt{5-2 \sqrt{5}} \sin (x)+\cos (x)\right)-\frac{1}{10} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \log \left(\cos (x)-\sqrt{5+2 \sqrt{5}} \sin (x)\right)+\frac{1}{10} \sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} \log \left(\sqrt{5+2 \sqrt{5}} \sin (x)+\cos (x)\right)",1,"(Sqrt[5 + Sqrt[5]]*ArcTanh[((5 + Sqrt[5])*Tan[x])/Sqrt[10 - 2*Sqrt[5]]] + Sqrt[5 - Sqrt[5]]*ArcTanh[((-5 + Sqrt[5])*Tan[x])/Sqrt[2*(5 + Sqrt[5])]])/(5*Sqrt[2])","A",1
120,1,81,85,0.0830001,"\int \cos (x) \sec (6 x) \, dx","Integrate[Cos[x]*Sec[6*x],x]","\frac{1}{6} \left(-\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin (x)\right)+\sqrt{2+\sqrt{3}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2-\sqrt{3}}}\right)+\sqrt{2-\sqrt{3}} \tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2+\sqrt{3}}}\right)\right)","-\frac{\tanh ^{-1}\left(\sqrt{2} \sin (x)\right)}{3 \sqrt{2}}+\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2-\sqrt{3}}}\right)}{6 \sqrt{2-\sqrt{3}}}+\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2+\sqrt{3}}}\right)}{6 \sqrt{2+\sqrt{3}}}",1,"(-(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[x]]) + Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[3]]] + Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[3]]])/6","A",1
121,1,10,10,0.0063395,"\int \cos (2 x) \sec (x) \, dx","Integrate[Cos[2*x]*Sec[x],x]","2 \sin (x)-\tanh ^{-1}(\sin (x))","2 \sin (x)-\tanh ^{-1}(\sin (x))",1,"-ArcTanh[Sin[x]] + 2*Sin[x]","A",1
122,1,14,14,0.007348,"\int \cos (4 x) \sec (2 x) \, dx","Integrate[Cos[4*x]*Sec[2*x],x]","\sin (2 x)-\frac{1}{2} \tanh ^{-1}(\sin (2 x))","\sin (2 x)-\frac{1}{2} \tanh ^{-1}(\sin (2 x))",1,"-1/2*ArcTanh[Sin[2*x]] + Sin[2*x]","A",1
123,1,21,7,0.0031372,"\int \cos (x) \csc (2 x) \, dx","Integrate[Cos[x]*Csc[2*x],x]","\frac{1}{2} \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)","-\frac{1}{2} \tanh ^{-1}(\cos (x))",1,"(-Log[Cos[x/2]] + Log[Sin[x/2]])/2","B",1
124,1,21,21,0.0080411,"\int \cos (x) \csc (3 x) \, dx","Integrate[Cos[x]*Csc[3*x],x]","\frac{1}{3} \log (\sin (x))-\frac{1}{6} \log \left(3-4 \sin ^2(x)\right)","\frac{1}{3} \log (\sin (x))-\frac{1}{6} \log \left(3-4 \sin ^2(x)\right)",1,"Log[Sin[x]]/3 - Log[3 - 4*Sin[x]^2]/6","A",1
125,1,66,26,0.0587576,"\int \cos (x) \csc (4 x) \, dx","Integrate[Cos[x]*Csc[4*x],x]","\frac{1}{4} \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)+(1+i) (-1)^{3/4} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)-1}{\sqrt{2}}\right)+\sqrt{2} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+1}{\sqrt{2}}\right)\right)","\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{2 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\cos (x))",1,"((1 + I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] + Sqrt[2]*ArcTanh[(1 + Tan[x/2])/Sqrt[2]] - Log[Cos[x/2]] + Log[Sin[x/2]])/4","C",1
126,1,57,62,0.0600161,"\int \cos (x) \csc (5 x) \, dx","Integrate[Cos[x]*Csc[5*x],x]","\frac{1}{20} \left(4 \log (\sin (x))-\left(\left(1+\sqrt{5}\right) \log \left(4 \cos (2 x)-\sqrt{5}+1\right)\right)+\left(\sqrt{5}-1\right) \log \left(4 \cos (2 x)+\sqrt{5}+1\right)\right)","-\frac{1}{20} \left(1+\sqrt{5}\right) \log \left(-8 \sin ^2(x)-\sqrt{5}+5\right)-\frac{1}{20} \left(1-\sqrt{5}\right) \log \left(-8 \sin ^2(x)+\sqrt{5}+5\right)+\frac{1}{5} \log (\sin (x))",1,"(-((1 + Sqrt[5])*Log[1 - Sqrt[5] + 4*Cos[2*x]]) + (-1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cos[2*x]] + 4*Log[Sin[x]])/20","A",1
127,1,83,36,0.0769741,"\int \cos (x) \csc (6 x) \, dx","Integrate[Cos[x]*Csc[6*x],x]","\frac{1}{12} \left(2 \log \left(\sin \left(\frac{x}{2}\right)\right)-2 \log \left(\cos \left(\frac{x}{2}\right)\right)+\log (1-2 \cos (x))-\log (2 \cos (x)+1)-2 \sqrt{3} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)-2}{\sqrt{3}}\right)+2 \sqrt{3} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+2}{\sqrt{3}}\right)\right)","-\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))+\frac{\tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{3}}\right)}{2 \sqrt{3}}",1,"(-2*Sqrt[3]*ArcTanh[(-2 + Tan[x/2])/Sqrt[3]] + 2*Sqrt[3]*ArcTanh[(2 + Tan[x/2])/Sqrt[3]] - 2*Log[Cos[x/2]] + Log[1 - 2*Cos[x]] - Log[1 + 2*Cos[x]] + 2*Log[Sin[x/2]])/12","B",1
128,1,33,33,0.0161299,"\int \cos ^3(6 x) \sin (x) \, dx","Integrate[Cos[6*x]^3*Sin[x],x]","\frac{3}{40} \cos (5 x)-\frac{3}{56} \cos (7 x)+\frac{1}{136} \cos (17 x)-\frac{1}{152} \cos (19 x)","\frac{3}{40} \cos (5 x)-\frac{3}{56} \cos (7 x)+\frac{1}{136} \cos (17 x)-\frac{1}{152} \cos (19 x)",1,"(3*Cos[5*x])/40 - (3*Cos[7*x])/56 + Cos[17*x]/136 - Cos[19*x]/152","A",1
129,1,33,33,0.01689,"\int \cos ^3(6 x) \sin (9 x) \, dx","Integrate[Cos[6*x]^3*Sin[9*x],x]","-\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x)","-\frac{1}{8} \cos (3 x)+\frac{1}{72} \cos (9 x)-\frac{1}{40} \cos (15 x)-\frac{1}{216} \cos (27 x)",1,"-1/8*Cos[3*x] + Cos[9*x]/72 - Cos[15*x]/40 - Cos[27*x]/216","A",1
130,1,25,25,0.0156546,"\int \cos (2 x) \sin ^2(6 x) \, dx","Integrate[Cos[2*x]*Sin[6*x]^2,x]","\frac{1}{4} \sin (2 x)-\frac{1}{40} \sin (10 x)-\frac{1}{56} \sin (14 x)","\frac{1}{4} \sin (2 x)-\frac{1}{40} \sin (10 x)-\frac{1}{56} \sin (14 x)",1,"Sin[2*x]/4 - Sin[10*x]/40 - Sin[14*x]/56","A",1
131,1,23,23,0.0117154,"\int \cos (x) \sin ^2(6 x) \, dx","Integrate[Cos[x]*Sin[6*x]^2,x]","\frac{\sin (x)}{2}-\frac{1}{44} \sin (11 x)-\frac{1}{52} \sin (13 x)","\frac{\sin (x)}{2}-\frac{1}{44} \sin (11 x)-\frac{1}{52} \sin (13 x)",1,"Sin[x]/2 - Sin[11*x]/44 - Sin[13*x]/52","A",1
132,1,33,33,0.0135886,"\int \cos (x) \sin ^3(6 x) \, dx","Integrate[Cos[x]*Sin[6*x]^3,x]","-\frac{3}{40} \cos (5 x)-\frac{3}{56} \cos (7 x)+\frac{1}{136} \cos (17 x)+\frac{1}{152} \cos (19 x)","-\frac{3}{40} \cos (5 x)-\frac{3}{56} \cos (7 x)+\frac{1}{136} \cos (17 x)+\frac{1}{152} \cos (19 x)",1,"(-3*Cos[5*x])/40 - (3*Cos[7*x])/56 + Cos[17*x]/136 + Cos[19*x]/152","A",1
133,1,31,31,0.0158462,"\int \cos (7 x) \sin ^3(6 x) \, dx","Integrate[Cos[7*x]*Sin[6*x]^3,x]","\frac{3 \cos (x)}{8}+\frac{1}{88} \cos (11 x)-\frac{3}{104} \cos (13 x)+\frac{1}{200} \cos (25 x)","\frac{3 \cos (x)}{8}+\frac{1}{88} \cos (11 x)-\frac{3}{104} \cos (13 x)+\frac{1}{200} \cos (25 x)",1,"(3*Cos[x])/8 + Cos[11*x]/88 - (3*Cos[13*x])/104 + Cos[25*x]/200","A",1
134,1,41,41,0.0179373,"\int \cos ^2(3 x) \sin ^3(2 x) \, dx","Integrate[Cos[3*x]^2*Sin[2*x]^3,x]","-\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x)","-\frac{3}{16} \cos (2 x)+\frac{3}{64} \cos (4 x)+\frac{1}{48} \cos (6 x)-\frac{3}{128} \cos (8 x)+\frac{1}{192} \cos (12 x)",1,"(-3*Cos[2*x])/16 + (3*Cos[4*x])/64 + Cos[6*x]/48 - (3*Cos[8*x])/128 + Cos[12*x]/192","A",1
135,1,26,27,0.0474727,"\int \sin (a+b x) \sin (c+b x) \, dx","Integrate[Sin[a + b*x]*Sin[c + b*x],x]","-\frac{\sin (a+2 b x+c)-2 b x \cos (a-c)}{4 b}","\frac{1}{2} x \cos (a-c)-\frac{\sin (a+2 b x+c)}{4 b}",1,"-1/4*(-2*b*x*Cos[a - c] + Sin[a + c + 2*b*x])/b","A",1
136,1,26,27,0.0341481,"\int \sin (c-b x) \sin (a+b x) \, dx","Integrate[Sin[c - b*x]*Sin[a + b*x],x]","\frac{\sin (a+2 b x-c)-2 b x \cos (a+c)}{4 b}","\frac{\sin (a+2 b x-c)}{4 b}-\frac{1}{2} x \cos (a+c)",1,"(-2*b*x*Cos[a + c] + Sin[a - c + 2*b*x])/(4*b)","A",1
137,1,26,27,0.0245573,"\int \cos (a+b x) \cos (c+b x) \, dx","Integrate[Cos[a + b*x]*Cos[c + b*x],x]","\frac{\sin (a+2 b x+c)+2 b x \cos (a-c)}{4 b}","\frac{\sin (a+2 b x+c)}{4 b}+\frac{1}{2} x \cos (a-c)",1,"(2*b*x*Cos[a - c] + Sin[a + c + 2*b*x])/(4*b)","A",1
138,1,26,27,0.0231,"\int \cos (c-b x) \cos (a+b x) \, dx","Integrate[Cos[c - b*x]*Cos[a + b*x],x]","\frac{\sin (a+2 b x-c)+2 b x \cos (a+c)}{4 b}","\frac{\sin (a+2 b x-c)}{4 b}+\frac{1}{2} x \cos (a+c)",1,"(2*b*x*Cos[a + c] + Sin[a - c + 2*b*x])/(4*b)","A",1
139,1,31,39,0.5179283,"\int \tan (a+b x) \tan (c+b x) \, dx","Integrate[Tan[a + b*x]*Tan[c + b*x],x]","\frac{\cot (a-c) (\log (\cos (b x+c))-\log (\cos (a+b x)))}{b}-x","-\frac{\cot (a-c) \log (\cos (a+b x))}{b}+\frac{\cot (a-c) \log (\cos (b x+c))}{b}-x",1,"-x + (Cot[a - c]*(-Log[Cos[a + b*x]] + Log[Cos[c + b*x]]))/b","A",1
140,1,28,34,0.5294052,"\int \tan (c-b x) \tan (a+b x) \, dx","Integrate[Tan[c - b*x]*Tan[a + b*x],x]","\frac{\cot (a+c) (\log (\cos (a+b x))-\log (\cos (c-b x)))}{b}+x","-\frac{\cot (a+c) \log (\cos (c-b x))}{b}+\frac{\cot (a+c) \log (\cos (a+b x))}{b}+x",1,"x + (Cot[a + c]*(-Log[Cos[c - b*x]] + Log[Cos[a + b*x]]))/b","A",1
141,1,31,39,0.5104872,"\int \cot (a+b x) \cot (c+b x) \, dx","Integrate[Cot[a + b*x]*Cot[c + b*x],x]","\frac{\cot (a-c) (\log (\sin (b x+c))-\log (\sin (a+b x)))}{b}-x","-\frac{\cot (a-c) \log (\sin (a+b x))}{b}+\frac{\cot (a-c) \log (\sin (b x+c))}{b}-x",1,"-x + (Cot[a - c]*(-Log[Sin[a + b*x]] + Log[Sin[c + b*x]]))/b","A",1
142,1,30,34,0.5014687,"\int \cot (c-b x) \cot (a+b x) \, dx","Integrate[Cot[c - b*x]*Cot[a + b*x],x]","\frac{\cot (a+c) (\log (-\sin (a+b x))-\log (\sin (c-b x)))}{b}+x","-\frac{\cot (a+c) \log (\sin (c-b x))}{b}+\frac{\cot (a+c) \log (\sin (a+b x))}{b}+x",1,"x + (Cot[a + c]*(-Log[Sin[c - b*x]] + Log[-Sin[a + b*x]]))/b","A",1
143,1,28,36,0.2287616,"\int \sec (a+b x) \sec (c+b x) \, dx","Integrate[Sec[a + b*x]*Sec[c + b*x],x]","-\frac{\csc (a-c) (\log (\cos (a+b x))-\log (\cos (b x+c)))}{b}","\frac{\csc (a-c) \log (\cos (b x+c))}{b}-\frac{\csc (a-c) \log (\cos (a+b x))}{b}",1,"-((Csc[a - c]*(Log[Cos[a + b*x]] - Log[Cos[c + b*x]]))/b)","A",1
144,1,26,33,0.2345408,"\int \sec (c-b x) \sec (a+b x) \, dx","Integrate[Sec[c - b*x]*Sec[a + b*x],x]","\frac{\csc (a+c) (\log (\cos (c-b x))-\log (\cos (a+b x)))}{b}","\frac{\csc (a+c) \log (\cos (c-b x))}{b}-\frac{\csc (a+c) \log (\cos (a+b x))}{b}",1,"(Csc[a + c]*(Log[Cos[c - b*x]] - Log[Cos[a + b*x]]))/b","A",1
145,1,28,36,0.2432581,"\int \csc (a+b x) \csc (c+b x) \, dx","Integrate[Csc[a + b*x]*Csc[c + b*x],x]","-\frac{\csc (a-c) (\log (\sin (a+b x))-\log (\sin (b x+c)))}{b}","\frac{\csc (a-c) \log (\sin (b x+c))}{b}-\frac{\csc (a-c) \log (\sin (a+b x))}{b}",1,"-((Csc[a - c]*(Log[Sin[a + b*x]] - Log[Sin[c + b*x]]))/b)","A",1
146,1,29,33,0.219591,"\int \csc (c-b x) \csc (a+b x) \, dx","Integrate[Csc[c - b*x]*Csc[a + b*x],x]","-\frac{\csc (a+c) (\log (\sin (c-b x))-\log (-\sin (a+b x)))}{b}","\frac{\csc (a+c) \log (\sin (a+b x))}{b}-\frac{\csc (a+c) \log (\sin (c-b x))}{b}",1,"-((Csc[a + c]*(Log[Sin[c - b*x]] - Log[-Sin[a + b*x]]))/b)","A",1
147,1,13,13,0.0776539,"\int \sqrt{\sin (x) \tan (x)} \, dx","Integrate[Sqrt[Sin[x]*Tan[x]],x]","-2 \cot (x) \sqrt{\sin (x) \tan (x)}","-2 \cot (x) \sqrt{\sin (x) \tan (x)}",1,"-2*Cot[x]*Sqrt[Sin[x]*Tan[x]]","A",1
148,1,23,31,0.0377306,"\int (\sin (x) \tan (x))^{3/2} \, dx","Integrate[(Sin[x]*Tan[x])^(3/2),x]","\frac{2}{3} \sin (x) \left(4 \csc ^2(x)-1\right) \sqrt{\sin (x) \tan (x)}","\frac{8}{3} \csc (x) \sqrt{\sin (x) \tan (x)}-\frac{2}{3} \sin (x) \sqrt{\sin (x) \tan (x)}",1,"(2*(-1 + 4*Csc[x]^2)*Sin[x]*Sqrt[Sin[x]*Tan[x]])/3","A",1
149,1,29,50,0.0980275,"\int (\sin (x) \tan (x))^{5/2} \, dx","Integrate[(Sin[x]*Tan[x])^(5/2),x]","\frac{2}{15} \tan (x) \sqrt{\sin (x) \tan (x)} \left(3 \cos ^2(x)+32 \cot ^2(x)+5\right)","-\frac{2}{5} \sin ^2(x) \tan (x) \sqrt{\sin (x) \tan (x)}+\frac{16}{15} \tan (x) \sqrt{\sin (x) \tan (x)}+\frac{64}{15} \cot (x) \sqrt{\sin (x) \tan (x)}",1,"(2*(5 + 3*Cos[x]^2 + 32*Cot[x]^2)*Tan[x]*Sqrt[Sin[x]*Tan[x]])/15","A",1
150,1,13,13,0.0677207,"\int \sqrt{\cos (x) \cot (x)} \, dx","Integrate[Sqrt[Cos[x]*Cot[x]],x]","2 \tan (x) \sqrt{\cos (x) \cot (x)}","2 \tan (x) \sqrt{\cos (x) \cot (x)}",1,"2*Sqrt[Cos[x]*Cot[x]]*Tan[x]","A",1
151,1,21,31,0.0364069,"\int (\cos (x) \cot (x))^{3/2} \, dx","Integrate[(Cos[x]*Cot[x])^(3/2),x]","\frac{2}{3} \left(\cos ^2(x)-4\right) \sec (x) \sqrt{\cos (x) \cot (x)}","\frac{2}{3} \cos (x) \sqrt{\cos (x) \cot (x)}-\frac{8}{3} \sec (x) \sqrt{\cos (x) \cot (x)}",1,"(2*(-4 + Cos[x]^2)*Sqrt[Cos[x]*Cot[x]]*Sec[x])/3","A",1
152,1,29,50,0.0981998,"\int (\cos (x) \cot (x))^{5/2} \, dx","Integrate[(Cos[x]*Cot[x])^(5/2),x]","-\frac{2}{15} \tan (x) \sqrt{\cos (x) \cot (x)} \left(3 \cos ^2(x)+5 \cot ^2(x)+32\right)","\frac{2}{5} \cos ^2(x) \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{16}{15} \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{64}{15} \tan (x) \sqrt{\cos (x) \cot (x)}",1,"(-2*Sqrt[Cos[x]*Cot[x]]*(32 + 3*Cos[x]^2 + 5*Cot[x]^2)*Tan[x])/15","A",1
153,1,56,58,0.139299,"\int \frac{x \cos (x)}{(a+b \sin (x))^2} \, dx","Integrate[(x*Cos[x])/(a + b*Sin[x])^2,x]","\frac{\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{x}{a+b \sin (x)}}{b}","\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2}}-\frac{x}{b (a+b \sin (x))}",1,"((2*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - x/(a + b*Sin[x]))/b","A",1
154,1,84,85,0.2600679,"\int \frac{x \cos (x)}{(a+b \sin (x))^3} \, dx","Integrate[(x*Cos[x])/(a + b*Sin[x])^3,x]","\frac{a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right)^{3/2}}+\frac{\frac{\cos (x) (a+b \sin (x))}{(a-b) (a+b)}-\frac{x}{b}}{2 (a+b \sin (x))^2}","\frac{a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b \left(a^2-b^2\right)^{3/2}}+\frac{\cos (x)}{2 \left(a^2-b^2\right) (a+b \sin (x))}-\frac{x}{2 b (a+b \sin (x))^2}",1,"(a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)) + (-(x/b) + (Cos[x]*(a + b*Sin[x]))/((a - b)*(a + b)))/(2*(a + b*Sin[x])^2)","A",1
155,1,58,59,0.1024216,"\int \frac{x \sin (x)}{(a+b \cos (x))^2} \, dx","Integrate[(x*Sin[x])/(a + b*Cos[x])^2,x]","\frac{2 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{b \sqrt{b^2-a^2}}+\frac{x}{b (a+b \cos (x))}","\frac{x}{b (a+b \cos (x))}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{b \sqrt{a-b} \sqrt{a+b}}",1,"(2*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/(b*Sqrt[-a^2 + b^2]) + x/(b*(a + b*Cos[x]))","A",1
156,1,85,88,0.3152336,"\int \frac{x \sin (x)}{(a+b \cos (x))^3} \, dx","Integrate[(x*Sin[x])/(a + b*Cos[x])^3,x]","\frac{\frac{\sin (x) (a+b \cos (x))}{(a-b) (a+b)}+\frac{x}{b}}{2 (a+b \cos (x))^2}-\frac{a \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{b \left(b^2-a^2\right)^{3/2}}","\frac{\sin (x)}{2 \left(a^2-b^2\right) (a+b \cos (x))}+\frac{x}{2 b (a+b \cos (x))^2}-\frac{a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{b (a-b)^{3/2} (a+b)^{3/2}}",1,"-((a*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/(b*(-a^2 + b^2)^(3/2))) + (x/b + ((a + b*Cos[x])*Sin[x])/((a - b)*(a + b)))/(2*(a + b*Cos[x])^2)","A",1
157,1,48,50,0.1842012,"\int \frac{x \sec ^2(x)}{(a+b \tan (x))^2} \, dx","Integrate[(x*Sec[x]^2)/(a + b*Tan[x])^2,x]","\frac{a \log (a \cos (x)+b \sin (x))-b x}{a^3+a b^2}+\frac{x \sin (x)}{a^2 \cos (x)+a b \sin (x)}","\frac{a x}{b \left(a^2+b^2\right)}+\frac{\log (a \cos (x)+b \sin (x))}{a^2+b^2}-\frac{x}{b (a+b \tan (x))}",1,"(-(b*x) + a*Log[a*Cos[x] + b*Sin[x]])/(a^3 + a*b^2) + (x*Sin[x])/(a^2*Cos[x] + a*b*Sin[x])","A",1
158,1,48,50,0.1866216,"\int \frac{x \csc ^2(x)}{(a+b \cot (x))^2} \, dx","Integrate[(x*Csc[x]^2)/(a + b*Cot[x])^2,x]","\frac{b \log (a \sin (x)+b \cos (x))-a x}{a^2 b+b^3}+\frac{x \sin (x)}{a b \sin (x)+b^2 \cos (x)}","-\frac{a x}{b \left(a^2+b^2\right)}+\frac{\log (a \sin (x)+b \cos (x))}{a^2+b^2}+\frac{x}{b (a+b \cot (x))}",1,"(-(a*x) + b*Log[b*Cos[x] + a*Sin[x]])/(a^2*b + b^3) + (x*Sin[x])/(b^2*Cos[x] + a*b*Sin[x])","A",1
159,1,32,32,0.0932405,"\int \frac{\sec ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}",1,"ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)","A",1
160,1,512,211,6.4375459,"\int \frac{x \sec ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[(x*Sec[c + d*x]^2)/(a + b*Tan[c + d*x]^2),x]","\frac{x \left(-\sqrt{a} \text{Li}_2\left(\frac{\sqrt{b} (1-i \tan (c+d x))}{i \sqrt{-a}+\sqrt{b}}\right)-\sqrt{a} \text{Li}_2\left(\frac{\sqrt{b} (i \tan (c+d x)+1)}{i \sqrt{-a}+\sqrt{b}}\right)+\sqrt{a} \text{Li}_2\left(-\frac{\sqrt{b} (\tan (c+d x)-i)}{\sqrt{-a}+i \sqrt{b}}\right)+\sqrt{a} \text{Li}_2\left(\frac{\sqrt{b} (\tan (c+d x)+i)}{\sqrt{-a}+i \sqrt{b}}\right)+4 i \sqrt{-a} c \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)-\sqrt{a} \log (1+i \tan (c+d x)) \log \left(\frac{\sqrt{-a}-\sqrt{b} \tan (c+d x)}{\sqrt{-a}-i \sqrt{b}}\right)+\sqrt{a} \log (1-i \tan (c+d x)) \log \left(\frac{\sqrt{-a}-\sqrt{b} \tan (c+d x)}{\sqrt{-a}+i \sqrt{b}}\right)-\sqrt{a} \log (1-i \tan (c+d x)) \log \left(\frac{\sqrt{-a}+\sqrt{b} \tan (c+d x)}{\sqrt{-a}-i \sqrt{b}}\right)+\sqrt{a} \log (1+i \tan (c+d x)) \log \left(\frac{\sqrt{-a}+\sqrt{b} \tan (c+d x)}{\sqrt{-a}+i \sqrt{b}}\right)\right)}{2 \sqrt{-a^2} \sqrt{b} d (\log (1-i \tan (c+d x))-\log (1+i \tan (c+d x))+2 i c)}","-\frac{\text{Li}_2\left(-\frac{(a-b) e^{2 i (c+d x)}}{\left(\sqrt{a}-\sqrt{b}\right)^2}\right)}{4 \sqrt{a} \sqrt{b} d^2}+\frac{\text{Li}_2\left(-\frac{(a-b) e^{2 i (c+d x)}}{\left(\sqrt{a}+\sqrt{b}\right)^2}\right)}{4 \sqrt{a} \sqrt{b} d^2}-\frac{i x \log \left(1+\frac{(a-b) e^{2 i (c+d x)}}{\left(\sqrt{a}-\sqrt{b}\right)^2}\right)}{2 \sqrt{a} \sqrt{b} d}+\frac{i x \log \left(1+\frac{(a-b) e^{2 i (c+d x)}}{\left(\sqrt{a}+\sqrt{b}\right)^2}\right)}{2 \sqrt{a} \sqrt{b} d}",1,"(x*((4*I)*Sqrt[-a]*c*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]] - Sqrt[a]*Log[1 + I*Tan[c + d*x]]*Log[(Sqrt[-a] - Sqrt[b]*Tan[c + d*x])/(Sqrt[-a] - I*Sqrt[b])] + Sqrt[a]*Log[1 - I*Tan[c + d*x]]*Log[(Sqrt[-a] - Sqrt[b]*Tan[c + d*x])/(Sqrt[-a] + I*Sqrt[b])] - Sqrt[a]*Log[1 - I*Tan[c + d*x]]*Log[(Sqrt[-a] + Sqrt[b]*Tan[c + d*x])/(Sqrt[-a] - I*Sqrt[b])] + Sqrt[a]*Log[1 + I*Tan[c + d*x]]*Log[(Sqrt[-a] + Sqrt[b]*Tan[c + d*x])/(Sqrt[-a] + I*Sqrt[b])] - Sqrt[a]*PolyLog[2, (Sqrt[b]*(1 - I*Tan[c + d*x]))/(I*Sqrt[-a] + Sqrt[b])] - Sqrt[a]*PolyLog[2, (Sqrt[b]*(1 + I*Tan[c + d*x]))/(I*Sqrt[-a] + Sqrt[b])] + Sqrt[a]*PolyLog[2, -((Sqrt[b]*(-I + Tan[c + d*x]))/(Sqrt[-a] + I*Sqrt[b]))] + Sqrt[a]*PolyLog[2, (Sqrt[b]*(I + Tan[c + d*x]))/(Sqrt[-a] + I*Sqrt[b])]))/(2*Sqrt[-a^2]*Sqrt[b]*d*((2*I)*c + Log[1 - I*Tan[c + d*x]] - Log[1 + I*Tan[c + d*x]]))","B",0
161,1,294,337,1.0892964,"\int \frac{x^2 \sec ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Integrate[(x^2*Sec[c + d*x]^2)/(a + b*Tan[c + d*x]^2),x]","\frac{i \left(2 d^2 x^2 \log \left(1+\frac{\left(\sqrt{a}-\sqrt{b}\right) e^{2 i (c+d x)}}{\sqrt{a}+\sqrt{b}}\right)-2 d^2 x^2 \log \left(1+\frac{\left(\sqrt{a}+\sqrt{b}\right) e^{2 i (c+d x)}}{\sqrt{a}-\sqrt{b}}\right)-2 i d x \text{Li}_2\left(\frac{\left(\sqrt{b}-\sqrt{a}\right) e^{2 i (c+d x)}}{\sqrt{a}+\sqrt{b}}\right)+2 i d x \text{Li}_2\left(-\frac{\left(\sqrt{a}+\sqrt{b}\right) e^{2 i (c+d x)}}{\sqrt{a}-\sqrt{b}}\right)+\text{Li}_3\left(\frac{\left(\sqrt{b}-\sqrt{a}\right) e^{2 i (c+d x)}}{\sqrt{a}+\sqrt{b}}\right)-\text{Li}_3\left(-\frac{\left(\sqrt{a}+\sqrt{b}\right) e^{2 i (c+d x)}}{\sqrt{a}-\sqrt{b}}\right)\right)}{4 \sqrt{a} \sqrt{b} d^3}","\frac{i \text{Li}_3\left(-\frac{\left(\sqrt{a}-\sqrt{b}\right) e^{2 i (c+d x)}}{\sqrt{a}+\sqrt{b}}\right)}{4 \sqrt{a} \sqrt{b} d^3}-\frac{i \text{Li}_3\left(-\frac{\left(\sqrt{a}+\sqrt{b}\right) e^{2 i (c+d x)}}{\sqrt{a}-\sqrt{b}}\right)}{4 \sqrt{a} \sqrt{b} d^3}-\frac{x \text{Li}_2\left(-\frac{(a-b) e^{2 i (c+d x)}}{\left(\sqrt{a}-\sqrt{b}\right)^2}\right)}{2 \sqrt{a} \sqrt{b} d^2}+\frac{x \text{Li}_2\left(-\frac{(a-b) e^{2 i (c+d x)}}{\left(\sqrt{a}+\sqrt{b}\right)^2}\right)}{2 \sqrt{a} \sqrt{b} d^2}-\frac{i x^2 \log \left(1+\frac{(a-b) e^{2 i (c+d x)}}{\left(\sqrt{a}-\sqrt{b}\right)^2}\right)}{2 \sqrt{a} \sqrt{b} d}+\frac{i x^2 \log \left(1+\frac{(a-b) e^{2 i (c+d x)}}{\left(\sqrt{a}+\sqrt{b}\right)^2}\right)}{2 \sqrt{a} \sqrt{b} d}",1,"((I/4)*(2*d^2*x^2*Log[1 + ((Sqrt[a] - Sqrt[b])*E^((2*I)*(c + d*x)))/(Sqrt[a] + Sqrt[b])] - 2*d^2*x^2*Log[1 + ((Sqrt[a] + Sqrt[b])*E^((2*I)*(c + d*x)))/(Sqrt[a] - Sqrt[b])] - (2*I)*d*x*PolyLog[2, ((-Sqrt[a] + Sqrt[b])*E^((2*I)*(c + d*x)))/(Sqrt[a] + Sqrt[b])] + (2*I)*d*x*PolyLog[2, -(((Sqrt[a] + Sqrt[b])*E^((2*I)*(c + d*x)))/(Sqrt[a] - Sqrt[b]))] + PolyLog[3, ((-Sqrt[a] + Sqrt[b])*E^((2*I)*(c + d*x)))/(Sqrt[a] + Sqrt[b])] - PolyLog[3, -(((Sqrt[a] + Sqrt[b])*E^((2*I)*(c + d*x)))/(Sqrt[a] - Sqrt[b]))]))/(Sqrt[a]*Sqrt[b]*d^3)","A",1
162,1,40,40,0.2549378,"\int \frac{\sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b+c} \tan (c+d x)}{\sqrt{a+c}}\right)}{d \sqrt{a+c} \sqrt{b+c}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b+c} \tan (c+d x)}{\sqrt{a+c}}\right)}{d \sqrt{a+c} \sqrt{b+c}}",1,"ArcTan[(Sqrt[b + c]*Tan[c + d*x])/Sqrt[a + c]]/(Sqrt[a + c]*Sqrt[b + c]*d)","A",1
163,1,751,267,4.225774,"\int \frac{x \sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx","Integrate[(x*Sec[c + d*x]^2)/(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2),x]","\frac{x \left(\sqrt{a+c}-\sqrt{-b-c} \tan (c+d x)\right) \left(\sqrt{a+c}+\sqrt{-b-c} \tan (c+d x)\right) \left(i \sqrt{b+c} \text{Li}_2\left(\frac{\sqrt{-b-c} (1-i \tan (c+d x))}{\sqrt{-b-c}-i \sqrt{a+c}}\right)-i \sqrt{b+c} \text{Li}_2\left(\frac{\sqrt{-b-c} (1-i \tan (c+d x))}{\sqrt{-b-c}+i \sqrt{a+c}}\right)+i \sqrt{b+c} \text{Li}_2\left(\frac{\sqrt{-b-c} (i \tan (c+d x)+1)}{\sqrt{-b-c}-i \sqrt{a+c}}\right)-i \sqrt{b+c} \text{Li}_2\left(\frac{\sqrt{-b-c} (i \tan (c+d x)+1)}{\sqrt{-b-c}+i \sqrt{a+c}}\right)+4 c \sqrt{-b-c} \tan ^{-1}\left(\frac{\sqrt{b+c} \tan (c+d x)}{\sqrt{a+c}}\right)-i \sqrt{b+c} \log (1+i \tan (c+d x)) \log \left(\frac{i \left(\sqrt{a+c}-\sqrt{-b-c} \tan (c+d x)\right)}{\sqrt{-b-c}+i \sqrt{a+c}}\right)+i \sqrt{b+c} \log (1-i \tan (c+d x)) \log \left(\frac{i \left(\sqrt{-b-c} \tan (c+d x)-\sqrt{a+c}\right)}{\sqrt{-b-c}-i \sqrt{a+c}}\right)+i \sqrt{b+c} \log (1+i \tan (c+d x)) \log \left(-\frac{i \left(\sqrt{a+c}+\sqrt{-b-c} \tan (c+d x)\right)}{\sqrt{-b-c}-i \sqrt{a+c}}\right)-i \sqrt{b+c} \log (1-i \tan (c+d x)) \log \left(\frac{i \left(\sqrt{a+c}+\sqrt{-b-c} \tan (c+d x)\right)}{\sqrt{-b-c}+i \sqrt{a+c}}\right)\right)}{2 d \sqrt{a+c} \sqrt{-(b+c)^2} (-i \log (1-i \tan (c+d x))+i \log (1+i \tan (c+d x))+2 c) \left(a+b \tan ^2(c+d x)+c \sec ^2(c+d x)\right)}","-\frac{\text{Li}_2\left(-\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 c-2 \sqrt{a+c} \sqrt{b+c}}\right)}{4 d^2 \sqrt{a+c} \sqrt{b+c}}+\frac{\text{Li}_2\left(-\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 \left(c+\sqrt{a+c} \sqrt{b+c}\right)}\right)}{4 d^2 \sqrt{a+c} \sqrt{b+c}}-\frac{i x \log \left(1+\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right)}{2 d \sqrt{a+c} \sqrt{b+c}}+\frac{i x \log \left(1+\frac{(a-b) e^{2 i (c+d x)}}{2 \left(\sqrt{a+c} \sqrt{b+c}+c\right)+a+b}\right)}{2 d \sqrt{a+c} \sqrt{b+c}}",1,"(x*(4*Sqrt[-b - c]*c*ArcTan[(Sqrt[b + c]*Tan[c + d*x])/Sqrt[a + c]] - I*Sqrt[b + c]*Log[1 + I*Tan[c + d*x]]*Log[(I*(Sqrt[a + c] - Sqrt[-b - c]*Tan[c + d*x]))/(Sqrt[-b - c] + I*Sqrt[a + c])] + I*Sqrt[b + c]*Log[1 - I*Tan[c + d*x]]*Log[(I*(-Sqrt[a + c] + Sqrt[-b - c]*Tan[c + d*x]))/(Sqrt[-b - c] - I*Sqrt[a + c])] + I*Sqrt[b + c]*Log[1 + I*Tan[c + d*x]]*Log[((-I)*(Sqrt[a + c] + Sqrt[-b - c]*Tan[c + d*x]))/(Sqrt[-b - c] - I*Sqrt[a + c])] - I*Sqrt[b + c]*Log[1 - I*Tan[c + d*x]]*Log[(I*(Sqrt[a + c] + Sqrt[-b - c]*Tan[c + d*x]))/(Sqrt[-b - c] + I*Sqrt[a + c])] + I*Sqrt[b + c]*PolyLog[2, (Sqrt[-b - c]*(1 - I*Tan[c + d*x]))/(Sqrt[-b - c] - I*Sqrt[a + c])] - I*Sqrt[b + c]*PolyLog[2, (Sqrt[-b - c]*(1 - I*Tan[c + d*x]))/(Sqrt[-b - c] + I*Sqrt[a + c])] + I*Sqrt[b + c]*PolyLog[2, (Sqrt[-b - c]*(1 + I*Tan[c + d*x]))/(Sqrt[-b - c] - I*Sqrt[a + c])] - I*Sqrt[b + c]*PolyLog[2, (Sqrt[-b - c]*(1 + I*Tan[c + d*x]))/(Sqrt[-b - c] + I*Sqrt[a + c])])*(Sqrt[a + c] - Sqrt[-b - c]*Tan[c + d*x])*(Sqrt[a + c] + Sqrt[-b - c]*Tan[c + d*x]))/(2*Sqrt[a + c]*Sqrt[-(b + c)^2]*d*(2*c - I*Log[1 - I*Tan[c + d*x]] + I*Log[1 + I*Tan[c + d*x]])*(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2))","B",0
164,1,499,407,2.2138368,"\int \frac{x^2 \sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx","Integrate[(x^2*Sec[c + d*x]^2)/(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2),x]","-\frac{i e^{2 i c} \left(2 d^2 x^2 \log \left(1+\frac{(a-b) e^{2 i (2 c+d x)}}{-2 \sqrt{e^{4 i c} (a+c) (b+c)}+a e^{2 i c}+b e^{2 i c}+2 c e^{2 i c}}\right)-2 d^2 x^2 \log \left(1+\frac{(a-b) e^{2 i (2 c+d x)}}{2 \sqrt{e^{4 i c} (a+c) (b+c)}+a e^{2 i c}+b e^{2 i c}+2 c e^{2 i c}}\right)-2 i d x \text{Li}_2\left(\frac{(b-a) e^{2 i (2 c+d x)}}{e^{2 i c} a+b e^{2 i c}+2 c e^{2 i c}-2 \sqrt{(a+c) (b+c) e^{4 i c}}}\right)+2 i d x \text{Li}_2\left(\frac{(b-a) e^{2 i (2 c+d x)}}{e^{2 i c} a+b e^{2 i c}+2 c e^{2 i c}+2 \sqrt{(a+c) (b+c) e^{4 i c}}}\right)+\text{Li}_3\left(\frac{(b-a) e^{2 i (2 c+d x)}}{e^{2 i c} a+b e^{2 i c}+2 c e^{2 i c}-2 \sqrt{(a+c) (b+c) e^{4 i c}}}\right)-\text{Li}_3\left(\frac{(b-a) e^{2 i (2 c+d x)}}{e^{2 i c} a+b e^{2 i c}+2 c e^{2 i c}+2 \sqrt{(a+c) (b+c) e^{4 i c}}}\right)\right)}{4 d^3 \sqrt{e^{4 i c} (a+c) (b+c)}}","-\frac{i \text{Li}_3\left(-\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 c-2 \sqrt{a+c} \sqrt{b+c}}\right)}{4 d^3 \sqrt{a+c} \sqrt{b+c}}+\frac{i \text{Li}_3\left(-\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 \left(c+\sqrt{a+c} \sqrt{b+c}\right)}\right)}{4 d^3 \sqrt{a+c} \sqrt{b+c}}-\frac{x \text{Li}_2\left(-\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 c-2 \sqrt{a+c} \sqrt{b+c}}\right)}{2 d^2 \sqrt{a+c} \sqrt{b+c}}+\frac{x \text{Li}_2\left(-\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 \left(c+\sqrt{a+c} \sqrt{b+c}\right)}\right)}{2 d^2 \sqrt{a+c} \sqrt{b+c}}-\frac{i x^2 \log \left(1+\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right)}{2 d \sqrt{a+c} \sqrt{b+c}}+\frac{i x^2 \log \left(1+\frac{(a-b) e^{2 i (c+d x)}}{2 \left(\sqrt{a+c} \sqrt{b+c}+c\right)+a+b}\right)}{2 d \sqrt{a+c} \sqrt{b+c}}",1,"((-1/4*I)*E^((2*I)*c)*(2*d^2*x^2*Log[1 + ((a - b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*c) - 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])] - 2*d^2*x^2*Log[1 + ((a - b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*c) + 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])] - (2*I)*d*x*PolyLog[2, ((-a + b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*c) - 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])] + (2*I)*d*x*PolyLog[2, ((-a + b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*c) + 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])] + PolyLog[3, ((-a + b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*c) - 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])] - PolyLog[3, ((-a + b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*c) + 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])]))/(d^3*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])","A",1
165,1,61,155,0.5121772,"\int x^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \, dx","Integrate[x^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]],x]","\frac{\left(f x \left(f^2 x^2-6\right) \tan (e+f x)+3 f^2 x^2-6\right) \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)}}{f^4}","-\frac{6 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^4}-\frac{6 x \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^3}+\frac{3 x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}+\frac{x^3 \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f}",1,"(Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(-6 + 3*f^2*x^2 + f*x*(-6 + f^2*x^2)*Tan[e + f*x]))/f^4","A",1
166,1,54,118,0.3329473,"\int x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \, dx","Integrate[x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]],x]","\frac{\left(\left(f^2 x^2-2\right) \tan (e+f x)+2 f x\right) \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)}}{f^3}","-\frac{2 \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^3}+\frac{2 x \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}+\frac{x^2 \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f}",1,"(Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(2*f*x + (-2 + f^2*x^2)*Tan[e + f*x]))/f^3","A",1
167,1,44,74,0.2026206,"\int x \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \, dx","Integrate[x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]],x]","\frac{(f x \tan (e+f x)+1) \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)}}{f^2}","\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}+\frac{x \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f}",1,"(Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(1 + f*x*Tan[e + f*x]))/f^2","A",1
168,1,52,86,0.2144866,"\int \frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x} \, dx","Integrate[(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x,x]","\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)} (\cos (e) \text{Ci}(f x)-\sin (e) \text{Si}(f x))","\cos (e) \text{Ci}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}",1,"Sec[e + f*x]*Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(Cos[e]*CosIntegral[f*x] - Sin[e]*SinIntegral[f*x])","A",1
169,1,65,123,0.2432634,"\int \frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x^2} \, dx","Integrate[(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x^2,x]","-\frac{\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)} (f x \sin (e) \text{Ci}(f x)+f x \cos (e) \text{Si}(f x)+\cos (e+f x))}{x}","-f \sin (e) \text{Ci}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-f \cos (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{x}",1,"-((Sec[e + f*x]*Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(Cos[e + f*x] + f*x*CosIntegral[f*x]*Sin[e] + f*x*Cos[e]*SinIntegral[f*x]))/x)","A",1
170,1,87,176,0.2841957,"\int \frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x^3} \, dx","Integrate[(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x^3,x]","\frac{\sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)} \left(-f^2 x^2 \cos (e) \text{Ci}(f x)+f^2 x^2 \sin (e) \text{Si}(f x)+f x \sin (e+f x)-\cos (e+f x)\right)}{2 x^2}","-\frac{1}{2} f^2 \cos (e) \text{Ci}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+\frac{1}{2} f^2 \sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x^2}+\frac{f \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x}",1,"(Sec[e + f*x]*Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(-Cos[e + f*x] - f^2*x^2*Cos[e]*CosIntegral[f*x] + f*x*Sin[e + f*x] + f^2*x^2*Sin[e]*SinIntegral[f*x]))/(2*x^2)","A",1
171,1,113,393,1.1555235,"\int x^3 \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2} \, dx","Integrate[x^3*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2),x]","\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)} \left(\left(6 f^2 x^2-3\right) \sin (e+f x)+8 \left(f x \left(f^2 x^2-6\right) \tan (e+f x)+3 f^2 x^2-6\right)-f x \left(2 f^2 x^2-3\right) \cos (2 (e+f x)) \sec (e+f x)\right)}{8 f^4}","-\frac{3 c \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{8 f^4}-\frac{6 c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^4}-\frac{3 c x \sin (e+f x) \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f^3}-\frac{6 c x \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^3}+\frac{3 c x \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{8 f^3}+\frac{3 c x^2 \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f^2}+\frac{3 c x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}+\frac{x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c \sin (e+f x)+c)^{5/2}}{2 c f}-\frac{3 c x^3 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f}",1,"(c*Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(-(f*x*(-3 + 2*f^2*x^2)*Cos[2*(e + f*x)]*Sec[e + f*x]) + (-3 + 6*f^2*x^2)*Sin[e + f*x] + 8*(-6 + 3*f^2*x^2 + f*x*(-6 + f^2*x^2)*Tan[e + f*x])))/(8*f^4)","A",1
172,1,95,265,0.8084717,"\int x^2 \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2} \, dx","Integrate[x^2*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2),x]","\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)} \left(8 \left(f^2 x^2-2\right) \tan (e+f x)-\left(2 f^2 x^2-1\right) \cos (2 (e+f x)) \sec (e+f x)+4 f x \sin (e+f x)+16 f x\right)}{8 f^3}","-\frac{c \sin (e+f x) \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f^3}-\frac{2 c \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^3}+\frac{2 c x \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}+\frac{c x \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 f^2}+\frac{x^2 \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c \sin (e+f x)+c)^{5/2}}{2 c f}-\frac{3 c x^2 \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f}",1,"(c*Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(16*f*x - (-1 + 2*f^2*x^2)*Cos[2*(e + f*x)]*Sec[e + f*x] + 4*f*x*Sin[e + f*x] + 8*(-2 + f^2*x^2)*Tan[e + f*x]))/(8*f^3)","A",1
173,1,73,168,0.638678,"\int x \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2} \, dx","Integrate[x*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2),x]","\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)} (\sin (e+f x)+4 f x \tan (e+f x)-f x \cos (2 (e+f x)) \sec (e+f x)+4)}{4 f^2}","\frac{c \sin (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f^2}+\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}+\frac{x \sec (e+f x) \sqrt{a-a \sin (e+f x)} (c \sin (e+f x)+c)^{5/2}}{2 c f}-\frac{3 c x \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 f}",1,"(c*Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]]*(4 - f*x*Cos[2*(e + f*x)]*Sec[e + f*x] + Sin[e + f*x] + 4*f*x*Tan[e + f*x]))/(4*f^2)","A",1
174,1,150,186,1.2231213,"\int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x} \, dx","Integrate[(Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2))/x,x]","\frac{c e^{-i (e-f x)} \sqrt{-i c e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2} \left(2 e^{i e} \text{Ei}(-i f x)+2 e^{3 i e} \text{Ei}(i f x)+i \left(\text{Ei}(-2 i f x)-e^{4 i e} \text{Ei}(2 i f x)\right)\right) \sqrt{a-a \sin (e+f x)}}{2 \sqrt{2} \left(1+e^{2 i (e+f x)}\right)}","\frac{1}{2} c \sin (2 e) \text{Ci}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+c \cos (e) \text{Ci}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c \sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+\frac{1}{2} c \cos (2 e) \text{Si}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}",1,"(c*Sqrt[((-I)*c*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*(2*E^(I*e)*ExpIntegralEi[(-I)*f*x] + 2*E^((3*I)*e)*ExpIntegralEi[I*f*x] + I*(ExpIntegralEi[(-2*I)*f*x] - E^((4*I)*e)*ExpIntegralEi[(2*I)*f*x]))*Sqrt[a - a*Sin[e + f*x]])/(2*Sqrt[2]*E^(I*(e - f*x))*(1 + E^((2*I)*(e + f*x))))","C",1
175,1,231,273,1.4376121,"\int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx","Integrate[(Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2))/x^2,x]","\frac{c e^{-i (e+f x)} \sqrt{-i c e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2} \left(-2 i f x e^{i (e+2 f x)} \text{Ei}(-i f x)+2 i f x e^{3 i e+2 i f x} \text{Ei}(i f x)+2 f x e^{2 i (2 e+f x)} \text{Ei}(2 i f x)-2 e^{i (e+f x)}-2 e^{3 i (e+f x)}+i e^{4 i (e+f x)}+2 f x e^{2 i f x} \text{Ei}(-2 i f x)-i\right) \sqrt{a-a \sin (e+f x)}}{2 \sqrt{2} x \left(1+e^{2 i (e+f x)}\right)}","-c f \sin (e) \text{Ci}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+c f \cos (2 e) \text{Ci}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c f \sin (2 e) \text{Si}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c f \cos (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{x}-\frac{c \sin (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x}",1,"(c*Sqrt[((-I)*c*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*(-I - 2*E^(I*(e + f*x)) - 2*E^((3*I)*(e + f*x)) + I*E^((4*I)*(e + f*x)) - (2*I)*E^(I*(e + 2*f*x))*f*x*ExpIntegralEi[(-I)*f*x] + (2*I)*E^((3*I)*e + (2*I)*f*x)*f*x*ExpIntegralEi[I*f*x] + 2*E^((2*I)*f*x)*f*x*ExpIntegralEi[(-2*I)*f*x] + 2*E^((2*I)*(2*e + f*x))*f*x*ExpIntegralEi[(2*I)*f*x])*Sqrt[a - a*Sin[e + f*x]])/(2*Sqrt[2]*E^(I*(e + f*x))*(1 + E^((2*I)*(e + f*x)))*x)","C",1
176,1,317,385,1.8382349,"\int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^3} \, dx","Integrate[(Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2))/x^3,x]","\frac{c^2 e^{-2 i (e+f x)} \left(e^{i (e+f x)}+i\right) \left(2 i f^2 x^2 e^{i (e+2 f x)} \text{Ei}(-i f x)+2 i f^2 x^2 e^{3 i e+2 i f x} \text{Ei}(i f x)+4 f^2 x^2 e^{2 i (2 e+f x)} \text{Ei}(2 i f x)+2 f x e^{i (e+f x)}-2 f x e^{3 i (e+f x)}+2 i f x e^{4 i (e+f x)}+2 i e^{i (e+f x)}+2 i e^{3 i (e+f x)}+e^{4 i (e+f x)}-4 f^2 x^2 e^{2 i f x} \text{Ei}(-2 i f x)+2 i f x-1\right) \sqrt{a-a \sin (e+f x)}}{4 \sqrt{2} x^2 \left(e^{i (e+f x)}-i\right) \sqrt{-i c e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2}}","-c f^2 \sin (2 e) \text{Ci}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{1}{2} c f^2 \cos (e) \text{Ci}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}+\frac{1}{2} c f^2 \sin (e) \text{Si}(f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-c f^2 \cos (2 e) \text{Si}(2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}-\frac{c \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x^2}-\frac{c \sin (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{4 x^2}+\frac{c f \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x}-\frac{c f \cos (2 e+2 f x) \sec (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{2 x}",1,"(c^2*(I + E^(I*(e + f*x)))*(-1 + (2*I)*E^(I*(e + f*x)) + (2*I)*E^((3*I)*(e + f*x)) + E^((4*I)*(e + f*x)) + (2*I)*f*x + 2*E^(I*(e + f*x))*f*x - 2*E^((3*I)*(e + f*x))*f*x + (2*I)*E^((4*I)*(e + f*x))*f*x + (2*I)*E^(I*(e + 2*f*x))*f^2*x^2*ExpIntegralEi[(-I)*f*x] + (2*I)*E^((3*I)*e + (2*I)*f*x)*f^2*x^2*ExpIntegralEi[I*f*x] - 4*E^((2*I)*f*x)*f^2*x^2*ExpIntegralEi[(-2*I)*f*x] + 4*E^((2*I)*(2*e + f*x))*f^2*x^2*ExpIntegralEi[(2*I)*f*x])*Sqrt[a - a*Sin[e + f*x]])/(4*Sqrt[2]*E^((2*I)*(e + f*x))*(-I + E^(I*(e + f*x)))*Sqrt[((-I)*c*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*x^2)","C",1
177,1,247,767,2.6583106,"\int \frac{(g+h x)^3 \sqrt{a-a \sin (e+f x)}}{\sqrt{c+c \sin (e+f x)}} \, dx","Integrate[((g + h*x)^3*Sqrt[a - a*Sin[e + f*x]])/Sqrt[c + c*Sin[e + f*x]],x]","\frac{\left(\frac{1}{4}+\frac{i}{4}\right) e^{-\frac{1}{2} i (e+f x)} \left(e^{i (e+f x)}+i\right) \sqrt{a-a \sin (e+f x)} \left(\frac{24 h \left(f^2 (g+h x)^2 \text{Li}_2\left(-i e^{-i (e+f x)}\right)-2 h \left(i f (g+h x) \text{Li}_3\left(-i e^{-i (e+f x)}\right)+h \text{Li}_4\left(-i e^{-i (e+f x)}\right)\right)\right)}{f^4}-\frac{8 i (g+h x)^3 \log \left(1+i e^{-i (e+f x)}\right)}{f}+\frac{(g+h x)^4}{h}\right)}{\sqrt{2} \sqrt{-i c e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","-\frac{6 i a h^3 \text{Li}_4\left(-i e^{i (e+f x)}\right) \cos (e+f x)}{f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{6 i a h^3 \text{Li}_4\left(i e^{i (e+f x)}\right) \cos (e+f x)}{f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{3 i a h^3 \text{Li}_4\left(-e^{2 i (e+f x)}\right) \cos (e+f x)}{4 f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{6 a h^2 (g+h x) \text{Li}_3\left(-i e^{i (e+f x)}\right) \cos (e+f x)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{6 a h^2 (g+h x) \text{Li}_3\left(i e^{i (e+f x)}\right) \cos (e+f x)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{3 a h^2 (g+h x) \text{Li}_3\left(-e^{2 i (e+f x)}\right) \cos (e+f x)}{2 f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{3 i a h (g+h x)^2 \text{Li}_2\left(-i e^{i (e+f x)}\right) \cos (e+f x)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{3 i a h (g+h x)^2 \text{Li}_2\left(i e^{i (e+f x)}\right) \cos (e+f x)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{3 i a h (g+h x)^2 \text{Li}_2\left(-e^{2 i (e+f x)}\right) \cos (e+f x)}{2 f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a (g+h x)^4 \cos (e+f x)}{4 h \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a (g+h x)^3 \log \left(1+e^{2 i (e+f x)}\right) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a (g+h x)^3 \cos (e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}",1,"((1/4 + I/4)*(I + E^(I*(e + f*x)))*((g + h*x)^4/h - ((8*I)*(g + h*x)^3*Log[1 + I/E^(I*(e + f*x))])/f + (24*h*(f^2*(g + h*x)^2*PolyLog[2, (-I)/E^(I*(e + f*x))] - 2*h*(I*f*(g + h*x)*PolyLog[3, (-I)/E^(I*(e + f*x))] + h*PolyLog[4, (-I)/E^(I*(e + f*x))])))/f^4)*Sqrt[a - a*Sin[e + f*x]])/(Sqrt[2]*E^((I/2)*(e + f*x))*Sqrt[((-I)*c*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))","A",1
178,1,194,555,1.8877121,"\int \frac{(g+h x)^2 \sqrt{a-a \sin (e+f x)}}{\sqrt{c+c \sin (e+f x)}} \, dx","Integrate[((g + h*x)^2*Sqrt[a - a*Sin[e + f*x]])/Sqrt[c + c*Sin[e + f*x]],x]","\frac{\sqrt{2} \left(e^{i (e+f x)}+i\right) \sqrt{a-a \sin (e+f x)} \left(f^2 (g+h x)^2 \left(f (g+h x)-6 i h \log \left(1+i e^{-i (e+f x)}\right)\right)+12 f h^2 (g+h x) \text{Li}_2\left(-i e^{-i (e+f x)}\right)-12 i h^3 \text{Li}_3\left(-i e^{-i (e+f x)}\right)\right)}{3 f^3 h \left(e^{i (e+f x)}-i\right) \sqrt{-i c e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2}}","-\frac{2 a h^2 \text{Li}_3\left(-i e^{i (e+f x)}\right) \cos (e+f x)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a h^2 \text{Li}_3\left(i e^{i (e+f x)}\right) \cos (e+f x)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a h^2 \text{Li}_3\left(-e^{2 i (e+f x)}\right) \cos (e+f x)}{2 f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 i a h (g+h x) \text{Li}_2\left(-i e^{i (e+f x)}\right) \cos (e+f x)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a h (g+h x) \text{Li}_2\left(i e^{i (e+f x)}\right) \cos (e+f x)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h (g+h x) \text{Li}_2\left(-e^{2 i (e+f x)}\right) \cos (e+f x)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a (g+h x)^3 \cos (e+f x)}{3 h \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a (g+h x)^2 \log \left(1+e^{2 i (e+f x)}\right) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a (g+h x)^2 \cos (e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}",1,"(Sqrt[2]*(I + E^(I*(e + f*x)))*(f^2*(g + h*x)^2*(f*(g + h*x) - (6*I)*h*Log[1 + I/E^(I*(e + f*x))]) + 12*f*h^2*(g + h*x)*PolyLog[2, (-I)/E^(I*(e + f*x))] - (12*I)*h^3*PolyLog[3, (-I)/E^(I*(e + f*x))])*Sqrt[a - a*Sin[e + f*x]])/(3*(-I + E^(I*(e + f*x)))*Sqrt[((-I)*c*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*f^3*h)","A",1
179,1,154,355,1.2199574,"\int \frac{(g+h x) \sqrt{a-a \sin (e+f x)}}{\sqrt{c+c \sin (e+f x)}} \, dx","Integrate[((g + h*x)*Sqrt[a - a*Sin[e + f*x]])/Sqrt[c + c*Sin[e + f*x]],x]","\frac{\left(e^{i (e+f x)}+i\right) \sqrt{a-a \sin (e+f x)} \left(f \left(f x (2 g+h x)-4 i (g+h x) \log \left(1+i e^{-i (e+f x)}\right)\right)+4 h \text{Li}_2\left(-i e^{-i (e+f x)}\right)\right)}{\sqrt{2} f^2 \left(e^{i (e+f x)}-i\right) \sqrt{-i c e^{-i (e+f x)} \left(e^{i (e+f x)}+i\right)^2}}","\frac{i a h \text{Li}_2\left(-i e^{i (e+f x)}\right) \cos (e+f x)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h \text{Li}_2\left(i e^{i (e+f x)}\right) \cos (e+f x)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h \text{Li}_2\left(-e^{2 i (e+f x)}\right) \cos (e+f x)}{2 f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a (g+h x)^2 \cos (e+f x)}{2 h \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a (g+h x) \log \left(1+e^{2 i (e+f x)}\right) \cos (e+f x)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a (g+h x) \cos (e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right)}{f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}",1,"((I + E^(I*(e + f*x)))*(f*(f*x*(2*g + h*x) - (4*I)*(g + h*x)*Log[1 + I/E^(I*(e + f*x))]) + 4*h*PolyLog[2, (-I)/E^(I*(e + f*x))])*Sqrt[a - a*Sin[e + f*x]])/(Sqrt[2]*(-I + E^(I*(e + f*x)))*Sqrt[((-I)*c*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*f^2)","A",1
180,0,0,110,3.543916,"\int \frac{\sqrt{a-a \sin (e+f x)}}{(g+h x) \sqrt{c+c \sin (e+f x)}} \, dx","Integrate[Sqrt[a - a*Sin[e + f*x]]/((g + h*x)*Sqrt[c + c*Sin[e + f*x]]),x]","\int \frac{\sqrt{a-a \sin (e+f x)}}{(g+h x) \sqrt{c+c \sin (e+f x)}} \, dx","\frac{a \cos (e+f x) \text{Int}\left(\frac{\sec (e+f x)}{g+h x},x\right)}{\sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{a \cos (e+f x) \text{Int}\left(\frac{\tan (e+f x)}{g+h x},x\right)}{\sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}",0,"Integrate[Sqrt[a - a*Sin[e + f*x]]/((g + h*x)*Sqrt[c + c*Sin[e + f*x]]), x]","A",-1
181,1,193,536,2.0817265,"\int \frac{x^3 \sqrt{a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx","Integrate[(x^3*Sqrt[a - a*Sin[e + f*x]])/(c + c*Sin[e + f*x])^(3/2),x]","-\frac{\sqrt{a-a \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(f x \left(-12 \log \left(1-i e^{i (e+f x)}\right)+3 f x \cos (e+f x)+3 i \left(f x+4 i \log \left(1-i e^{i (e+f x)}\right)\right) \sin (e+f x)+f^2 x^2+3 i f x\right)+12 i \text{Li}_2\left(i e^{i (e+f x)}\right) (\sin (e+f x)+1)\right)}{f^4 (c (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{6 i a \text{Li}_2\left(-i e^{i (e+f x)}\right) \cos (e+f x)}{c f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{6 i a \text{Li}_2\left(i e^{i (e+f x)}\right) \cos (e+f x)}{c f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{3 i a \text{Li}_2\left(-e^{2 i (e+f x)}\right) \cos (e+f x)}{c f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{6 a x \log \left(1+e^{2 i (e+f x)}\right) \cos (e+f x)}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{12 i a x \cos (e+f x) \tan ^{-1}\left(e^{i (e+f x)}\right)}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{3 a x^2 \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{3 a x^2}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{3 i a x^2 \cos (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a x^3 \tan (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{a x^3 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}",1,"-(((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[a - a*Sin[e + f*x]]*((12*I)*PolyLog[2, I*E^(I*(e + f*x))]*(1 + Sin[e + f*x]) + f*x*((3*I)*f*x + f^2*x^2 + 3*f*x*Cos[e + f*x] - 12*Log[1 - I*E^(I*(e + f*x))] + (3*I)*(f*x + (4*I)*Log[1 - I*E^(I*(e + f*x))])*Sin[e + f*x])))/(f^4*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c*(1 + Sin[e + f*x]))^(3/2)))","A",1
182,1,154,280,1.555977,"\int \frac{x^2 \sqrt{a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx","Integrate[(x^2*Sqrt[a - a*Sin[e + f*x]])/(c + c*Sin[e + f*x])^(3/2),x]","-\frac{\sqrt{a-a \sin (e+f x)} \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right) \left(-4 \log \left(e^{i (e+f x)}+i\right)+2 f x \cos (e+f x)+\left(2 i f x-4 \log \left(e^{i (e+f x)}+i\right)\right) \sin (e+f x)+f^2 x^2+2 i f x\right)}{f^3 (c (\sin (e+f x)+1))^{3/2} \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}","\frac{2 a \cos (e+f x) \log (\cos (e+f x))}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a \cos (e+f x) \tanh ^{-1}(\sin (e+f x))}{c f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a x \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 a x}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a x^2 \tan (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{a x^2 \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}",1,"-(((Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sqrt[a - a*Sin[e + f*x]]*((2*I)*f*x + f^2*x^2 + 2*f*x*Cos[e + f*x] - 4*Log[I + E^(I*(e + f*x))] + ((2*I)*f*x - 4*Log[I + E^(I*(e + f*x))])*Sin[e + f*x]))/(f^3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(c*(1 + Sin[e + f*x]))^(3/2)))","C",1
183,1,150,171,0.6028374,"\int \frac{x \sqrt{a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx","Integrate[(x*Sqrt[a - a*Sin[e + f*x]])/(c + c*Sin[e + f*x])^(3/2),x]","-\frac{\sqrt{a-a \sin (e+f x)} \sqrt{c (\sin (e+f x)+1)} \left(f x \sin \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}+f x\right)+\cos \left(\frac{e}{2}\right) (f x-1)+\cos \left(\frac{e}{2}+f x\right)+\sin \left(\frac{e}{2}\right)\right)}{c^2 f^2 \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}","\frac{a \sin (e+f x)}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{a}{c f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a x \tan (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{a x \sec (e+f x)}{c f \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}",1,"-((((-1 + f*x)*Cos[e/2] + Cos[e/2 + f*x] + Sin[e/2] + f*x*Sin[e/2] - Sin[e/2 + f*x])*Sqrt[c*(1 + Sin[e + f*x])]*Sqrt[a - a*Sin[e + f*x]])/(c^2*f^2*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3))","A",1
184,1,85,300,0.0895969,"\int \frac{z^2 \sqrt{1+\cos (z)}}{\sqrt{1-\cos (z)}} \, dz","Integrate[(z^2*Sqrt[1 + Cos[z]])/Sqrt[1 - Cos[z]],z]","\frac{\sqrt{\cos (z)+1} \tan \left(\frac{z}{2}\right) \left(12 i z \text{Li}_2\left(e^{-i z}\right)+12 \text{Li}_3\left(e^{-i z}\right)+i z^3+6 z^2 \log \left(1-e^{-i z}\right)-i \pi ^3\right)}{3 \sqrt{1-\cos (z)}}","\frac{2 i z \text{Li}_2\left(-e^{i z}\right) \sin (z)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{2 i z \text{Li}_2\left(e^{i z}\right) \sin (z)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{i z \text{Li}_2\left(e^{2 i z}\right) \sin (z)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{2 \text{Li}_3\left(-e^{i z}\right) \sin (z)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}+\frac{2 \text{Li}_3\left(e^{i z}\right) \sin (z)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}+\frac{\text{Li}_3\left(e^{2 i z}\right) \sin (z)}{2 \sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{i z^3 \sin (z)}{3 \sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}+\frac{z^2 \log \left(1-e^{2 i z}\right) \sin (z)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{2 z^2 \sin (z) \tanh ^{-1}\left(e^{i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}",1,"(Sqrt[1 + Cos[z]]*((-I)*Pi^3 + I*z^3 + 6*z^2*Log[1 - E^((-I)*z)] + (12*I)*z*PolyLog[2, E^((-I)*z)] + 12*PolyLog[3, E^((-I)*z)])*Tan[z/2])/(3*Sqrt[1 - Cos[z]])","A",1
185,1,51,18,0.014892,"\int (a+a \cos (x)) (A+B \sec (x)) \, dx","Integrate[(a + a*Cos[x])*(A + B*Sec[x]),x]","a A x+a A \sin (x)+a B x-a B \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+a B \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","a x (A+B)+a A \sin (x)+a B \tanh ^{-1}(\sin (x))",1,"a*A*x + a*B*x - a*B*Log[Cos[x/2] - Sin[x/2]] + a*B*Log[Cos[x/2] + Sin[x/2]] + a*A*Sin[x]","B",1
186,1,67,57,0.0811632,"\int (a+a \cos (x))^2 (A+B \sec (x)) \, dx","Integrate[(a + a*Cos[x])^2*(A + B*Sec[x]),x]","\frac{1}{4} a^2 \left(4 (2 A+B) \sin (x)+6 A x+A \sin (2 x)+8 B x-4 B \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+4 B \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)","\frac{1}{2} a^2 x (3 A+4 B)+\frac{1}{2} a^2 (3 A+2 B) \sin (x)+\frac{1}{2} A \sin (x) \left(a^2 \cos (x)+a^2\right)+a^2 B \tanh ^{-1}(\sin (x))",1,"(a^2*(6*A*x + 8*B*x - 4*B*Log[Cos[x/2] - Sin[x/2]] + 4*B*Log[Cos[x/2] + Sin[x/2]] + 4*(2*A + B)*Sin[x] + A*Sin[2*x]))/4","A",1
187,1,80,75,0.102441,"\int (a+a \cos (x))^3 (A+B \sec (x)) \, dx","Integrate[(a + a*Cos[x])^3*(A + B*Sec[x]),x]","\frac{1}{12} a^3 \left(9 (5 A+4 B) \sin (x)+3 (3 A+B) \sin (2 x)+30 A x+A \sin (3 x)+42 B x-12 B \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+12 B \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)","\frac{1}{2} a^3 x (5 A+7 B)+\frac{5}{2} a^3 (A+B) \sin (x)+\frac{1}{6} (5 A+3 B) \sin (x) \left(a^3 \cos (x)+a^3\right)+a^3 B \tanh ^{-1}(\sin (x))+\frac{1}{3} a A \sin (x) (a \cos (x)+a)^2",1,"(a^3*(30*A*x + 42*B*x - 12*B*Log[Cos[x/2] - Sin[x/2]] + 12*B*Log[Cos[x/2] + Sin[x/2]] + 9*(5*A + 4*B)*Sin[x] + 3*(3*A + B)*Sin[2*x] + A*Sin[3*x]))/12","A",1
188,1,97,104,0.1234233,"\int (a+a \cos (x))^4 (A+B \sec (x)) \, dx","Integrate[(a + a*Cos[x])^4*(A + B*Sec[x]),x]","\frac{1}{96} a^4 \left(24 (28 A+27 B) \sin (x)+24 (7 A+4 B) \sin (2 x)+420 A x+32 A \sin (3 x)+3 A \sin (4 x)+576 B x+8 B \sin (3 x)-96 B \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+96 B \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)","\frac{1}{8} a^4 x (35 A+48 B)+\frac{5}{8} a^4 (7 A+8 B) \sin (x)+\frac{1}{24} (35 A+32 B) \sin (x) \left(a^4 \cos (x)+a^4\right)+a^4 B \tanh ^{-1}(\sin (x))+\frac{1}{12} (7 A+4 B) \sin (x) \left(a^2 \cos (x)+a^2\right)^2+\frac{1}{4} a A \sin (x) (a \cos (x)+a)^3",1,"(a^4*(420*A*x + 576*B*x - 96*B*Log[Cos[x/2] - Sin[x/2]] + 96*B*Log[Cos[x/2] + Sin[x/2]] + 24*(28*A + 27*B)*Sin[x] + 24*(7*A + 4*B)*Sin[2*x] + 32*A*Sin[3*x] + 8*B*Sin[3*x] + 3*A*Sin[4*x]))/96","A",1
189,1,71,25,0.0842085,"\int \frac{A+B \sec (x)}{a+a \cos (x)} \, dx","Integrate[(A + B*Sec[x])/(a + a*Cos[x]),x]","-\frac{2 \cos \left(\frac{x}{2}\right) \left((B-A) \sin \left(\frac{x}{2}\right)+B \cos \left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)\right)}{a (\cos (x)+1)}","\frac{(A-B) \sin (x)}{a \cos (x)+a}+\frac{B \tanh ^{-1}(\sin (x))}{a}",1,"(-2*Cos[x/2]*(B*Cos[x/2]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + (-A + B)*Sin[x/2]))/(a*(1 + Cos[x]))","B",1
190,1,76,48,0.2065019,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^2} \, dx","Integrate[(A + B*Sec[x])/(a + a*Cos[x])^2,x]","\frac{\sin (x) ((A-4 B) \cos (x)+2 A-5 B)-12 B \cos ^4\left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{3 a^2 (\cos (x)+1)^2}","\frac{(A-4 B) \sin (x)}{3 a^2 (\cos (x)+1)}+\frac{B \tanh ^{-1}(\sin (x))}{a^2}+\frac{(A-B) \sin (x)}{3 (a \cos (x)+a)^2}",1,"(-12*B*Cos[x/2]^4*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + (2*A - 5*B + (A - 4*B)*Cos[x])*Sin[x])/(3*a^2*(1 + Cos[x])^2)","A",1
191,1,88,75,0.3466155,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^3} \, dx","Integrate[(A + B*Sec[x])/(a + a*Cos[x])^3,x]","\frac{\sin (x) ((6 A-51 B) \cos (x)+(A-11 B) \cos (2 x)+8 A-43 B)-120 B \cos ^6\left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{15 a^3 (\cos (x)+1)^3}","\frac{2 (A-11 B) \sin (x)}{15 \left(a^3 \cos (x)+a^3\right)}+\frac{B \tanh ^{-1}(\sin (x))}{a^3}+\frac{(2 A-7 B) \sin (x)}{15 a (a \cos (x)+a)^2}+\frac{(A-B) \sin (x)}{5 (a \cos (x)+a)^3}",1,"(-120*B*Cos[x/2]^6*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + (8*A - 43*B + (6*A - 51*B)*Cos[x] + (A - 11*B)*Cos[2*x])*Sin[x])/(15*a^3*(1 + Cos[x])^3)","A",1
192,1,104,96,0.7495717,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^4} \, dx","Integrate[(A + B*Sec[x])/(a + a*Cos[x])^4,x]","\frac{\sin (x) ((87 A-1480 B) \cos (x)+(24 A-535 B) \cos (2 x)+3 A \cos (3 x)+96 A-80 B \cos (3 x)-1055 B)-3360 B \cos ^8\left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{210 a^4 (\cos (x)+1)^4}","\frac{2 (3 A-80 B) \sin (x)}{105 a^4 (\cos (x)+1)}+\frac{(6 A-55 B) \sin (x)}{105 a^4 (\cos (x)+1)^2}+\frac{B \tanh ^{-1}(\sin (x))}{a^4}+\frac{(3 A-10 B) \sin (x)}{35 a (a \cos (x)+a)^3}+\frac{(A-B) \sin (x)}{7 (a \cos (x)+a)^4}",1,"(-3360*B*Cos[x/2]^8*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + (96*A - 1055*B + (87*A - 1480*B)*Cos[x] + (24*A - 535*B)*Cos[2*x] + 3*A*Cos[3*x] - 80*B*Cos[3*x])*Sin[x])/(210*a^4*(1 + Cos[x])^4)","A",1
193,1,78,98,0.1577562,"\int (a+a \cos (x))^{5/2} (A+B \sec (x)) \, dx","Integrate[(a + a*Cos[x])^(5/2)*(A + B*Sec[x]),x]","\frac{1}{30} a^2 \sec \left(\frac{x}{2}\right) \sqrt{a (\cos (x)+1)} \left(2 \sin \left(\frac{x}{2}\right) (2 (14 A+5 B) \cos (x)+3 A \cos (2 x)+89 A+80 B)+30 \sqrt{2} B \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{x}{2}\right)\right)\right)","2 a^{5/2} B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)+\frac{2 a^3 (32 A+35 B) \sin (x)}{15 \sqrt{a \cos (x)+a}}+\frac{2}{15} a^2 (8 A+5 B) \sin (x) \sqrt{a \cos (x)+a}+\frac{2}{5} a A \sin (x) (a \cos (x)+a)^{3/2}",1,"(a^2*Sqrt[a*(1 + Cos[x])]*Sec[x/2]*(30*Sqrt[2]*B*ArcTanh[Sqrt[2]*Sin[x/2]] + 2*(89*A + 80*B + 2*(14*A + 5*B)*Cos[x] + 3*A*Cos[2*x])*Sin[x/2]))/30","A",1
194,1,62,72,0.1013394,"\int (a+a \cos (x))^{3/2} (A+B \sec (x)) \, dx","Integrate[(a + a*Cos[x])^(3/2)*(A + B*Sec[x]),x]","\frac{1}{3} a \sec \left(\frac{x}{2}\right) \sqrt{a (\cos (x)+1)} \left(2 \sin \left(\frac{x}{2}\right) (A \cos (x)+5 A+3 B)+3 \sqrt{2} B \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{x}{2}\right)\right)\right)","2 a^{3/2} B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)+\frac{2 a^2 (4 A+3 B) \sin (x)}{3 \sqrt{a \cos (x)+a}}+\frac{2}{3} a A \sin (x) \sqrt{a \cos (x)+a}",1,"(a*Sqrt[a*(1 + Cos[x])]*Sec[x/2]*(3*Sqrt[2]*B*ArcTanh[Sqrt[2]*Sin[x/2]] + 2*(5*A + 3*B + A*Cos[x])*Sin[x/2]))/3","A",1
195,1,47,44,0.0359678,"\int \sqrt{a+a \cos (x)} (A+B \sec (x)) \, dx","Integrate[Sqrt[a + a*Cos[x]]*(A + B*Sec[x]),x]","\sec \left(\frac{x}{2}\right) \sqrt{a (\cos (x)+1)} \left(2 A \sin \left(\frac{x}{2}\right)+\sqrt{2} B \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{x}{2}\right)\right)\right)","\frac{2 a A \sin (x)}{\sqrt{a \cos (x)+a}}+2 \sqrt{a} B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)",1,"Sqrt[a*(1 + Cos[x])]*Sec[x/2]*(Sqrt[2]*B*ArcTanh[Sqrt[2]*Sin[x/2]] + 2*A*Sin[x/2])","A",1
196,1,52,68,0.0479136,"\int \frac{A+B \sec (x)}{\sqrt{a+a \cos (x)}} \, dx","Integrate[(A + B*Sec[x])/Sqrt[a + a*Cos[x]],x]","\frac{2 \cos \left(\frac{x}{2}\right) \left((A-B) \tanh ^{-1}\left(\sin \left(\frac{x}{2}\right)\right)+\sqrt{2} B \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{x}{2}\right)\right)\right)}{\sqrt{a (\cos (x)+1)}}","\frac{\sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{a \cos (x)+a}}\right)}{\sqrt{a}}+\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)}{\sqrt{a}}",1,"(2*((A - B)*ArcTanh[Sin[x/2]] + Sqrt[2]*B*ArcTanh[Sqrt[2]*Sin[x/2]])*Cos[x/2])/Sqrt[a*(1 + Cos[x])]","A",1
197,1,73,92,0.3696307,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^{3/2}} \, dx","Integrate[(A + B*Sec[x])/(a + a*Cos[x])^(3/2),x]","\frac{\frac{1}{2} (A-B) \sin (x)+(A-5 B) \cos ^3\left(\frac{x}{2}\right) \tanh ^{-1}\left(\sin \left(\frac{x}{2}\right)\right)+4 \sqrt{2} B \cos ^3\left(\frac{x}{2}\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{x}{2}\right)\right)}{(a (\cos (x)+1))^{3/2}}","\frac{(A-5 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{a \cos (x)+a}}\right)}{2 \sqrt{2} a^{3/2}}+\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)}{a^{3/2}}+\frac{(A-B) \sin (x)}{2 (a \cos (x)+a)^{3/2}}",1,"((A - 5*B)*ArcTanh[Sin[x/2]]*Cos[x/2]^3 + 4*Sqrt[2]*B*ArcTanh[Sqrt[2]*Sin[x/2]]*Cos[x/2]^3 + ((A - B)*Sin[x])/2)/(a*(1 + Cos[x]))^(3/2)","A",1
198,1,95,120,0.5000936,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx","Integrate[(A + B*Sec[x])/(a + a*Cos[x])^(5/2),x]","\frac{\tan \left(\frac{x}{2}\right) (3 A \cos (x)+7 A-11 B \cos (x)-15 B)+2 (3 A-43 B) \cos ^3\left(\frac{x}{2}\right) \tanh ^{-1}\left(\sin \left(\frac{x}{2}\right)\right)+64 \sqrt{2} B \cos ^3\left(\frac{x}{2}\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{x}{2}\right)\right)}{16 a (a (\cos (x)+1))^{3/2}}","\frac{(3 A-43 B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{2} \sqrt{a \cos (x)+a}}\right)}{16 \sqrt{2} a^{5/2}}+\frac{2 B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)}{a^{5/2}}+\frac{(3 A-11 B) \sin (x)}{16 a (a \cos (x)+a)^{3/2}}+\frac{(A-B) \sin (x)}{4 (a \cos (x)+a)^{5/2}}",1,"(2*(3*A - 43*B)*ArcTanh[Sin[x/2]]*Cos[x/2]^3 + 64*Sqrt[2]*B*ArcTanh[Sqrt[2]*Sin[x/2]]*Cos[x/2]^3 + (7*A - 15*B + 3*A*Cos[x] - 11*B*Cos[x])*Tan[x/2])/(16*a*(a*(1 + Cos[x]))^(3/2))","A",1
199,1,25,25,0.1988652,"\int \frac{x (b+a \sin (x))}{(a+b \sin (x))^2} \, dx","Integrate[(x*(b + a*Sin[x]))/(a + b*Sin[x])^2,x]","\frac{\log (a+b \sin (x))}{b}-\frac{x \cos (x)}{a+b \sin (x)}","\frac{\log (a+b \sin (x))}{b}-\frac{x \cos (x)}{a+b \sin (x)}",1,"Log[a + b*Sin[x]]/b - (x*Cos[x])/(a + b*Sin[x])","A",1
200,1,24,24,0.1264698,"\int \frac{x (b+a \cos (x))}{(a+b \cos (x))^2} \, dx","Integrate[(x*(b + a*Cos[x]))/(a + b*Cos[x])^2,x]","\frac{\log (a+b \cos (x))}{b}+\frac{x \sin (x)}{a+b \cos (x)}","\frac{\log (a+b \cos (x))}{b}+\frac{x \sin (x)}{a+b \cos (x)}",1,"Log[a + b*Cos[x]]/b + (x*Sin[x])/(a + b*Cos[x])","A",1
201,1,8,8,0.007919,"\int \frac{1+\sin ^2(x)}{1-\sin ^2(x)} \, dx","Integrate[(1 + Sin[x]^2)/(1 - Sin[x]^2),x]","2 \tan (x)-x","2 \tan (x)-x",1,"-x + 2*Tan[x]","A",1
202,1,24,36,0.0282828,"\int \frac{1-\sin ^2(x)}{1+\sin ^2(x)} \, dx","Integrate[(1 - Sin[x]^2)/(1 + Sin[x]^2),x]","-2 \left(\frac{x}{2}-\frac{\tan ^{-1}\left(\sqrt{2} \tan (x)\right)}{\sqrt{2}}\right)","\sqrt{2} x-x+\sqrt{2} \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)",1,"-2*(x/2 - ArcTan[Sqrt[2]*Tan[x]]/Sqrt[2])","A",1
203,1,8,8,0.0074498,"\int \frac{1+\cos ^2(x)}{1-\cos ^2(x)} \, dx","Integrate[(1 + Cos[x]^2)/(1 - Cos[x]^2),x]","-x-2 \cot (x)","-x-2 \cot (x)",1,"-x - 2*Cot[x]","A",1
204,1,23,37,0.0309921,"\int \frac{1-\cos ^2(x)}{1+\cos ^2(x)} \, dx","Integrate[(1 - Cos[x]^2)/(1 + Cos[x]^2),x]","2 \left(\frac{\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)}{\sqrt{2}}-\frac{x}{2}\right)","\sqrt{2} x-x-\sqrt{2} \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)",1,"2*(-1/2*x + ArcTan[Tan[x]/Sqrt[2]]/Sqrt[2])","A",1
205,1,14,14,0.0101458,"\int \frac{-1+\frac{c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx","Integrate[(-1 + c^2/d^2 + Sin[x]^2)/(c + d*Cos[x]),x]","\frac{c x}{d^2}-\frac{\sin (x)}{d}","\frac{c x}{d^2}-\frac{\sin (x)}{d}",1,"(c*x)/d^2 - Sin[x]/d","A",1
206,1,73,105,0.1524378,"\int \frac{a+b \sin ^2(x)}{c+d \cos (x)} \, dx","Integrate[(a + b*Sin[x]^2)/(c + d*Cos[x]),x]","\frac{-\frac{2 \left(a d^2+b \left(d^2-c^2\right)\right) \tanh ^{-1}\left(\frac{(c-d) \tan \left(\frac{x}{2}\right)}{\sqrt{d^2-c^2}}\right)}{\sqrt{d^2-c^2}}+b c x-b d \sin (x)}{d^2}","\frac{2 a \tan ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{x}{2}\right)}{\sqrt{c+d}}\right)}{\sqrt{c-d} \sqrt{c+d}}+\frac{b c x}{d^2}-\frac{2 b \sqrt{c-d} \sqrt{c+d} \tan ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{x}{2}\right)}{\sqrt{c+d}}\right)}{d^2}-\frac{b \sin (x)}{d}",1,"(b*c*x - (2*(a*d^2 + b*(-c^2 + d^2))*ArcTanh[((c - d)*Tan[x/2])/Sqrt[-c^2 + d^2]])/Sqrt[-c^2 + d^2] - b*d*Sin[x])/d^2","A",1
207,1,34,57,0.0849298,"\int \frac{a+b \sin ^2(x)}{c+c \cos ^2(x)} \, dx","Integrate[(a + b*Sin[x]^2)/(c + c*Cos[x]^2),x]","-\frac{(-a-2 b) \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)}{\sqrt{2} c}-\frac{b x}{c}","\frac{x (a+2 b)}{\sqrt{2} c}-\frac{(a+2 b) \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{\sqrt{2} c}-\frac{b x}{c}",1,"-((b*x)/c) - ((-a - 2*b)*ArcTan[Tan[x]/Sqrt[2]])/(Sqrt[2]*c)","A",1
208,1,15,15,0.0125544,"\int \frac{a+b \sin ^2(x)}{c-c \cos ^2(x)} \, dx","Integrate[(a + b*Sin[x]^2)/(c - c*Cos[x]^2),x]","\frac{b x}{c}-\frac{a \cot (x)}{c}","\frac{b x}{c}-\frac{a \cot (x)}{c}",1,"(b*x)/c - (a*Cot[x])/c","A",1
209,1,47,49,0.1613741,"\int \frac{a+b \sin ^2(x)}{c+d \cos ^2(x)} \, dx","Integrate[(a + b*Sin[x]^2)/(c + d*Cos[x]^2),x]","\frac{\frac{(a d+b (c+d)) \tan ^{-1}\left(\frac{\sqrt{c} \tan (x)}{\sqrt{c+d}}\right)}{\sqrt{c} \sqrt{c+d}}-b x}{d}","\frac{(a d+b (c+d)) \tan ^{-1}\left(\frac{\sqrt{c} \tan (x)}{\sqrt{c+d}}\right)}{\sqrt{c} d \sqrt{c+d}}-\frac{b x}{d}",1,"(-(b*x) + ((a*d + b*(c + d))*ArcTan[(Sqrt[c]*Tan[x])/Sqrt[c + d]])/(Sqrt[c]*Sqrt[c + d]))/d","A",1
210,1,13,13,0.0106692,"\int \frac{-1+\frac{c^2}{d^2}+\cos ^2(x)}{c+d \sin (x)} \, dx","Integrate[(-1 + c^2/d^2 + Cos[x]^2)/(c + d*Sin[x]),x]","\frac{c x}{d^2}+\frac{\cos (x)}{d}","\frac{c x}{d^2}+\frac{\cos (x)}{d}",1,"(c*x)/d^2 + Cos[x]/d","A",1
211,1,72,100,0.1828375,"\int \frac{a+b \cos ^2(x)}{c+d \sin (x)} \, dx","Integrate[(a + b*Cos[x]^2)/(c + d*Sin[x]),x]","\frac{\frac{2 \left(a d^2+b \left(d^2-c^2\right)\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{x}{2}\right)+d}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}+b (c x+d \cos (x))}{d^2}","\frac{2 a \tan ^{-1}\left(\frac{c \tan \left(\frac{x}{2}\right)+d}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}-\frac{2 b \sqrt{c^2-d^2} \tan ^{-1}\left(\frac{c \tan \left(\frac{x}{2}\right)+d}{\sqrt{c^2-d^2}}\right)}{d^2}+\frac{b c x}{d^2}+\frac{b \cos (x)}{d}",1,"((2*(a*d^2 + b*(-c^2 + d^2))*ArcTan[(d + c*Tan[x/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] + b*(c*x + d*Cos[x]))/d^2","A",1
212,1,31,56,0.0810068,"\int \frac{a+b \cos ^2(x)}{c+c \sin ^2(x)} \, dx","Integrate[(a + b*Cos[x]^2)/(c + c*Sin[x]^2),x]","\frac{(a+2 b) \tan ^{-1}\left(\sqrt{2} \tan (x)\right)}{\sqrt{2} c}-\frac{b x}{c}","\frac{x (a+2 b)}{\sqrt{2} c}+\frac{(a+2 b) \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)}{\sqrt{2} c}-\frac{b x}{c}",1,"-((b*x)/c) + ((a + 2*b)*ArcTan[Sqrt[2]*Tan[x]])/(Sqrt[2]*c)","A",1
213,1,14,14,0.0124653,"\int \frac{a+b \cos ^2(x)}{c-c \sin ^2(x)} \, dx","Integrate[(a + b*Cos[x]^2)/(c - c*Sin[x]^2),x]","\frac{a \tan (x)}{c}+\frac{b x}{c}","\frac{a \tan (x)}{c}+\frac{b x}{c}",1,"(b*x)/c + (a*Tan[x])/c","A",1
214,1,47,49,0.1513783,"\int \frac{a+b \cos ^2(x)}{c+d \sin ^2(x)} \, dx","Integrate[(a + b*Cos[x]^2)/(c + d*Sin[x]^2),x]","\frac{\frac{(a d+b (c+d)) \tan ^{-1}\left(\frac{\sqrt{c+d} \tan (x)}{\sqrt{c}}\right)}{\sqrt{c} \sqrt{c+d}}-b x}{d}","\frac{(a d+b (c+d)) \tan ^{-1}\left(\frac{\sqrt{c+d} \tan (x)}{\sqrt{c}}\right)}{\sqrt{c} d \sqrt{c+d}}-\frac{b x}{d}",1,"(-(b*x) + ((a*d + b*(c + d))*ArcTan[(Sqrt[c + d]*Tan[x])/Sqrt[c]])/(Sqrt[c]*Sqrt[c + d]))/d","A",1
215,1,98,74,0.451117,"\int \frac{a+b \sec ^2(x)}{c+d \cos (x)} \, dx","Integrate[(a + b*Sec[x]^2)/(c + d*Cos[x]),x]","\frac{-\frac{2 \left(a c^2+b d^2\right) \tanh ^{-1}\left(\frac{(c-d) \tan \left(\frac{x}{2}\right)}{\sqrt{d^2-c^2}}\right)}{\sqrt{d^2-c^2}}+b c \tan (x)+b d \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{c^2}","\frac{2 \left(a c^2+b d^2\right) \tan ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{x}{2}\right)}{\sqrt{c+d}}\right)}{c^2 \sqrt{c-d} \sqrt{c+d}}-\frac{b d \tanh ^{-1}(\sin (x))}{c^2}+\frac{b \tan (x)}{c}",1,"((-2*(a*c^2 + b*d^2)*ArcTanh[((c - d)*Tan[x/2])/Sqrt[-c^2 + d^2]])/Sqrt[-c^2 + d^2] + b*d*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + b*c*Tan[x])/c^2","A",1
216,1,102,72,0.5296569,"\int \frac{a+b \csc ^2(x)}{c+d \sin (x)} \, dx","Integrate[(a + b*Csc[x]^2)/(c + d*Sin[x]),x]","\frac{\csc \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \left(\frac{2 \sin (x) \left(a c^2+b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{x}{2}\right)+d}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}-b \left(c \cos (x)+d \sin (x) \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)\right)\right)}{2 c^2}","\frac{2 \left(a c^2+b d^2\right) \tan ^{-1}\left(\frac{c \tan \left(\frac{x}{2}\right)+d}{\sqrt{c^2-d^2}}\right)}{c^2 \sqrt{c^2-d^2}}+\frac{b d \tanh ^{-1}(\cos (x))}{c^2}-\frac{b \cot (x)}{c}",1,"(Csc[x/2]*Sec[x/2]*((2*(a*c^2 + b*d^2)*ArcTan[(d + c*Tan[x/2])/Sqrt[c^2 - d^2]]*Sin[x])/Sqrt[c^2 - d^2] - b*(c*Cos[x] + d*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x])))/(2*c^2)","A",1
217,1,94,136,0.2376059,"\int (a \cos (c+d x)+b \sin (c+d x))^n \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^n,x]","-\frac{\sin \left(2 \left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)\right) \sin ^2\left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)^{-\frac{n}{2}-\frac{1}{2}} (a \cos (c+d x)+b \sin (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\cos ^2\left(c+d x+\tan ^{-1}\left(\frac{a}{b}\right)\right)\right)}{2 d}","-\frac{\sin \left(-\tan ^{-1}(a,b)+c+d x\right) (a \cos (c+d x)+b \sin (c+d x))^n \left(\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}\right)^{-n} \cos ^{n+1}\left(-\tan ^{-1}(a,b)+c+d x\right) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2\left(c+d x-\tan ^{-1}(a,b)\right)\right)}{d (n+1) \sqrt{\sin ^2\left(-\tan ^{-1}(a,b)+c+d x\right)}}",1,"-1/2*(Hypergeometric2F1[1/2, (1 - n)/2, 3/2, Cos[c + d*x + ArcTan[a/b]]^2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^n*(Sin[c + d*x + ArcTan[a/b]]^2)^(-1/2 - n/2)*Sin[2*(c + d*x + ArcTan[a/b])])/d","A",1
218,1,88,95,0.1710308,"\int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx","Integrate[(2*Cos[c + d*x] + 3*Sin[c + d*x])^n,x]","-\frac{\sin \left(2 \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right) \sin ^2\left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)^{-\frac{n}{2}-\frac{1}{2}} (3 \sin (c+d x)+2 \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\cos ^2\left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right)}{2 d}","-\frac{13^{n/2} \sin \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right) \cos ^{n+1}\left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2\left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)}{d (n+1) \sqrt{\sin ^2\left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}}",1,"-1/2*(Hypergeometric2F1[1/2, (1 - n)/2, 3/2, Cos[c + d*x + ArcTan[2/3]]^2]*(2*Cos[c + d*x] + 3*Sin[c + d*x])^n*(Sin[c + d*x + ArcTan[2/3]]^2)^(-1/2 - n/2)*Sin[2*(c + d*x + ArcTan[2/3])])/d","A",1
219,1,246,127,1.0221202,"\int (a \cos (c+d x)+b \sin (c+d x))^7 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^7,x]","\frac{1225 a \left(a^2+b^2\right)^3 \sin (c+d x)+245 a \left(a^2-3 b^2\right) \left(a^2+b^2\right)^2 \sin (3 (c+d x))-1225 b \left(a^2+b^2\right)^3 \cos (c+d x)+245 b \left(b^2-3 a^2\right) \left(a^2+b^2\right)^2 \cos (3 (c+d x))+49 a \left(a^6-9 a^4 b^2-5 a^2 b^4+5 b^6\right) \sin (5 (c+d x))+5 a \left(a^6-21 a^4 b^2+35 a^2 b^4-7 b^6\right) \sin (7 (c+d x))-49 b \left(5 a^6-5 a^4 b^2-9 a^2 b^4+b^6\right) \cos (5 (c+d x))+5 b \left(-7 a^6+35 a^4 b^2-21 a^2 b^4+b^6\right) \cos (7 (c+d x))}{2240 d}","-\frac{3 \left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x))^5}{5 d}+\frac{\left(a^2+b^2\right)^2 (b \cos (c+d x)-a \sin (c+d x))^3}{d}-\frac{\left(a^2+b^2\right)^3 (b \cos (c+d x)-a \sin (c+d x))}{d}+\frac{(b \cos (c+d x)-a \sin (c+d x))^7}{7 d}",1,"(-1225*b*(a^2 + b^2)^3*Cos[c + d*x] + 245*b*(-3*a^2 + b^2)*(a^2 + b^2)^2*Cos[3*(c + d*x)] - 49*b*(5*a^6 - 5*a^4*b^2 - 9*a^2*b^4 + b^6)*Cos[5*(c + d*x)] + 5*b*(-7*a^6 + 35*a^4*b^2 - 21*a^2*b^4 + b^6)*Cos[7*(c + d*x)] + 1225*a*(a^2 + b^2)^3*Sin[c + d*x] + 245*a*(a^2 - 3*b^2)*(a^2 + b^2)^2*Sin[3*(c + d*x)] + 49*a*(a^6 - 9*a^4*b^2 - 5*a^2*b^4 + 5*b^6)*Sin[5*(c + d*x)] + 5*a*(a^6 - 21*a^4*b^2 + 35*a^2*b^4 - 7*b^6)*Sin[7*(c + d*x)])/(2240*d)","A",1
220,1,192,161,0.6909705,"\int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^6,x]","\frac{-36 a b \left(a^4-b^4\right) \cos (4 (c+d x))+60 \left(a^2+b^2\right)^3 (c+d x)+45 \left(a^2-b^2\right) \left(a^2+b^2\right)^2 \sin (2 (c+d x))-90 a b \left(a^2+b^2\right)^2 \cos (2 (c+d x))-2 a b \left(3 a^4-10 a^2 b^2+3 b^4\right) \cos (6 (c+d x))+9 \left(a^6-5 a^4 b^2-5 a^2 b^4+b^6\right) \sin (4 (c+d x))+\left(a^6-15 a^4 b^2+15 a^2 b^4-b^6\right) \sin (6 (c+d x))}{192 d}","-\frac{5 \left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{24 d}-\frac{5 \left(a^2+b^2\right)^2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{16 d}+\frac{5}{16} x \left(a^2+b^2\right)^3-\frac{(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^5}{6 d}",1,"(60*(a^2 + b^2)^3*(c + d*x) - 90*a*b*(a^2 + b^2)^2*Cos[2*(c + d*x)] - 36*a*b*(a^4 - b^4)*Cos[4*(c + d*x)] - 2*a*b*(3*a^4 - 10*a^2*b^2 + 3*b^4)*Cos[6*(c + d*x)] + 45*(a^2 - b^2)*(a^2 + b^2)^2*Sin[2*(c + d*x)] + 9*(a^6 - 5*a^4*b^2 - 5*a^2*b^4 + b^6)*Sin[4*(c + d*x)] + (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*Sin[6*(c + d*x)])/(192*d)","A",1
221,1,156,94,0.4552814,"\int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{150 a \left(a^2+b^2\right)^2 \sin (c+d x)-150 b \left(a^2+b^2\right)^2 \cos (c+d x)+25 a \left(a^4-2 a^2 b^2-3 b^4\right) \sin (3 (c+d x))+3 a \left(a^4-10 a^2 b^2+5 b^4\right) \sin (5 (c+d x))+25 b \left(-3 a^4-2 a^2 b^2+b^4\right) \cos (3 (c+d x))-3 b \left(5 a^4-10 a^2 b^2+b^4\right) \cos (5 (c+d x))}{240 d}","\frac{2 \left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x))^3}{3 d}-\frac{\left(a^2+b^2\right)^2 (b \cos (c+d x)-a \sin (c+d x))}{d}-\frac{(b \cos (c+d x)-a \sin (c+d x))^5}{5 d}",1,"(-150*b*(a^2 + b^2)^2*Cos[c + d*x] + 25*b*(-3*a^4 - 2*a^2*b^2 + b^4)*Cos[3*(c + d*x)] - 3*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[5*(c + d*x)] + 150*a*(a^2 + b^2)^2*Sin[c + d*x] + 25*a*(a^4 - 2*a^2*b^2 - 3*b^4)*Sin[3*(c + d*x)] + 3*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[5*(c + d*x)])/(240*d)","A",1
222,1,107,108,0.3029195,"\int (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{8 \left(a^4-b^4\right) \sin (2 (c+d x))+12 \left(a^2+b^2\right)^2 (c+d x)-16 a b \left(a^2+b^2\right) \cos (2 (c+d x))-4 a b \left(a^2-b^2\right) \cos (4 (c+d x))+\left(a^4-6 a^2 b^2+b^4\right) \sin (4 (c+d x))}{32 d}","-\frac{3 \left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{8 d}+\frac{3}{8} x \left(a^2+b^2\right)^2-\frac{(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{4 d}",1,"(12*(a^2 + b^2)^2*(c + d*x) - 16*a*b*(a^2 + b^2)*Cos[2*(c + d*x)] - 4*a*b*(a^2 - b^2)*Cos[4*(c + d*x)] + 8*(a^4 - b^4)*Sin[2*(c + d*x)] + (a^4 - 6*a^2*b^2 + b^4)*Sin[4*(c + d*x)])/(32*d)","A",1
223,1,81,58,0.3231857,"\int (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\left(b^3-3 a^2 b\right) \cos (3 (c+d x))-9 b \left(a^2+b^2\right) \cos (c+d x)+2 a \sin (c+d x) \left(\left(a^2-3 b^2\right) \cos (2 (c+d x))+5 a^2+3 b^2\right)}{12 d}","\frac{(b \cos (c+d x)-a \sin (c+d x))^3}{3 d}-\frac{\left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x))}{d}",1,"(-9*b*(a^2 + b^2)*Cos[c + d*x] + (-3*a^2*b + b^3)*Cos[3*(c + d*x)] + 2*a*(5*a^2 + 3*b^2 + (a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(12*d)","A",1
224,1,52,55,0.1022261,"\int (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{2 \left(a^2+b^2\right) (c+d x)+\left(a^2-b^2\right) \sin (2 (c+d x))-2 a b \cos (2 (c+d x))}{4 d}","\frac{1}{2} x \left(a^2+b^2\right)-\frac{(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{2 d}",1,"(2*(a^2 + b^2)*(c + d*x) - 2*a*b*Cos[2*(c + d*x)] + (a^2 - b^2)*Sin[2*(c + d*x)])/(4*d)","A",1
225,1,46,24,0.0118271,"\int (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[a*Cos[c + d*x] + b*Sin[c + d*x],x]","\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d}","\frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d}",1,"-((b*Cos[c]*Cos[d*x])/d) + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d + (b*Sin[c]*Sin[d*x])/d","A",1
226,1,45,47,0.060199,"\int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-1),x]","\frac{2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{d \sqrt{a^2+b^2}}","-\frac{\tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \sqrt{a^2+b^2}}",1,"(2*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d)","A",1
227,1,32,32,0.0358615,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-2),x]","\frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))}","\frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))}",1,"Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","A",1
228,1,132,103,0.2944793,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-3),x]","\frac{\left(a^2+b^2\right) (a \sin (c+d x)-b \cos (c+d x))+2 \sqrt{a^2+b^2} (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{2 d (a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x))^2}","-\frac{b \cos (c+d x)-a \sin (c+d x)}{2 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{\tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 d \left(a^2+b^2\right)^{3/2}}",1,"((a^2 + b^2)*(-(b*Cos[c + d*x]) + a*Sin[c + d*x]) + 2*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*(a - I*b)^2*(a + I*b)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)","C",1
229,1,85,98,0.2899028,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-4),x]","\frac{\sin (c+d x) \left(\left(a^2-b^2\right) \cos (2 (c+d x))+2 a^2+b^2\right)-a b \cos (3 (c+d x))}{3 a d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^3}","\frac{2 \sin (c+d x)}{3 a d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))}-\frac{b \cos (c+d x)-a \sin (c+d x)}{3 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^3}",1,"(-(a*b*Cos[3*(c + d*x)]) + (2*a^2 + b^2 + (a^2 - b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(3*a*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)","A",1
230,1,157,156,1.1470273,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-5),x]","\frac{\frac{6 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}+\frac{\left(3 b^3-9 a^2 b\right) \cos (3 (c+d x))-11 b \left(a^2+b^2\right) \cos (c+d x)+2 a \sin (c+d x) \left(3 \left(a^2-3 b^2\right) \cos (2 (c+d x))+7 a^2+b^2\right)}{4 \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^4}}{8 d}","-\frac{3 (b \cos (c+d x)-a \sin (c+d x))}{8 d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{b \cos (c+d x)-a \sin (c+d x)}{4 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{3 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{8 d \left(a^2+b^2\right)^{5/2}}",1,"((6*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (-11*b*(a^2 + b^2)*Cos[c + d*x] + (-9*a^2*b + 3*b^3)*Cos[3*(c + d*x)] + 2*a*(7*a^2 + b^2 + 3*(a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(4*(a^2 + b^2)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4))/(8*d)","A",1
231,1,182,151,0.5230737,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^6} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-6),x]","\frac{10 a^4 \sin (c+d x)+5 a^4 \sin (3 (c+d x))+a^4 \sin (5 (c+d x))+\left(4 a b^3-4 a^3 b\right) \cos (5 (c+d x))+20 a^2 b^2 \sin (c+d x)-6 a^2 b^2 \sin (5 (c+d x))-10 a b \left(a^2+b^2\right) \cos (3 (c+d x))+10 b^4 \sin (c+d x)-5 b^4 \sin (3 (c+d x))+b^4 \sin (5 (c+d x))}{30 a d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^5}","\frac{8 \sin (c+d x)}{15 a d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))}-\frac{4 (b \cos (c+d x)-a \sin (c+d x))}{15 d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{b \cos (c+d x)-a \sin (c+d x)}{5 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^5}",1,"(-10*a*b*(a^2 + b^2)*Cos[3*(c + d*x)] + (-4*a^3*b + 4*a*b^3)*Cos[5*(c + d*x)] + 10*a^4*Sin[c + d*x] + 20*a^2*b^2*Sin[c + d*x] + 10*b^4*Sin[c + d*x] + 5*a^4*Sin[3*(c + d*x)] - 5*b^4*Sin[3*(c + d*x)] + a^4*Sin[5*(c + d*x)] - 6*a^2*b^2*Sin[5*(c + d*x)] + b^4*Sin[5*(c + d*x)])/(30*a*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5)","A",1
232,1,205,186,1.8159029,"\int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(7/2),x]","\frac{\frac{20 \left(a^2+b^2\right)^2 \tan \left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right) \sqrt{\cos ^2\left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(c+d x+\tan ^{-1}\left(\frac{a}{b}\right)\right)\right)}{\sqrt{b \sqrt{\frac{a^2}{b^2}+1} \sin \left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)}}+\sqrt{a \cos (c+d x)+b \sin (c+d x)} \left(\left(3 b^3-9 a^2 b\right) \cos (3 (c+d x))-23 b \left(a^2+b^2\right) \cos (c+d x)+2 a \sin (c+d x) \left(3 \left(a^2-3 b^2\right) \cos (2 (c+d x))+13 a^2+7 b^2\right)\right)}{42 d}","\frac{10 \left(a^2+b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}} F\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}(a,b)\right)\right|2\right)}{21 d \sqrt{a \cos (c+d x)+b \sin (c+d x)}}-\frac{10 \left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x)) \sqrt{a \cos (c+d x)+b \sin (c+d x)}}{21 d}-\frac{2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^{5/2}}{7 d}",1,"(Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]]*(-23*b*(a^2 + b^2)*Cos[c + d*x] + (-9*a^2*b + 3*b^3)*Cos[3*(c + d*x)] + 2*a*(13*a^2 + 7*b^2 + 3*(a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]) + (20*(a^2 + b^2)^2*Sqrt[Cos[c + d*x + ArcTan[a/b]]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[a/b]]^2]*Tan[c + d*x + ArcTan[a/b]])/Sqrt[Sqrt[1 + a^2/b^2]*b*Sin[c + d*x + ArcTan[a/b]]])/(42*d)","C",1
233,1,256,131,1.5996395,"\int (a \cos (c+d x)+b \sin (c+d x))^{5/2} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(5/2),x]","\frac{\sqrt{a \cos (c+d x)+b \sin (c+d x)} \left(b \left(a^2-b^2\right) \sin (2 (c+d x))+6 a \left(a^2+b^2\right)-2 a b^2 \cos (2 (c+d x))\right)-\frac{3 \left(a^2+b^2\right)^2 \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right) \left(b \sin \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(c+d x-\tan ^{-1}\left(\frac{b}{a}\right)\right)\right)+\sqrt{\sin ^2\left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)} \left(2 a \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)-b \sin \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)\right)\right)}{\sqrt{\sin ^2\left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)} \left(a \sqrt{\frac{b^2}{a^2}+1} \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)\right)^{3/2}}}{5 b d}","\frac{6 \left(a^2+b^2\right) \sqrt{a \cos (c+d x)+b \sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}(a,b)\right)\right|2\right)}{5 d \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}}}-\frac{2 (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^{3/2}}{5 d}",1,"(Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]]*(6*a*(a^2 + b^2) - 2*a*b^2*Cos[2*(c + d*x)] + b*(a^2 - b^2)*Sin[2*(c + d*x)]) - (3*(a^2 + b^2)^2*Cos[c + d*x - ArcTan[b/a]]*(b*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[c + d*x - ArcTan[b/a]]^2]*Sin[c + d*x - ArcTan[b/a]] + Sqrt[Sin[c + d*x - ArcTan[b/a]]^2]*(2*a*Cos[c + d*x - ArcTan[b/a]] - b*Sin[c + d*x - ArcTan[b/a]])))/((a*Sqrt[1 + b^2/a^2]*Cos[c + d*x - ArcTan[b/a]])^(3/2)*Sqrt[Sin[c + d*x - ArcTan[b/a]]^2]))/(5*b*d)","C",1
234,1,143,131,1.2895089,"\int (a \cos (c+d x)+b \sin (c+d x))^{3/2} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(3/2),x]","\frac{2 \left(\frac{\left(a^2+b^2\right) \tan \left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right) \sqrt{\cos ^2\left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(c+d x+\tan ^{-1}\left(\frac{a}{b}\right)\right)\right)}{\sqrt{b \sqrt{\frac{a^2}{b^2}+1} \sin \left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)}}+\sqrt{a \cos (c+d x)+b \sin (c+d x)} (a \sin (c+d x)-b \cos (c+d x))\right)}{3 d}","\frac{2 \left(a^2+b^2\right) \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}} F\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}(a,b)\right)\right|2\right)}{3 d \sqrt{a \cos (c+d x)+b \sin (c+d x)}}-\frac{2 (b \cos (c+d x)-a \sin (c+d x)) \sqrt{a \cos (c+d x)+b \sin (c+d x)}}{3 d}",1,"(2*((-(b*Cos[c + d*x]) + a*Sin[c + d*x])*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]] + ((a^2 + b^2)*Sqrt[Cos[c + d*x + ArcTan[a/b]]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[a/b]]^2]*Tan[c + d*x + ArcTan[a/b]])/Sqrt[Sqrt[1 + a^2/b^2]*b*Sin[c + d*x + ArcTan[a/b]]]))/(3*d)","C",1
235,1,268,75,1.0957129,"\int \sqrt{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]],x]","\frac{\cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right) \left(\sqrt{\sin ^2\left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)} \left(b \left(a^2+b^2\right) \sin \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)-2 a \left(a^2+b^2\right) \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)+2 a^2 \sqrt{\frac{b^2}{a^2}+1} \sqrt{a \cos (c+d x)+b \sin (c+d x)} \sqrt{a \sqrt{\frac{b^2}{a^2}+1} \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)}\right)-b \left(a^2+b^2\right) \sin \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(c+d x-\tan ^{-1}\left(\frac{b}{a}\right)\right)\right)\right)}{b d \sqrt{\sin ^2\left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)} \left(a \sqrt{\frac{b^2}{a^2}+1} \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)\right)^{3/2}}","\frac{2 \sqrt{a \cos (c+d x)+b \sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}(a,b)\right)\right|2\right)}{d \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}}}",1,"(Cos[c + d*x - ArcTan[b/a]]*(-(b*(a^2 + b^2)*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[c + d*x - ArcTan[b/a]]^2]*Sin[c + d*x - ArcTan[b/a]]) + Sqrt[Sin[c + d*x - ArcTan[b/a]]^2]*(-2*a*(a^2 + b^2)*Cos[c + d*x - ArcTan[b/a]] + 2*a^2*Sqrt[1 + b^2/a^2]*Sqrt[a*Sqrt[1 + b^2/a^2]*Cos[c + d*x - ArcTan[b/a]]]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]] + b*(a^2 + b^2)*Sin[c + d*x - ArcTan[b/a]])))/(b*d*(a*Sqrt[1 + b^2/a^2]*Cos[c + d*x - ArcTan[b/a]])^(3/2)*Sqrt[Sin[c + d*x - ArcTan[b/a]]^2])","C",1
236,1,92,75,0.1877956,"\int \frac{1}{\sqrt{a \cos (c+d x)+b \sin (c+d x)}} \, dx","Integrate[1/Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]],x]","\frac{2 \tan \left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right) \sqrt{\cos ^2\left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(c+d x+\tan ^{-1}\left(\frac{a}{b}\right)\right)\right)}{d \sqrt{b \sqrt{\frac{a^2}{b^2}+1} \sin \left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)}}","\frac{2 \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}} F\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}(a,b)\right)\right|2\right)}{d \sqrt{a \cos (c+d x)+b \sin (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x + ArcTan[a/b]]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[a/b]]^2]*Tan[c + d*x + ArcTan[a/b]])/(d*Sqrt[Sqrt[1 + a^2/b^2]*b*Sin[c + d*x + ArcTan[a/b]]])","C",1
237,1,219,138,3.0582005,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^{3/2}} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-3/2),x]","\frac{\frac{\tan \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right) \sqrt{a \sqrt{\frac{b^2}{a^2}+1} \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(c+d x-\tan ^{-1}\left(\frac{b}{a}\right)\right)\right)}{\sqrt{\sin ^2\left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)}}-\tan \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right) \sqrt{a \sqrt{\frac{b^2}{a^2}+1} \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)}-\frac{2 b \cos (c+d x)}{\sqrt{a \cos (c+d x)+b \sin (c+d x)}}+\frac{2 a \sin (c+d x)}{\sqrt{a \cos (c+d x)+b \sin (c+d x)}}}{d \left(a^2+b^2\right)}","-\frac{2 \sqrt{a \cos (c+d x)+b \sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}(a,b)\right)\right|2\right)}{d \left(a^2+b^2\right) \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}}}-\frac{2 (b \cos (c+d x)-a \sin (c+d x))}{d \left(a^2+b^2\right) \sqrt{a \cos (c+d x)+b \sin (c+d x)}}",1,"((-2*b*Cos[c + d*x])/Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]] + (2*a*Sin[c + d*x])/Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]] - Sqrt[a*Sqrt[1 + b^2/a^2]*Cos[c + d*x - ArcTan[b/a]]]*Tan[c + d*x - ArcTan[b/a]] + (Sqrt[a*Sqrt[1 + b^2/a^2]*Cos[c + d*x - ArcTan[b/a]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[c + d*x - ArcTan[b/a]]^2]*Tan[c + d*x - ArcTan[b/a]])/Sqrt[Sin[c + d*x - ArcTan[b/a]]^2])/((a^2 + b^2)*d)","C",1
238,1,145,142,1.6835326,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^{5/2}} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-5/2),x]","\frac{2 \left(\frac{\tan \left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right) \sqrt{\cos ^2\left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(c+d x+\tan ^{-1}\left(\frac{a}{b}\right)\right)\right)}{\sqrt{b \sqrt{\frac{a^2}{b^2}+1} \sin \left(\tan ^{-1}\left(\frac{a}{b}\right)+c+d x\right)}}+\frac{a \sin (c+d x)-b \cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^{3/2}}\right)}{3 d \left(a^2+b^2\right)}","\frac{2 \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}} F\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}(a,b)\right)\right|2\right)}{3 d \left(a^2+b^2\right) \sqrt{a \cos (c+d x)+b \sin (c+d x)}}-\frac{2 (b \cos (c+d x)-a \sin (c+d x))}{3 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^{3/2}}",1,"(2*((-(b*Cos[c + d*x]) + a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x])^(3/2) + (Sqrt[Cos[c + d*x + ArcTan[a/b]]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[a/b]]^2]*Tan[c + d*x + ArcTan[a/b]])/Sqrt[Sqrt[1 + a^2/b^2]*b*Sin[c + d*x + ArcTan[a/b]]]))/(3*(a^2 + b^2)*d)","C",1
239,1,277,197,2.4358657,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^{7/2}} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-7/2),x]","\frac{\frac{\cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right) \left(3 b \sin \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(c+d x-\tan ^{-1}\left(\frac{b}{a}\right)\right)\right)-3 \sqrt{\sin ^2\left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)} \left(b \sin \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)-2 a \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)\right)\right)}{\sqrt{\sin ^2\left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)} \left(a \sqrt{\frac{b^2}{a^2}+1} \cos \left(-\tan ^{-1}\left(\frac{b}{a}\right)+c+d x\right)\right)^{3/2}}-\frac{2 \left(3 a^2 \cos ^3(c+d x)-a b \sin (c+d x)+6 a b \sin (c+d x) \cos ^2(c+d x)+b^2 \left(3 \sin ^2(c+d x)+1\right) \cos (c+d x)\right)}{(a \cos (c+d x)+b \sin (c+d x))^{5/2}}}{5 b d \left(a^2+b^2\right)}","-\frac{6 \sqrt{a \cos (c+d x)+b \sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}(a,b)\right)\right|2\right)}{5 d \left(a^2+b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b \sin (c+d x)}{\sqrt{a^2+b^2}}}}-\frac{6 (b \cos (c+d x)-a \sin (c+d x))}{5 d \left(a^2+b^2\right)^2 \sqrt{a \cos (c+d x)+b \sin (c+d x)}}-\frac{2 (b \cos (c+d x)-a \sin (c+d x))}{5 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^{5/2}}",1,"((-2*(3*a^2*Cos[c + d*x]^3 - a*b*Sin[c + d*x] + 6*a*b*Cos[c + d*x]^2*Sin[c + d*x] + b^2*Cos[c + d*x]*(1 + 3*Sin[c + d*x]^2)))/(a*Cos[c + d*x] + b*Sin[c + d*x])^(5/2) + (Cos[c + d*x - ArcTan[b/a]]*(3*b*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[c + d*x - ArcTan[b/a]]^2]*Sin[c + d*x - ArcTan[b/a]] - 3*Sqrt[Sin[c + d*x - ArcTan[b/a]]^2]*(-2*a*Cos[c + d*x - ArcTan[b/a]] + b*Sin[c + d*x - ArcTan[b/a]])))/((a*Sqrt[1 + b^2/a^2]*Cos[c + d*x - ArcTan[b/a]])^(3/2)*Sqrt[Sin[c + d*x - ArcTan[b/a]]^2]))/(5*b*(a^2 + b^2)*d)","C",1
240,1,153,120,0.4761777,"\int (2 \cos (c+d x)+3 \sin (c+d x))^{7/2} \, dx","Integrate[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(7/2),x]","\frac{260\ 13^{3/4} \sqrt{-\left(\left(\sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)-1\right) \sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right)} \sqrt{\sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)+1} \sec \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right)-\sqrt{3 \sin (c+d x)+2 \cos (c+d x)} (-598 \sin (c+d x)+138 \sin (3 (c+d x))+897 \cos (c+d x)+27 \cos (3 (c+d x)))}{42 d}","-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x)) (3 \sin (c+d x)+2 \cos (c+d x))^{5/2}}{7 d}-\frac{130 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt{3 \sin (c+d x)+2 \cos (c+d x)}}{21 d}+\frac{130\ 13^{3/4} F\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right|2\right)}{21 d}",1,"(-(Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]]*(897*Cos[c + d*x] + 27*Cos[3*(c + d*x)] - 598*Sin[c + d*x] + 138*Sin[3*(c + d*x)])) + 260*13^(3/4)*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[2/3]]^2]*Sec[c + d*x + ArcTan[2/3]]*Sqrt[-((-1 + Sin[c + d*x + ArcTan[2/3]])*Sin[c + d*x + ArcTan[2/3]])]*Sqrt[1 + Sin[c + d*x + ArcTan[2/3]]])/(42*d)","C",0
241,1,199,75,0.8199545,"\int (2 \cos (c+d x)+3 \sin (c+d x))^{5/2} \, dx","Integrate[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(5/2),x]","\frac{-\frac{39 \sqrt[4]{13} \sin \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)}{\sqrt{-\left(\left(\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)-1\right) \cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)} \sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)+1}}+\sqrt{3 \sin (c+d x)+2 \cos (c+d x)} (-5 \sin (2 (c+d x))-12 \cos (2 (c+d x))+52)-\frac{13 \sqrt[4]{13} \left(4 \cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)-3 \sin \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)}{\sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}}}{5 d}","\frac{78 \sqrt[4]{13} E\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right|2\right)}{5 d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x)) (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}}{5 d}",1,"(Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]]*(52 - 12*Cos[2*(c + d*x)] - 5*Sin[2*(c + d*x)]) - (13*13^(1/4)*(4*Cos[c + d*x - ArcTan[3/2]] - 3*Sin[c + d*x - ArcTan[3/2]]))/Sqrt[Cos[c + d*x - ArcTan[3/2]]] - (39*13^(1/4)*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[c + d*x - ArcTan[3/2]]^2]*Sin[c + d*x - ArcTan[3/2]])/(Sqrt[-((-1 + Cos[c + d*x - ArcTan[3/2]])*Cos[c + d*x - ArcTan[3/2]])]*Sqrt[1 + Cos[c + d*x - ArcTan[3/2]]]))/(5*d)","C",0
242,1,133,75,0.2989155,"\int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx","Integrate[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2),x]","\frac{2\ 13^{3/4} \sqrt{-\left(\left(\sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)-1\right) \sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right)} \sqrt{\sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)+1} \sec \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right)+2 (2 \sin (c+d x)-3 \cos (c+d x)) \sqrt{3 \sin (c+d x)+2 \cos (c+d x)}}{3 d}","\frac{2\ 13^{3/4} F\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right|2\right)}{3 d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x)) \sqrt{3 \sin (c+d x)+2 \cos (c+d x)}}{3 d}",1,"(2*(-3*Cos[c + d*x] + 2*Sin[c + d*x])*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]] + 2*13^(3/4)*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[2/3]]^2]*Sec[c + d*x + ArcTan[2/3]]*Sqrt[-((-1 + Sin[c + d*x + ArcTan[2/3]])*Sin[c + d*x + ArcTan[2/3]])]*Sqrt[1 + Sin[c + d*x + ArcTan[2/3]]])/(3*d)","C",0
243,1,184,27,0.8105501,"\int \sqrt{2 \cos (c+d x)+3 \sin (c+d x)} \, dx","Integrate[Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]],x]","\frac{-\frac{3 \sqrt[4]{13} \sin \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)}{\sqrt{-\left(\left(\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)-1\right) \cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)} \sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)+1}}+4 \sqrt{3 \sin (c+d x)+2 \cos (c+d x)}-4 \sqrt[4]{13} \sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}+\frac{3 \sqrt[4]{13} \sin \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}{\sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}}}{3 d}","\frac{2 \sqrt[4]{13} E\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right|2\right)}{d}",1,"(-4*13^(1/4)*Sqrt[Cos[c + d*x - ArcTan[3/2]]] + 4*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]] + (3*13^(1/4)*Sin[c + d*x - ArcTan[3/2]])/Sqrt[Cos[c + d*x - ArcTan[3/2]]] - (3*13^(1/4)*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[c + d*x - ArcTan[3/2]]^2]*Sin[c + d*x - ArcTan[3/2]])/(Sqrt[-((-1 + Cos[c + d*x - ArcTan[3/2]])*Cos[c + d*x - ArcTan[3/2]])]*Sqrt[1 + Cos[c + d*x - ArcTan[3/2]]]))/(3*d)","C",0
244,1,88,27,0.0940987,"\int \frac{1}{\sqrt{2 \cos (c+d x)+3 \sin (c+d x)}} \, dx","Integrate[1/Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]],x]","\frac{2 \sqrt{-\left(\left(\sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)-1\right) \sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right)} \sqrt{\sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)+1} \sec \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right)}{\sqrt[4]{13} d}","\frac{2 F\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right|2\right)}{\sqrt[4]{13} d}",1,"(2*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[2/3]]^2]*Sec[c + d*x + ArcTan[2/3]]*Sqrt[-((-1 + Sin[c + d*x + ArcTan[2/3]])*Sin[c + d*x + ArcTan[2/3]])]*Sqrt[1 + Sin[c + d*x + ArcTan[2/3]]])/(13^(1/4)*d)","C",0
245,1,190,73,1.031734,"\int \frac{1}{(2 \cos (c+d x)+3 \sin (c+d x))^{3/2}} \, dx","Integrate[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(-3/2),x]","\frac{\frac{3 \sin \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)}{13^{3/4} \sqrt{-\left(\left(\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)-1\right) \cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)} \sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)+1}}-\frac{2 \cos (c+d x)}{\sqrt{3 \sin (c+d x)+2 \cos (c+d x)}}+\frac{4 \sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}}{13^{3/4}}-\frac{3 \sin \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}{13^{3/4} \sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}}}{3 d}","-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x))}{13 d \sqrt{3 \sin (c+d x)+2 \cos (c+d x)}}-\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right|2\right)}{13^{3/4} d}",1,"((4*Sqrt[Cos[c + d*x - ArcTan[3/2]]])/13^(3/4) - (2*Cos[c + d*x])/Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]] - (3*Sin[c + d*x - ArcTan[3/2]])/(13^(3/4)*Sqrt[Cos[c + d*x - ArcTan[3/2]]]) + (3*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[c + d*x - ArcTan[3/2]]^2]*Sin[c + d*x - ArcTan[3/2]])/(13^(3/4)*Sqrt[-((-1 + Cos[c + d*x - ArcTan[3/2]])*Cos[c + d*x - ArcTan[3/2]])]*Sqrt[1 + Cos[c + d*x - ArcTan[3/2]]]))/(3*d)","C",0
246,1,157,75,0.7062569,"\int \frac{1}{(2 \cos (c+d x)+3 \sin (c+d x))^{5/2}} \, dx","Integrate[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(-5/2),x]","\frac{\sqrt{2} 13^{3/4} \sqrt{\sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)+1} (3 \sin (c+d x)+2 \cos (c+d x))^{3/2} \sec \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right) \sqrt{2 \sin \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)+\cos \left(2 \left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right)-1} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(c+d x+\tan ^{-1}\left(\frac{2}{3}\right)\right)\right)+52 \sin (c+d x)-78 \cos (c+d x)}{507 d (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}}","\frac{2 F\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right|2\right)}{39 \sqrt[4]{13} d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}}",1,"(-78*Cos[c + d*x] + 52*Sin[c + d*x] + Sqrt[2]*13^(3/4)*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcTan[2/3]]^2]*Sec[c + d*x + ArcTan[2/3]]*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2)*Sqrt[1 + Sin[c + d*x + ArcTan[2/3]]]*Sqrt[-1 + Cos[2*(c + d*x + ArcTan[2/3])] + 2*Sin[c + d*x + ArcTan[2/3]]])/(507*d*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2))","C",0
247,1,224,120,1.9507297,"\int \frac{1}{(2 \cos (c+d x)+3 \sin (c+d x))^{7/2}} \, dx","Integrate[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(-7/2),x]","\frac{\frac{3 \sin \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)}{13^{3/4} \sqrt{-\left(\left(\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)-1\right) \cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right)} \sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)+1}}+\frac{-4 (\sin (c+d x)+3 \sin (3 (c+d x)))-33 \cos (c+d x)+5 \cos (3 (c+d x))}{2 (3 \sin (c+d x)+2 \cos (c+d x))^{5/2}}+\frac{4 \sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}}{13^{3/4}}-\frac{3 \sin \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}{13^{3/4} \sqrt{\cos \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)}}}{65 d}","-\frac{6 (3 \cos (c+d x)-2 \sin (c+d x))}{845 d \sqrt{3 \sin (c+d x)+2 \cos (c+d x)}}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x))}{65 d (3 \sin (c+d x)+2 \cos (c+d x))^{5/2}}-\frac{6 E\left(\left.\frac{1}{2} \left(c+d x-\tan ^{-1}\left(\frac{3}{2}\right)\right)\right|2\right)}{65\ 13^{3/4} d}",1,"((4*Sqrt[Cos[c + d*x - ArcTan[3/2]]])/13^(3/4) + (-33*Cos[c + d*x] + 5*Cos[3*(c + d*x)] - 4*(Sin[c + d*x] + 3*Sin[3*(c + d*x)]))/(2*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(5/2)) - (3*Sin[c + d*x - ArcTan[3/2]])/(13^(3/4)*Sqrt[Cos[c + d*x - ArcTan[3/2]]]) + (3*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[c + d*x - ArcTan[3/2]]^2]*Sin[c + d*x - ArcTan[3/2]])/(13^(3/4)*Sqrt[-((-1 + Cos[c + d*x - ArcTan[3/2]])*Cos[c + d*x - ArcTan[3/2]])]*Sqrt[1 + Cos[c + d*x - ArcTan[3/2]]]))/(65*d)","C",0
248,1,31,32,0.0853615,"\int (a \cos (c+d x)+i a \sin (c+d x))^n \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n,x]","-\frac{i (a (\cos (c+d x)+i \sin (c+d x)))^n}{d n}","-\frac{i (a \cos (c+d x)+i a \sin (c+d x))^n}{d n}",1,"((-I)*(a*(Cos[c + d*x] + I*Sin[c + d*x]))^n)/(d*n)","A",1
249,1,31,31,0.1107098,"\int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4,x]","-\frac{i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d}","-\frac{i (a \cos (c+d x)+i a \sin (c+d x))^4}{4 d}",1,"((-1/4*I)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4)/d","A",1
250,1,31,31,0.076502,"\int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","-\frac{i (a \cos (c+d x)+i a \sin (c+d x))^3}{3 d}","-\frac{i (a \cos (c+d x)+i a \sin (c+d x))^3}{3 d}",1,"((-1/3*I)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3)/d","A",1
251,1,31,31,0.0517805,"\int (a \cos (c+d x)+i a \sin (c+d x))^2 \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{i (a \cos (c+d x)+i a \sin (c+d x))^2}{2 d}","-\frac{i (a \cos (c+d x)+i a \sin (c+d x))^2}{2 d}",1,"((-1/2*I)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2)/d","A",1
252,1,51,26,0.0117136,"\int (a \cos (c+d x)+i a \sin (c+d x)) \, dx","Integrate[a*Cos[c + d*x] + I*a*Sin[c + d*x],x]","\frac{i a \sin (c) \sin (d x)}{d}-\frac{i a \cos (c) \cos (d x)}{d}+\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}","\frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d}",1,"((-I)*a*Cos[c]*Cos[d*x])/d + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d + (I*a*Sin[c]*Sin[d*x])/d","A",1
253,1,29,29,0.032499,"\int \frac{1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-1),x]","\frac{i}{d (a \cos (c+d x)+i a \sin (c+d x))}","\frac{i}{d (a \cos (c+d x)+i a \sin (c+d x))}",1,"I/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x]))","A",1
254,1,31,31,0.0404703,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-2),x]","\frac{i}{2 d (a \cos (c+d x)+i a \sin (c+d x))^2}","\frac{i}{2 d (a \cos (c+d x)+i a \sin (c+d x))^2}",1,"(I/2)/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2)","A",1
255,1,31,31,0.0448076,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-3),x]","\frac{i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3}","\frac{i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3}",1,"(I/3)/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3)","A",1
256,1,31,31,0.0468505,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^4} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-4),x]","\frac{i}{4 d (a \cos (c+d x)+i a \sin (c+d x))^4}","\frac{i}{4 d (a \cos (c+d x)+i a \sin (c+d x))^4}",1,"(I/4)/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4)","A",1
257,1,32,33,0.0302833,"\int (a \cos (c+d x)+i a \sin (c+d x))^{5/2} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2),x]","-\frac{2 i (a (\cos (c+d x)+i \sin (c+d x)))^{5/2}}{5 d}","-\frac{2 i (a \cos (c+d x)+i a \sin (c+d x))^{5/2}}{5 d}",1,"(((-2*I)/5)*(a*(Cos[c + d*x] + I*Sin[c + d*x]))^(5/2))/d","A",1
258,1,32,33,0.0291221,"\int (a \cos (c+d x)+i a \sin (c+d x))^{3/2} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2),x]","-\frac{2 i (a (\cos (c+d x)+i \sin (c+d x)))^{3/2}}{3 d}","-\frac{2 i (a \cos (c+d x)+i a \sin (c+d x))^{3/2}}{3 d}",1,"(((-2*I)/3)*(a*(Cos[c + d*x] + I*Sin[c + d*x]))^(3/2))/d","A",1
259,1,30,31,0.022211,"\int \sqrt{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Sqrt[a*Cos[c + d*x] + I*a*Sin[c + d*x]],x]","-\frac{2 i \sqrt{a (\cos (c+d x)+i \sin (c+d x))}}{d}","-\frac{2 i \sqrt{a \cos (c+d x)+i a \sin (c+d x)}}{d}",1,"((-2*I)*Sqrt[a*(Cos[c + d*x] + I*Sin[c + d*x])])/d","A",1
260,1,30,31,0.0337157,"\int \frac{1}{\sqrt{a \cos (c+d x)+i a \sin (c+d x)}} \, dx","Integrate[1/Sqrt[a*Cos[c + d*x] + I*a*Sin[c + d*x]],x]","\frac{2 i}{d \sqrt{a (\cos (c+d x)+i \sin (c+d x))}}","\frac{2 i}{d \sqrt{a \cos (c+d x)+i a \sin (c+d x)}}",1,"(2*I)/(d*Sqrt[a*(Cos[c + d*x] + I*Sin[c + d*x])])","A",1
261,1,32,33,0.0313219,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^{3/2}} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-3/2),x]","\frac{2 i}{3 d (a (\cos (c+d x)+i \sin (c+d x)))^{3/2}}","\frac{2 i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^{3/2}}",1,"((2*I)/3)/(d*(a*(Cos[c + d*x] + I*Sin[c + d*x]))^(3/2))","A",1
262,1,32,33,0.0344416,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^{5/2}} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-5/2),x]","\frac{2 i}{5 d (a (\cos (c+d x)+i \sin (c+d x)))^{5/2}}","\frac{2 i}{5 d (a \cos (c+d x)+i a \sin (c+d x))^{5/2}}",1,"((2*I)/5)/(d*(a*(Cos[c + d*x] + I*Sin[c + d*x]))^(5/2))","A",1
263,1,303,149,1.2391911,"\int (a \sec (x)+b \tan (x))^5 \, dx","Integrate[(a*Sec[x] + b*Tan[x])^5,x]","-\frac{\left(a^2-b^2\right)^2 \left((a+b)^3 \left(3 a^2-9 a b+8 b^2\right) \log (1-\sin (x))-(a-b)^3 \left(3 a^2+9 a b+8 b^2\right) \log (\sin (x)+1)\right)+4 \left(b^2-a^2\right) \sec ^4(x) (a \sin (x)-b) (a+b \sin (x))^6-2 a b^6 \left(3 a^2+5 b^2\right) \sin ^5(x)-10 a b^4 \left(9 a^4+8 a^2 b^2-b^4\right) \sin ^3(x)+4 b^5 \left(-9 a^4-12 a^2 b^2+b^4\right) \sin ^4(x)+2 \sec ^2(x) (a+b \sin (x))^6 \left(-3 a^3 \sin (x)+6 a^2 b-5 a b^2 \sin (x)+2 b^3\right)-10 a b^2 \left(9 a^6-6 a^4 b^2+8 a^2 b^4-3 b^6\right) \sin (x)+8 b^3 \left(-15 a^6-4 a^4 b^2-2 a^2 b^4+b^6\right) \sin ^2(x)}{16 \left(a^2-b^2\right)^2}","-\frac{1}{16} (a+b)^3 \left(3 a^2-9 a b+8 b^2\right) \log (1-\sin (x))+\frac{1}{16} (a-b)^3 \left(3 a^2+9 a b+8 b^2\right) \log (\sin (x)+1)+\frac{1}{8} \sec ^2(x) (a+b \sin (x))^2 \left(a \left(3 a^2-5 b^2\right) \sin (x)+2 b \left(a^2-2 b^2\right)\right)-\frac{1}{8} a b^4 \left(7-\frac{3 a^2}{b^2}\right) \sin (x)+\frac{1}{4} \sec ^4(x) (a \sin (x)+b) (a+b \sin (x))^4",1,"-1/16*((a^2 - b^2)^2*((a + b)^3*(3*a^2 - 9*a*b + 8*b^2)*Log[1 - Sin[x]] - (a - b)^3*(3*a^2 + 9*a*b + 8*b^2)*Log[1 + Sin[x]]) - 10*a*b^2*(9*a^6 - 6*a^4*b^2 + 8*a^2*b^4 - 3*b^6)*Sin[x] + 8*b^3*(-15*a^6 - 4*a^4*b^2 - 2*a^2*b^4 + b^6)*Sin[x]^2 - 10*a*b^4*(9*a^4 + 8*a^2*b^2 - b^4)*Sin[x]^3 + 4*b^5*(-9*a^4 - 12*a^2*b^2 + b^4)*Sin[x]^4 - 2*a*b^6*(3*a^2 + 5*b^2)*Sin[x]^5 + 4*(-a^2 + b^2)*Sec[x]^4*(-b + a*Sin[x])*(a + b*Sin[x])^6 + 2*Sec[x]^2*(a + b*Sin[x])^6*(6*a^2*b + 2*b^3 - 3*a^3*Sin[x] - 5*a*b^2*Sin[x]))/(a^2 - b^2)^2","B",1
264,1,96,100,0.1965163,"\int (a \sec (x)+b \tan (x))^4 \, dx","Integrate[(a*Sec[x] + b*Tan[x])^4,x]","\frac{1}{12} \sec ^3(x) \left(6 a^4 \sin (x)+2 a^4 \sin (3 x)+16 a^3 b+18 a^2 b^2 \sin (x)-6 a^2 b^2 \sin (3 x)-24 a b^3 \cos (2 x)-8 a b^3-4 b^4 \sin (3 x)+9 b^4 x \cos (x)+3 b^4 x \cos (3 x)\right)","\frac{4}{3} a b \left(a^2-2 b^2\right) \cos (x)+\frac{1}{3} b^2 \left(2 a^2-3 b^2\right) \sin (x) \cos (x)-\frac{1}{3} \sec (x) (a+b \sin (x))^2 \left(a b-\left(2 a^2-3 b^2\right) \sin (x)\right)+\frac{1}{3} \sec ^3(x) (a \sin (x)+b) (a+b \sin (x))^3+b^4 x",1,"(Sec[x]^3*(16*a^3*b - 8*a*b^3 + 9*b^4*x*Cos[x] - 24*a*b^3*Cos[2*x] + 3*b^4*x*Cos[3*x] + 6*a^4*Sin[x] + 18*a^2*b^2*Sin[x] + 2*a^4*Sin[3*x] - 6*a^2*b^2*Sin[3*x] - 4*b^4*Sin[3*x]))/12","A",1
265,1,123,75,0.5720361,"\int (a \sec (x)+b \tan (x))^3 \, dx","Integrate[(a*Sec[x] + b*Tan[x])^3,x]","\frac{2 a^4 b \sec ^2(x)+\left(a^2-b^2\right) \left((a-2 b) (a+b)^2 \log (1-\sin (x))-(a-b)^2 (a+2 b) \log (\sin (x)+1)\right)+\left(-8 a^4 b+4 a^2 b^3+2 b^5\right) \tan ^2(x)-2 a \left(a^4+2 a^2 b^2-3 b^4\right) \tan (x) \sec (x)}{4 \left(b^2-a^2\right)}","\frac{1}{2} a b^2 \sin (x)+\frac{1}{4} (a+2 b) (a-b)^2 \log (\sin (x)+1)-\frac{1}{4} (a-2 b) (a+b)^2 \log (1-\sin (x))+\frac{1}{2} \sec ^2(x) (a \sin (x)+b) (a+b \sin (x))^2",1,"((a^2 - b^2)*((a - 2*b)*(a + b)^2*Log[1 - Sin[x]] - (a - b)^2*(a + 2*b)*Log[1 + Sin[x]]) + 2*a^4*b*Sec[x]^2 - 2*a*(a^4 + 2*a^2*b^2 - 3*b^4)*Sec[x]*Tan[x] + (-8*a^4*b + 4*a^2*b^3 + 2*b^5)*Tan[x]^2)/(4*(-a^2 + b^2))","A",1
266,1,25,27,0.0473432,"\int (a \sec (x)+b \tan (x))^2 \, dx","Integrate[(a*Sec[x] + b*Tan[x])^2,x]","\left(a^2+b^2\right) \tan (x)+2 a b \sec (x)+b^2 \left(-\tan ^{-1}(\tan (x))\right)","a b \cos (x)+\sec (x) (a \sin (x)+b) (a+b \sin (x))+b^2 (-x)",1,"-(b^2*ArcTan[Tan[x]]) + 2*a*b*Sec[x] + (a^2 + b^2)*Tan[x]","A",1
267,1,42,12,0.0052762,"\int (a \sec (x)+b \tan (x)) \, dx","Integrate[a*Sec[x] + b*Tan[x],x]","-a \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+a \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-b \log (\cos (x))","a \tanh ^{-1}(\sin (x))-b \log (\cos (x))",1,"-(b*Log[Cos[x]]) - a*Log[Cos[x/2] - Sin[x/2]] + a*Log[Cos[x/2] + Sin[x/2]]","B",1
268,1,11,11,0.0067185,"\int \frac{1}{a \sec (x)+b \tan (x)} \, dx","Integrate[(a*Sec[x] + b*Tan[x])^(-1),x]","\frac{\log (a+b \sin (x))}{b}","\frac{\log (a+b \sin (x))}{b}",1,"Log[a + b*Sin[x]]/b","A",1
269,1,344,66,2.0210401,"\int \frac{1}{(a \sec (x)+b \tan (x))^2} \, dx","Integrate[(a*Sec[x] + b*Tan[x])^(-2),x]","\frac{(\sin (x)+1) \cos (x) \left(2 a (a-b) \sqrt{1-\sin (x)} (a+b \sin (x)) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (x)+1)}{a-b}}}{\sqrt{a+b} \sqrt{-\frac{b (\sin (x)-1)}{a+b}}}\right)+\sqrt{a+b} \left(-(b-a) \sqrt{\frac{b-b \sin (x)}{a+b}} \left(\sqrt{a-b} (a+b) \sqrt{1-\sin (x)} \sqrt{-\frac{b (\sin (x)+1)}{a-b}}+2 \sqrt{b} (a+b \sin (x)) \sinh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (x)+1)}{a-b}}}{\sqrt{2} \sqrt{b}}\right)\right)-2 a \sqrt{a-b} \sqrt{1-\sin (x)} (a+b \sin (x)) \tanh ^{-1}\left(\frac{\sqrt{\frac{b (\sin (x)+1)}{b-a}}}{\sqrt{\frac{b-b \sin (x)}{a+b}}}\right)\right)\right)}{(a-b)^{5/2} (a+b)^{3/2} \sqrt{1-\sin (x)} \left(-\frac{b (\sin (x)+1)}{a-b}\right)^{3/2} \sqrt{\frac{b-b \sin (x)}{a+b}} (a+b \sin (x))}","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 \sqrt{a^2-b^2}}-\frac{\cos (x)}{b (a+b \sin (x))}-\frac{x}{b^2}",1,"(Cos[x]*(1 + Sin[x])*(2*a*(a - b)*ArcTanh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[x]))/(a - b))])/(Sqrt[a + b]*Sqrt[-((b*(-1 + Sin[x]))/(a + b))])]*Sqrt[1 - Sin[x]]*(a + b*Sin[x]) + Sqrt[a + b]*(-2*a*Sqrt[a - b]*ArcTanh[Sqrt[(b*(1 + Sin[x]))/(-a + b)]/Sqrt[(b - b*Sin[x])/(a + b)]]*Sqrt[1 - Sin[x]]*(a + b*Sin[x]) - (-a + b)*Sqrt[(b - b*Sin[x])/(a + b)]*(Sqrt[a - b]*(a + b)*Sqrt[1 - Sin[x]]*Sqrt[-((b*(1 + Sin[x]))/(a - b))] + 2*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[x]))/(a - b))])/(Sqrt[2]*Sqrt[b])]*(a + b*Sin[x])))))/((a - b)^(5/2)*(a + b)^(3/2)*Sqrt[1 - Sin[x]]*(-((b*(1 + Sin[x]))/(a - b)))^(3/2)*Sqrt[(b - b*Sin[x])/(a + b)]*(a + b*Sin[x]))","B",1
270,1,40,51,0.1729142,"\int \frac{1}{(a \sec (x)+b \tan (x))^3} \, dx","Integrate[(a*Sec[x] + b*Tan[x])^(-3),x]","-\frac{\frac{3 a^2+4 a b \sin (x)+b^2}{2 (a+b \sin (x))^2}+\log (a+b \sin (x))}{b^3}","\frac{a^2-b^2}{2 b^3 (a+b \sin (x))^2}-\frac{2 a}{b^3 (a+b \sin (x))}-\frac{\log (a+b \sin (x))}{b^3}",1,"-((Log[a + b*Sin[x]] + (3*a^2 + b^2 + 4*a*b*Sin[x])/(2*(a + b*Sin[x])^2))/b^3)","A",1
271,1,2661,156,6.3591739,"\int \frac{1}{(a \sec (x)+b \tan (x))^4} \, dx","Integrate[(a*Sec[x] + b*Tan[x])^(-4),x]","\text{Result too large to show}","\frac{a \cos ^3(x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^2}-\frac{a \left(2 a^2-3 b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 \left(a^2-b^2\right)^{3/2}}+\frac{\cos (x) \left(2 \left(a^2-b^2\right)+a b \sin (x)\right)}{2 b^3 \left(a^2-b^2\right) (a+b \sin (x))}-\frac{\cos ^3(x)}{3 b (a+b \sin (x))^3}+\frac{x}{b^4}",1,"(Sec[x]*(a + b*Sin[x])^4*(-1/3*(b*(-(b/(a - b)) - (b*Sin[x])/(a - b))^(5/2)*(b/(a + b) - (b*Sin[x])/(a + b))^(5/2))/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[x])^3) - ((a*b^3*(-(b/(a - b)) - (b*Sin[x])/(a - b))^(5/2)*(b/(a + b) - (b*Sin[x])/(a + b))^(5/2))/(2*(a^2 - b^2)*((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[x])^2) - (-((((-3*a^2*b^5)/((a - b)^2*(a + b)^2) + (2*b^5*(3*a^2 - 2*b^2))/((a - b)^2*(a + b)^2))*(-(b/(a - b)) - (b*Sin[x])/(a - b))^(5/2)*(b/(a + b) - (b*Sin[x])/(a + b))^(5/2))/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[x]))) - ((16*Sqrt[2]*b^6*(3*a^2 - 4*b^2)*(-(b/(a - b)) - (b*Sin[x])/(a - b))^(5/2)*Sqrt[b/(a + b) - (b*Sin[x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^(5/2)*((5*(1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^(-1)))/8 - (15*b^3*(((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/b - ((a - b)^2*(-(b/(a - b)) - (b*Sin[x])/(a - b))^2)/(3*b^2) - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b)])))/(32*(a - b)^3*(-(b/(a - b)) - (b*Sin[x])/(a - b))^3*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^2)))/(5*(a - b)^2*(a + b)^4*Sqrt[((a + b)*(b/(a + b) - (b*Sin[x])/(a + b)))/b]) + ((-((a*b^7*(6*a^2 - 7*b^2))/((a - b)^3*(a + b)^3)) + (4*a*b^7*(3*a^2 - 4*b^2))/((a - b)^3*(a + b)^3))*((4*Sqrt[2]*(-(b/(a - b)) - (b*Sin[x])/(a - b))^(3/2)*Sqrt[b/(a + b) - (b*Sin[x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^(5/2)*((3/(4*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^(-1))/2 + (3*b^2*(((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/b - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b)])))/(8*(a - b)^2*(-(b/(a - b)) - (b*Sin[x])/(a - b))^2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^2)))/(3*(a + b)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*(-(((-((a*b)/(a - b)) + b^2/(a - b))*(-(((-((a*b)/(a + b)) - b^2/(a + b))*((2*Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)])/(Sqrt[a + b]*Sqrt[b/(a + b) - (b*Sin[x])/(a + b)])])/(b*Sqrt[a + b]) - (2*Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*ArcTanh[(Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)])/(Sqrt[-((a*b)/(a - b)) + b^2/(a - b)]*Sqrt[b/(a + b) - (b*Sin[x])/(a + b)])])/(b*Sqrt[-((a*b)/(a - b)) + b^2/(a - b)])))/b) + (2*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^(3/2)*((Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^(3/2)) + 1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b)))))/(b*(a + b)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[x])/(a + b)))/b])))/b) + (4*Sqrt[2]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^(5/2)*((3*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(4*Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^(5/2)) + (3/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[x])/(a - b)))/(2*b))^(-1))/4))/((a + b)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[x])/(a + b)))/b])))/b))/b)/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))))/(2*((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))))/(3*((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b)))))/((1 - (a + b*Sin[x])/(a - b))^(3/2)*(1 - (a + b*Sin[x])/(a + b))^(3/2)*(a*Sec[x] + b*Tan[x])^4)","B",0
272,1,86,101,0.338468,"\int \frac{1}{(a \sec (x)+b \tan (x))^5} \, dx","Integrate[(a*Sec[x] + b*Tan[x])^(-5),x]","\frac{\frac{25 a^4+12 b^2 \left(9 a^2+b^2\right) \sin ^2(x)+8 a b \left(11 a^2+b^2\right) \sin (x)+2 a^2 b^2+48 a b^3 \sin ^3(x)-3 b^4}{12 (a+b \sin (x))^4}+\log (a+b \sin (x))}{b^5}","-\frac{\left(a^2-b^2\right)^2}{4 b^5 (a+b \sin (x))^4}+\frac{4 a \left(a^2-b^2\right)}{3 b^5 (a+b \sin (x))^3}-\frac{3 a^2-b^2}{b^5 (a+b \sin (x))^2}+\frac{4 a}{b^5 (a+b \sin (x))}+\frac{\log (a+b \sin (x))}{b^5}",1,"(Log[a + b*Sin[x]] + (25*a^4 + 2*a^2*b^2 - 3*b^4 + 8*a*b*(11*a^2 + b^2)*Sin[x] + 12*b^2*(9*a^2 + b^2)*Sin[x]^2 + 48*a*b^3*Sin[x]^3)/(12*(a + b*Sin[x])^4))/b^5","A",1
273,1,54,30,0.1063559,"\int (\sec (x)+\tan (x))^5 \, dx","Integrate[(Sec[x] + Tan[x])^5,x]","\frac{11 \tan ^4(x)}{4}-\frac{\tan ^2(x)}{2}+\frac{5 \sec ^4(x)}{4}+\tanh ^{-1}(\sin (x))-\log (\cos (x))-\tan (x) \sec ^3(x)+5 \tan ^3(x) \sec (x)+\tan (x) \sec (x)","-\frac{4}{1-\sin (x)}+\frac{2}{(1-\sin (x))^2}-\log (1-\sin (x))",1,"ArcTanh[Sin[x]] - Log[Cos[x]] + (5*Sec[x]^4)/4 + Sec[x]*Tan[x] - Sec[x]^3*Tan[x] - Tan[x]^2/2 + 5*Sec[x]*Tan[x]^3 + (11*Tan[x]^4)/4","A",1
274,1,64,30,0.1326331,"\int (\sec (x)+\tan (x))^4 \, dx","Integrate[(Sec[x] + Tan[x])^4,x]","-\frac{-3 (3 x+8) \cos \left(\frac{x}{2}\right)+(3 x+16) \cos \left(\frac{3 x}{2}\right)+6 \sin \left(\frac{x}{2}\right) (2 x+x \cos (x)+4)}{6 \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^3}","x+\frac{2 \cos ^3(x)}{3 (1-\sin (x))^3}-\frac{2 \cos (x)}{1-\sin (x)}",1,"-1/6*(-3*(8 + 3*x)*Cos[x/2] + (16 + 3*x)*Cos[(3*x)/2] + 6*(4 + 2*x + x*Cos[x])*Sin[x/2])/(Cos[x/2] - Sin[x/2])^3","B",1
275,1,31,18,0.0229296,"\int (\sec (x)+\tan (x))^3 \, dx","Integrate[(Sec[x] + Tan[x])^3,x]","\frac{\tan ^2(x)}{2}+\frac{3 \sec ^2(x)}{2}-\tanh ^{-1}(\sin (x))+\log (\cos (x))+2 \tan (x) \sec (x)","\frac{2}{1-\sin (x)}+\log (1-\sin (x))",1,"-ArcTanh[Sin[x]] + Log[Cos[x]] + (3*Sec[x]^2)/2 + 2*Sec[x]*Tan[x] + Tan[x]^2/2","A",1
276,1,14,16,0.0110678,"\int (\sec (x)+\tan (x))^2 \, dx","Integrate[(Sec[x] + Tan[x])^2,x]","-\tan ^{-1}(\tan (x))+2 \tan (x)+2 \sec (x)","\frac{2 \cos (x)}{1-\sin (x)}-x",1,"-ArcTan[Tan[x]] + 2*Sec[x] + 2*Tan[x]","A",1
277,1,38,13,0.0048654,"\int (\sec (x)+\tan (x)) \, dx","Integrate[Sec[x] + Tan[x],x]","-\log (\cos (x))-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","-2 \log \left(\cos \left(\frac{1}{4} (2 x+\pi )\right)\right)",1,"-Log[Cos[x]] - Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]]","B",1
278,1,16,5,0.0131249,"\int \frac{1}{\sec (x)+\tan (x)} \, dx","Integrate[(Sec[x] + Tan[x])^(-1),x]","2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\log (\sin (x)+1)",1,"2*Log[Cos[x/2] + Sin[x/2]]","B",1
279,1,27,14,0.0242948,"\int \frac{1}{(\sec (x)+\tan (x))^2} \, dx","Integrate[(Sec[x] + Tan[x])^(-2),x]","\frac{4 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}-x","-x-\frac{2 \cos (x)}{\sin (x)+1}",1,"-x + (4*Sin[x/2])/(Cos[x/2] + Sin[x/2])","A",1
280,1,34,16,0.0193775,"\int \frac{1}{(\sec (x)+\tan (x))^3} \, dx","Integrate[(Sec[x] + Tan[x])^(-3),x]","-\frac{2}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}-2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","-\frac{2}{\sin (x)+1}-\log (\sin (x)+1)",1,"-2*Log[Cos[x/2] + Sin[x/2]] - 2/(Cos[x/2] + Sin[x/2])^2","B",1
281,1,62,26,0.0676935,"\int \frac{1}{(\sec (x)+\tan (x))^4} \, dx","Integrate[(Sec[x] + Tan[x])^(-4),x]","\frac{3 (3 x-8) \cos \left(\frac{x}{2}\right)+(16-3 x) \cos \left(\frac{3 x}{2}\right)+6 \sin \left(\frac{x}{2}\right) (2 x+x \cos (x)-4)}{6 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3}","x-\frac{2 \cos ^3(x)}{3 (\sin (x)+1)^3}+\frac{2 \cos (x)}{\sin (x)+1}",1,"(3*(-8 + 3*x)*Cos[x/2] + (16 - 3*x)*Cos[(3*x)/2] + 6*(-4 + 2*x + x*Cos[x])*Sin[x/2])/(6*(Cos[x/2] + Sin[x/2])^3)","B",1
282,1,39,22,0.0462283,"\int \frac{1}{(\sec (x)+\tan (x))^5} \, dx","Integrate[(Sec[x] + Tan[x])^(-5),x]","\frac{4 \sin (x)+2}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4}+2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\frac{4}{\sin (x)+1}-\frac{2}{(\sin (x)+1)^2}+\log (\sin (x)+1)",1,"2*Log[Cos[x/2] + Sin[x/2]] + (2 + 4*Sin[x])/(Cos[x/2] + Sin[x/2])^4","A",1
283,1,143,152,0.6963463,"\int (a \cot (x)+b \csc (x))^5 \, dx","Integrate[(a*Cot[x] + b*Csc[x])^5,x]","\frac{1}{64} \left(8 (a+b)^3 \left(8 a^2-9 a b+3 b^2\right) \log \left(\sin \left(\frac{x}{2}\right)\right)+8 \left(8 a^2+9 a b+3 b^2\right) (a-b)^3 \log \left(\cos \left(\frac{x}{2}\right)\right)-(a+b)^5 \csc ^4\left(\frac{x}{2}\right)+2 (7 a-3 b) (a+b)^4 \csc ^2\left(\frac{x}{2}\right)+(a-b)^5 \left(-\sec ^4\left(\frac{x}{2}\right)\right)+2 (7 a+3 b) (a-b)^4 \sec ^2\left(\frac{x}{2}\right)\right)","\frac{1}{8} a^2 b \left(7 a^2-3 b^2\right) \cos (x)+\frac{1}{16} (a+b)^3 \left(8 a^2-9 a b+3 b^2\right) \log (1-\cos (x))+\frac{1}{16} (a-b)^3 \left(8 a^2+9 a b+3 b^2\right) \log (\cos (x)+1)+\frac{1}{8} \csc ^2(x) (a \cos (x)+b)^2 \left(b \left(5 a^2-3 b^2\right) \cos (x)+2 a \left(2 a^2-b^2\right)\right)-\frac{1}{4} \csc ^4(x) (a \cos (x)+b)^4 (a+b \cos (x))",1,"(2*(7*a - 3*b)*(a + b)^4*Csc[x/2]^2 - (a + b)^5*Csc[x/2]^4 + 8*(a - b)^3*(8*a^2 + 9*a*b + 3*b^2)*Log[Cos[x/2]] + 8*(a + b)^3*(8*a^2 - 9*a*b + 3*b^2)*Log[Sin[x/2]] + 2*(a - b)^4*(7*a + 3*b)*Sec[x/2]^2 - (a - b)^5*Sec[x/2]^4)/64","A",1
284,1,95,101,0.2608222,"\int (a \cot (x)+b \csc (x))^4 \, dx","Integrate[(a*Cot[x] + b*Csc[x])^4,x]","-\frac{1}{12} \csc ^3(x) \left(-9 a^4 x \sin (x)+3 a^4 x \sin (3 x)+4 a^4 \cos (3 x)+24 a^3 b \cos (2 x)-8 a^3 b+6 a^2 b^2 \cos (3 x)+6 b^2 \left(3 a^2+b^2\right) \cos (x)+16 a b^3-2 b^4 \cos (3 x)\right)","a^4 x+\frac{4}{3} a b \left(2 a^2-b^2\right) \sin (x)+\frac{1}{3} a^2 \left(3 a^2-2 b^2\right) \sin (x) \cos (x)+\frac{1}{3} \csc (x) (a \cos (x)+b)^2 \left(\left(3 a^2-2 b^2\right) \cos (x)+a b\right)-\frac{1}{3} \csc ^3(x) (a \cos (x)+b)^3 (a+b \cos (x))",1,"-1/12*(Csc[x]^3*(-8*a^3*b + 16*a*b^3 + 6*b^2*(3*a^2 + b^2)*Cos[x] + 24*a^3*b*Cos[2*x] + 4*a^4*Cos[3*x] + 6*a^2*b^2*Cos[3*x] - 2*b^4*Cos[3*x] - 9*a^4*x*Sin[x] + 3*a^4*x*Sin[3*x]))","A",1
285,1,79,77,0.288478,"\int (a \cot (x)+b \csc (x))^3 \, dx","Integrate[(a*Cot[x] + b*Csc[x])^3,x]","\frac{1}{8} \left(-(a+b)^3 \csc ^2\left(\frac{x}{2}\right)+(a-b)^3 \left(-\sec ^2\left(\frac{x}{2}\right)\right)-4 (2 a-b) (a+b)^2 \log \left(\sin \left(\frac{x}{2}\right)\right)-4 (2 a+b) (a-b)^2 \log \left(\cos \left(\frac{x}{2}\right)\right)\right)","-\frac{1}{2} a^2 b \cos (x)-\frac{1}{4} (2 a-b) (a+b)^2 \log (1-\cos (x))-\frac{1}{4} (a-b)^2 (2 a+b) \log (\cos (x)+1)-\frac{1}{2} \csc ^2(x) (a \cos (x)+b)^2 (a+b \cos (x))",1,"(-((a + b)^3*Csc[x/2]^2) - 4*(a - b)^2*(2*a + b)*Log[Cos[x/2]] - 4*(2*a - b)*(a + b)^2*Log[Sin[x/2]] - (a - b)^3*Sec[x/2]^2)/8","A",1
286,1,24,29,0.1342479,"\int (a \cot (x)+b \csc (x))^2 \, dx","Integrate[(a*Cot[x] + b*Csc[x])^2,x]","-\left(\left(a^2+b^2\right) \cot (x)\right)-a (a x+2 b \csc (x))","a^2 (-x)-a b \sin (x)-\csc (x) (a \cos (x)+b) (a+b \cos (x))",1,"-((a^2 + b^2)*Cot[x]) - a*(a*x + 2*b*Csc[x])","A",1
287,1,25,12,0.0073553,"\int (a \cot (x)+b \csc (x)) \, dx","Integrate[a*Cot[x] + b*Csc[x],x]","a \log (\sin (x))+b \log \left(\sin \left(\frac{x}{2}\right)\right)-b \log \left(\cos \left(\frac{x}{2}\right)\right)","a \log (\sin (x))-b \tanh ^{-1}(\cos (x))",1,"-(b*Log[Cos[x/2]]) + b*Log[Sin[x/2]] + a*Log[Sin[x]]","B",1
288,1,12,12,0.0157225,"\int \frac{1}{a \cot (x)+b \csc (x)} \, dx","Integrate[(a*Cot[x] + b*Csc[x])^(-1),x]","-\frac{\log (a \cos (x)+b)}{a}","-\frac{\log (a \cos (x)+b)}{a}",1,"-(Log[b + a*Cos[x]]/a)","A",1
289,1,71,67,0.2687866,"\int \frac{1}{(a \cot (x)+b \csc (x))^2} \, dx","Integrate[(a*Cot[x] + b*Csc[x])^(-2),x]","-\frac{\frac{2 b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{-a \sin (x)+a x \cos (x)+b x}{a \cos (x)+b}}{a^2}","\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{a^2 \sqrt{a-b} \sqrt{a+b}}-\frac{x}{a^2}+\frac{\sin (x)}{a (a \cos (x)+b)}",1,"-(((2*b*ArcTanh[((-a + b)*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (b*x + a*x*Cos[x] - a*Sin[x])/(b + a*Cos[x]))/a^2)","A",1
290,1,77,50,0.1120768,"\int \frac{1}{(a \cot (x)+b \csc (x))^3} \, dx","Integrate[(a*Cot[x] + b*Csc[x])^(-3),x]","\frac{a^2 \cos (2 x) \log (a \cos (x)+b)+a^2 \log (a \cos (x)+b)+a^2+2 b^2 \log (a \cos (x)+b)+4 a b \cos (x) (\log (a \cos (x)+b)+1)+3 b^2}{2 a^3 (a \cos (x)+b)^2}","\frac{2 b}{a^3 (a \cos (x)+b)}+\frac{\log (a \cos (x)+b)}{a^3}+\frac{a^2-b^2}{2 a^3 (a \cos (x)+b)^2}",1,"(a^2 + 3*b^2 + a^2*Log[b + a*Cos[x]] + 2*b^2*Log[b + a*Cos[x]] + a^2*Cos[2*x]*Log[b + a*Cos[x]] + 4*a*b*Cos[x]*(1 + Log[b + a*Cos[x]]))/(2*a^3*(b + a*Cos[x])^2)","A",1
291,1,150,159,0.4840047,"\int \frac{1}{(a \cot (x)+b \csc (x))^4} \, dx","Integrate[(a*Cot[x] + b*Csc[x])^(-4),x]","\frac{\sin (x) \left(-\frac{a \left(8 a^2-11 b^2\right) (a \cos (x)+b)^2}{(a-b) (a+b)}-\frac{6 b \left(2 b^2-3 a^2\right) \csc (x) (a \cos (x)+b)^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+2 a \left(a^2-b^2\right)+7 a b (a \cos (x)+b)+6 x \csc (x) (a \cos (x)+b)^3\right)}{6 a^4 (a \cos (x)+b)^3}","\frac{x}{a^4}+\frac{b \sin ^3(x)}{2 a \left(a^2-b^2\right) (a \cos (x)+b)^2}-\frac{b \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{a^4 (a-b)^{3/2} (a+b)^{3/2}}-\frac{\sin (x) \left(2 \left(a^2-b^2\right)-a b \cos (x)\right)}{2 a^3 \left(a^2-b^2\right) (a \cos (x)+b)}+\frac{\sin ^3(x)}{3 a (a \cos (x)+b)^3}",1,"((2*a*(a^2 - b^2) + 7*a*b*(b + a*Cos[x]) - (a*(8*a^2 - 11*b^2)*(b + a*Cos[x])^2)/((a - b)*(a + b)) + 6*x*(b + a*Cos[x])^3*Csc[x] - (6*b*(-3*a^2 + 2*b^2)*ArcTanh[((-a + b)*Tan[x/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[x])^3*Csc[x])/(a^2 - b^2)^(3/2))*Sin[x])/(6*a^4*(b + a*Cos[x])^3)","A",1
292,1,138,100,0.3352916,"\int \frac{1}{(a \cot (x)+b \csc (x))^5} \, dx","Integrate[(a*Cot[x] + b*Csc[x])^(-5),x]","-\frac{12 a^4 \cos ^4(x) \log (a \cos (x)+b)-3 a^4+48 a^3 b \cos ^3(x) (\log (a \cos (x)+b)+1)+12 a^2 \cos ^2(x) \left(a^2+6 b^2 \log (a \cos (x)+b)+9 b^2\right)+8 a b \cos (x) \left(a^2+6 b^2 \log (a \cos (x)+b)+11 b^2\right)+2 a^2 b^2+12 b^4 \log (a \cos (x)+b)+25 b^4}{12 a^5 (a \cos (x)+b)^4}","-\frac{4 b}{a^5 (a \cos (x)+b)}-\frac{\log (a \cos (x)+b)}{a^5}+\frac{\left(a^2-b^2\right)^2}{4 a^5 (a \cos (x)+b)^4}+\frac{4 b \left(a^2-b^2\right)}{3 a^5 (a \cos (x)+b)^3}-\frac{a^2-3 b^2}{a^5 (a \cos (x)+b)^2}",1,"-1/12*(-3*a^4 + 2*a^2*b^2 + 25*b^4 + 12*b^4*Log[b + a*Cos[x]] + 12*a^4*Cos[x]^4*Log[b + a*Cos[x]] + 48*a^3*b*Cos[x]^3*(1 + Log[b + a*Cos[x]]) + 12*a^2*Cos[x]^2*(a^2 + 9*b^2 + 6*b^2*Log[b + a*Cos[x]]) + 8*a*b*Cos[x]*(a^2 + 11*b^2 + 6*b^2*Log[b + a*Cos[x]]))/(a^5*(b + a*Cos[x])^4)","A",1
293,1,32,28,0.0724142,"\int (\cot (x)+\csc (x))^5 \, dx","Integrate[(Cot[x] + Csc[x])^5,x]","-\frac{1}{2} \csc ^4\left(\frac{x}{2}\right)+2 \csc ^2\left(\frac{x}{2}\right)+2 \log \left(\sin \left(\frac{x}{2}\right)\right)","\frac{4}{1-\cos (x)}-\frac{2}{(1-\cos (x))^2}+\log (1-\cos (x))",1,"2*Csc[x/2]^2 - Csc[x/2]^4/2 + 2*Log[Sin[x/2]]","A",1
294,1,30,30,0.0443982,"\int (\cot (x)+\csc (x))^4 \, dx","Integrate[(Cot[x] + Csc[x])^4,x]","x+\frac{8}{3} \cot \left(\frac{x}{2}\right)-\frac{2}{3} \cot \left(\frac{x}{2}\right) \csc ^2\left(\frac{x}{2}\right)","x-\frac{2 \sin ^3(x)}{3 (1-\cos (x))^3}+\frac{2 \sin (x)}{1-\cos (x)}",1,"x + (8*Cot[x/2])/3 - (2*Cot[x/2]*Csc[x/2]^2)/3","A",1
295,1,20,20,0.0387496,"\int (\cot (x)+\csc (x))^3 \, dx","Integrate[(Cot[x] + Csc[x])^3,x]","-\csc ^2\left(\frac{x}{2}\right)-2 \log \left(\sin \left(\frac{x}{2}\right)\right)","-\frac{2}{1-\cos (x)}-\log (1-\cos (x))",1,"-Csc[x/2]^2 - 2*Log[Sin[x/2]]","A",1
296,1,12,16,0.0217973,"\int (\cot (x)+\csc (x))^2 \, dx","Integrate[(Cot[x] + Csc[x])^2,x]","-x-2 \cot \left(\frac{x}{2}\right)","-x-\frac{2 \sin (x)}{1-\cos (x)}",1,"-x - 2*Cot[x/2]","A",1
297,1,20,9,0.0044833,"\int (\cot (x)+\csc (x)) \, dx","Integrate[Cot[x] + Csc[x],x]","\log \left(\sin \left(\frac{x}{2}\right)\right)+\log (\sin (x))-\log \left(\cos \left(\frac{x}{2}\right)\right)","\log (\sin (x))-\tanh ^{-1}(\cos (x))",1,"-Log[Cos[x/2]] + Log[Sin[x/2]] + Log[Sin[x]]","B",1
298,1,9,7,0.0112407,"\int \frac{1}{\cot (x)+\csc (x)} \, dx","Integrate[(Cot[x] + Csc[x])^(-1),x]","-2 \log \left(\cos \left(\frac{x}{2}\right)\right)","-\log (\cos (x)+1)",1,"-2*Log[Cos[x/2]]","A",1
299,1,12,14,0.0123534,"\int \frac{1}{(\cot (x)+\csc (x))^2} \, dx","Integrate[(Cot[x] + Csc[x])^(-2),x]","2 \tan \left(\frac{x}{2}\right)-x","\frac{2 \sin (x)}{\cos (x)+1}-x",1,"-x + 2*Tan[x/2]","A",1
300,1,18,14,0.0126415,"\int \frac{1}{(\cot (x)+\csc (x))^3} \, dx","Integrate[(Cot[x] + Csc[x])^(-3),x]","\sec ^2\left(\frac{x}{2}\right)+2 \log \left(\cos \left(\frac{x}{2}\right)\right)","\frac{2}{\cos (x)+1}+\log (\cos (x)+1)",1,"2*Log[Cos[x/2]] + Sec[x/2]^2","A",1
301,1,30,26,0.013066,"\int \frac{1}{(\cot (x)+\csc (x))^4} \, dx","Integrate[(Cot[x] + Csc[x])^(-4),x]","x-\frac{8}{3} \tan \left(\frac{x}{2}\right)+\frac{2}{3} \tan \left(\frac{x}{2}\right) \sec ^2\left(\frac{x}{2}\right)","x+\frac{2 \sin ^3(x)}{3 (\cos (x)+1)^3}-\frac{2 \sin (x)}{\cos (x)+1}",1,"x - (8*Tan[x/2])/3 + (2*Sec[x/2]^2*Tan[x/2])/3","A",1
302,1,32,24,0.0129956,"\int \frac{1}{(\cot (x)+\csc (x))^5} \, dx","Integrate[(Cot[x] + Csc[x])^(-5),x]","\frac{1}{2} \sec ^4\left(\frac{x}{2}\right)-2 \sec ^2\left(\frac{x}{2}\right)-2 \log \left(\cos \left(\frac{x}{2}\right)\right)","-\frac{4}{\cos (x)+1}+\frac{2}{(\cos (x)+1)^2}-\log (\cos (x)+1)",1,"-2*Log[Cos[x/2]] - 2*Sec[x/2]^2 + Sec[x/2]^4/2","A",1
303,1,38,44,0.0300416,"\int (\csc (x)-\sin (x))^4 \, dx","Integrate[(Csc[x] - Sin[x])^4,x]","\frac{35 x}{8}+\frac{3}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)+\frac{10 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x)","\frac{35 x}{8}-\frac{35 \cot ^3(x)}{24}+\frac{35 \cot (x)}{8}+\frac{1}{4} \cos ^4(x) \cot ^3(x)+\frac{7}{8} \cos ^2(x) \cot ^3(x)",1,"(35*x)/8 + (10*Cot[x])/3 - (Cot[x]*Csc[x]^2)/3 + (3*Sin[2*x])/4 + Sin[4*x]/32","A",1
304,1,61,34,0.0199819,"\int (\csc (x)-\sin (x))^3 \, dx","Integrate[(Csc[x] - Sin[x])^3,x]","-\frac{9 \cos (x)}{4}-\frac{1}{12} \cos (3 x)-\frac{1}{8} \csc ^2\left(\frac{x}{2}\right)+\frac{1}{8} \sec ^2\left(\frac{x}{2}\right)-\frac{5}{2} \log \left(\sin \left(\frac{x}{2}\right)\right)+\frac{5}{2} \log \left(\cos \left(\frac{x}{2}\right)\right)","-\frac{5 \cos ^3(x)}{6}-\frac{5 \cos (x)}{2}-\frac{1}{2} \cos ^3(x) \cot ^2(x)+\frac{5}{2} \tanh ^{-1}(\cos (x))",1,"(-9*Cos[x])/4 - Cos[3*x]/12 - Csc[x/2]^2/8 + (5*Log[Cos[x/2]])/2 - (5*Log[Sin[x/2]])/2 + Sec[x/2]^2/8","A",1
305,1,18,22,0.0154781,"\int (\csc (x)-\sin (x))^2 \, dx","Integrate[(Csc[x] - Sin[x])^2,x]","-\frac{3 x}{2}-\frac{1}{4} \sin (2 x)-\cot (x)","-\frac{3 x}{2}-\frac{3 \cot (x)}{2}+\frac{1}{2} \cos ^2(x) \cot (x)",1,"(-3*x)/2 - Cot[x] - Sin[2*x]/4","A",1
306,1,19,8,0.0035096,"\int (\csc (x)-\sin (x)) \, dx","Integrate[Csc[x] - Sin[x],x]","\cos (x)+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)","\cos (x)-\tanh ^{-1}(\cos (x))",1,"Cos[x] - Log[Cos[x/2]] + Log[Sin[x/2]]","B",1
307,1,2,2,0.0038064,"\int \frac{1}{\csc (x)-\sin (x)} \, dx","Integrate[(Csc[x] - Sin[x])^(-1),x]","\sec (x)","\sec (x)",1,"Sec[x]","A",1
308,1,8,8,0.0028632,"\int \frac{1}{(\csc (x)-\sin (x))^2} \, dx","Integrate[(Csc[x] - Sin[x])^(-2),x]","\frac{\tan ^3(x)}{3}","\frac{\tan ^3(x)}{3}",1,"Tan[x]^3/3","A",1
309,1,17,17,0.019488,"\int \frac{1}{(\csc (x)-\sin (x))^3} \, dx","Integrate[(Csc[x] - Sin[x])^(-3),x]","\frac{\sec ^5(x)}{5}-\frac{\sec ^3(x)}{3}","\frac{\sec ^5(x)}{5}-\frac{\sec ^3(x)}{3}",1,"-1/3*Sec[x]^3 + Sec[x]^5/5","A",1
310,1,37,17,0.0165378,"\int \frac{1}{(\csc (x)-\sin (x))^4} \, dx","Integrate[(Csc[x] - Sin[x])^(-4),x]","\frac{2 \tan (x)}{35}+\frac{1}{7} \tan (x) \sec ^6(x)-\frac{8}{35} \tan (x) \sec ^4(x)+\frac{1}{35} \tan (x) \sec ^2(x)","\frac{\tan ^7(x)}{7}+\frac{\tan ^5(x)}{5}",1,"(2*Tan[x])/35 + (Sec[x]^2*Tan[x])/35 - (8*Sec[x]^4*Tan[x])/35 + (Sec[x]^6*Tan[x])/7","B",1
311,1,25,25,0.0154412,"\int \frac{1}{(\csc (x)-\sin (x))^5} \, dx","Integrate[(Csc[x] - Sin[x])^(-5),x]","\frac{\sec ^9(x)}{9}-\frac{2 \sec ^7(x)}{7}+\frac{\sec ^5(x)}{5}","\frac{\sec ^9(x)}{9}-\frac{2 \sec ^7(x)}{7}+\frac{\sec ^5(x)}{5}",1,"Sec[x]^5/5 - (2*Sec[x]^7)/7 + Sec[x]^9/9","A",1
312,1,57,25,0.017182,"\int \frac{1}{(\csc (x)-\sin (x))^6} \, dx","Integrate[(Csc[x] - Sin[x])^(-6),x]","-\frac{8 \tan (x)}{693}+\frac{1}{11} \tan (x) \sec ^{10}(x)-\frac{23}{99} \tan (x) \sec ^8(x)+\frac{113}{693} \tan (x) \sec ^6(x)-\frac{1}{231} \tan (x) \sec ^4(x)-\frac{4}{693} \tan (x) \sec ^2(x)","\frac{\tan ^{11}(x)}{11}+\frac{2 \tan ^9(x)}{9}+\frac{\tan ^7(x)}{7}",1,"(-8*Tan[x])/693 - (4*Sec[x]^2*Tan[x])/693 - (Sec[x]^4*Tan[x])/231 + (113*Sec[x]^6*Tan[x])/693 - (23*Sec[x]^8*Tan[x])/99 + (Sec[x]^10*Tan[x])/11","B",1
313,1,33,33,0.015594,"\int \frac{1}{(\csc (x)-\sin (x))^7} \, dx","Integrate[(Csc[x] - Sin[x])^(-7),x]","\frac{\sec ^{13}(x)}{13}-\frac{3 \sec ^{11}(x)}{11}+\frac{\sec ^9(x)}{3}-\frac{\sec ^7(x)}{7}","\frac{\sec ^{13}(x)}{13}-\frac{3 \sec ^{11}(x)}{11}+\frac{\sec ^9(x)}{3}-\frac{\sec ^7(x)}{7}",1,"-1/7*Sec[x]^7 + Sec[x]^9/3 - (3*Sec[x]^11)/11 + Sec[x]^13/13","A",1
314,1,37,73,0.0799781,"\int (\csc (x)-\sin (x))^{7/2} \, dx","Integrate[(Csc[x] - Sin[x])^(7/2),x]","-\frac{1}{70} \sec (x) \sqrt{\cos (x) \cot (x)} \left(115 \cos ^2(x)+5 \cos (3 x) \cos (x)+28 \cot ^2(x)-512\right)","\frac{2}{7} \cos ^3(x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}+\frac{8}{7} \cos (x) \cot ^2(x) \sqrt{\cos (x) \cot (x)}-\frac{64}{35} \cot (x) \csc (x) \sqrt{\cos (x) \cot (x)}+\frac{256}{35} \sec (x) \sqrt{\cos (x) \cot (x)}",1,"-1/70*(Sqrt[Cos[x]*Cot[x]]*(-512 + 115*Cos[x]^2 + 5*Cos[x]*Cos[3*x] + 28*Cot[x]^2)*Sec[x])","A",1
315,1,29,50,0.076815,"\int (\csc (x)-\sin (x))^{5/2} \, dx","Integrate[(Csc[x] - Sin[x])^(5/2),x]","-\frac{2}{15} \tan (x) \sqrt{\cos (x) \cot (x)} \left(3 \cos ^2(x)+5 \cot ^2(x)+32\right)","\frac{2}{5} \cos ^2(x) \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{16}{15} \cot (x) \sqrt{\cos (x) \cot (x)}-\frac{64}{15} \tan (x) \sqrt{\cos (x) \cot (x)}",1,"(-2*Sqrt[Cos[x]*Cot[x]]*(32 + 3*Cos[x]^2 + 5*Cot[x]^2)*Tan[x])/15","A",1
316,1,21,31,0.0363692,"\int (\csc (x)-\sin (x))^{3/2} \, dx","Integrate[(Csc[x] - Sin[x])^(3/2),x]","\frac{2}{3} \left(\cos ^2(x)-4\right) \sec (x) \sqrt{\cos (x) \cot (x)}","\frac{2}{3} \cos (x) \sqrt{\cos (x) \cot (x)}-\frac{8}{3} \sec (x) \sqrt{\cos (x) \cot (x)}",1,"(2*(-4 + Cos[x]^2)*Sqrt[Cos[x]*Cot[x]]*Sec[x])/3","A",1
317,1,13,13,0.024548,"\int \sqrt{\csc (x)-\sin (x)} \, dx","Integrate[Sqrt[Csc[x] - Sin[x]],x]","2 \tan (x) \sqrt{\cos (x) \cot (x)}","2 \tan (x) \sqrt{\cos (x) \cot (x)}",1,"2*Sqrt[Cos[x]*Cot[x]]*Tan[x]","A",1
318,1,44,60,0.2673045,"\int \frac{1}{\sqrt{\csc (x)-\sin (x)}} \, dx","Integrate[1/Sqrt[Csc[x] - Sin[x]],x]","-\frac{\sin (x) \tan (x) \sqrt{\cos (x) \cot (x)} \left(\tan ^{-1}\left(\sqrt[4]{\sin ^2(x)}\right)-\tanh ^{-1}\left(\sqrt[4]{\sin ^2(x)}\right)\right)}{\sin ^2(x)^{3/4}}","\frac{\cos (x) \tan ^{-1}\left(\sqrt{-\sin (x)}\right)}{\sqrt{-\sin (x)} \sqrt{\cos (x) \cot (x)}}-\frac{\cos (x) \tanh ^{-1}\left(\sqrt{-\sin (x)}\right)}{\sqrt{-\sin (x)} \sqrt{\cos (x) \cot (x)}}",1,"-(((ArcTan[(Sin[x]^2)^(1/4)] - ArcTanh[(Sin[x]^2)^(1/4)])*Sqrt[Cos[x]*Cot[x]]*Sin[x]*Tan[x])/(Sin[x]^2)^(3/4))","A",1
319,1,60,80,0.1509204,"\int \frac{1}{(\csc (x)-\sin (x))^{3/2}} \, dx","Integrate[(Csc[x] - Sin[x])^(-3/2),x]","\frac{2 \sqrt[4]{\sin ^2(x)} \sec (x)+\cos (x) \left(-\tan ^{-1}\left(\sqrt[4]{\sin ^2(x)}\right)\right)-\cos (x) \tanh ^{-1}\left(\sqrt[4]{\sin ^2(x)}\right)}{4 \sqrt[4]{\sin ^2(x)} \sqrt{\cos (x) \cot (x)}}","\frac{\sec (x)}{2 \sqrt{\cos (x) \cot (x)}}+\frac{\sqrt{-\sin (x)} \cot (x) \tan ^{-1}\left(\sqrt{-\sin (x)}\right)}{4 \sqrt{\cos (x) \cot (x)}}+\frac{\sqrt{-\sin (x)} \cot (x) \tanh ^{-1}\left(\sqrt{-\sin (x)}\right)}{4 \sqrt{\cos (x) \cot (x)}}",1,"(-(ArcTan[(Sin[x]^2)^(1/4)]*Cos[x]) - ArcTanh[(Sin[x]^2)^(1/4)]*Cos[x] + 2*Sec[x]*(Sin[x]^2)^(1/4))/(4*Sqrt[Cos[x]*Cot[x]]*(Sin[x]^2)^(1/4))","A",1
320,1,69,99,0.5149994,"\int \frac{1}{(\csc (x)-\sin (x))^{5/2}} \, dx","Integrate[(Csc[x] - Sin[x])^(-5/2),x]","-\frac{\sin (x) \tan (x) \sqrt{\cos (x) \cot (x)} \left(-3 \tan ^{-1}\left(\sqrt[4]{\sin ^2(x)}\right)+3 \tanh ^{-1}\left(\sqrt[4]{\sin ^2(x)}\right)+\sin ^2(x)^{3/4} (3 \cos (2 x)-5) \sec ^4(x)\right)}{32 \sin ^2(x)^{3/4}}","-\frac{3 \tan (x)}{16 \sqrt{\cos (x) \cot (x)}}-\frac{3 \cos (x) \tan ^{-1}\left(\sqrt{-\sin (x)}\right)}{32 \sqrt{-\sin (x)} \sqrt{\cos (x) \cot (x)}}+\frac{3 \cos (x) \tanh ^{-1}\left(\sqrt{-\sin (x)}\right)}{32 \sqrt{-\sin (x)} \sqrt{\cos (x) \cot (x)}}+\frac{\tan (x) \sec ^2(x)}{4 \sqrt{\cos (x) \cot (x)}}",1,"-1/32*(Sqrt[Cos[x]*Cot[x]]*Sin[x]*(-3*ArcTan[(Sin[x]^2)^(1/4)] + 3*ArcTanh[(Sin[x]^2)^(1/4)] + (-5 + 3*Cos[2*x])*Sec[x]^4*(Sin[x]^2)^(3/4))*Tan[x])/(Sin[x]^2)^(3/4)","A",1
321,1,74,118,0.2630719,"\int \frac{1}{(\csc (x)-\sin (x))^{7/2}} \, dx","Integrate[(Csc[x] - Sin[x])^(-7/2),x]","\frac{2 \sqrt[4]{\sin ^2(x)} \sec (x) \left(32 \sec ^4(x)-52 \sec ^2(x)+5\right)+15 \cos (x) \tan ^{-1}\left(\sqrt[4]{\sin ^2(x)}\right)+15 \cos (x) \tanh ^{-1}\left(\sqrt[4]{\sin ^2(x)}\right)}{384 \sqrt[4]{\sin ^2(x)} \sqrt{\cos (x) \cot (x)}}","-\frac{5 \sec ^3(x)}{48 \sqrt{\cos (x) \cot (x)}}+\frac{5 \sec (x)}{192 \sqrt{\cos (x) \cot (x)}}-\frac{5 \sqrt{-\sin (x)} \cot (x) \tan ^{-1}\left(\sqrt{-\sin (x)}\right)}{128 \sqrt{\cos (x) \cot (x)}}-\frac{5 \sqrt{-\sin (x)} \cot (x) \tanh ^{-1}\left(\sqrt{-\sin (x)}\right)}{128 \sqrt{\cos (x) \cot (x)}}+\frac{\tan ^2(x) \sec ^3(x)}{6 \sqrt{\cos (x) \cot (x)}}",1,"(15*ArcTan[(Sin[x]^2)^(1/4)]*Cos[x] + 15*ArcTanh[(Sin[x]^2)^(1/4)]*Cos[x] + 2*Sec[x]*(5 - 52*Sec[x]^2 + 32*Sec[x]^4)*(Sin[x]^2)^(1/4))/(384*Sqrt[Cos[x]*Cot[x]]*(Sin[x]^2)^(1/4))","A",1
322,1,38,44,0.0318206,"\int (-\cos (x)+\sec (x))^4 \, dx","Integrate[(-Cos[x] + Sec[x])^4,x]","\frac{35 x}{8}-\frac{3}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)-\frac{10 \tan (x)}{3}+\frac{1}{3} \tan (x) \sec ^2(x)","\frac{35 x}{8}+\frac{35 \tan ^3(x)}{24}-\frac{35 \tan (x)}{8}-\frac{1}{4} \sin ^4(x) \tan ^3(x)-\frac{7}{8} \sin ^2(x) \tan ^3(x)",1,"(35*x)/8 - (3*Sin[2*x])/4 + Sin[4*x]/32 - (10*Tan[x])/3 + (Sec[x]^2*Tan[x])/3","A",1
323,1,38,34,0.0103162,"\int (-\cos (x)+\sec (x))^3 \, dx","Integrate[(-Cos[x] + Sec[x])^3,x]","-\frac{1}{3} \sin ^3(x) \tan ^2(x)-\frac{5}{3} \sin (x) \tan ^2(x)-\frac{5}{2} \tanh ^{-1}(\sin (x))+\frac{5}{2} \tan (x) \sec (x)","\frac{5 \sin ^3(x)}{6}+\frac{5 \sin (x)}{2}+\frac{1}{2} \sin ^3(x) \tan ^2(x)-\frac{5}{2} \tanh ^{-1}(\sin (x))",1,"(-5*ArcTanh[Sin[x]])/2 + (5*Sec[x]*Tan[x])/2 - (5*Sin[x]*Tan[x]^2)/3 - (Sin[x]^3*Tan[x]^2)/3","A",1
324,1,16,22,0.0168171,"\int (-\cos (x)+\sec (x))^2 \, dx","Integrate[(-Cos[x] + Sec[x])^2,x]","-\frac{3 x}{2}+\frac{1}{4} \sin (2 x)+\tan (x)","-\frac{3 x}{2}+\frac{3 \tan (x)}{2}-\frac{1}{2} \sin ^2(x) \tan (x)",1,"(-3*x)/2 + Sin[2*x]/4 + Tan[x]","A",1
325,1,37,8,0.0039992,"\int (-\cos (x)+\sec (x)) \, dx","Integrate[-Cos[x] + Sec[x],x]","-\sin (x)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\tanh ^{-1}(\sin (x))-\sin (x)",1,"-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]] - Sin[x]","B",1
326,1,4,4,0.0027645,"\int \frac{1}{-\cos (x)+\sec (x)} \, dx","Integrate[(-Cos[x] + Sec[x])^(-1),x]","-\csc (x)","-\csc (x)",1,"-Csc[x]","A",1
327,1,8,8,0.002555,"\int \frac{1}{(-\cos (x)+\sec (x))^2} \, dx","Integrate[(-Cos[x] + Sec[x])^(-2),x]","-\frac{1}{3} \cot ^3(x)","-\frac{1}{3} \cot ^3(x)",1,"-1/3*Cot[x]^3","A",1
328,1,17,17,0.0090579,"\int \frac{1}{(-\cos (x)+\sec (x))^3} \, dx","Integrate[(-Cos[x] + Sec[x])^(-3),x]","\frac{\csc ^3(x)}{3}-\frac{\csc ^5(x)}{5}","\frac{\csc ^3(x)}{3}-\frac{\csc ^5(x)}{5}",1,"Csc[x]^3/3 - Csc[x]^5/5","A",1
329,1,37,17,0.0217687,"\int \frac{1}{(-\cos (x)+\sec (x))^4} \, dx","Integrate[(-Cos[x] + Sec[x])^(-4),x]","-\frac{2 \cot (x)}{35}-\frac{1}{7} \cot (x) \csc ^6(x)+\frac{8}{35} \cot (x) \csc ^4(x)-\frac{1}{35} \cot (x) \csc ^2(x)","-\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5}",1,"(-2*Cot[x])/35 - (Cot[x]*Csc[x]^2)/35 + (8*Cot[x]*Csc[x]^4)/35 - (Cot[x]*Csc[x]^6)/7","B",1
330,1,25,25,0.0107935,"\int \frac{1}{(-\cos (x)+\sec (x))^5} \, dx","Integrate[(-Cos[x] + Sec[x])^(-5),x]","-\frac{1}{9} \csc ^9(x)+\frac{2 \csc ^7(x)}{7}-\frac{\csc ^5(x)}{5}","-\frac{1}{9} \csc ^9(x)+\frac{2 \csc ^7(x)}{7}-\frac{\csc ^5(x)}{5}",1,"-1/5*Csc[x]^5 + (2*Csc[x]^7)/7 - Csc[x]^9/9","A",1
331,1,57,25,0.0194741,"\int \frac{1}{(-\cos (x)+\sec (x))^6} \, dx","Integrate[(-Cos[x] + Sec[x])^(-6),x]","\frac{8 \cot (x)}{693}-\frac{1}{11} \cot (x) \csc ^{10}(x)+\frac{23}{99} \cot (x) \csc ^8(x)-\frac{113}{693} \cot (x) \csc ^6(x)+\frac{1}{231} \cot (x) \csc ^4(x)+\frac{4}{693} \cot (x) \csc ^2(x)","-\frac{1}{11} \cot ^{11}(x)-\frac{2 \cot ^9(x)}{9}-\frac{\cot ^7(x)}{7}",1,"(8*Cot[x])/693 + (4*Cot[x]*Csc[x]^2)/693 + (Cot[x]*Csc[x]^4)/231 - (113*Cot[x]*Csc[x]^6)/693 + (23*Cot[x]*Csc[x]^8)/99 - (Cot[x]*Csc[x]^10)/11","B",1
332,1,33,33,0.0131134,"\int \frac{1}{(-\cos (x)+\sec (x))^7} \, dx","Integrate[(-Cos[x] + Sec[x])^(-7),x]","-\frac{1}{13} \csc ^{13}(x)+\frac{3 \csc ^{11}(x)}{11}-\frac{\csc ^9(x)}{3}+\frac{\csc ^7(x)}{7}","-\frac{1}{13} \csc ^{13}(x)+\frac{3 \csc ^{11}(x)}{11}-\frac{\csc ^9(x)}{3}+\frac{\csc ^7(x)}{7}",1,"Csc[x]^7/7 - Csc[x]^9/3 + (3*Csc[x]^11)/11 - Csc[x]^13/13","A",1
333,1,37,73,0.2125618,"\int (-\cos (x)+\sec (x))^{7/2} \, dx","Integrate[(-Cos[x] + Sec[x])^(7/2),x]","\frac{1}{70} \sec (x) \sqrt{\sin (x) \tan (x)} (28 \tan (x)-512 \cot (x)-5 (\sin (3 x)-23 \sin (x)) \cos (x))","-\frac{2}{7} \sin ^3(x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{8}{7} \sin (x) \tan ^2(x) \sqrt{\sin (x) \tan (x)}-\frac{256}{35} \csc (x) \sqrt{\sin (x) \tan (x)}+\frac{64}{35} \tan (x) \sec (x) \sqrt{\sin (x) \tan (x)}",1,"(Sec[x]*Sqrt[Sin[x]*Tan[x]]*(-512*Cot[x] - 5*Cos[x]*(-23*Sin[x] + Sin[3*x]) + 28*Tan[x]))/70","A",1
334,1,29,50,0.0796988,"\int (-\cos (x)+\sec (x))^{5/2} \, dx","Integrate[(-Cos[x] + Sec[x])^(5/2),x]","\frac{2}{15} \tan (x) \sqrt{\sin (x) \tan (x)} \left(3 \cos ^2(x)+32 \cot ^2(x)+5\right)","-\frac{2}{5} \sin ^2(x) \tan (x) \sqrt{\sin (x) \tan (x)}+\frac{16}{15} \tan (x) \sqrt{\sin (x) \tan (x)}+\frac{64}{15} \cot (x) \sqrt{\sin (x) \tan (x)}",1,"(2*(5 + 3*Cos[x]^2 + 32*Cot[x]^2)*Tan[x]*Sqrt[Sin[x]*Tan[x]])/15","A",1
335,1,23,31,0.037941,"\int (-\cos (x)+\sec (x))^{3/2} \, dx","Integrate[(-Cos[x] + Sec[x])^(3/2),x]","\frac{2}{3} \sin (x) \left(4 \csc ^2(x)-1\right) \sqrt{\sin (x) \tan (x)}","\frac{8}{3} \csc (x) \sqrt{\sin (x) \tan (x)}-\frac{2}{3} \sin (x) \sqrt{\sin (x) \tan (x)}",1,"(2*(-1 + 4*Csc[x]^2)*Sin[x]*Sqrt[Sin[x]*Tan[x]])/3","A",1
336,1,13,13,0.0264857,"\int \sqrt{-\cos (x)+\sec (x)} \, dx","Integrate[Sqrt[-Cos[x] + Sec[x]],x]","-2 \cot (x) \sqrt{\sin (x) \tan (x)}","-2 \cot (x) \sqrt{\sin (x) \tan (x)}",1,"-2*Cot[x]*Sqrt[Sin[x]*Tan[x]]","A",1
337,1,43,52,0.249676,"\int \frac{1}{\sqrt{-\cos (x)+\sec (x)}} \, dx","Integrate[1/Sqrt[-Cos[x] + Sec[x]],x]","\frac{\cos (x) \cot (x) \sqrt{\sin (x) \tan (x)} \left(\tan ^{-1}\left(\sqrt[4]{\cos ^2(x)}\right)-\tanh ^{-1}\left(\sqrt[4]{\cos ^2(x)}\right)\right)}{\cos ^2(x)^{3/4}}","\frac{\sin (x) \tan ^{-1}\left(\sqrt{\cos (x)}\right)}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\sin (x) \tanh ^{-1}\left(\sqrt{\cos (x)}\right)}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}",1,"((ArcTan[(Cos[x]^2)^(1/4)] - ArcTanh[(Cos[x]^2)^(1/4)])*Cos[x]*Cot[x]*Sqrt[Sin[x]*Tan[x]])/(Cos[x]^2)^(3/4)","A",1
338,1,56,72,0.1753986,"\int \frac{1}{(-\cos (x)+\sec (x))^{3/2}} \, dx","Integrate[(-Cos[x] + Sec[x])^(-3/2),x]","\frac{\cot (x) \sqrt{\sin (x) \tan (x)} \left(\tan ^{-1}\left(\sqrt[4]{\cos ^2(x)}\right)-2 \sqrt[4]{\cos ^2(x)} \csc ^2(x)+\tanh ^{-1}\left(\sqrt[4]{\cos ^2(x)}\right)\right)}{4 \sqrt[4]{\cos ^2(x)}}","\frac{\sin (x) \tan ^{-1}\left(\sqrt{\cos (x)}\right)}{4 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\csc (x)}{2 \sqrt{\sin (x) \tan (x)}}+\frac{\sin (x) \tanh ^{-1}\left(\sqrt{\cos (x)}\right)}{4 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}",1,"(Cot[x]*(ArcTan[(Cos[x]^2)^(1/4)] + ArcTanh[(Cos[x]^2)^(1/4)] - 2*(Cos[x]^2)^(1/4)*Csc[x]^2)*Sqrt[Sin[x]*Tan[x]])/(4*(Cos[x]^2)^(1/4))","A",1
339,1,73,91,0.6485491,"\int \frac{1}{(-\cos (x)+\sec (x))^{5/2}} \, dx","Integrate[(-Cos[x] + Sec[x])^(-5/2),x]","-\frac{\cot (x) \sqrt{\sin (x) \tan (x)} \left(3 \cos (x) \tan ^{-1}\left(\sqrt[4]{\cos ^2(x)}\right)-3 \cos (x) \tanh ^{-1}\left(\sqrt[4]{\cos ^2(x)}\right)+\cos ^2(x)^{3/4} (3 \cos (2 x)+5) \cot (x) \csc ^3(x)\right)}{32 \cos ^2(x)^{3/4}}","-\frac{3 \sin (x) \tan ^{-1}\left(\sqrt{\cos (x)}\right)}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}+\frac{3 \cot (x)}{16 \sqrt{\sin (x) \tan (x)}}+\frac{3 \sin (x) \tanh ^{-1}\left(\sqrt{\cos (x)}\right)}{32 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\cot (x) \csc ^2(x)}{4 \sqrt{\sin (x) \tan (x)}}",1,"-1/32*(Cot[x]*(3*ArcTan[(Cos[x]^2)^(1/4)]*Cos[x] - 3*ArcTanh[(Cos[x]^2)^(1/4)]*Cos[x] + (Cos[x]^2)^(3/4)*(5 + 3*Cos[2*x])*Cot[x]*Csc[x]^3)*Sqrt[Sin[x]*Tan[x]])/(Cos[x]^2)^(3/4)","A",1
340,1,74,110,0.3580641,"\int \frac{1}{(-\cos (x)+\sec (x))^{7/2}} \, dx","Integrate[(-Cos[x] + Sec[x])^(-7/2),x]","-\frac{\cot (x) \sqrt{\sin (x) \tan (x)} \left(15 \tan ^{-1}\left(\sqrt[4]{\cos ^2(x)}\right)+2 \sqrt[4]{\cos ^2(x)} \left(32 \csc ^4(x)-52 \csc ^2(x)+5\right) \csc ^2(x)+15 \tanh ^{-1}\left(\sqrt[4]{\cos ^2(x)}\right)\right)}{384 \sqrt[4]{\cos ^2(x)}}","-\frac{5 \sin (x) \tan ^{-1}\left(\sqrt{\cos (x)}\right)}{128 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}+\frac{5 \csc ^3(x)}{48 \sqrt{\sin (x) \tan (x)}}-\frac{5 \csc (x)}{192 \sqrt{\sin (x) \tan (x)}}-\frac{5 \sin (x) \tanh ^{-1}\left(\sqrt{\cos (x)}\right)}{128 \sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\cot ^2(x) \csc ^3(x)}{6 \sqrt{\sin (x) \tan (x)}}",1,"-1/384*(Cot[x]*(15*ArcTan[(Cos[x]^2)^(1/4)] + 15*ArcTanh[(Cos[x]^2)^(1/4)] + 2*(Cos[x]^2)^(1/4)*Csc[x]^2*(5 - 52*Csc[x]^2 + 32*Csc[x]^4))*Sqrt[Sin[x]*Tan[x]])/(Cos[x]^2)^(1/4)","A",1
341,1,129,55,0.203034,"\int (\sin (x)+\tan (x))^4 \, dx","Integrate[(Sin[x] + Tan[x])^4,x]","\frac{1}{768} \sec ^3(x) \left(1395 \sin (x)+672 \sin (2 x)+1265 \sin (3 x)+129 \sin (5 x)+32 \sin (6 x)+3 \sin (7 x)-72 \cos (x) \left(61 x-16 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+16 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)-24 \cos (3 x) \left(61 x-16 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+16 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)\right)","-\frac{61 x}{8}-\frac{4 \sin ^3(x)}{3}+\frac{\tan ^3(x)}{3}+5 \tan (x)-2 \tanh ^{-1}(\sin (x))+\frac{1}{4} \sin (x) \cos ^3(x)+\frac{19}{8} \sin (x) \cos (x)+2 \tan (x) \sec (x)",1,"(Sec[x]^3*(-72*Cos[x]*(61*x - 16*Log[Cos[x/2] - Sin[x/2]] + 16*Log[Cos[x/2] + Sin[x/2]]) - 24*Cos[3*x]*(61*x - 16*Log[Cos[x/2] - Sin[x/2]] + 16*Log[Cos[x/2] + Sin[x/2]]) + 1395*Sin[x] + 672*Sin[2*x] + 1265*Sin[3*x] + 129*Sin[5*x] + 32*Sin[6*x] + 3*Sin[7*x]))/768","B",1
342,1,40,38,0.0389932,"\int (\sin (x)+\tan (x))^3 \, dx","Integrate[(Sin[x] + Tan[x])^3,x]","\frac{9 \cos (x)}{4}+\frac{3}{4} \cos (2 x)+\frac{1}{12} \cos (3 x)+\frac{\sec ^2(x)}{2}+3 \sec (x)-2 \log (\cos (x))","\frac{\cos ^3(x)}{3}+\frac{3 \cos ^2(x)}{2}+2 \cos (x)+\frac{\sec ^2(x)}{2}+3 \sec (x)-2 \log (\cos (x))",1,"(9*Cos[x])/4 + (3*Cos[2*x])/4 + Cos[3*x]/12 - 2*Log[Cos[x]] + 3*Sec[x] + Sec[x]^2/2","A",1
343,1,60,25,0.0945174,"\int (\sin (x)+\tan (x))^2 \, dx","Integrate[(Sin[x] + Tan[x])^2,x]","-\frac{x}{2}-2 \sin (x)+\frac{7 \tan (x)}{8}-\frac{1}{8} \sin (3 x) \sec (x)-2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","-\frac{x}{2}-2 \sin (x)+\tan (x)+2 \tanh ^{-1}(\sin (x))-\frac{1}{2} \sin (x) \cos (x)",1,"-1/2*x - 2*Log[Cos[x/2] - Sin[x/2]] + 2*Log[Cos[x/2] + Sin[x/2]] - 2*Sin[x] - (Sec[x]*Sin[3*x])/8 + (7*Tan[x])/8","B",1
344,1,10,10,0.0028869,"\int (\sin (x)+\tan (x)) \, dx","Integrate[Sin[x] + Tan[x],x]","-\cos (x)-\log (\cos (x))","-\cos (x)-\log (\cos (x))",1,"-Cos[x] - Log[Cos[x]]","A",1
345,1,35,24,0.0136566,"\int \frac{1}{\sin (x)+\tan (x)} \, dx","Integrate[(Sin[x] + Tan[x])^(-1),x]","-\frac{1}{4} \sec ^2\left(\frac{x}{2}\right)+\frac{1}{2} \log \left(\sin \left(\frac{x}{2}\right)\right)-\frac{1}{2} \log \left(\cos \left(\frac{x}{2}\right)\right)","-\frac{1}{2} \csc ^2(x)-\frac{1}{2} \tanh ^{-1}(\cos (x))+\frac{1}{2} \cot (x) \csc (x)",1,"-1/2*Log[Cos[x/2]] + Log[Sin[x/2]]/2 - Sec[x/2]^2/4","A",1
346,1,57,33,0.0151087,"\int \frac{1}{(\sin (x)+\tan (x))^2} \, dx","Integrate[(Sin[x] + Tan[x])^(-2),x]","-\frac{7}{120} \tan \left(\frac{x}{2}\right)-\frac{1}{8} \cot \left(\frac{x}{2}\right)+\frac{1}{40} \tan \left(\frac{x}{2}\right) \sec ^4\left(\frac{x}{2}\right)-\frac{11}{120} \tan \left(\frac{x}{2}\right) \sec ^2\left(\frac{x}{2}\right)","-\frac{2}{5} \cot ^5(x)-\frac{\cot ^3(x)}{3}+\frac{2 \csc ^5(x)}{5}-\frac{2 \csc ^3(x)}{3}",1,"-1/8*Cot[x/2] - (7*Tan[x/2])/120 - (11*Sec[x/2]^2*Tan[x/2])/120 + (Sec[x/2]^4*Tan[x/2])/40","A",1
347,1,83,60,0.0168327,"\int \frac{1}{(\sin (x)+\tan (x))^3} \, dx","Integrate[(Sin[x] + Tan[x])^(-3),x]","-\frac{1}{64} \csc ^2\left(\frac{x}{2}\right)-\frac{1}{256} \sec ^8\left(\frac{x}{2}\right)+\frac{1}{48} \sec ^6\left(\frac{x}{2}\right)-\frac{3}{128} \sec ^4\left(\frac{x}{2}\right)-\frac{1}{32} \sec ^2\left(\frac{x}{2}\right)-\frac{1}{32} \log \left(\sin \left(\frac{x}{2}\right)\right)+\frac{1}{32} \log \left(\cos \left(\frac{x}{2}\right)\right)","-\frac{1}{32 (1-\cos (x))}-\frac{1}{16 (\cos (x)+1)}-\frac{3}{32 (\cos (x)+1)^2}+\frac{1}{6 (\cos (x)+1)^3}-\frac{1}{16 (\cos (x)+1)^4}+\frac{1}{32} \tanh ^{-1}(\cos (x))",1,"-1/64*Csc[x/2]^2 + Log[Cos[x/2]]/32 - Log[Sin[x/2]]/32 - Sec[x/2]^2/32 - (3*Sec[x/2]^4)/128 + Sec[x/2]^6/48 - Sec[x/2]^8/256","A",1
348,1,129,65,0.0191277,"\int \frac{1}{(\sin (x)+\tan (x))^4} \, dx","Integrate[(Sin[x] + Tan[x])^(-4),x]","-\frac{2749 \tan \left(\frac{x}{2}\right)}{110880}+\frac{1}{96} \cot \left(\frac{x}{2}\right)-\frac{1}{384} \cot \left(\frac{x}{2}\right) \csc ^2\left(\frac{x}{2}\right)+\frac{\tan \left(\frac{x}{2}\right) \sec ^{10}\left(\frac{x}{2}\right)}{1408}-\frac{7 \tan \left(\frac{x}{2}\right) \sec ^8\left(\frac{x}{2}\right)}{1584}+\frac{641 \tan \left(\frac{x}{2}\right) \sec ^6\left(\frac{x}{2}\right)}{88704}+\frac{179 \tan \left(\frac{x}{2}\right) \sec ^4\left(\frac{x}{2}\right)}{73920}-\frac{2033 \tan \left(\frac{x}{2}\right) \sec ^2\left(\frac{x}{2}\right)}{443520}","-\frac{8}{11} \cot ^{11}(x)-\frac{16 \cot ^9(x)}{9}-\frac{9 \cot ^7(x)}{7}-\frac{\cot ^5(x)}{5}+\frac{8 \csc ^{11}(x)}{11}-\frac{20 \csc ^9(x)}{9}+\frac{16 \csc ^7(x)}{7}-\frac{4 \csc ^5(x)}{5}",1,"Cot[x/2]/96 - (Cot[x/2]*Csc[x/2]^2)/384 - (2749*Tan[x/2])/110880 - (2033*Sec[x/2]^2*Tan[x/2])/443520 + (179*Sec[x/2]^4*Tan[x/2])/73920 + (641*Sec[x/2]^6*Tan[x/2])/88704 - (7*Sec[x/2]^8*Tan[x/2])/1584 + (Sec[x/2]^10*Tan[x/2])/1408","A",1
349,1,68,74,0.1968139,"\int \frac{A+C \sin (x)}{b \cos (x)+c \sin (x)} \, dx","Integrate[(A + C*Sin[x])/(b*Cos[x] + c*Sin[x]),x]","\frac{2 A \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)}{\sqrt{b^2+c^2}}+\frac{C (c x-b \log (b \cos (x)+c \sin (x)))}{b^2+c^2}","-\frac{A \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{\sqrt{b^2+c^2}}+\frac{c C x}{b^2+c^2}-\frac{b C \log (b \cos (x)+c \sin (x))}{b^2+c^2}",1,"(2*A*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]])/Sqrt[b^2 + c^2] + (C*(c*x - b*Log[b*Cos[x] + c*Sin[x]]))/(b^2 + c^2)","A",1
350,1,82,75,0.3142382,"\int \frac{A+C \sin (x)}{(b \cos (x)+c \sin (x))^2} \, dx","Integrate[(A + C*Sin[x])/(b*Cos[x] + c*Sin[x])^2,x]","\frac{A \left(b^2+c^2\right) \sin (x)+b^2 C}{b \left(b^2+c^2\right) (b \cos (x)+c \sin (x))}+\frac{2 c C \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)}{\left(b^2+c^2\right)^{3/2}}","\frac{A b \sin (x)-A c \cos (x)+b C}{\left(b^2+c^2\right) (b \cos (x)+c \sin (x))}-\frac{c C \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{\left(b^2+c^2\right)^{3/2}}",1,"(2*c*C*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2) + (b^2*C + A*(b^2 + c^2)*Sin[x])/(b*(b^2 + c^2)*(b*Cos[x] + c*Sin[x]))","A",1
351,1,132,116,0.3874832,"\int \frac{A+C \sin (x)}{(b \cos (x)+c \sin (x))^3} \, dx","Integrate[(A + C*Sin[x])/(b*Cos[x] + c*Sin[x])^3,x]","\frac{\left(b^2+c^2\right) \left(A b^2 \sin (x)-A b c \cos (x)+b C (b+c \sin (2 x))+2 c^2 C \sin ^2(x)\right)+2 A b \sqrt{b^2+c^2} (b \cos (x)+c \sin (x))^2 \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)}{2 b (b-i c)^2 (b+i c)^2 (b \cos (x)+c \sin (x))^2}","\frac{A b \sin (x)-A c \cos (x)+b C}{2 \left(b^2+c^2\right) (b \cos (x)+c \sin (x))^2}-\frac{A \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{2 \left(b^2+c^2\right)^{3/2}}-\frac{c^2 C \cos (x)-b c C \sin (x)}{\left(b^2+c^2\right)^2 (b \cos (x)+c \sin (x))}",1,"(2*A*b*Sqrt[b^2 + c^2]*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]]*(b*Cos[x] + c*Sin[x])^2 + (b^2 + c^2)*(-(A*b*c*Cos[x]) + A*b^2*Sin[x] + 2*c^2*C*Sin[x]^2 + b*C*(b + c*Sin[2*x])))/(2*b*(b - I*c)^2*(b + I*c)^2*(b*Cos[x] + c*Sin[x])^2)","C",1
352,1,67,73,0.1444737,"\int \frac{A+B \cos (x)}{b \cos (x)+c \sin (x)} \, dx","Integrate[(A + B*Cos[x])/(b*Cos[x] + c*Sin[x]),x]","\frac{2 A \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)}{\sqrt{b^2+c^2}}+\frac{B (c \log (b \cos (x)+c \sin (x))+b x)}{b^2+c^2}","-\frac{A \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{\sqrt{b^2+c^2}}+\frac{b B x}{b^2+c^2}+\frac{B c \log (b \cos (x)+c \sin (x))}{b^2+c^2}",1,"(2*A*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]])/Sqrt[b^2 + c^2] + (B*(b*x + c*Log[b*Cos[x] + c*Sin[x]]))/(b^2 + c^2)","A",1
353,1,82,76,0.2338568,"\int \frac{A+B \cos (x)}{(b \cos (x)+c \sin (x))^2} \, dx","Integrate[(A + B*Cos[x])/(b*Cos[x] + c*Sin[x])^2,x]","\frac{A \left(b^2+c^2\right) \sin (x)-b B c}{b \left(b^2+c^2\right) (b \cos (x)+c \sin (x))}+\frac{2 b B \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)}{\left(b^2+c^2\right)^{3/2}}","-\frac{-A b \sin (x)+A c \cos (x)+B c}{\left(b^2+c^2\right) (b \cos (x)+c \sin (x))}-\frac{b B \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{\left(b^2+c^2\right)^{3/2}}",1,"(2*b*B*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2) + (-(b*B*c) + A*(b^2 + c^2)*Sin[x])/(b*(b^2 + c^2)*(b*Cos[x] + c*Sin[x]))","A",1
354,1,118,116,0.3370698,"\int \frac{A+B \cos (x)}{(b \cos (x)+c \sin (x))^3} \, dx","Integrate[(A + B*Cos[x])/(b*Cos[x] + c*Sin[x])^3,x]","\frac{\left(b^2+c^2\right) (b \sin (x) (A+2 B \cos (x))-A c \cos (x)-B c \cos (2 x))+2 A \sqrt{b^2+c^2} (b \cos (x)+c \sin (x))^2 \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)}{2 (b-i c)^2 (b+i c)^2 (b \cos (x)+c \sin (x))^2}","-\frac{-A b \sin (x)+A c \cos (x)+B c}{2 \left(b^2+c^2\right) (b \cos (x)+c \sin (x))^2}-\frac{A \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{2 \left(b^2+c^2\right)^{3/2}}-\frac{b B c \cos (x)-b^2 B \sin (x)}{\left(b^2+c^2\right)^2 (b \cos (x)+c \sin (x))}",1,"(2*A*Sqrt[b^2 + c^2]*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]]*(b*Cos[x] + c*Sin[x])^2 + (b^2 + c^2)*(-(A*c*Cos[x]) - B*c*Cos[2*x] + b*(A + 2*B*Cos[x])*Sin[x]))/(2*(b - I*c)^2*(b + I*c)^2*(b*Cos[x] + c*Sin[x])^2)","C",1
355,1,238,246,1.4194168,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^4 \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4,x]","\frac{168 \left(b^4-c^4\right) \sin (2 (d+e x))+420 \left(b^2+c^2\right)^2 (d+e x)+672 b (b-i c) (b+i c) \sqrt{b^2+c^2} \sin (d+e x)+32 b \left(b^2-3 c^2\right) \sqrt{b^2+c^2} \sin (3 (d+e x))-336 b c \left(b^2+c^2\right) \cos (2 (d+e x))-672 c (b-i c) (b+i c) \sqrt{b^2+c^2} \cos (d+e x)+32 c \left(c^2-3 b^2\right) \sqrt{b^2+c^2} \cos (3 (d+e x))-12 b c \left(b^2-c^2\right) \cos (4 (d+e x))+3 \left(b^4-6 b^2 c^2+c^4\right) \sin (4 (d+e x))}{96 e}","\frac{35 b \left(b^2+c^2\right)^{3/2} \sin (d+e x)}{8 e}-\frac{35 c \left(b^2+c^2\right)^{3/2} \cos (d+e x)}{8 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^3}{4 e}-\frac{7 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^2}{12 e}-\frac{35 \left(b^2+c^2\right) (c \cos (d+e x)-b \sin (d+e x)) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)}{24 e}+\frac{35}{8} x \left(b^2+c^2\right)^2",1,"(420*(b^2 + c^2)^2*(d + e*x) - 672*(b - I*c)*(b + I*c)*c*Sqrt[b^2 + c^2]*Cos[d + e*x] - 336*b*c*(b^2 + c^2)*Cos[2*(d + e*x)] + 32*c*(-3*b^2 + c^2)*Sqrt[b^2 + c^2]*Cos[3*(d + e*x)] - 12*b*c*(b^2 - c^2)*Cos[4*(d + e*x)] + 672*b*(b - I*c)*(b + I*c)*Sqrt[b^2 + c^2]*Sin[d + e*x] + 168*(b^4 - c^4)*Sin[2*(d + e*x)] + 32*b*(b^2 - 3*c^2)*Sqrt[b^2 + c^2]*Sin[3*(d + e*x)] + 3*(b^4 - 6*b^2*c^2 + c^4)*Sin[4*(d + e*x)])/(96*e)","C",1
356,1,163,178,0.5978675,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^3 \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3,x]","\frac{30 (b-i c) (b+i c) \sqrt{b^2+c^2} (d+e x)+45 b \left(b^2+c^2\right) \sin (d+e x)+9 \left(b^2-c^2\right) \sqrt{b^2+c^2} \sin (2 (d+e x))+b \left(b^2-3 c^2\right) \sin (3 (d+e x))-45 c \left(b^2+c^2\right) \cos (d+e x)-18 b c \sqrt{b^2+c^2} \cos (2 (d+e x))+c \left(c^2-3 b^2\right) \cos (3 (d+e x))}{12 e}","\frac{5 b \left(b^2+c^2\right) \sin (d+e x)}{2 e}-\frac{5 c \left(b^2+c^2\right) \cos (d+e x)}{2 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^2}{3 e}-\frac{5 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)}{6 e}+\frac{5}{2} x \left(b^2+c^2\right)^{3/2}",1,"(30*(b - I*c)*(b + I*c)*Sqrt[b^2 + c^2]*(d + e*x) - 45*c*(b^2 + c^2)*Cos[d + e*x] - 18*b*c*Sqrt[b^2 + c^2]*Cos[2*(d + e*x)] + c*(-3*b^2 + c^2)*Cos[3*(d + e*x)] + 45*b*(b^2 + c^2)*Sin[d + e*x] + 9*(b^2 - c^2)*Sqrt[b^2 + c^2]*Sin[2*(d + e*x)] + b*(b^2 - 3*c^2)*Sin[3*(d + e*x)])/(12*e)","C",1
357,1,111,116,0.2123359,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^2 \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2,x]","\frac{8 b \sqrt{b^2+c^2} \sin (d+e x)-8 c \sqrt{b^2+c^2} \cos (d+e x)+b^2 \sin (2 (d+e x))+6 b^2 d+6 b^2 e x-2 b c \cos (2 (d+e x))-c^2 \sin (2 (d+e x))+6 c^2 d+6 c^2 e x}{4 e}","\frac{3 b \sqrt{b^2+c^2} \sin (d+e x)}{2 e}-\frac{3 c \sqrt{b^2+c^2} \cos (d+e x)}{2 e}-\frac{(c \cos (d+e x)-b \sin (d+e x)) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)}{2 e}+\frac{3}{2} x \left(b^2+c^2\right)",1,"(6*b^2*d + 6*c^2*d + 6*b^2*e*x + 6*c^2*e*x - 8*c*Sqrt[b^2 + c^2]*Cos[d + e*x] - 2*b*c*Cos[2*(d + e*x)] + 8*b*Sqrt[b^2 + c^2]*Sin[d + e*x] + b^2*Sin[2*(d + e*x)] - c^2*Sin[2*(d + e*x)])/(4*e)","A",1
358,1,36,37,0.0329775,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right) \, dx","Integrate[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x],x]","\frac{e x \sqrt{b^2+c^2}+b \sin (d+e x)-c \cos (d+e x)}{e}","x \sqrt{b^2+c^2}+\frac{b \sin (d+e x)}{e}-\frac{c \cos (d+e x)}{e}",1,"(Sqrt[b^2 + c^2]*e*x - c*Cos[d + e*x] + b*Sin[d + e*x])/e","A",1
359,1,49,49,0.0986926,"\int \frac{1}{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-1),x]","\frac{\sqrt{b^2+c^2} \sin (d+e x)-c}{c e (c \cos (d+e x)-b \sin (d+e x))}","-\frac{c-\sqrt{b^2+c^2} \sin (d+e x)}{c e (c \cos (d+e x)-b \sin (d+e x))}",1,"(-c + Sqrt[b^2 + c^2]*Sin[d + e*x])/(c*e*(c*Cos[d + e*x] - b*Sin[d + e*x]))","A",1
360,1,98,129,0.2472129,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^2} \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-2),x]","\frac{-2 c \sqrt{b^2+c^2}+b^2 \sin ^3(d+e x)+2 b c \cos ^3(d+e x)+2 c^2 \sin (d+e x)+c^2 \sin (d+e x) \cos ^2(d+e x)}{3 c e (c \cos (d+e x)-b \sin (d+e x))^3}","\frac{b \sin (d+e x)-c \cos (d+e x)}{3 e \sqrt{b^2+c^2} \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^2}-\frac{c-\sqrt{b^2+c^2} \sin (d+e x)}{3 c e \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}",1,"(-2*c*Sqrt[b^2 + c^2] + 2*b*c*Cos[d + e*x]^3 + 2*c^2*Sin[d + e*x] + c^2*Cos[d + e*x]^2*Sin[d + e*x] + b^2*Sin[d + e*x]^3)/(3*c*e*(c*Cos[d + e*x] - b*Sin[d + e*x])^3)","A",1
361,1,420,191,2.5912562,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^3} \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3),x]","\frac{20 c \left(c^4-b^4\right) \cos (2 (d+e x))-76 b^4 c-40 b^3 c^2 \sin (2 (d+e x))-152 b^2 c^3+110 b^2 c^2 \sqrt{b^2+c^2} \sin (d+e x)-6 b^2 c^2 \sqrt{b^2+c^2} \sin (5 (d+e x))+90 b c \left(b^2+c^2\right)^{3/2} \cos (d+e x)+100 c^4 \sqrt{b^2+c^2} \sin (d+e x)+5 c^4 \sqrt{b^2+c^2} \sin (3 (d+e x))+c^4 \sqrt{b^2+c^2} \sin (5 (d+e x))+10 b c^3 \sqrt{b^2+c^2} \cos (3 (d+e x))+4 b c^3 \sqrt{b^2+c^2} \cos (5 (d+e x))+10 b^4 \sqrt{b^2+c^2} \sin (d+e x)-5 b^4 \sqrt{b^2+c^2} \sin (3 (d+e x))+b^4 \sqrt{b^2+c^2} \sin (5 (d+e x))+10 b^3 c \sqrt{b^2+c^2} \cos (3 (d+e x))-4 b^3 c \sqrt{b^2+c^2} \cos (5 (d+e x))-40 b c^4 \sin (2 (d+e x))-76 c^5}{120 c e \left(b^2+c^2\right) (c \cos (d+e x)-b \sin (d+e x))^5}","-\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{15 e \left(b^2+c^2\right) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^2}+\frac{b \sin (d+e x)-c \cos (d+e x)}{5 e \sqrt{b^2+c^2} \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^3}-\frac{2 \left(c-\sqrt{b^2+c^2} \sin (d+e x)\right)}{15 c e \left(b^2+c^2\right) (c \cos (d+e x)-b \sin (d+e x))}",1,"(-76*b^4*c - 152*b^2*c^3 - 76*c^5 + 90*b*c*(b^2 + c^2)^(3/2)*Cos[d + e*x] + 20*c*(-b^4 + c^4)*Cos[2*(d + e*x)] + 10*b^3*c*Sqrt[b^2 + c^2]*Cos[3*(d + e*x)] + 10*b*c^3*Sqrt[b^2 + c^2]*Cos[3*(d + e*x)] - 4*b^3*c*Sqrt[b^2 + c^2]*Cos[5*(d + e*x)] + 4*b*c^3*Sqrt[b^2 + c^2]*Cos[5*(d + e*x)] + 10*b^4*Sqrt[b^2 + c^2]*Sin[d + e*x] + 110*b^2*c^2*Sqrt[b^2 + c^2]*Sin[d + e*x] + 100*c^4*Sqrt[b^2 + c^2]*Sin[d + e*x] - 40*b^3*c^2*Sin[2*(d + e*x)] - 40*b*c^4*Sin[2*(d + e*x)] - 5*b^4*Sqrt[b^2 + c^2]*Sin[3*(d + e*x)] + 5*c^4*Sqrt[b^2 + c^2]*Sin[3*(d + e*x)] + b^4*Sqrt[b^2 + c^2]*Sin[5*(d + e*x)] - 6*b^2*c^2*Sqrt[b^2 + c^2]*Sin[5*(d + e*x)] + c^4*Sqrt[b^2 + c^2]*Sin[5*(d + e*x)])/(120*c*(b^2 + c^2)*e*(c*Cos[d + e*x] - b*Sin[d + e*x])^5)","B",1
362,1,533,259,2.0303007,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^4} \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]","\frac{-35 b^6 \sin (d+e x)+21 b^6 \sin (3 (d+e x))-7 b^6 \sin (5 (d+e x))+b^6 \sin (7 (d+e x))-112 b^5 c \cos (3 (d+e x))+28 b^5 c \cos (5 (d+e x))-6 b^5 c \cos (7 (d+e x))-1295 b^4 c^2 \sin (d+e x)-189 b^4 c^2 \sin (3 (d+e x))+35 b^4 c^2 \sin (5 (d+e x))-15 b^4 c^2 \sin (7 (d+e x))+56 b^3 c^3 \cos (3 (d+e x))+20 b^3 c^3 \cos (7 (d+e x))-2485 b^2 c^4 \sin (d+e x)-161 b^2 c^4 \sin (3 (d+e x))+35 b^2 c^4 \sin (5 (d+e x))+15 b^2 c^4 \sin (7 (d+e x))-1190 b c \left(b^2+c^2\right)^2 \cos (d+e x)+832 c^5 \sqrt{b^2+c^2}+896 b c^4 \sqrt{b^2+c^2} \sin (2 (d+e x))+1664 b^2 c^3 \sqrt{b^2+c^2}+832 b^4 c \sqrt{b^2+c^2}+448 c \sqrt{b^2+c^2} \left(b^4-c^4\right) \cos (2 (d+e x))+896 b^3 c^2 \sqrt{b^2+c^2} \sin (2 (d+e x))+168 b c^5 \cos (3 (d+e x))-28 b c^5 \cos (5 (d+e x))-6 b c^5 \cos (7 (d+e x))-1225 c^6 \sin (d+e x)+49 c^6 \sin (3 (d+e x))-7 c^6 \sin (5 (d+e x))-c^6 \sin (7 (d+e x))}{1120 c e \left(b^2+c^2\right) (b \sin (d+e x)-c \cos (d+e x))^7}","-\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left(b^2+c^2\right)^{3/2} \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^2}-\frac{3 (c \cos (d+e x)-b \sin (d+e x))}{35 e \left(b^2+c^2\right) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^3}+\frac{b \sin (d+e x)-c \cos (d+e x)}{7 e \sqrt{b^2+c^2} \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^4}-\frac{2 \left(c-\sqrt{b^2+c^2} \sin (d+e x)\right)}{35 c e \left(b^2+c^2\right)^{3/2} (c \cos (d+e x)-b \sin (d+e x))}",1,"(832*b^4*c*Sqrt[b^2 + c^2] + 1664*b^2*c^3*Sqrt[b^2 + c^2] + 832*c^5*Sqrt[b^2 + c^2] - 1190*b*c*(b^2 + c^2)^2*Cos[d + e*x] + 448*c*Sqrt[b^2 + c^2]*(b^4 - c^4)*Cos[2*(d + e*x)] - 112*b^5*c*Cos[3*(d + e*x)] + 56*b^3*c^3*Cos[3*(d + e*x)] + 168*b*c^5*Cos[3*(d + e*x)] + 28*b^5*c*Cos[5*(d + e*x)] - 28*b*c^5*Cos[5*(d + e*x)] - 6*b^5*c*Cos[7*(d + e*x)] + 20*b^3*c^3*Cos[7*(d + e*x)] - 6*b*c^5*Cos[7*(d + e*x)] - 35*b^6*Sin[d + e*x] - 1295*b^4*c^2*Sin[d + e*x] - 2485*b^2*c^4*Sin[d + e*x] - 1225*c^6*Sin[d + e*x] + 896*b^3*c^2*Sqrt[b^2 + c^2]*Sin[2*(d + e*x)] + 896*b*c^4*Sqrt[b^2 + c^2]*Sin[2*(d + e*x)] + 21*b^6*Sin[3*(d + e*x)] - 189*b^4*c^2*Sin[3*(d + e*x)] - 161*b^2*c^4*Sin[3*(d + e*x)] + 49*c^6*Sin[3*(d + e*x)] - 7*b^6*Sin[5*(d + e*x)] + 35*b^4*c^2*Sin[5*(d + e*x)] + 35*b^2*c^4*Sin[5*(d + e*x)] - 7*c^6*Sin[5*(d + e*x)] + b^6*Sin[7*(d + e*x)] - 15*b^4*c^2*Sin[7*(d + e*x)] + 15*b^2*c^4*Sin[7*(d + e*x)] - c^6*Sin[7*(d + e*x)])/(1120*c*(b^2 + c^2)*e*(-(c*Cos[d + e*x]) + b*Sin[d + e*x])^7)","B",1
363,1,135,157,0.4128477,"\int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^3,x]","\frac{2 \left(6 a \left(5 a^2+3 c^2\right) (d+e x)+9 a \left(5 a^2+c^2\right) \sin (d+e x)+9 a \left(a^2-c^2\right) \sin (2 (d+e x))+a \left(a^2-3 c^2\right) \sin (3 (d+e x))-9 c \left(5 a^2+c^2\right) \cos (d+e x)+c \left(c^2-3 a^2\right) \cos (3 (d+e x))-18 a^2 c \cos (2 (d+e x))\right)}{3 e}","\frac{4 a \left(15 a^2+4 c^2\right) \sin (d+e x)}{3 e}-\frac{4 c \left(15 a^2+4 c^2\right) \cos (d+e x)}{3 e}+4 a x \left(5 a^2+3 c^2\right)-\frac{20 \left(a c \cos (d+e x)-a^2 \sin (d+e x)\right) (a \cos (d+e x)+a+c \sin (d+e x))}{3 e}-\frac{8 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))^2}{3 e}",1,"(2*(6*a*(5*a^2 + 3*c^2)*(d + e*x) - 9*c*(5*a^2 + c^2)*Cos[d + e*x] - 18*a^2*c*Cos[2*(d + e*x)] + c*(-3*a^2 + c^2)*Cos[3*(d + e*x)] + 9*a*(5*a^2 + c^2)*Sin[d + e*x] + 9*a*(a^2 - c^2)*Sin[2*(d + e*x)] + a*(a^2 - 3*c^2)*Sin[3*(d + e*x)]))/(3*e)","A",1
364,1,92,81,0.1440847,"\int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2,x]","4 \left(\frac{\left(3 a^2+c^2\right) (d+e x)}{2 e}+\frac{\left(a^2-c^2\right) \sin (2 (d+e x))}{4 e}+\frac{2 a^2 \sin (d+e x)}{e}-\frac{2 a c \cos (d+e x)}{e}-\frac{a c \cos (2 (d+e x))}{2 e}\right)","2 x \left(3 a^2+c^2\right)+\frac{6 a^2 \sin (d+e x)}{e}-\frac{6 a c \cos (d+e x)}{e}-\frac{2 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+a+c \sin (d+e x))}{e}",1,"4*(((3*a^2 + c^2)*(d + e*x))/(2*e) - (2*a*c*Cos[d + e*x])/e - (a*c*Cos[2*(d + e*x)])/(2*e) + (2*a^2*Sin[d + e*x])/e + ((a^2 - c^2)*Sin[2*(d + e*x)])/(4*e))","A",1
365,1,53,29,0.0151958,"\int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x)) \, dx","Integrate[2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x],x]","\frac{2 a \sin (d) \cos (e x)}{e}+\frac{2 a \cos (d) \sin (e x)}{e}+2 a x+\frac{2 c \sin (d) \sin (e x)}{e}-\frac{2 c \cos (d) \cos (e x)}{e}","\frac{2 a \sin (d+e x)}{e}+2 a x-\frac{2 c \cos (d+e x)}{e}",1,"2*a*x - (2*c*Cos[d]*Cos[e*x])/e + (2*a*Cos[e*x]*Sin[d])/e + (2*a*Cos[d]*Sin[e*x])/e + (2*c*Sin[d]*Sin[e*x])/e","A",1
366,1,57,25,0.049758,"\int \frac{1}{2 a+2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-1),x]","\frac{1}{2} \left(\frac{\log \left(a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)\right)}{c e}-\frac{\log \left(\cos \left(\frac{1}{2} (d+e x)\right)\right)}{c e}\right)","\frac{\log \left(a+c \tan \left(\frac{1}{2} (d+e x)\right)\right)}{2 c e}",1,"(-(Log[Cos[(d + e*x)/2]]/(c*e)) + Log[a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2]]/(c*e))/2","B",1
367,1,115,75,0.5526147,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-2),x]","\frac{\frac{c \left(a^2+c^2\right) \sin \left(\frac{1}{2} (d+e x)\right)}{a \left(a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)\right)}+2 a \left(\log \left(\cos \left(\frac{1}{2} (d+e x)\right)\right)-\log \left(a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)\right)\right)+c \tan \left(\frac{1}{2} (d+e x)\right)}{8 c^3 e}","-\frac{a \log \left(a+c \tan \left(\frac{1}{2} (d+e x)\right)\right)}{4 c^3 e}-\frac{c \cos (d+e x)-a \sin (d+e x)}{4 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))}",1,"(2*a*(Log[Cos[(d + e*x)/2]] - Log[a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2]]) + (c*(a^2 + c^2)*Sin[(d + e*x)/2])/(a*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])) + c*Tan[(d + e*x)/2])/(8*c^3*e)","A",1
368,1,186,134,2.9962427,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-3),x]","-\frac{4 \left(3 a^2+c^2\right) \log \left(\cos \left(\frac{1}{2} (d+e x)\right)\right)+\frac{c^2 \left(a^2+c^2\right)}{\left(a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)\right)^2}+\frac{6 c \left(a^2+c^2\right) \sin \left(\frac{1}{2} (d+e x)\right)}{a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)}-4 \left(3 a^2+c^2\right) \log \left(a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)\right)+6 a c \tan \left(\frac{1}{2} (d+e x)\right)+c^2 \left(-\sec ^2\left(\frac{1}{2} (d+e x)\right)\right)}{64 c^5 e}","\frac{3 \left(a c \cos (d+e x)-a^2 \sin (d+e x)\right)}{16 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))}+\frac{\left(3 a^2+c^2\right) \log \left(a+c \tan \left(\frac{1}{2} (d+e x)\right)\right)}{16 c^5 e}-\frac{c \cos (d+e x)-a \sin (d+e x)}{16 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^2}",1,"-1/64*(4*(3*a^2 + c^2)*Log[Cos[(d + e*x)/2]] - 4*(3*a^2 + c^2)*Log[a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2]] - c^2*Sec[(d + e*x)/2]^2 + (c^2*(a^2 + c^2))/(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])^2 + (6*c*(a^2 + c^2)*Sin[(d + e*x)/2])/(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2]) + 6*a*c*Tan[(d + e*x)/2])/(c^5*e)","A",1
369,1,492,207,1.7266709,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]","\frac{\cos \left(\frac{1}{2} (d+e x)\right) \left(a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)\right) \left(192 \left(5 a^3+3 a c^2\right) \cos ^3\left(\frac{1}{2} (d+e x)\right) \log \left(\cos \left(\frac{1}{2} (d+e x)\right)\right) \left(a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)\right)^3-192 \left(5 a^3+3 a c^2\right) \cos ^3\left(\frac{1}{2} (d+e x)\right) \left(a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)\right)^3 \log \left(a \cos \left(\frac{1}{2} (d+e x)\right)+c \sin \left(\frac{1}{2} (d+e x)\right)\right)+\frac{c \left(150 a^6 \sin (d+e x)+120 a^6 \sin (2 (d+e x))+30 a^6 \sin (3 (d+e x))-75 a^5 c \cos (3 (d+e x))+150 a^5 c+255 a^4 c^2 \sin (d+e x)+72 a^4 c^2 \sin (2 (d+e x))-37 a^4 c^2 \sin (3 (d+e x))-35 a^3 c^3 \cos (3 (d+e x))+130 a^3 c^3+129 a^2 c^4 \sin (d+e x)+36 a^2 c^4 \sin (2 (d+e x))-27 a^2 c^4 \sin (3 (d+e x))-6 \left(25 a^5 c+15 a^3 c^3+4 a c^5\right) \cos (2 (d+e x))+3 a c \left(25 a^4+25 a^2 c^2-4 c^4\right) \cos (d+e x)-4 a c^5 \cos (3 (d+e x))+24 a c^5+12 c^6 \sin (d+e x)-4 c^6 \sin (3 (d+e x))\right)}{a}\right)}{384 c^7 e (a \cos (d+e x)+a+c \sin (d+e x))^4}","\frac{5 \left(a c \cos (d+e x)-a^2 \sin (d+e x)\right)}{96 c^4 e (a \cos (d+e x)+a+c \sin (d+e x))^2}-\frac{a \left(5 a^2+3 c^2\right) \log \left(a+c \tan \left(\frac{1}{2} (d+e x)\right)\right)}{32 c^7 e}-\frac{c \left(15 a^2+4 c^2\right) \cos (d+e x)-a \left(15 a^2+4 c^2\right) \sin (d+e x)}{96 c^6 e (a \cos (d+e x)+a+c \sin (d+e x))}-\frac{c \cos (d+e x)-a \sin (d+e x)}{48 c^2 e (a \cos (d+e x)+a+c \sin (d+e x))^3}",1,"(Cos[(d + e*x)/2]*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])*(192*(5*a^3 + 3*a*c^2)*Cos[(d + e*x)/2]^3*Log[Cos[(d + e*x)/2]]*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])^3 - 192*(5*a^3 + 3*a*c^2)*Cos[(d + e*x)/2]^3*Log[a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2]]*(a*Cos[(d + e*x)/2] + c*Sin[(d + e*x)/2])^3 + (c*(150*a^5*c + 130*a^3*c^3 + 24*a*c^5 + 3*a*c*(25*a^4 + 25*a^2*c^2 - 4*c^4)*Cos[d + e*x] - 6*(25*a^5*c + 15*a^3*c^3 + 4*a*c^5)*Cos[2*(d + e*x)] - 75*a^5*c*Cos[3*(d + e*x)] - 35*a^3*c^3*Cos[3*(d + e*x)] - 4*a*c^5*Cos[3*(d + e*x)] + 150*a^6*Sin[d + e*x] + 255*a^4*c^2*Sin[d + e*x] + 129*a^2*c^4*Sin[d + e*x] + 12*c^6*Sin[d + e*x] + 120*a^6*Sin[2*(d + e*x)] + 72*a^4*c^2*Sin[2*(d + e*x)] + 36*a^2*c^4*Sin[2*(d + e*x)] + 30*a^6*Sin[3*(d + e*x)] - 37*a^4*c^2*Sin[3*(d + e*x)] - 27*a^2*c^4*Sin[3*(d + e*x)] - 4*c^6*Sin[3*(d + e*x)]))/a))/(384*c^7*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x])^4)","B",1
370,1,50,23,0.0299831,"\int \frac{1}{2 a+2 a \cos (d+e x)+2 a \sin (d+e x)} \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-1),x]","\frac{\frac{\log \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)}{e}-\frac{\log \left(\cos \left(\frac{1}{2} (d+e x)\right)\right)}{e}}{2 a}","\frac{\log \left(\tan \left(\frac{1}{2} (d+e x)\right)+1\right)}{2 a e}",1,"(-(Log[Cos[(d + e*x)/2]]/e) + Log[Cos[(d + e*x)/2] + Sin[(d + e*x)/2]]/e)/(2*a)","B",1
371,1,93,75,0.1862596,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^2} \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-2),x]","\frac{\tan \left(\frac{1}{2} (d+e x)\right)+2 \log \left(\cos \left(\frac{1}{2} (d+e x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (d+e x)\right)}{\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)}-2 \log \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)}{8 a^2 e}","-\frac{a \cos (d+e x)-a \sin (d+e x)}{4 e \left(a^3 \sin (d+e x)+a^3 \cos (d+e x)+a^3\right)}-\frac{\log \left(\tan \left(\frac{1}{2} (d+e x)\right)+1\right)}{4 a^2 e}",1,"(2*Log[Cos[(d + e*x)/2]] - 2*Log[Cos[(d + e*x)/2] + Sin[(d + e*x)/2]] + (2*Sin[(d + e*x)/2])/(Cos[(d + e*x)/2] + Sin[(d + e*x)/2]) + Tan[(d + e*x)/2])/(8*a^2*e)","A",1
372,1,135,123,0.5706192,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^3} \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-3),x]","\frac{\sec ^2\left(\frac{1}{2} (d+e x)\right)+2 \left(-3 \tan \left(\frac{1}{2} (d+e x)\right)-8 \log \left(\cos \left(\frac{1}{2} (d+e x)\right)\right)-\frac{6 \sin \left(\frac{1}{2} (d+e x)\right)}{\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)}-\frac{1}{\left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)^2}+8 \log \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)\right)}{64 a^3 e}","\frac{\log \left(\tan \left(\frac{1}{2} (d+e x)\right)+1\right)}{4 a^3 e}+\frac{3 (\cos (d+e x)-\sin (d+e x))}{16 e \left(a^3 \sin (d+e x)+a^3 \cos (d+e x)+a^3\right)}-\frac{a \cos (d+e x)-a \sin (d+e x)}{16 e \left(a^2 \sin (d+e x)+a^2 \cos (d+e x)+a^2\right)^2}",1,"(Sec[(d + e*x)/2]^2 + 2*(-8*Log[Cos[(d + e*x)/2]] + 8*Log[Cos[(d + e*x)/2] + Sin[(d + e*x)/2]] - (Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^(-2) - (6*Sin[(d + e*x)/2])/(Cos[(d + e*x)/2] + Sin[(d + e*x)/2]) - 3*Tan[(d + e*x)/2]))/(64*a^3*e)","A",1
373,1,247,168,0.9843307,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx","Integrate[(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-4),x]","\frac{19 \tan \left(\frac{1}{2} (d+e x)\right)}{192 a^4 e}-\frac{\sec ^2\left(\frac{1}{2} (d+e x)\right)}{64 a^4 e}+\frac{\log \left(\cos \left(\frac{1}{2} (d+e x)\right)\right)}{4 a^4 e}+\frac{19 \sin \left(\frac{1}{2} (d+e x)\right)}{96 a^4 e \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)}+\frac{5}{192 a^4 e \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)^2}+\frac{\sin \left(\frac{1}{2} (d+e x)\right)}{96 a^4 e \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)^3}+\frac{\tan \left(\frac{1}{2} (d+e x)\right) \sec ^2\left(\frac{1}{2} (d+e x)\right)}{384 a^4 e}-\frac{\log \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)}{4 a^4 e}","-\frac{19 (a \cos (d+e x)-a \sin (d+e x))}{96 e \left(a^5 \sin (d+e x)+a^5 \cos (d+e x)+a^5\right)}-\frac{\log \left(\tan \left(\frac{1}{2} (d+e x)\right)+1\right)}{4 a^4 e}+\frac{5 (\cos (d+e x)-\sin (d+e x))}{96 e \left(a^2 \sin (d+e x)+a^2 \cos (d+e x)+a^2\right)^2}-\frac{\cos (d+e x)-\sin (d+e x)}{48 a e (a \sin (d+e x)+a \cos (d+e x)+a)^3}",1,"Log[Cos[(d + e*x)/2]]/(4*a^4*e) - Log[Cos[(d + e*x)/2] + Sin[(d + e*x)/2]]/(4*a^4*e) - Sec[(d + e*x)/2]^2/(64*a^4*e) + Sin[(d + e*x)/2]/(96*a^4*e*(Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^3) + 5/(192*a^4*e*(Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^2) + (19*Sin[(d + e*x)/2])/(96*a^4*e*(Cos[(d + e*x)/2] + Sin[(d + e*x)/2])) + (19*Tan[(d + e*x)/2])/(192*a^4*e) + (Sec[(d + e*x)/2]^2*Tan[(d + e*x)/2])/(384*a^4*e)","A",1
374,1,136,157,0.4344835,"\int (2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx","Integrate[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^3,x]","\frac{2 \left(6 a \left(5 a^2+3 c^2\right) (d+e x)-9 a \left(5 a^2+c^2\right) \sin (d+e x)+9 a \left(a^2-c^2\right) \sin (2 (d+e x))-a \left(a^2-3 c^2\right) \sin (3 (d+e x))-9 c \left(5 a^2+c^2\right) \cos (d+e x)+c \left(c^2-3 a^2\right) \cos (3 (d+e x))+18 a^2 c \cos (2 (d+e x))\right)}{3 e}","-\frac{4 a \left(15 a^2+4 c^2\right) \sin (d+e x)}{3 e}-\frac{4 c \left(15 a^2+4 c^2\right) \cos (d+e x)}{3 e}+4 a x \left(5 a^2+3 c^2\right)-\frac{20 \left(a^2 \sin (d+e x)+a c \cos (d+e x)\right) (a (-\cos (d+e x))+a+c \sin (d+e x))}{3 e}-\frac{8 (a \sin (d+e x)+c \cos (d+e x)) (a (-\cos (d+e x))+a+c \sin (d+e x))^2}{3 e}",1,"(2*(6*a*(5*a^2 + 3*c^2)*(d + e*x) - 9*c*(5*a^2 + c^2)*Cos[d + e*x] + 18*a^2*c*Cos[2*(d + e*x)] + c*(-3*a^2 + c^2)*Cos[3*(d + e*x)] - 9*a*(5*a^2 + c^2)*Sin[d + e*x] + 9*a*(a^2 - c^2)*Sin[2*(d + e*x)] - a*(a^2 - 3*c^2)*Sin[3*(d + e*x)]))/(3*e)","A",1
375,1,92,81,0.1453112,"\int (2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx","Integrate[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2,x]","4 \left(\frac{\left(3 a^2+c^2\right) (d+e x)}{2 e}+\frac{\left(a^2-c^2\right) \sin (2 (d+e x))}{4 e}-\frac{2 a^2 \sin (d+e x)}{e}-\frac{2 a c \cos (d+e x)}{e}+\frac{a c \cos (2 (d+e x))}{2 e}\right)","2 x \left(3 a^2+c^2\right)-\frac{6 a^2 \sin (d+e x)}{e}-\frac{6 a c \cos (d+e x)}{e}-\frac{2 (a \sin (d+e x)+c \cos (d+e x)) (a (-\cos (d+e x))+a+c \sin (d+e x))}{e}",1,"4*(((3*a^2 + c^2)*(d + e*x))/(2*e) - (2*a*c*Cos[d + e*x])/e + (a*c*Cos[2*(d + e*x)])/(2*e) - (2*a^2*Sin[d + e*x])/e + ((a^2 - c^2)*Sin[2*(d + e*x)])/(4*e))","A",1
376,1,53,29,0.0119416,"\int (2 a-2 a \cos (d+e x)+2 c \sin (d+e x)) \, dx","Integrate[2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x],x]","-\frac{2 a \sin (d) \cos (e x)}{e}-\frac{2 a \cos (d) \sin (e x)}{e}+2 a x+\frac{2 c \sin (d) \sin (e x)}{e}-\frac{2 c \cos (d) \cos (e x)}{e}","-\frac{2 a \sin (d+e x)}{e}+2 a x-\frac{2 c \cos (d+e x)}{e}",1,"2*a*x - (2*c*Cos[d]*Cos[e*x])/e - (2*a*Cos[e*x]*Sin[d])/e - (2*a*Cos[d]*Sin[e*x])/e + (2*c*Sin[d]*Sin[e*x])/e","A",1
377,1,50,25,0.1509432,"\int \frac{1}{2 a-2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx","Integrate[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-1),x]","\frac{\log \left(\sin \left(\frac{1}{2} (d+e x)\right)\right)-\log \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)}{2 c e}","-\frac{\log \left(a+c \cot \left(\frac{1}{2} (d+e x)\right)\right)}{2 c e}",1,"(Log[Sin[(d + e*x)/2]] - Log[c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2]])/(2*c*e)","A",1
378,1,229,75,0.4170538,"\int \frac{1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx","Integrate[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-2),x]","-\frac{\sin \left(\frac{1}{2} (d+e x)\right) \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right) \left(\cos (d+e x) \left(2 a^2 \log \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)-2 a^2 \log \left(\sin \left(\frac{1}{2} (d+e x)\right)\right)+a^2+2 c^2\right)+a \left(a \left(-2 \log \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)+2 \log \left(\sin \left(\frac{1}{2} (d+e x)\right)\right)-1\right)+c \sin (d+e x) \left(-2 \log \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)+2 \log \left(\sin \left(\frac{1}{2} (d+e x)\right)\right)+1\right)\right)\right)}{4 c^3 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}","\frac{a \log \left(a+c \cot \left(\frac{1}{2} (d+e x)\right)\right)}{4 c^3 e}-\frac{a \sin (d+e x)+c \cos (d+e x)}{4 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))}",1,"-1/4*(Sin[(d + e*x)/2]*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])*(Cos[d + e*x]*(a^2 + 2*c^2 - 2*a^2*Log[Sin[(d + e*x)/2]] + 2*a^2*Log[c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2]]) + a*(a*(-1 + 2*Log[Sin[(d + e*x)/2]] - 2*Log[c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2]]) + c*(1 + 2*Log[Sin[(d + e*x)/2]] - 2*Log[c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2]])*Sin[d + e*x])))/(c^3*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^2)","B",1
379,1,350,134,0.6124048,"\int \frac{1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx","Integrate[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-3),x]","\frac{\sin \left(\frac{1}{2} (d+e x)\right) \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right) \left(-6 a \left(a^2+c^2\right) \sin ^3\left(\frac{1}{2} (d+e x)\right) \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)+4 \left(3 a^2+c^2\right) \sin ^2\left(\frac{1}{2} (d+e x)\right) \log \left(\sin \left(\frac{1}{2} (d+e x)\right)\right) \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)^2-4 \left(3 a^2+c^2\right) \sin ^2\left(\frac{1}{2} (d+e x)\right) \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)^2 \log \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)+c^2 (c-i a) (c+i a) \sin ^2\left(\frac{1}{2} (d+e x)\right)-c^2 \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)^2+3 a c \sin (d+e x) \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)^2\right)}{8 c^5 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}","\frac{3 \left(a^2 \sin (d+e x)+a c \cos (d+e x)\right)}{16 c^4 e (a (-\cos (d+e x))+a+c \sin (d+e x))}-\frac{\left(3 a^2+c^2\right) \log \left(a+c \cot \left(\frac{1}{2} (d+e x)\right)\right)}{16 c^5 e}-\frac{a \sin (d+e x)+c \cos (d+e x)}{16 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}",1,"(Sin[(d + e*x)/2]*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])*(c^2*((-I)*a + c)*(I*a + c)*Sin[(d + e*x)/2]^2 - 6*a*(a^2 + c^2)*Sin[(d + e*x)/2]^3*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2]) - c^2*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])^2 + 4*(3*a^2 + c^2)*Log[Sin[(d + e*x)/2]]*Sin[(d + e*x)/2]^2*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])^2 - 4*(3*a^2 + c^2)*Log[c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2]]*Sin[(d + e*x)/2]^2*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])^2 + 3*a*c*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])^2*Sin[d + e*x]))/(8*c^5*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^3)","C",1
380,1,494,207,1.171524,"\int \frac{1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx","Integrate[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]","\frac{\sin \left(\frac{1}{2} (d+e x)\right) \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right) \left(-225 a^6 \cos (d+e x)+90 a^6 \cos (2 (d+e x))-15 a^6 \cos (3 (d+e x))+150 a^6+75 a^5 c \sin (d+e x)-60 a^5 c \sin (2 (d+e x))+15 a^5 c \sin (3 (d+e x))-255 a^4 c^2 \cos (d+e x)+174 a^4 c^2 \cos (2 (d+e x))-49 a^4 c^2 \cos (3 (d+e x))+130 a^4 c^2+75 a^3 c^3 \sin (d+e x)-156 a^3 c^3 \sin (2 (d+e x))+79 a^3 c^3 \sin (3 (d+e x))-192 \left(5 a^3+3 a c^2\right) \sin ^3\left(\frac{1}{2} (d+e x)\right) \log \left(\sin \left(\frac{1}{2} (d+e x)\right)\right) \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)^3+192 \left(5 a^3+3 a c^2\right) \sin ^3\left(\frac{1}{2} (d+e x)\right) \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)^3 \log \left(a \sin \left(\frac{1}{2} (d+e x)\right)+c \cos \left(\frac{1}{2} (d+e x)\right)\right)-42 a^2 c^4 \cos (d+e x)+18 a^2 c^4 \cos (3 (d+e x))+24 a^2 c^4-12 a c^5 \sin (d+e x)-12 a c^5 \sin (2 (d+e x))+20 a c^5 \sin (3 (d+e x))-24 c^6 \cos (d+e x)+8 c^6 \cos (3 (d+e x))\right)}{384 c^7 e (a (-\cos (d+e x))+a+c \sin (d+e x))^4}","\frac{5 \left(a^2 \sin (d+e x)+a c \cos (d+e x)\right)}{96 c^4 e (a (-\cos (d+e x))+a+c \sin (d+e x))^2}+\frac{a \left(5 a^2+3 c^2\right) \log \left(a+c \cot \left(\frac{1}{2} (d+e x)\right)\right)}{32 c^7 e}-\frac{a \left(15 a^2+4 c^2\right) \sin (d+e x)+c \left(15 a^2+4 c^2\right) \cos (d+e x)}{96 c^6 e (a (-\cos (d+e x))+a+c \sin (d+e x))}-\frac{a \sin (d+e x)+c \cos (d+e x)}{48 c^2 e (a (-\cos (d+e x))+a+c \sin (d+e x))^3}",1,"(Sin[(d + e*x)/2]*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])*(150*a^6 + 130*a^4*c^2 + 24*a^2*c^4 - 225*a^6*Cos[d + e*x] - 255*a^4*c^2*Cos[d + e*x] - 42*a^2*c^4*Cos[d + e*x] - 24*c^6*Cos[d + e*x] + 90*a^6*Cos[2*(d + e*x)] + 174*a^4*c^2*Cos[2*(d + e*x)] - 15*a^6*Cos[3*(d + e*x)] - 49*a^4*c^2*Cos[3*(d + e*x)] + 18*a^2*c^4*Cos[3*(d + e*x)] + 8*c^6*Cos[3*(d + e*x)] - 192*(5*a^3 + 3*a*c^2)*Log[Sin[(d + e*x)/2]]*Sin[(d + e*x)/2]^3*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])^3 + 192*(5*a^3 + 3*a*c^2)*Log[c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2]]*Sin[(d + e*x)/2]^3*(c*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2])^3 + 75*a^5*c*Sin[d + e*x] + 75*a^3*c^3*Sin[d + e*x] - 12*a*c^5*Sin[d + e*x] - 60*a^5*c*Sin[2*(d + e*x)] - 156*a^3*c^3*Sin[2*(d + e*x)] - 12*a*c^5*Sin[2*(d + e*x)] + 15*a^5*c*Sin[3*(d + e*x)] + 79*a^3*c^3*Sin[3*(d + e*x)] + 20*a*c^5*Sin[3*(d + e*x)]))/(384*c^7*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^4)","B",1
381,1,135,157,0.442786,"\int (2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3 \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^3,x]","\frac{2 \left(6 a \left(5 a^2+3 b^2\right) (d+e x)-9 a \left(a^2-b^2\right) \sin (2 (d+e x))+9 b \left(5 a^2+b^2\right) \sin (d+e x)+b \left(b^2-3 a^2\right) \sin (3 (d+e x))-9 a \left(5 a^2+b^2\right) \cos (d+e x)+a \left(a^2-3 b^2\right) \cos (3 (d+e x))-18 a^2 b \cos (2 (d+e x))\right)}{3 e}","\frac{4 b \left(15 a^2+4 b^2\right) \sin (d+e x)}{3 e}-\frac{4 a \left(15 a^2+4 b^2\right) \cos (d+e x)}{3 e}+4 a x \left(5 a^2+3 b^2\right)-\frac{20 \left(a^2 \cos (d+e x)-a b \sin (d+e x)\right) (a \sin (d+e x)+a+b \cos (d+e x))}{3 e}-\frac{8 (a \cos (d+e x)-b \sin (d+e x)) (a \sin (d+e x)+a+b \cos (d+e x))^2}{3 e}",1,"(2*(6*a*(5*a^2 + 3*b^2)*(d + e*x) - 9*a*(5*a^2 + b^2)*Cos[d + e*x] - 18*a^2*b*Cos[2*(d + e*x)] + a*(a^2 - 3*b^2)*Cos[3*(d + e*x)] + 9*b*(5*a^2 + b^2)*Sin[d + e*x] - 9*a*(a^2 - b^2)*Sin[2*(d + e*x)] + b*(-3*a^2 + b^2)*Sin[3*(d + e*x)]))/(3*e)","A",1
382,1,92,81,0.143637,"\int (2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^2 \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^2,x]","4 \left(\frac{\left(3 a^2+b^2\right) (d+e x)}{2 e}-\frac{\left(a^2-b^2\right) \sin (2 (d+e x))}{4 e}-\frac{2 a^2 \cos (d+e x)}{e}+\frac{2 a b \sin (d+e x)}{e}-\frac{a b \cos (2 (d+e x))}{2 e}\right)","2 x \left(3 a^2+b^2\right)-\frac{6 a^2 \cos (d+e x)}{e}+\frac{6 a b \sin (d+e x)}{e}-\frac{2 (a \sin (d+e x)+a+b \cos (d+e x)) (a \cos (d+e x)-b \sin (d+e x))}{e}",1,"4*(((3*a^2 + b^2)*(d + e*x))/(2*e) - (2*a^2*Cos[d + e*x])/e - (a*b*Cos[2*(d + e*x)])/(2*e) + (2*a*b*Sin[d + e*x])/e - ((a^2 - b^2)*Sin[2*(d + e*x)])/(4*e))","A",1
383,1,53,29,0.0142491,"\int (2 a+2 b \cos (d+e x)+2 a \sin (d+e x)) \, dx","Integrate[2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x],x]","\frac{2 a \sin (d) \sin (e x)}{e}-\frac{2 a \cos (d) \cos (e x)}{e}+2 a x+\frac{2 b \sin (d) \cos (e x)}{e}+\frac{2 b \cos (d) \sin (e x)}{e}","-\frac{2 a \cos (d+e x)}{e}+2 a x+\frac{2 b \sin (d+e x)}{e}",1,"2*a*x - (2*a*Cos[d]*Cos[e*x])/e + (2*b*Cos[e*x]*Sin[d])/e + (2*b*Cos[d]*Sin[e*x])/e + (2*a*Sin[d]*Sin[e*x])/e","A",1
384,1,93,33,0.0667412,"\int \frac{1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-1),x]","\frac{1}{2} \left(\frac{\log \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)}{b e}-\frac{\log \left(a \sin \left(\frac{1}{2} (d+e x)\right)+a \cos \left(\frac{1}{2} (d+e x)\right)-b \sin \left(\frac{1}{2} (d+e x)\right)+b \cos \left(\frac{1}{2} (d+e x)\right)\right)}{b e}\right)","-\frac{\log \left(a+b \cot \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{2 b e}",1,"(Log[Cos[(d + e*x)/2] + Sin[(d + e*x)/2]]/(b*e) - Log[a*Cos[(d + e*x)/2] + b*Cos[(d + e*x)/2] + a*Sin[(d + e*x)/2] - b*Sin[(d + e*x)/2]]/(b*e))/2","B",1
385,1,162,83,0.5518747,"\int \frac{1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^2} \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-2),x]","\frac{\frac{b \left(a^2+b^2\right) \sin \left(\frac{1}{2} (d+e x)\right)}{(a+b) \left((a-b) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)}+a \log \left((a-b) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)-a \log \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)+\frac{b \sin \left(\frac{1}{2} (d+e x)\right)}{\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)}}{4 b^3 e}","\frac{a \log \left(a+b \cot \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{4 b^3 e}-\frac{a \cos (d+e x)-b \sin (d+e x)}{4 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))}",1,"(-(a*Log[Cos[(d + e*x)/2] + Sin[(d + e*x)/2]]) + a*Log[(a + b)*Cos[(d + e*x)/2] + (a - b)*Sin[(d + e*x)/2]] + (b*Sin[(d + e*x)/2])/(Cos[(d + e*x)/2] + Sin[(d + e*x)/2]) + (b*(a^2 + b^2)*Sin[(d + e*x)/2])/((a + b)*((a + b)*Cos[(d + e*x)/2] + (a - b)*Sin[(d + e*x)/2])))/(4*b^3*e)","A",1
386,1,255,142,2.4344259,"\int \frac{1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-3),x]","-\frac{-\frac{b^2 \left(a^2+b^2\right)}{\left((a-b) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)^2}+\frac{6 a b \left(a^2+b^2\right) \sin \left(\frac{1}{2} (d+e x)\right)}{(a+b) \left((a-b) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)}-2 \left(3 a^2+b^2\right) \log \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)+2 \left(3 a^2+b^2\right) \log \left((a-b) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)+\frac{6 a b \sin \left(\frac{1}{2} (d+e x)\right)}{\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)}+\frac{b^2}{\left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)^2}}{32 b^5 e}","\frac{3 \left(a^2 \cos (d+e x)-a b \sin (d+e x)\right)}{16 b^4 e (a \sin (d+e x)+a+b \cos (d+e x))}-\frac{\left(3 a^2+b^2\right) \log \left(a+b \cot \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{16 b^5 e}-\frac{a \cos (d+e x)-b \sin (d+e x)}{16 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^2}",1,"-1/32*(-2*(3*a^2 + b^2)*Log[Cos[(d + e*x)/2] + Sin[(d + e*x)/2]] + 2*(3*a^2 + b^2)*Log[(a + b)*Cos[(d + e*x)/2] + (a - b)*Sin[(d + e*x)/2]] + b^2/(Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^2 + (6*a*b*Sin[(d + e*x)/2])/(Cos[(d + e*x)/2] + Sin[(d + e*x)/2]) - (b^2*(a^2 + b^2))/((a + b)*Cos[(d + e*x)/2] + (a - b)*Sin[(d + e*x)/2])^2 + (6*a*b*(a^2 + b^2)*Sin[(d + e*x)/2])/((a + b)*((a + b)*Cos[(d + e*x)/2] + (a - b)*Sin[(d + e*x)/2])))/(b^5*e)","A",1
387,1,632,215,3.0135533,"\int \frac{1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-4),x]","\frac{-12 a \left(5 a^2+3 b^2\right) \log \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)+12 a \left(5 a^2+3 b^2\right) \log \left((a-b) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)+\frac{b \left(225 a^6 \sin (d+e x)-60 a^6 \sin (2 (d+e x))-15 a^6 \sin (3 (d+e x))+15 a^6 \cos (3 (d+e x))+150 a^6+75 a^5 b \sin (d+e x)+120 a^5 b \sin (2 (d+e x))-45 a^5 b \sin (3 (d+e x))-30 a^5 b \cos (3 (d+e x))+180 a^4 b^2 \sin (d+e x)+54 a^4 b^2 \sin (2 (d+e x))-4 a^4 b^2 \sin (3 (d+e x))-41 a^4 b^2 \cos (3 (d+e x))+130 a^4 b^2+15 a^3 b^3 \sin (d+e x)+102 a^3 b^3 \sin (2 (d+e x))+3 a^3 b^3 \sin (3 (d+e x))-38 a^3 b^3 \cos (3 (d+e x))+27 a^2 b^4 \sin (d+e x)+6 a^2 b^4 \sin (2 (d+e x))+15 a^2 b^4 \sin (3 (d+e x))-12 a^2 b^4 \cos (3 (d+e x))+24 a^2 b^4-3 a^2 \left(25 a^4-50 a^3 b+5 a^2 b^2-30 a b^3+4 b^4\right) \cos (d+e x)-6 a^2 \left(15 a^4+20 a^3 b+9 a^2 b^2+2 a b^3-2 b^4\right) \cos (2 (d+e x))+12 a b^5 \sin (d+e x)+6 a b^5 \sin (2 (d+e x))+4 a b^5 \sin (3 (d+e x))-8 a b^5 \cos (3 (d+e x))+12 b^6 \sin (d+e x)+4 b^6 \sin (3 (d+e x))\right)}{(a+b) \left(\sin \left(\frac{1}{2} (d+e x)\right)+\cos \left(\frac{1}{2} (d+e x)\right)\right)^3 \left((a-b) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)^3}}{384 b^7 e}","\frac{5 \left(a^2 \cos (d+e x)-a b \sin (d+e x)\right)}{96 b^4 e (a \sin (d+e x)+a+b \cos (d+e x))^2}+\frac{a \left(5 a^2+3 b^2\right) \log \left(a+b \cot \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{32 b^7 e}-\frac{a \left(15 a^2+4 b^2\right) \cos (d+e x)-b \left(15 a^2+4 b^2\right) \sin (d+e x)}{96 b^6 e (a \sin (d+e x)+a+b \cos (d+e x))}-\frac{a \cos (d+e x)-b \sin (d+e x)}{48 b^2 e (a \sin (d+e x)+a+b \cos (d+e x))^3}",1,"(-12*a*(5*a^2 + 3*b^2)*Log[Cos[(d + e*x)/2] + Sin[(d + e*x)/2]] + 12*a*(5*a^2 + 3*b^2)*Log[(a + b)*Cos[(d + e*x)/2] + (a - b)*Sin[(d + e*x)/2]] + (b*(150*a^6 + 130*a^4*b^2 + 24*a^2*b^4 - 3*a^2*(25*a^4 - 50*a^3*b + 5*a^2*b^2 - 30*a*b^3 + 4*b^4)*Cos[d + e*x] - 6*a^2*(15*a^4 + 20*a^3*b + 9*a^2*b^2 + 2*a*b^3 - 2*b^4)*Cos[2*(d + e*x)] + 15*a^6*Cos[3*(d + e*x)] - 30*a^5*b*Cos[3*(d + e*x)] - 41*a^4*b^2*Cos[3*(d + e*x)] - 38*a^3*b^3*Cos[3*(d + e*x)] - 12*a^2*b^4*Cos[3*(d + e*x)] - 8*a*b^5*Cos[3*(d + e*x)] + 225*a^6*Sin[d + e*x] + 75*a^5*b*Sin[d + e*x] + 180*a^4*b^2*Sin[d + e*x] + 15*a^3*b^3*Sin[d + e*x] + 27*a^2*b^4*Sin[d + e*x] + 12*a*b^5*Sin[d + e*x] + 12*b^6*Sin[d + e*x] - 60*a^6*Sin[2*(d + e*x)] + 120*a^5*b*Sin[2*(d + e*x)] + 54*a^4*b^2*Sin[2*(d + e*x)] + 102*a^3*b^3*Sin[2*(d + e*x)] + 6*a^2*b^4*Sin[2*(d + e*x)] + 6*a*b^5*Sin[2*(d + e*x)] - 15*a^6*Sin[3*(d + e*x)] - 45*a^5*b*Sin[3*(d + e*x)] - 4*a^4*b^2*Sin[3*(d + e*x)] + 3*a^3*b^3*Sin[3*(d + e*x)] + 15*a^2*b^4*Sin[3*(d + e*x)] + 4*a*b^5*Sin[3*(d + e*x)] + 4*b^6*Sin[3*(d + e*x)]))/((a + b)*(Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^3*((a + b)*Cos[(d + e*x)/2] + (a - b)*Sin[(d + e*x)/2])^3))/(384*b^7*e)","B",1
388,1,136,157,0.4362527,"\int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^3,x]","\frac{2 \left(6 a \left(5 a^2+3 b^2\right) (d+e x)-9 a \left(a^2-b^2\right) \sin (2 (d+e x))+9 b \left(5 a^2+b^2\right) \sin (d+e x)+b \left(b^2-3 a^2\right) \sin (3 (d+e x))+9 a \left(5 a^2+b^2\right) \cos (d+e x)-a \left(a^2-3 b^2\right) \cos (3 (d+e x))+18 a^2 b \cos (2 (d+e x))\right)}{3 e}","\frac{4 b \left(15 a^2+4 b^2\right) \sin (d+e x)}{3 e}+\frac{4 a \left(15 a^2+4 b^2\right) \cos (d+e x)}{3 e}+4 a x \left(5 a^2+3 b^2\right)+\frac{20 \left(a^2 \cos (d+e x)+a b \sin (d+e x)\right) (a (-\sin (d+e x))+a+b \cos (d+e x))}{3 e}+\frac{8 (a \cos (d+e x)+b \sin (d+e x)) (a (-\sin (d+e x))+a+b \cos (d+e x))^2}{3 e}",1,"(2*(6*a*(5*a^2 + 3*b^2)*(d + e*x) + 9*a*(5*a^2 + b^2)*Cos[d + e*x] + 18*a^2*b*Cos[2*(d + e*x)] - a*(a^2 - 3*b^2)*Cos[3*(d + e*x)] + 9*b*(5*a^2 + b^2)*Sin[d + e*x] - 9*a*(a^2 - b^2)*Sin[2*(d + e*x)] + b*(-3*a^2 + b^2)*Sin[3*(d + e*x)]))/(3*e)","A",1
389,1,92,81,0.1494145,"\int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^2 \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^2,x]","4 \left(\frac{\left(3 a^2+b^2\right) (d+e x)}{2 e}-\frac{\left(a^2-b^2\right) \sin (2 (d+e x))}{4 e}+\frac{2 a^2 \cos (d+e x)}{e}+\frac{2 a b \sin (d+e x)}{e}+\frac{a b \cos (2 (d+e x))}{2 e}\right)","2 x \left(3 a^2+b^2\right)+\frac{6 a^2 \cos (d+e x)}{e}+\frac{6 a b \sin (d+e x)}{e}+\frac{2 (a (-\sin (d+e x))+a+b \cos (d+e x)) (a \cos (d+e x)+b \sin (d+e x))}{e}",1,"4*(((3*a^2 + b^2)*(d + e*x))/(2*e) + (2*a^2*Cos[d + e*x])/e + (a*b*Cos[2*(d + e*x)])/(2*e) + (2*a*b*Sin[d + e*x])/e - ((a^2 - b^2)*Sin[2*(d + e*x)])/(4*e))","A",1
390,1,53,29,0.0119255,"\int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx","Integrate[2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x],x]","-\frac{2 a \sin (d) \sin (e x)}{e}+\frac{2 a \cos (d) \cos (e x)}{e}+2 a x+\frac{2 b \sin (d) \cos (e x)}{e}+\frac{2 b \cos (d) \sin (e x)}{e}","\frac{2 a \cos (d+e x)}{e}+2 a x+\frac{2 b \sin (d+e x)}{e}",1,"2*a*x + (2*a*Cos[d]*Cos[e*x])/e + (2*b*Cos[e*x]*Sin[d])/e + (2*b*Cos[d]*Sin[e*x])/e - (2*a*Sin[d]*Sin[e*x])/e","A",1
391,1,96,33,0.1010147,"\int \frac{1}{2 a+2 b \cos (d+e x)-2 a \sin (d+e x)} \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^(-1),x]","\frac{\log \left(-a \sin \left(\frac{1}{2} (d+e x)\right)+a \cos \left(\frac{1}{2} (d+e x)\right)+b \sin \left(\frac{1}{2} (d+e x)\right)+b \cos \left(\frac{1}{2} (d+e x)\right)\right)}{2 b e}-\frac{\log \left(\cos \left(\frac{1}{2} (d+e x)\right)-\sin \left(\frac{1}{2} (d+e x)\right)\right)}{2 b e}","\frac{\log \left(a+b \tan \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{2 b e}",1,"-1/2*Log[Cos[(d + e*x)/2] - Sin[(d + e*x)/2]]/(b*e) + Log[a*Cos[(d + e*x)/2] + b*Cos[(d + e*x)/2] - a*Sin[(d + e*x)/2] + b*Sin[(d + e*x)/2]]/(2*b*e)","B",1
392,1,166,83,0.6174265,"\int \frac{1}{(2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^2} \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^(-2),x]","\frac{\frac{b \left(a^2+b^2\right) \sin \left(\frac{1}{2} (d+e x)\right)}{(a+b) \left((b-a) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)}-a \log \left((b-a) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)+a \log \left(\cos \left(\frac{1}{2} (d+e x)\right)-\sin \left(\frac{1}{2} (d+e x)\right)\right)+\frac{b \sin \left(\frac{1}{2} (d+e x)\right)}{\cos \left(\frac{1}{2} (d+e x)\right)-\sin \left(\frac{1}{2} (d+e x)\right)}}{4 b^3 e}","\frac{a \cos (d+e x)+b \sin (d+e x)}{4 b^2 e (a (-\sin (d+e x))+a+b \cos (d+e x))}-\frac{a \log \left(a+b \tan \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{4 b^3 e}",1,"(a*Log[Cos[(d + e*x)/2] - Sin[(d + e*x)/2]] - a*Log[(a + b)*Cos[(d + e*x)/2] + (-a + b)*Sin[(d + e*x)/2]] + (b*Sin[(d + e*x)/2])/(Cos[(d + e*x)/2] - Sin[(d + e*x)/2]) + (b*(a^2 + b^2)*Sin[(d + e*x)/2])/((a + b)*((a + b)*Cos[(d + e*x)/2] + (-a + b)*Sin[(d + e*x)/2])))/(4*b^3*e)","A",1
393,1,261,142,2.7706997,"\int \frac{1}{(2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3} \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^(-3),x]","-\frac{\frac{b^2 \left(a^2+b^2\right)}{\left((b-a) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)^2}+\frac{6 a b \left(a^2+b^2\right) \sin \left(\frac{1}{2} (d+e x)\right)}{(a+b) \left((b-a) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)}+2 \left(3 a^2+b^2\right) \log \left(\cos \left(\frac{1}{2} (d+e x)\right)-\sin \left(\frac{1}{2} (d+e x)\right)\right)-2 \left(3 a^2+b^2\right) \log \left((b-a) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)+\frac{6 a b \sin \left(\frac{1}{2} (d+e x)\right)}{\cos \left(\frac{1}{2} (d+e x)\right)-\sin \left(\frac{1}{2} (d+e x)\right)}-\frac{b^2}{\left(\cos \left(\frac{1}{2} (d+e x)\right)-\sin \left(\frac{1}{2} (d+e x)\right)\right)^2}}{32 b^5 e}","-\frac{3 \left(a^2 \cos (d+e x)+a b \sin (d+e x)\right)}{16 b^4 e (a (-\sin (d+e x))+a+b \cos (d+e x))}+\frac{\left(3 a^2+b^2\right) \log \left(a+b \tan \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{16 b^5 e}+\frac{a \cos (d+e x)+b \sin (d+e x)}{16 b^2 e (a (-\sin (d+e x))+a+b \cos (d+e x))^2}",1,"-1/32*(2*(3*a^2 + b^2)*Log[Cos[(d + e*x)/2] - Sin[(d + e*x)/2]] - 2*(3*a^2 + b^2)*Log[(a + b)*Cos[(d + e*x)/2] + (-a + b)*Sin[(d + e*x)/2]] - b^2/(Cos[(d + e*x)/2] - Sin[(d + e*x)/2])^2 + (6*a*b*Sin[(d + e*x)/2])/(Cos[(d + e*x)/2] - Sin[(d + e*x)/2]) + (b^2*(a^2 + b^2))/((a + b)*Cos[(d + e*x)/2] + (-a + b)*Sin[(d + e*x)/2])^2 + (6*a*b*(a^2 + b^2)*Sin[(d + e*x)/2])/((a + b)*((a + b)*Cos[(d + e*x)/2] + (-a + b)*Sin[(d + e*x)/2])))/(b^5*e)","A",0
394,1,636,215,1.9201614,"\int \frac{1}{(2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^4} \, dx","Integrate[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^(-4),x]","\frac{12 a \left(5 a^2+3 b^2\right) \log \left(\cos \left(\frac{1}{2} (d+e x)\right)-\sin \left(\frac{1}{2} (d+e x)\right)\right)-12 a \left(5 a^2+3 b^2\right) \log \left((b-a) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)+\frac{b \left(225 a^6 \sin (d+e x)-60 a^6 \sin (2 (d+e x))-15 a^6 \sin (3 (d+e x))-15 a^6 \cos (3 (d+e x))-150 a^6+75 a^5 b \sin (d+e x)+120 a^5 b \sin (2 (d+e x))-45 a^5 b \sin (3 (d+e x))+30 a^5 b \cos (3 (d+e x))+180 a^4 b^2 \sin (d+e x)+54 a^4 b^2 \sin (2 (d+e x))-4 a^4 b^2 \sin (3 (d+e x))+41 a^4 b^2 \cos (3 (d+e x))-130 a^4 b^2+15 a^3 b^3 \sin (d+e x)+102 a^3 b^3 \sin (2 (d+e x))+3 a^3 b^3 \sin (3 (d+e x))+38 a^3 b^3 \cos (3 (d+e x))+27 a^2 b^4 \sin (d+e x)+6 a^2 b^4 \sin (2 (d+e x))+15 a^2 b^4 \sin (3 (d+e x))+12 a^2 b^4 \cos (3 (d+e x))-24 a^2 b^4+3 a^2 \left(25 a^4-50 a^3 b+5 a^2 b^2-30 a b^3+4 b^4\right) \cos (d+e x)+6 a^2 \left(15 a^4+20 a^3 b+9 a^2 b^2+2 a b^3-2 b^4\right) \cos (2 (d+e x))+12 a b^5 \sin (d+e x)+6 a b^5 \sin (2 (d+e x))+4 a b^5 \sin (3 (d+e x))+8 a b^5 \cos (3 (d+e x))+12 b^6 \sin (d+e x)+4 b^6 \sin (3 (d+e x))\right)}{(a+b) \left(\cos \left(\frac{1}{2} (d+e x)\right)-\sin \left(\frac{1}{2} (d+e x)\right)\right)^3 \left((b-a) \sin \left(\frac{1}{2} (d+e x)\right)+(a+b) \cos \left(\frac{1}{2} (d+e x)\right)\right)^3}}{384 b^7 e}","-\frac{5 \left(a^2 \cos (d+e x)+a b \sin (d+e x)\right)}{96 b^4 e (a (-\sin (d+e x))+a+b \cos (d+e x))^2}-\frac{a \left(5 a^2+3 b^2\right) \log \left(a+b \tan \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{32 b^7 e}+\frac{b \left(15 a^2+4 b^2\right) \sin (d+e x)+a \left(15 a^2+4 b^2\right) \cos (d+e x)}{96 b^6 e (a (-\sin (d+e x))+a+b \cos (d+e x))}+\frac{a \cos (d+e x)+b \sin (d+e x)}{48 b^2 e (a (-\sin (d+e x))+a+b \cos (d+e x))^3}",1,"(12*a*(5*a^2 + 3*b^2)*Log[Cos[(d + e*x)/2] - Sin[(d + e*x)/2]] - 12*a*(5*a^2 + 3*b^2)*Log[(a + b)*Cos[(d + e*x)/2] + (-a + b)*Sin[(d + e*x)/2]] + (b*(-150*a^6 - 130*a^4*b^2 - 24*a^2*b^4 + 3*a^2*(25*a^4 - 50*a^3*b + 5*a^2*b^2 - 30*a*b^3 + 4*b^4)*Cos[d + e*x] + 6*a^2*(15*a^4 + 20*a^3*b + 9*a^2*b^2 + 2*a*b^3 - 2*b^4)*Cos[2*(d + e*x)] - 15*a^6*Cos[3*(d + e*x)] + 30*a^5*b*Cos[3*(d + e*x)] + 41*a^4*b^2*Cos[3*(d + e*x)] + 38*a^3*b^3*Cos[3*(d + e*x)] + 12*a^2*b^4*Cos[3*(d + e*x)] + 8*a*b^5*Cos[3*(d + e*x)] + 225*a^6*Sin[d + e*x] + 75*a^5*b*Sin[d + e*x] + 180*a^4*b^2*Sin[d + e*x] + 15*a^3*b^3*Sin[d + e*x] + 27*a^2*b^4*Sin[d + e*x] + 12*a*b^5*Sin[d + e*x] + 12*b^6*Sin[d + e*x] - 60*a^6*Sin[2*(d + e*x)] + 120*a^5*b*Sin[2*(d + e*x)] + 54*a^4*b^2*Sin[2*(d + e*x)] + 102*a^3*b^3*Sin[2*(d + e*x)] + 6*a^2*b^4*Sin[2*(d + e*x)] + 6*a*b^5*Sin[2*(d + e*x)] - 15*a^6*Sin[3*(d + e*x)] - 45*a^5*b*Sin[3*(d + e*x)] - 4*a^4*b^2*Sin[3*(d + e*x)] + 3*a^3*b^3*Sin[3*(d + e*x)] + 15*a^2*b^4*Sin[3*(d + e*x)] + 4*a*b^5*Sin[3*(d + e*x)] + 4*b^6*Sin[3*(d + e*x)]))/((a + b)*(Cos[(d + e*x)/2] - Sin[(d + e*x)/2])^3*((a + b)*Cos[(d + e*x)/2] + (-a + b)*Sin[(d + e*x)/2])^3))/(384*b^7*e)","B",1
395,1,237,260,1.0519196,"\int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^4,x]","\frac{96 a b \left(4 a^2+3 \left(b^2+c^2\right)\right) \sin (d+e x)+24 \left(b^2-c^2\right) \left(6 a^2+b^2+c^2\right) \sin (2 (d+e x))-96 a c \left(4 a^2+3 \left(b^2+c^2\right)\right) \cos (d+e x)-48 b c \left(6 a^2+b^2+c^2\right) \cos (2 (d+e x))+12 \left(8 a^4+24 a^2 \left(b^2+c^2\right)+3 \left(b^2+c^2\right)^2\right) (d+e x)+32 a b \left(b^2-3 c^2\right) \sin (3 (d+e x))+32 a c \left(c^2-3 b^2\right) \cos (3 (d+e x))-12 b c \left(b^2-c^2\right) \cos (4 (d+e x))+3 \left(b^4-6 b^2 c^2+c^4\right) \sin (4 (d+e x))}{96 e}","\frac{5 a b \left(10 a^2+11 \left(b^2+c^2\right)\right) \sin (d+e x)}{24 e}-\frac{5 a c \left(10 a^2+11 \left(b^2+c^2\right)\right) \cos (d+e x)}{24 e}-\frac{\left(c \left(26 a^2+9 \left(b^2+c^2\right)\right) \cos (d+e x)-b \left(26 a^2+9 \left(b^2+c^2\right)\right) \sin (d+e x)\right) (a+b \cos (d+e x)+c \sin (d+e x))}{24 e}+\frac{1}{8} x \left(8 a^4+24 a^2 \left(b^2+c^2\right)+3 \left(b^2+c^2\right)^2\right)-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^3}{4 e}-\frac{7 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{12 e}",1,"(12*(8*a^4 + 24*a^2*(b^2 + c^2) + 3*(b^2 + c^2)^2)*(d + e*x) - 96*a*c*(4*a^2 + 3*(b^2 + c^2))*Cos[d + e*x] - 48*b*c*(6*a^2 + b^2 + c^2)*Cos[2*(d + e*x)] + 32*a*c*(-3*b^2 + c^2)*Cos[3*(d + e*x)] - 12*b*c*(b^2 - c^2)*Cos[4*(d + e*x)] + 96*a*b*(4*a^2 + 3*(b^2 + c^2))*Sin[d + e*x] + 24*(b^2 - c^2)*(6*a^2 + b^2 + c^2)*Sin[2*(d + e*x)] + 32*a*b*(b^2 - 3*c^2)*Sin[3*(d + e*x)] + 3*(b^4 - 6*b^2*c^2 + c^4)*Sin[4*(d + e*x)])/(96*e)","A",1
396,1,144,170,0.4329863,"\int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3,x]","\frac{6 a \left(2 a^2+3 \left(b^2+c^2\right)\right) (d+e x)+9 b \left(4 a^2+b^2+c^2\right) \sin (d+e x)-9 c \left(4 a^2+b^2+c^2\right) \cos (d+e x)+9 a \left(b^2-c^2\right) \sin (2 (d+e x))-18 a b c \cos (2 (d+e x))+b \left(b^2-3 c^2\right) \sin (3 (d+e x))+c \left(c^2-3 b^2\right) \cos (3 (d+e x))}{12 e}","\frac{b \left(11 a^2+4 \left(b^2+c^2\right)\right) \sin (d+e x)}{6 e}-\frac{c \left(11 a^2+4 \left(b^2+c^2\right)\right) \cos (d+e x)}{6 e}+\frac{1}{2} a x \left(2 a^2+3 \left(b^2+c^2\right)\right)-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^2}{3 e}-\frac{5 (a c \cos (d+e x)-a b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{6 e}",1,"(6*a*(2*a^2 + 3*(b^2 + c^2))*(d + e*x) - 9*c*(4*a^2 + b^2 + c^2)*Cos[d + e*x] - 18*a*b*c*Cos[2*(d + e*x)] + c*(-3*b^2 + c^2)*Cos[3*(d + e*x)] + 9*b*(4*a^2 + b^2 + c^2)*Sin[d + e*x] + 9*a*(b^2 - c^2)*Sin[2*(d + e*x)] + b*(b^2 - 3*c^2)*Sin[3*(d + e*x)])/(12*e)","A",1
397,1,77,91,0.1729588,"\int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2,x]","\frac{2 \left(2 a^2+b^2+c^2\right) (d+e x)+8 a b \sin (d+e x)-8 a c \cos (d+e x)+\left(b^2-c^2\right) \sin (2 (d+e x))-2 b c \cos (2 (d+e x))}{4 e}","\frac{1}{2} x \left(2 a^2+b^2+c^2\right)-\frac{(c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))}{2 e}+\frac{3 a b \sin (d+e x)}{2 e}-\frac{3 a c \cos (d+e x)}{2 e}",1,"(2*(2*a^2 + b^2 + c^2)*(d + e*x) - 8*a*c*Cos[d + e*x] - 2*b*c*Cos[2*(d + e*x)] + 8*a*b*Sin[d + e*x] + (b^2 - c^2)*Sin[2*(d + e*x)])/(4*e)","A",1
398,1,49,27,0.0118468,"\int (a+b \cos (d+e x)+c \sin (d+e x)) \, dx","Integrate[a + b*Cos[d + e*x] + c*Sin[d + e*x],x]","a x+\frac{b \sin (d) \cos (e x)}{e}+\frac{b \cos (d) \sin (e x)}{e}+\frac{c \sin (d) \sin (e x)}{e}-\frac{c \cos (d) \cos (e x)}{e}","a x+\frac{b \sin (d+e x)}{e}-\frac{c \cos (d+e x)}{e}",1,"a*x - (c*Cos[d]*Cos[e*x])/e + (b*Cos[e*x]*Sin[d])/e + (b*Cos[d]*Sin[e*x])/e + (c*Sin[d]*Sin[e*x])/e","A",1
399,1,57,61,0.121249,"\int \frac{1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-1),x]","-\frac{2 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{e \sqrt{-a^2+b^2+c^2}}","\frac{2 \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{e \sqrt{a^2-b^2-c^2}}",1,"(-2*ArcTanh[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2 + c^2]])/(Sqrt[-a^2 + b^2 + c^2]*e)","A",1
400,1,116,121,0.358916,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^2} \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-2),x]","\frac{\frac{a c+\left(b^2+c^2\right) \sin (d+e x)}{b \left(-a^2+b^2+c^2\right) (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{2 a \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{3/2}}}{e}","\frac{2 a \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{e \left(a^2-b^2-c^2\right)^{3/2}}+\frac{c \cos (d+e x)-b \sin (d+e x)}{e \left(a^2-b^2-c^2\right) (a+b \cos (d+e x)+c \sin (d+e x))}",1,"((2*a*ArcTanh[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(3/2) + (a*c + (b^2 + c^2)*Sin[d + e*x])/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])))/e","A",1
401,1,200,197,0.9524798,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^3} \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3),x]","\frac{\frac{a c+\left(b^2+c^2\right) \sin (d+e x)}{b \left(-a^2+b^2+c^2\right) (a+b \cos (d+e x)+c \sin (d+e x))^2}-\frac{c \left(2 a^2+b^2+c^2\right)+3 a \left(b^2+c^2\right) \sin (d+e x)}{b \left(-a^2+b^2+c^2\right)^2 (a+b \cos (d+e x)+c \sin (d+e x))}-\frac{2 \left(2 a^2+b^2+c^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{5/2}}}{2 e}","\frac{\left(2 a^2+b^2+c^2\right) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{e \left(a^2-b^2-c^2\right)^{5/2}}+\frac{3 (a c \cos (d+e x)-a b \sin (d+e x))}{2 e \left(a^2-b^2-c^2\right)^2 (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{c \cos (d+e x)-b \sin (d+e x)}{2 e \left(a^2-b^2-c^2\right) (a+b \cos (d+e x)+c \sin (d+e x))^2}",1,"((-2*(2*a^2 + b^2 + c^2)*ArcTanh[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(5/2) + (a*c + (b^2 + c^2)*Sin[d + e*x])/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (c*(2*a^2 + b^2 + c^2) + 3*a*(b^2 + c^2)*Sin[d + e*x])/(b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[d + e*x] + c*Sin[d + e*x])))/(2*e)","A",1
402,1,606,292,2.1127364,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]","\frac{\frac{24 a \left(2 a^2+3 \left(b^2+c^2\right)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{7/2}}+\frac{44 a^5 c+72 a^4 b^2 \sin (d+e x)+132 a^4 c^2 \sin (d+e x)+54 a^3 b^3 \sin (2 (d+e x))+82 a^3 b^2 c+78 a^3 b c^2 \sin (2 (d+e x))+82 a^3 c^3-9 a^2 b^4 \sin (d+e x)+11 a^2 b^4 \sin (3 (d+e x))-22 a^2 b^3 c \cos (3 (d+e x))+72 a^2 b^2 c^2 \sin (d+e x)+30 a^2 b c \left(2 a^2+3 \left(b^2+c^2\right)\right) \cos (d+e x)-6 a c \left(a^2 \left(7 b^2+11 c^2\right)-2 b^4+2 b^2 c^2+4 c^4\right) \cos (2 (d+e x))-22 a^2 b c^3 \cos (3 (d+e x))+81 a^2 c^4 \sin (d+e x)-11 a^2 c^4 \sin (3 (d+e x))+6 a b^5 \sin (2 (d+e x))+24 a b^4 c+48 a b^3 c^2 \sin (2 (d+e x))+48 a b^2 c^3+42 a b c^4 \sin (2 (d+e x))+24 a c^5+12 b^6 \sin (d+e x)+4 b^6 \sin (3 (d+e x))-8 b^5 c \cos (3 (d+e x))+36 b^4 c^2 \sin (d+e x)+4 b^4 c^2 \sin (3 (d+e x))-16 b^3 c^3 \cos (3 (d+e x))+36 b^2 c^4 \sin (d+e x)-4 b^2 c^4 \sin (3 (d+e x))-8 b c^5 \cos (3 (d+e x))+12 c^6 \sin (d+e x)-4 c^6 \sin (3 (d+e x))}{b \left(-a^2+b^2+c^2\right)^3 (a+b \cos (d+e x)+c \sin (d+e x))^3}}{24 e}","\frac{a \left(2 a^2+3 \left(b^2+c^2\right)\right) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{e \left(a^2-b^2-c^2\right)^{7/2}}+\frac{c \left(11 a^2+4 \left(b^2+c^2\right)\right) \cos (d+e x)-b \left(11 a^2+4 \left(b^2+c^2\right)\right) \sin (d+e x)}{6 e \left(a^2-b^2-c^2\right)^3 (a+b \cos (d+e x)+c \sin (d+e x))}+\frac{5 (a c \cos (d+e x)-a b \sin (d+e x))}{6 e \left(a^2-b^2-c^2\right)^2 (a+b \cos (d+e x)+c \sin (d+e x))^2}+\frac{c \cos (d+e x)-b \sin (d+e x)}{3 e \left(a^2-b^2-c^2\right) (a+b \cos (d+e x)+c \sin (d+e x))^3}",1,"((24*a*(2*a^2 + 3*(b^2 + c^2))*ArcTanh[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(7/2) + (44*a^5*c + 82*a^3*b^2*c + 24*a*b^4*c + 82*a^3*c^3 + 48*a*b^2*c^3 + 24*a*c^5 + 30*a^2*b*c*(2*a^2 + 3*(b^2 + c^2))*Cos[d + e*x] - 6*a*c*(-2*b^4 + 2*b^2*c^2 + 4*c^4 + a^2*(7*b^2 + 11*c^2))*Cos[2*(d + e*x)] - 22*a^2*b^3*c*Cos[3*(d + e*x)] - 8*b^5*c*Cos[3*(d + e*x)] - 22*a^2*b*c^3*Cos[3*(d + e*x)] - 16*b^3*c^3*Cos[3*(d + e*x)] - 8*b*c^5*Cos[3*(d + e*x)] + 72*a^4*b^2*Sin[d + e*x] - 9*a^2*b^4*Sin[d + e*x] + 12*b^6*Sin[d + e*x] + 132*a^4*c^2*Sin[d + e*x] + 72*a^2*b^2*c^2*Sin[d + e*x] + 36*b^4*c^2*Sin[d + e*x] + 81*a^2*c^4*Sin[d + e*x] + 36*b^2*c^4*Sin[d + e*x] + 12*c^6*Sin[d + e*x] + 54*a^3*b^3*Sin[2*(d + e*x)] + 6*a*b^5*Sin[2*(d + e*x)] + 78*a^3*b*c^2*Sin[2*(d + e*x)] + 48*a*b^3*c^2*Sin[2*(d + e*x)] + 42*a*b*c^4*Sin[2*(d + e*x)] + 11*a^2*b^4*Sin[3*(d + e*x)] + 4*b^6*Sin[3*(d + e*x)] + 4*b^4*c^2*Sin[3*(d + e*x)] - 11*a^2*c^4*Sin[3*(d + e*x)] - 4*b^2*c^4*Sin[3*(d + e*x)] - 4*c^6*Sin[3*(d + e*x)])/(b*(-a^2 + b^2 + c^2)^3*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3))/(24*e)","B",1
403,1,399,185,5.9060457,"\int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx","Integrate[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2),x]","\frac{1276 \sqrt{\frac{10}{3}} \sqrt{\sqrt{34} \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+2} \sqrt{\cos ^2\left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)} \sec \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right) F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)-\frac{1990 \sqrt{30} \sqrt{\sin ^2\left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)} \csc \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right) F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)}{\sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}}-2 \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2} (550 \cos (d+e x)+3 (-110 \sin (d+e x)+40 \sin (2 (d+e x))+75 \cos (2 (d+e x))-398))-2388 \sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}+\frac{1990 \sin \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)}{\sqrt{\frac{\cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)}{\sqrt{34}}+\frac{1}{17}}}}{75 e}","-\frac{2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}-\frac{32 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}}{15 e}+\frac{64 F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{\sqrt{2+\sqrt{34}} e}+\frac{796 \sqrt{2+\sqrt{34}} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{15 e}",1,"(-2388*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] - 2*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]]*(550*Cos[d + e*x] + 3*(-398 + 75*Cos[2*(d + e*x)] - 110*Sin[d + e*x] + 40*Sin[2*(d + e*x)])) + 1276*Sqrt[10/3]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]] + (1990*Sin[d + e*x - ArcTan[5/3]])/Sqrt[1/17 + Cos[d + e*x - ArcTan[5/3]]/Sqrt[34]] - (1990*Sqrt[30]*AppellF1[-1/2, -1/2, -1/2, 1/2, (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(17 + Sqrt[34])]*Csc[d + e*x - ArcTan[5/3]]*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])/Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]])/(75*e)","C",0
404,1,349,139,3.4202645,"\int (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2} \, dx","Integrate[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2),x]","\frac{2 \left(\sqrt{\sin ^2\left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)} \left(23 \sqrt{30} \sqrt{\sqrt{34} \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+2} \sqrt{\cos ^2\left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)} \sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2} \sec \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right) F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)-15 (-18 \sin (d+e x)+8 \sin (2 (d+e x))+30 \cos (d+e x)+15 \cos (2 (d+e x)))\right)-60 \sqrt{30} \sin \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right) F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)\right)}{45 e \sqrt{\sin ^2\left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)} \sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}}","-\frac{2 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}+\frac{20 F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{\sqrt{2+\sqrt{34}} e}+\frac{16 \sqrt{2+\sqrt{34}} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{3 e}",1,"(2*(-60*Sqrt[30]*AppellF1[-1/2, -1/2, -1/2, 1/2, (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(17 + Sqrt[34])]*Sin[d + e*x - ArcTan[5/3]] + (-15*(30*Cos[d + e*x] + 15*Cos[2*(d + e*x)] - 18*Sin[d + e*x] + 8*Sin[2*(d + e*x)]) + 23*Sqrt[30]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]])*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2]))/(45*e*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]]*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])","C",0
405,1,326,45,2.2732417,"\int \sqrt{2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx","Integrate[Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]],x]","\frac{\sqrt{\sin ^2\left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)} \left(2 \sqrt{30} \sqrt{\sqrt{34} \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+2} \sqrt{\cos ^2\left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)} \sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2} \sec \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right) F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)+45 \sin (d+e x)-75 \cos (d+e x)\right)-15 \sqrt{30} \sin \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right) F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)}{15 e \sqrt{\sin ^2\left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)} \sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}}","\frac{2 \sqrt{2+\sqrt{34}} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{e}",1,"(-15*Sqrt[30]*AppellF1[-1/2, -1/2, -1/2, 1/2, (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(17 + Sqrt[34])]*Sin[d + e*x - ArcTan[5/3]] + (-75*Cos[d + e*x] + 45*Sin[d + e*x] + 2*Sqrt[30]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]])*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])/(15*e*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]]*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])","C",0
406,1,128,45,0.2577329,"\int \frac{1}{\sqrt{2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx","Integrate[1/Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]],x]","\frac{\sqrt{\frac{2}{15}} \sqrt{\sqrt{34} \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+2} \sqrt{\cos ^2\left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)} \sec \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right) F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)}{e}","\frac{2 F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{\sqrt{2+\sqrt{34}} e}",1,"(Sqrt[2/15]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]])/e","C",0
407,1,390,94,6.0349228,"\int \frac{1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx","Integrate[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-3/2),x]","\frac{-2 \sqrt{30} \sqrt{\sqrt{34} \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+2} \sqrt{\cos ^2\left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)} \sec \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right) F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)+\frac{15 \sqrt{30} \sqrt{\sin ^2\left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)} \csc \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right) F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)}{\sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}}+\frac{20 (17 \sin (d+e x)+5)}{\sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}}-68 \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}+18 \sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}-\frac{15 \sin \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)}{\sqrt{\frac{\cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)}{\sqrt{34}}+\frac{1}{17}}}}{450 e}","-\frac{5 \cos (d+e x)-3 \sin (d+e x)}{15 e \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}}-\frac{\sqrt{2+\sqrt{34}} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{15 e}",1,"(18*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] - 68*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]] + (20*(5 + 17*Sin[d + e*x]))/Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]] - 2*Sqrt[30]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]] - (15*Sin[d + e*x - ArcTan[5/3]])/Sqrt[1/17 + Cos[d + e*x - ArcTan[5/3]]/Sqrt[34]] + (15*Sqrt[30]*AppellF1[-1/2, -1/2, -1/2, 1/2, (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(17 + Sqrt[34])]*Csc[d + e*x - ArcTan[5/3]]*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])/Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]])/(450*e)","C",0
408,1,430,187,3.1817903,"\int \frac{1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx","Integrate[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-5/2),x]","\frac{23 \sqrt{\frac{10}{3}} \sqrt{\sqrt{34} \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+2} \sqrt{\cos ^2\left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)} \sec \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right) F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)-\frac{20 \sqrt{30} \sqrt{\sin ^2\left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)} \csc \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right) F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)}{\sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}}+\frac{100 (17 \sin (d+e x)+5)}{(5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}-\frac{10 (136 \sin (d+e x)+115)}{3 \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}}+\frac{272}{3} \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}-24 \sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}+\frac{20 \sin \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)}{\sqrt{\frac{\cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)}{\sqrt{34}}+\frac{1}{17}}}}{6750 e}","\frac{4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}}-\frac{5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}+\frac{F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{45 \sqrt{2+\sqrt{34}} e}+\frac{4 \sqrt{2+\sqrt{34}} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{675 e}",1,"(-24*Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] + (272*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])/3 + (100*(5 + 17*Sin[d + e*x]))/(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2) - (10*(115 + 136*Sin[d + e*x]))/(3*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]]) + 23*Sqrt[10/3]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]] + (20*Sin[d + e*x - ArcTan[5/3]])/Sqrt[1/17 + Cos[d + e*x - ArcTan[5/3]]/Sqrt[34]] - (20*Sqrt[30]*AppellF1[-1/2, -1/2, -1/2, 1/2, (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(17 + Sqrt[34])]*Csc[d + e*x - ArcTan[5/3]]*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])/Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]])/(6750*e)","C",0
409,1,436,233,3.838095,"\int \frac{1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx","Integrate[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-7/2),x]","\frac{-638 \sqrt{30} \sqrt{\sqrt{34} \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+2} \sqrt{\cos ^2\left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)} \sec \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right) F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \sin \left(d+e x+\tan ^{-1}\left(\frac{3}{5}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)+\frac{2985 \sqrt{30} \sqrt{\sin ^2\left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)} \csc \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right) F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{-17+\sqrt{34}},\frac{17 \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+\sqrt{34}}{17+\sqrt{34}}\right)}{\sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}}+\frac{27000 (17 \sin (d+e x)+5)}{(5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}-\frac{300 (272 \sin (d+e x)+305)}{(5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}+\frac{20 (3383 \sin (d+e x)+1595)}{\sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}}-13532 \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}+\frac{597 (15 \sin (d+e x)+43 \cos (d+e x)+12)}{\sqrt{\sqrt{34} \cos \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)+2}}}{3037500 e}","-\frac{199 (5 \cos (d+e x)-3 \sin (d+e x))}{101250 e \sqrt{5 \sin (d+e x)+3 \cos (d+e x)+2}}+\frac{8 (5 \cos (d+e x)-3 \sin (d+e x))}{3375 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}-\frac{5 \cos (d+e x)-3 \sin (d+e x)}{75 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{5/2}}-\frac{8 F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{3375 \sqrt{2+\sqrt{34}} e}-\frac{199 \sqrt{2+\sqrt{34}} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}\left(\frac{5}{3}\right)\right)|\frac{2}{15} \left(17-\sqrt{34}\right)\right)}{101250 e}",1,"(-13532*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]] + (597*(12 + 43*Cos[d + e*x] + 15*Sin[d + e*x]))/Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]] + (27000*(5 + 17*Sin[d + e*x]))/(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2) - (300*(305 + 272*Sin[d + e*x]))/(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2) + (20*(1595 + 3383*Sin[d + e*x]))/Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]] - 638*Sqrt[30]*AppellF1[1/2, 1/2, 1/2, 3/2, (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Sin[d + e*x + ArcTan[3/5]])/(17 + Sqrt[34])]*Sqrt[Cos[d + e*x + ArcTan[3/5]]^2]*Sec[d + e*x + ArcTan[3/5]]*Sqrt[2 + Sqrt[34]*Sin[d + e*x + ArcTan[3/5]]] + (2985*Sqrt[30]*AppellF1[-1/2, -1/2, -1/2, 1/2, (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(-17 + Sqrt[34]), (Sqrt[34] + 17*Cos[d + e*x - ArcTan[5/3]])/(17 + Sqrt[34])]*Csc[d + e*x - ArcTan[5/3]]*Sqrt[Sin[d + e*x - ArcTan[5/3]]^2])/Sqrt[2 + Sqrt[34]*Cos[d + e*x - ArcTan[5/3]]])/(3037500*e)","C",0
410,1,3767,347,6.6187013,"\int (a+b \cos (d+e x)+c \sin (d+e x))^{5/2} \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2),x]","\text{Result too large to show}","-\frac{16 a \left(a^2-b^2-c^2\right) \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{15 e \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{2 \left(23 a^2+9 \left(b^2+c^2\right)\right) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{15 e \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}{5 e}-\frac{16 (a c \cos (d+e x)-a b \sin (d+e x)) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}{15 e}",1,"(Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]*((2*b*(23*a^2 + 9*b^2 + 9*c^2))/(15*c) - (22*a*c*Cos[d + e*x])/15 - (2*b*c*Cos[2*(d + e*x)])/5 + (22*a*b*Sin[d + e*x])/15 + ((b^2 - c^2)*Sin[2*(d + e*x)])/5))/e + (2*a^3*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*e) + (34*a*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(15*Sqrt[1 + b^2/c^2]*c*e) + (34*a*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(15*Sqrt[1 + b^2/c^2]*e) + (23*a^2*b^2*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(15*c*e) + (3*b^4*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(5*c*e) + (23*a^2*c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(15*e) + (6*b^2*c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(5*e) + (3*c^3*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(5*e)","C",0
411,1,2190,283,6.2644234,"\int (a+b \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \left(a^2-b^2-c^2\right) \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{3 e \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{8 a \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{3 e \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}",1,"(((8*a*b)/(3*c) - (2*c*Cos[d + e*x])/3 + (2*b*Sin[d + e*x])/3)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/e + (2*a^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*e) + (2*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(3*Sqrt[1 + b^2/c^2]*c*e) + (2*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(3*Sqrt[1 + b^2/c^2]*e) + (4*a*b^2*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(3*c*e) + (4*a*c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(3*e)","C",0
412,1,1408,108,6.2310293,"\int \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} \, dx","Integrate[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","\frac{\left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(1-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}\right)},-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}-1\right)}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \sqrt{\frac{b \sqrt{\frac{b^2+c^2}{b^2}}-b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{a+b \sqrt{\frac{b^2+c^2}{b^2}}}} \sqrt{a+b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)} \sqrt{\frac{\sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right) b+\sqrt{\frac{b^2+c^2}{b^2}} b}{b \sqrt{\frac{b^2+c^2}{b^2}}-a}}}-\frac{\frac{2 b \left(a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)\right)}{b^2+c^2}-\frac{c \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1}}}{\sqrt{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}}\right) b^2}{c e}+\frac{2 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} b}{c e}+\frac{c \left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(1-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}\right)},-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}-1\right)}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \sqrt{\frac{b \sqrt{\frac{b^2+c^2}{b^2}}-b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{a+b \sqrt{\frac{b^2+c^2}{b^2}}}} \sqrt{a+b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)} \sqrt{\frac{\sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right) b+\sqrt{\frac{b^2+c^2}{b^2}} b}{b \sqrt{\frac{b^2+c^2}{b^2}}-a}}}-\frac{\frac{2 b \left(a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)\right)}{b^2+c^2}-\frac{c \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1}}}{\sqrt{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}}\right)}{e}+\frac{2 a F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{a+\sqrt{\frac{b^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{\sqrt{\frac{b^2}{c^2}+1} \left(1-\frac{a}{\sqrt{\frac{b^2}{c^2}+1} c}\right) c},-\frac{a+\sqrt{\frac{b^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{\sqrt{\frac{b^2}{c^2}+1} \left(-\frac{a}{\sqrt{\frac{b^2}{c^2}+1} c}-1\right) c}\right) \sec \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right) \sqrt{\frac{c \sqrt{\frac{b^2+c^2}{c^2}}-c \sqrt{\frac{b^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{a+c \sqrt{\frac{b^2+c^2}{c^2}}}} \sqrt{a+c \sqrt{\frac{b^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)} \sqrt{\frac{\sqrt{\frac{b^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right) c+\sqrt{\frac{b^2+c^2}{c^2}} c}{c \sqrt{\frac{b^2+c^2}{c^2}}-a}}}{\sqrt{\frac{b^2}{c^2}+1} c e}","\frac{2 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{e \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}",1,"(2*b*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(c*e) + (2*a*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*e) + (b^2*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(c*e) + (c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/e","C",0
413,1,285,108,0.5624602,"\int \frac{1}{\sqrt{a+b \cos (d+e x)+c \sin (d+e x)}} \, dx","Integrate[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","\frac{2 \sec \left(\tan ^{-1}\left(\frac{b}{c}\right)+d+e x\right) \sqrt{-\frac{c \sqrt{\frac{b^2}{c^2}+1} \left(\sin \left(\tan ^{-1}\left(\frac{b}{c}\right)+d+e x\right)-1\right)}{a+c \sqrt{\frac{b^2}{c^2}+1}}} \sqrt{\frac{c \sqrt{\frac{b^2}{c^2}+1} \left(\sin \left(\tan ^{-1}\left(\frac{b}{c}\right)+d+e x\right)+1\right)}{c \sqrt{\frac{b^2}{c^2}+1}-a}} \sqrt{a+c \sqrt{\frac{b^2}{c^2}+1} \sin \left(\tan ^{-1}\left(\frac{b}{c}\right)+d+e x\right)} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{a+\sqrt{\frac{b^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{a-\sqrt{\frac{b^2}{c^2}+1} c},\frac{a+\sqrt{\frac{b^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{a+\sqrt{\frac{b^2}{c^2}+1} c}\right)}{c e \sqrt{\frac{b^2}{c^2}+1}}","\frac{2 \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{e \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}",1,"(2*AppellF1[1/2, 1/2, 1/2, 3/2, (a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(a - Sqrt[1 + b^2/c^2]*c), (a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(a + Sqrt[1 + b^2/c^2]*c)]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[-((Sqrt[1 + b^2/c^2]*c*(-1 + Sin[d + e*x + ArcTan[b/c]]))/(a + Sqrt[1 + b^2/c^2]*c))]*Sqrt[(Sqrt[1 + b^2/c^2]*c*(1 + Sin[d + e*x + ArcTan[b/c]]))/(-a + Sqrt[1 + b^2/c^2]*c)]*Sqrt[a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]]])/(Sqrt[1 + b^2/c^2]*c*e)","C",0
414,1,1540,186,6.3538101,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3/2),x]","-\frac{\left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(1-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}\right)},-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}-1\right)}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \sqrt{\frac{b \sqrt{\frac{b^2+c^2}{b^2}}-b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{a+b \sqrt{\frac{b^2+c^2}{b^2}}}} \sqrt{a+b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)} \sqrt{\frac{\sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right) b+\sqrt{\frac{b^2+c^2}{b^2}} b}{b \sqrt{\frac{b^2+c^2}{b^2}}-a}}}-\frac{\frac{2 b \left(a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)\right)}{b^2+c^2}-\frac{c \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1}}}{\sqrt{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}}\right) b^2}{c \left(-a^2+b^2+c^2\right) e}+\frac{\sqrt{a+b \cos (d+e x)+c \sin (d+e x)} \left(\frac{2 \left(\sin (d+e x) b^2+a c+c^2 \sin (d+e x)\right)}{b \left(-a^2+b^2+c^2\right) (a+b \cos (d+e x)+c \sin (d+e x))}-\frac{2 \left(b^2+c^2\right)}{b c \left(-a^2+b^2+c^2\right)}\right)}{e}-\frac{c \left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(1-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}\right)},-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}-1\right)}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \sqrt{\frac{b \sqrt{\frac{b^2+c^2}{b^2}}-b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{a+b \sqrt{\frac{b^2+c^2}{b^2}}}} \sqrt{a+b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)} \sqrt{\frac{\sqrt{\frac{b^2+c^2}{b^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right) b+\sqrt{\frac{b^2+c^2}{b^2}} b}{b \sqrt{\frac{b^2+c^2}{b^2}}-a}}}-\frac{\frac{2 b \left(a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)\right)}{b^2+c^2}-\frac{c \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1}}}{\sqrt{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}}\right)}{\left(-a^2+b^2+c^2\right) e}-\frac{2 a F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{a+\sqrt{\frac{b^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{\sqrt{\frac{b^2}{c^2}+1} \left(1-\frac{a}{\sqrt{\frac{b^2}{c^2}+1} c}\right) c},-\frac{a+\sqrt{\frac{b^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{\sqrt{\frac{b^2}{c^2}+1} \left(-\frac{a}{\sqrt{\frac{b^2}{c^2}+1} c}-1\right) c}\right) \sec \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right) \sqrt{\frac{c \sqrt{\frac{b^2+c^2}{c^2}}-c \sqrt{\frac{b^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{a+c \sqrt{\frac{b^2+c^2}{c^2}}}} \sqrt{a+c \sqrt{\frac{b^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right)} \sqrt{\frac{\sqrt{\frac{b^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{b}{c}\right)\right) c+\sqrt{\frac{b^2+c^2}{c^2}} c}{c \sqrt{\frac{b^2+c^2}{c^2}}-a}}}{\sqrt{\frac{b^2}{c^2}+1} c \left(-a^2+b^2+c^2\right) e}","\frac{2 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{e \left(a^2-b^2-c^2\right) \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{e \left(a^2-b^2-c^2\right) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}",1,"(Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]*((-2*(b^2 + c^2))/(b*c*(-a^2 + b^2 + c^2)) + (2*(a*c + b^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))))/e - (2*a*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)*e) - (b^2*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(c*(-a^2 + b^2 + c^2)*e) - (c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/((-a^2 + b^2 + c^2)*e)","C",0
415,1,2408,382,6.4093247,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-5/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{3 e \left(a^2-b^2-c^2\right) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{8 a \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{3 e \left(a^2-b^2-c^2\right)^2 \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}+\frac{8 (a c \cos (d+e x)-a b \sin (d+e x))}{3 e \left(a^2-b^2-c^2\right)^2 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{3 e \left(a^2-b^2-c^2\right) (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}",1,"(Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]*((8*a*(b^2 + c^2))/(3*b*c*(a^2 - b^2 - c^2)^2) + (2*(a*c + b^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(3*b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (2*(3*a^2*c + b^2*c + c^3 + 4*a*b^2*Sin[d + e*x] + 4*a*c^2*Sin[d + e*x]))/(3*b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))))/e + (2*a^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)^2*e) + (2*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(3*Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)^2*e) + (2*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(3*Sqrt[1 + b^2/c^2]*(-a^2 + b^2 + c^2)^2*e) + (4*a*b^2*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(3*c*(-a^2 + b^2 + c^2)^2*e) + (4*a*c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(3*(-a^2 + b^2 + c^2)^2*e)","C",0
416,1,4116,490,6.6603914,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx","Integrate[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-7/2),x]","\text{Result too large to show}","-\frac{16 a \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{15 e \left(a^2-b^2-c^2\right)^2 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{2 \left(23 a^2+9 \left(b^2+c^2\right)\right) \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{15 e \left(a^2-b^2-c^2\right)^3 \sqrt{\frac{a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 \left(c \left(23 a^2+9 \left(b^2+c^2\right)\right) \cos (d+e x)-b \left(23 a^2+9 \left(b^2+c^2\right)\right) \sin (d+e x)\right)}{15 e \left(a^2-b^2-c^2\right)^3 \sqrt{a+b \cos (d+e x)+c \sin (d+e x)}}+\frac{16 (a c \cos (d+e x)-a b \sin (d+e x))}{15 e \left(a^2-b^2-c^2\right)^2 (a+b \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{5 e \left(a^2-b^2-c^2\right) (a+b \cos (d+e x)+c \sin (d+e x))^{5/2}}",1,"(Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]*((-2*(b^2 + c^2)*(23*a^2 + 9*b^2 + 9*c^2))/(15*b*c*(-a^2 + b^2 + c^2)^3) + (2*(a*c + b^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(5*b*(-a^2 + b^2 + c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3) - (2*(5*a^2*c + 3*b^2*c + 3*c^3 + 8*a*b^2*Sin[d + e*x] + 8*a*c^2*Sin[d + e*x]))/(15*b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (2*(15*a^3*c + 17*a*b^2*c + 17*a*c^3 + 23*a^2*b^2*Sin[d + e*x] + 9*b^4*Sin[d + e*x] + 23*a^2*c^2*Sin[d + e*x] + 18*b^2*c^2*Sin[d + e*x] + 9*c^4*Sin[d + e*x]))/(15*b*(-a^2 + b^2 + c^2)^3*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))))/e - (2*a^3*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)^3*e) - (34*a*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(15*Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)^3*e) - (34*a*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(15*Sqrt[1 + b^2/c^2]*(-a^2 + b^2 + c^2)^3*e) - (23*a^2*b^2*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(15*c*(-a^2 + b^2 + c^2)^3*e) - (3*b^4*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(5*c*(-a^2 + b^2 + c^2)^3*e) - (23*a^2*c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(15*(-a^2 + b^2 + c^2)^3*e) - (6*b^2*c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(5*(-a^2 + b^2 + c^2)^3*e) - (3*c^3*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[d + e*x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[d + e*x - ArcTan[c/b]]]))/(5*(-a^2 + b^2 + c^2)^3*e)","C",0
417,1,130,139,0.5952853,"\int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx","Integrate[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2),x]","-\frac{(3 \sin (d+e x)+4 \cos (d+e x)+5)^{5/2} \left(3750 \cos \left(\frac{1}{2} (d+e x)\right)+1625 \cos \left(\frac{3}{2} (d+e x)\right)+3 \left(-3750 \sin \left(\frac{1}{2} (d+e x)\right)-375 \sin \left(\frac{3}{2} (d+e x)\right)+3 \sin \left(\frac{5}{2} (d+e x)\right)+79 \cos \left(\frac{5}{2} (d+e x)\right)\right)\right)}{30 e \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right)^5}","-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}{5 e}-\frac{16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}}{3 e}-\frac{320 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}}",1,"-1/30*((5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2)*(3750*Cos[(d + e*x)/2] + 1625*Cos[(3*(d + e*x))/2] + 3*(79*Cos[(5*(d + e*x))/2] - 3750*Sin[(d + e*x)/2] - 375*Sin[(3*(d + e*x))/2] + 3*Sin[(5*(d + e*x))/2])))/(e*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^5)","A",1
418,1,104,93,0.3325281,"\int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx","Integrate[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2),x]","\frac{(3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2} \left(9 \left(15 \sin \left(\frac{1}{2} (d+e x)\right)+\sin \left(\frac{3}{2} (d+e x)\right)\right)-45 \cos \left(\frac{1}{2} (d+e x)\right)-13 \cos \left(\frac{3}{2} (d+e x)\right)\right)}{3 e \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right)^3}","-\frac{2 \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5} (3 \cos (d+e x)-4 \sin (d+e x))}{3 e}-\frac{40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}}",1,"((5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2)*(-45*Cos[(d + e*x)/2] - 13*Cos[(3*(d + e*x))/2] + 9*(15*Sin[(d + e*x)/2] + Sin[(3*(d + e*x))/2])))/(3*e*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^3)","A",1
419,1,75,44,0.0404819,"\int \sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx","Integrate[Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]","-\frac{2 \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}}{e \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right)}","-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}}",1,"(-2*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(e*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2]))","A",1
420,1,101,48,0.1046402,"\int \frac{1}{\sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx","Integrate[1/Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]","-\frac{\left(\frac{2}{5}+\frac{6 i}{5}\right) \sqrt{\frac{4}{5}+\frac{3 i}{5}} \tan ^{-1}\left(\left(\frac{1}{10}+\frac{3 i}{10}\right) \sqrt{\frac{4}{5}+\frac{3 i}{5}} \left(3 \tan \left(\frac{1}{4} (d+e x)\right)-1\right)\right) \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right)}{e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}}","\frac{\sqrt{\frac{2}{5}} \tanh ^{-1}\left(\frac{\sin \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)}{\sqrt{2} \sqrt{\cos \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)+1}}\right)}{e}",1,"((-2/5 - (6*I)/5)*Sqrt[4/5 + (3*I)/5]*ArcTan[(1/10 + (3*I)/10)*Sqrt[4/5 + (3*I)/5]*(-1 + 3*Tan[(d + e*x)/4])]*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2]))/(e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])","C",1
421,1,154,96,0.2949567,"\int \frac{1}{(5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx","Integrate[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-3/2),x]","-\frac{\left(\frac{1}{250}-\frac{i}{125}\right) \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right) \left((5+10 i) \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right)-(1-i) \sqrt{20+15 i} \tan ^{-1}\left(\left(\frac{1}{10}+\frac{3 i}{10}\right) \sqrt{\frac{4}{5}+\frac{3 i}{5}} \left(3 \tan \left(\frac{1}{4} (d+e x)\right)-1\right)\right) \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right)^2\right)}{e (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sin \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)}{\sqrt{2} \sqrt{\cos \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)+1}}\right)}{10 \sqrt{10} e}-\frac{3 \cos (d+e x)-4 \sin (d+e x)}{10 e (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}",1,"((-1/250 + I/125)*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])*((5 + 10*I)*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2]) - (1 - I)*Sqrt[20 + 15*I]*ArcTan[(1/10 + (3*I)/10)*Sqrt[4/5 + (3*I)/5]*(-1 + 3*Tan[(d + e*x)/4])]*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^2))/(e*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))","C",1
422,1,180,142,0.4051086,"\int \frac{1}{(5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx","Integrate[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-5/2),x]","-\frac{\left(\frac{1}{20000}-\frac{i}{10000}\right) \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right) \left((5+10 i) \left(-165 \sin \left(\frac{1}{2} (d+e x)\right)-27 \sin \left(\frac{3}{2} (d+e x)\right)+55 \cos \left(\frac{1}{2} (d+e x)\right)+39 \cos \left(\frac{3}{2} (d+e x)\right)\right)-(6-6 i) \sqrt{20+15 i} \tan ^{-1}\left(\left(\frac{1}{10}+\frac{3 i}{10}\right) \sqrt{\frac{4}{5}+\frac{3 i}{5}} \left(3 \tan \left(\frac{1}{4} (d+e x)\right)-1\right)\right) \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right)^4\right)}{e (3 \sin (d+e x)+4 \cos (d+e x)+5)^{5/2}}","-\frac{3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}-\frac{3 \cos (d+e x)-4 \sin (d+e x)}{20 e (3 \sin (d+e x)+4 \cos (d+e x)+5)^{5/2}}+\frac{3 \tanh ^{-1}\left(\frac{\sin \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)}{\sqrt{2} \sqrt{\cos \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)+1}}\right)}{400 \sqrt{10} e}",1,"((-1/20000 + I/10000)*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])*((-6 + 6*I)*Sqrt[20 + 15*I]*ArcTan[(1/10 + (3*I)/10)*Sqrt[4/5 + (3*I)/5]*(-1 + 3*Tan[(d + e*x)/4])]*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])^4 + (5 + 10*I)*(55*Cos[(d + e*x)/2] + 39*Cos[(3*(d + e*x))/2] - 165*Sin[(d + e*x)/2] - 27*Sin[(3*(d + e*x))/2])))/(e*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2))","C",1
423,1,151,185,1.8182875,"\int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx","Integrate[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(7/2),x]","\frac{(3 \sin (d+e x)+4 \cos (d+e x)-5)^{7/2} \left(30625 \sin \left(\frac{1}{2} (d+e x)\right)-15925 \sin \left(\frac{3}{2} (d+e x)\right)+3871 \sin \left(\frac{5}{2} (d+e x)\right)-307 \sin \left(\frac{7}{2} (d+e x)\right)+91875 \cos \left(\frac{1}{2} (d+e x)\right)-11025 \cos \left(\frac{3}{2} (d+e x)\right)-147 \cos \left(\frac{5}{2} (d+e x)\right)+249 \cos \left(\frac{7}{2} (d+e x)\right)\right)}{28 e \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right)^7}","-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}{7 e}+\frac{24 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}{7 e}-\frac{320 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}{7 e}+\frac{6400 (3 \cos (d+e x)-4 \sin (d+e x))}{7 e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}",1,"((-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(7/2)*(91875*Cos[(d + e*x)/2] - 11025*Cos[(3*(d + e*x))/2] - 147*Cos[(5*(d + e*x))/2] + 249*Cos[(7*(d + e*x))/2] + 30625*Sin[(d + e*x)/2] - 15925*Sin[(3*(d + e*x))/2] + 3871*Sin[(5*(d + e*x))/2] - 307*Sin[(7*(d + e*x))/2]))/(28*e*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])^7)","A",1
424,1,127,139,0.4891559,"\int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx","Integrate[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2),x]","\frac{(3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2} \left(3750 \sin \left(\frac{1}{2} (d+e x)\right)-1625 \sin \left(\frac{3}{2} (d+e x)\right)+237 \sin \left(\frac{5}{2} (d+e x)\right)+11250 \cos \left(\frac{1}{2} (d+e x)\right)-1125 \cos \left(\frac{3}{2} (d+e x)\right)-9 \cos \left(\frac{5}{2} (d+e x)\right)\right)}{30 e \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right)^5}","-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}{5 e}+\frac{16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}{3 e}-\frac{320 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}",1,"((-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2)*(11250*Cos[(d + e*x)/2] - 1125*Cos[(3*(d + e*x))/2] - 9*Cos[(5*(d + e*x))/2] + 3750*Sin[(d + e*x)/2] - 1625*Sin[(3*(d + e*x))/2] + 237*Sin[(5*(d + e*x))/2]))/(30*e*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])^5)","A",1
425,1,103,93,0.2225777,"\int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx","Integrate[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2),x]","\frac{(3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2} \left(45 \sin \left(\frac{1}{2} (d+e x)\right)-13 \sin \left(\frac{3}{2} (d+e x)\right)+135 \cos \left(\frac{1}{2} (d+e x)\right)-9 \cos \left(\frac{3}{2} (d+e x)\right)\right)}{3 e \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right)^3}","\frac{40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}{3 e}",1,"((-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2)*(135*Cos[(d + e*x)/2] - 9*Cos[(3*(d + e*x))/2] + 45*Sin[(d + e*x)/2] - 13*Sin[(3*(d + e*x))/2]))/(3*e*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])^3)","A",1
426,1,75,44,0.0437016,"\int \sqrt{-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx","Integrate[Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]","\frac{2 \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}{e \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right)}","-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}",1,"(2*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(e*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2]))","A",1
427,1,99,49,0.0933217,"\int \frac{1}{\sqrt{-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx","Integrate[1/Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]","\frac{\left(\frac{2}{5}+\frac{6 i}{5}\right) \sqrt{-\frac{4}{5}-\frac{3 i}{5}} \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right) \tanh ^{-1}\left(\left(\frac{1}{10}+\frac{3 i}{10}\right) \sqrt{-\frac{4}{5}-\frac{3 i}{5}} \left(\tan \left(\frac{1}{4} (d+e x)\right)+3\right)\right)}{e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)-5}}","-\frac{\sqrt{\frac{2}{5}} \tan ^{-1}\left(\frac{\sin \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)}{\sqrt{2} \sqrt{\cos \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)-1}}\right)}{e}",1,"((2/5 + (6*I)/5)*Sqrt[-4/5 - (3*I)/5]*ArcTanh[(1/10 + (3*I)/10)*Sqrt[-4/5 - (3*I)/5]*(3 + Tan[(d + e*x)/4])]*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2]))/(e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])","C",1
428,1,152,96,0.3125463,"\int \frac{1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx","Integrate[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-3/2),x]","\frac{\left(\frac{1}{250}-\frac{i}{125}\right) \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right) \left((5+10 i) \left(\sin \left(\frac{1}{2} (d+e x)\right)+3 \cos \left(\frac{1}{2} (d+e x)\right)\right)-(1-i) \sqrt{-20-15 i} \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{10}+\frac{3 i}{10}\right) \sqrt{-\frac{4}{5}-\frac{3 i}{5}} \left(\tan \left(\frac{1}{4} (d+e x)\right)+3\right)\right)\right)}{e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}","\frac{3 \cos (d+e x)-4 \sin (d+e x)}{10 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sin \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)}{\sqrt{2} \sqrt{\cos \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)-1}}\right)}{10 \sqrt{10} e}",1,"((1/250 - I/125)*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])*((-1 + I)*Sqrt[-20 - 15*I]*ArcTanh[(1/10 + (3*I)/10)*Sqrt[-4/5 - (3*I)/5]*(3 + Tan[(d + e*x)/4])]*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])^2 + (5 + 10*I)*(3*Cos[(d + e*x)/2] + Sin[(d + e*x)/2])))/(e*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))","C",1
429,1,178,142,0.3792712,"\int \frac{1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx","Integrate[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-5/2),x]","\frac{\left(\frac{1}{10000}+\frac{i}{20000}\right) \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right) \left((10-5 i) \left(55 \sin \left(\frac{1}{2} (d+e x)\right)-39 \sin \left(\frac{3}{2} (d+e x)\right)+165 \cos \left(\frac{1}{2} (d+e x)\right)-27 \cos \left(\frac{3}{2} (d+e x)\right)\right)+(6+6 i) \sqrt{-20-15 i} \left(\cos \left(\frac{1}{2} (d+e x)\right)-3 \sin \left(\frac{1}{2} (d+e x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{10}+\frac{3 i}{10}\right) \sqrt{-\frac{4}{5}-\frac{3 i}{5}} \left(\tan \left(\frac{1}{4} (d+e x)\right)+3\right)\right)\right)}{e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}","-\frac{3 (3 \cos (d+e x)-4 \sin (d+e x))}{400 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}+\frac{3 \cos (d+e x)-4 \sin (d+e x)}{20 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2}}-\frac{3 \tan ^{-1}\left(\frac{\sin \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)}{\sqrt{2} \sqrt{\cos \left(d+e x-\tan ^{-1}\left(\frac{3}{4}\right)\right)-1}}\right)}{400 \sqrt{10} e}",1,"((1/10000 + I/20000)*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])*((6 + 6*I)*Sqrt[-20 - 15*I]*ArcTanh[(1/10 + (3*I)/10)*Sqrt[-4/5 - (3*I)/5]*(3 + Tan[(d + e*x)/4])]*(Cos[(d + e*x)/2] - 3*Sin[(d + e*x)/2])^4 + (10 - 5*I)*(165*Cos[(d + e*x)/2] - 27*Cos[(3*(d + e*x))/2] + 55*Sin[(d + e*x)/2] - 39*Sin[(3*(d + e*x))/2])))/(e*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2))","C",1
430,1,11888,258,33.4559154,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{7/2} \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(7/2),x]","\text{Result too large to show}","-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2}}{7 e}-\frac{24 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}}{35 e}-\frac{64 \left(b^2+c^2\right) (c \cos (d+e x)-b \sin (d+e x)) \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{35 e}-\frac{256 \left(b^2+c^2\right)^{3/2} (c \cos (d+e x)-b \sin (d+e x))}{35 e \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}",1,"Result too large to show","C",0
431,1,11771,190,34.2556347,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2} \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}}{5 e}-\frac{16 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{15 e}-\frac{64 \left(b^2+c^2\right) (c \cos (d+e x)-b \sin (d+e x))}{15 e \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}",1,"Result too large to show","C",0
432,1,11679,126,33.0145071,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2} \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} (c \cos (d+e x)-b \sin (d+e x))}{3 e}-\frac{8 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}",1,"Result too large to show","C",0
433,1,11586,55,32.708037,"\int \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx","Integrate[Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","\text{Result too large to show}","-\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}",1,"Result too large to show","C",0
434,1,63264,88,33.826928,"\int \frac{1}{\sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx","Integrate[1/Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","\text{Result too large to show}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt[4]{b^2+c^2} \sin \left(-\tan ^{-1}(b,c)+d+e x\right)}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left(-\tan ^{-1}(b,c)+d+e x\right)+\sqrt{b^2+c^2}}}\right)}{e \sqrt[4]{b^2+c^2}}",1,"Result too large to show","C",0
435,-1,0,160,180.0153517,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}} \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3/2),x]","\text{\$Aborted}","\frac{\tanh ^{-1}\left(\frac{\sqrt[4]{b^2+c^2} \sin \left(-\tan ^{-1}(b,c)+d+e x\right)}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left(-\tan ^{-1}(b,c)+d+e x\right)+\sqrt{b^2+c^2}}}\right)}{2 \sqrt{2} e \left(b^2+c^2\right)^{3/4}}-\frac{c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt{b^2+c^2} \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}}",1,"$Aborted","F",-1
436,-1,0,226,180.031864,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2}} \, dx","Integrate[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-5/2),x]","\text{\$Aborted}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt[4]{b^2+c^2} \sin \left(-\tan ^{-1}(b,c)+d+e x\right)}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left(-\tan ^{-1}(b,c)+d+e x\right)+\sqrt{b^2+c^2}}}\right)}{16 \sqrt{2} e \left(b^2+c^2\right)^{5/4}}-\frac{3 (c \cos (d+e x)-b \sin (d+e x))}{16 e \left(b^2+c^2\right) \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}}-\frac{c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt{b^2+c^2} \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2}}",1,"$Aborted","F",-1
437,1,11602,196,34.1894678,"\int \left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2} \, dx","Integrate[(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}}{5 e}+\frac{16 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x)) \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{15 e}-\frac{64 \left(b^2+c^2\right) (c \cos (d+e x)-b \sin (d+e x))}{15 e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}",1,"Result too large to show","C",0
438,1,11512,130,21.6110663,"\int \left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2} \, dx","Integrate[(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2),x]","\text{Result too large to show}","\frac{8 \sqrt{b^2+c^2} (c \cos (d+e x)-b \sin (d+e x))}{3 e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}-\frac{2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}{3 e}",1,"Result too large to show","C",0
439,1,11415,57,21.9477963,"\int \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx","Integrate[Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","\text{Result too large to show}","-\frac{2 (c \cos (d+e x)-b \sin (d+e x))}{e \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}}",1,"Result too large to show","C",0
440,1,61904,91,34.4697327,"\int \frac{1}{\sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx","Integrate[1/Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","\text{Result too large to show}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt[4]{b^2+c^2} \sin \left(-\tan ^{-1}(b,c)+d+e x\right)}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left(-\tan ^{-1}(b,c)+d+e x\right)-\sqrt{b^2+c^2}}}\right)}{e \sqrt[4]{b^2+c^2}}",1,"Result too large to show","C",0
441,-1,0,164,180.0010161,"\int \frac{1}{\left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}} \, dx","Integrate[(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3/2),x]","\text{\$Aborted}","\frac{\tan ^{-1}\left(\frac{\sqrt[4]{b^2+c^2} \sin \left(-\tan ^{-1}(b,c)+d+e x\right)}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left(-\tan ^{-1}(b,c)+d+e x\right)-\sqrt{b^2+c^2}}}\right)}{2 \sqrt{2} e \left(b^2+c^2\right)^{3/4}}+\frac{c \cos (d+e x)-b \sin (d+e x)}{2 e \sqrt{b^2+c^2} \left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}}",1,"$Aborted","F",-1
442,-1,0,232,180.0612735,"\int \frac{1}{\left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2}} \, dx","Integrate[(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-5/2),x]","\text{\$Aborted}","-\frac{3 \tan ^{-1}\left(\frac{\sqrt[4]{b^2+c^2} \sin \left(-\tan ^{-1}(b,c)+d+e x\right)}{\sqrt{2} \sqrt{\sqrt{b^2+c^2} \cos \left(-\tan ^{-1}(b,c)+d+e x\right)-\sqrt{b^2+c^2}}}\right)}{16 \sqrt{2} e \left(b^2+c^2\right)^{5/4}}-\frac{3 (c \cos (d+e x)-b \sin (d+e x))}{16 e \left(b^2+c^2\right) \left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}}+\frac{c \cos (d+e x)-b \sin (d+e x)}{4 e \sqrt{b^2+c^2} \left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2}}",1,"$Aborted","F",-1
443,1,80,101,0.216941,"\int \frac{\sin (x)}{a+b \cos (x)+c \sin (x)} \, dx","Integrate[Sin[x]/(a + b*Cos[x] + c*Sin[x]),x]","\frac{\frac{2 a c \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\sqrt{-a^2+b^2+c^2}}-b \log (a+b \cos (x)+c \sin (x))+c x}{b^2+c^2}","-\frac{2 a c \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(b^2+c^2\right) \sqrt{a^2-b^2-c^2}}-\frac{b \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}+\frac{c x}{b^2+c^2}",1,"(c*x + (2*a*c*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/Sqrt[-a^2 + b^2 + c^2] - b*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",1
444,1,22,22,0.0407968,"\int \frac{\sin (x)}{1+\cos (x)+\sin (x)} \, dx","Integrate[Sin[x]/(1 + Cos[x] + Sin[x]),x]","\frac{x}{2}-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\frac{x}{2}-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)",1,"x/2 - Log[Cos[x/2] + Sin[x/2]]","A",1
445,1,79,97,0.1890739,"\int \frac{1}{a+c \sec (x)+b \tan (x)} \, dx","Integrate[(a + c*Sec[x] + b*Tan[x])^(-1),x]","\frac{\frac{2 a c \tanh ^{-1}\left(\frac{(c-a) \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2+b^2-c^2}}\right)}{\sqrt{a^2+b^2-c^2}}+b \log (a \cos (x)+b \sin (x)+c)+a x}{a^2+b^2}","\frac{2 a c \tanh ^{-1}\left(\frac{b-(a-c) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\left(a^2+b^2\right) \sqrt{a^2+b^2-c^2}}+\frac{b \log (a \cos (x)+b \sin (x)+c)}{a^2+b^2}+\frac{a x}{a^2+b^2}",1,"(a*x + (2*a*c*ArcTanh[(b + (-a + c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/Sqrt[a^2 + b^2 - c^2] + b*Log[c + a*Cos[x] + b*Sin[x]])/(a^2 + b^2)","A",1
446,1,50,51,0.0435017,"\int \frac{\sec (x)}{a+c \sec (x)+b \tan (x)} \, dx","Integrate[Sec[x]/(a + c*Sec[x] + b*Tan[x]),x]","-\frac{2 \tanh ^{-1}\left(\frac{(c-a) \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2+b^2-c^2}}\right)}{\sqrt{a^2+b^2-c^2}}","-\frac{2 \tanh ^{-1}\left(\frac{b-(a-c) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\sqrt{a^2+b^2-c^2}}",1,"(-2*ArcTanh[(b + (-a + c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/Sqrt[a^2 + b^2 - c^2]","A",1
447,1,120,142,0.2840635,"\int \frac{\sec ^2(x)}{a+c \sec (x)+b \tan (x)} \, dx","Integrate[Sec[x]^2/(a + c*Sec[x] + b*Tan[x]),x]","\frac{\frac{2 a c \tanh ^{-1}\left(\frac{(c-a) \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2+b^2-c^2}}\right)}{\sqrt{a^2+b^2-c^2}}-b \log (a \cos (x)+b \sin (x)+c)+(b-c) \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+(b+c) \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{(c-b) (b+c)}","-\frac{2 a c \tanh ^{-1}\left(\frac{b-(a-c) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\left(b^2-c^2\right) \sqrt{a^2+b^2-c^2}}+\frac{b \log \left(-\left((a-c) \tan ^2\left(\frac{x}{2}\right)\right)+a+2 b \tan \left(\frac{x}{2}\right)+c\right)}{b^2-c^2}-\frac{\log \left(1-\tan \left(\frac{x}{2}\right)\right)}{b+c}-\frac{\log \left(\tan \left(\frac{x}{2}\right)+1\right)}{b-c}",1,"((2*a*c*ArcTanh[(b + (-a + c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/Sqrt[a^2 + b^2 - c^2] + (b - c)*Log[Cos[x/2] - Sin[x/2]] + (b + c)*Log[Cos[x/2] + Sin[x/2]] - b*Log[c + a*Cos[x] + b*Sin[x]])/((-b + c)*(b + c))","A",1
448,1,2490,371,6.4554646,"\int \frac{(a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{\sec ^{\frac{3}{2}}(d+e x)} \, dx","Integrate[(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)/Sec[d + e*x]^(3/2),x]","\text{Result too large to show}","\frac{2 \left(a^2-b^2+c^2\right) \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \sec ^{\frac{3}{2}}(d+e x) (a \cos (d+e x)+b+c \sin (d+e x))^2}+\frac{8 b (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \sec ^{\frac{3}{2}}(d+e x) (a \cos (d+e x)+b+c \sin (d+e x)) \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}}}-\frac{2 (c \cos (d+e x)-a \sin (d+e x)) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e \sec ^{\frac{3}{2}}(d+e x) (a \cos (d+e x)+b+c \sin (d+e x))}",1,"(((8*a*b)/(3*c) - (2*c*Cos[d + e*x])/3 + (2*a*Sin[d + e*x])/3)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(e*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])) + (2*a^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[a/c]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*Sqrt[1 + a^2/c^2]*c*e*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + (2*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[a/c]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(Sqrt[1 + a^2/c^2]*c*e*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + (2*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[a/c]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*Sqrt[1 + a^2/c^2]*e*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + (4*a^2*b*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]])*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*c*e*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + (4*b*c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]])*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*e*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))","C",0
449,1,1580,118,6.2488713,"\int \frac{\sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}{\sqrt{\sec (d+e x)}} \, dx","Integrate[Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]]/Sqrt[Sec[d + e*x]],x]","\frac{\left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(1-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}\right)},-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}-1\right)}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \sqrt{\frac{a \sqrt{\frac{a^2+c^2}{a^2}}-a \sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{\sqrt{\frac{a^2+c^2}{a^2}} a+b}} \sqrt{b+a \sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right) a+\sqrt{\frac{a^2+c^2}{a^2}} a}{a \sqrt{\frac{a^2+c^2}{a^2}}-b}}}-\frac{\frac{2 a \left(b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)\right)}{a^2+c^2}-\frac{c \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1}}}{\sqrt{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}}\right) \sqrt{a+b \sec (d+e x)+c \tan (d+e x)} a^2}{c e \sqrt{\sec (d+e x)} \sqrt{b+a \cos (d+e x)+c \sin (d+e x)}}+\frac{2 \sqrt{a+b \sec (d+e x)+c \tan (d+e x)} a}{c e \sqrt{\sec (d+e x)}}+\frac{c \left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(1-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}\right)},-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}-1\right)}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \sqrt{\frac{a \sqrt{\frac{a^2+c^2}{a^2}}-a \sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{\sqrt{\frac{a^2+c^2}{a^2}} a+b}} \sqrt{b+a \sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right) a+\sqrt{\frac{a^2+c^2}{a^2}} a}{a \sqrt{\frac{a^2+c^2}{a^2}}-b}}}-\frac{\frac{2 a \left(b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)\right)}{a^2+c^2}-\frac{c \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1}}}{\sqrt{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}}\right) \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}{e \sqrt{\sec (d+e x)} \sqrt{b+a \cos (d+e x)+c \sin (d+e x)}}+\frac{2 b F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(1-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}\right) c},-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}-1\right) c}\right) \sec \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right) \sqrt{\frac{c \sqrt{\frac{a^2+c^2}{c^2}}-c \sqrt{\frac{a^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)}{b+c \sqrt{\frac{a^2+c^2}{c^2}}}} \sqrt{b+c \sqrt{\frac{a^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right) c+\sqrt{\frac{a^2+c^2}{c^2}} c}{c \sqrt{\frac{a^2+c^2}{c^2}}-b}} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}{\sqrt{\frac{a^2}{c^2}+1} c e \sqrt{\sec (d+e x)} \sqrt{b+a \cos (d+e x)+c \sin (d+e x)}}","\frac{2 \sqrt{a+b \sec (d+e x)+c \tan (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \sqrt{\sec (d+e x)} \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}}}",1,"(2*a*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])/(c*e*Sqrt[Sec[d + e*x]]) + (2*b*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sec[d + e*x + ArcTan[a/c]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])/(Sqrt[1 + a^2/c^2]*c*e*Sqrt[Sec[d + e*x]]*Sqrt[b + a*Cos[d + e*x] + c*Sin[d + e*x]]) + (a^2*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]])*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])/(c*e*Sqrt[Sec[d + e*x]]*Sqrt[b + a*Cos[d + e*x] + c*Sin[d + e*x]]) + (c*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]])*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])/(e*Sqrt[Sec[d + e*x]]*Sqrt[b + a*Cos[d + e*x] + c*Sin[d + e*x]])","C",0
450,1,339,118,0.8664062,"\int \frac{\sqrt{\sec (d+e x)}}{\sqrt{a+b \sec (d+e x)+c \tan (d+e x)}} \, dx","Integrate[Sqrt[Sec[d + e*x]]/Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]],x]","\frac{2 \sqrt{\sec (d+e x)} \sec \left(\tan ^{-1}\left(\frac{a}{c}\right)+d+e x\right) \sqrt{-\frac{c \sqrt{\frac{a^2}{c^2}+1} \left(\sin \left(\tan ^{-1}\left(\frac{a}{c}\right)+d+e x\right)-1\right)}{c \sqrt{\frac{a^2}{c^2}+1}+b}} \sqrt{\frac{c \sqrt{\frac{a^2}{c^2}+1} \left(\sin \left(\tan ^{-1}\left(\frac{a}{c}\right)+d+e x\right)+1\right)}{c \sqrt{\frac{a^2}{c^2}+1}-b}} \sqrt{c \sqrt{\frac{a^2}{c^2}+1} \sin \left(\tan ^{-1}\left(\frac{a}{c}\right)+d+e x\right)+b} \sqrt{a \cos (d+e x)+b+c \sin (d+e x)} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)}{b-\sqrt{\frac{a^2}{c^2}+1} c},\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)}{b+\sqrt{\frac{a^2}{c^2}+1} c}\right)}{c e \sqrt{\frac{a^2}{c^2}+1} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}","\frac{2 \sqrt{\sec (d+e x)} \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}",1,"(2*AppellF1[1/2, 1/2, 1/2, 3/2, (b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(b - Sqrt[1 + a^2/c^2]*c), (b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(b + Sqrt[1 + a^2/c^2]*c)]*Sqrt[Sec[d + e*x]]*Sec[d + e*x + ArcTan[a/c]]*Sqrt[b + a*Cos[d + e*x] + c*Sin[d + e*x]]*Sqrt[-((Sqrt[1 + a^2/c^2]*c*(-1 + Sin[d + e*x + ArcTan[a/c]]))/(b + Sqrt[1 + a^2/c^2]*c))]*Sqrt[(Sqrt[1 + a^2/c^2]*c*(1 + Sin[d + e*x + ArcTan[a/c]]))/(-b + Sqrt[1 + a^2/c^2]*c)]*Sqrt[b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]]])/(Sqrt[1 + a^2/c^2]*c*e*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])","C",0
451,1,1732,240,6.4015793,"\int \frac{\sec ^{\frac{3}{2}}(d+e x)}{(a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx","Integrate[Sec[d + e*x]^(3/2)/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2),x]","-\frac{\sec ^{\frac{3}{2}}(d+e x) (b+a \cos (d+e x)+c \sin (d+e x))^{3/2} \left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(1-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}\right)},-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}-1\right)}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \sqrt{\frac{a \sqrt{\frac{a^2+c^2}{a^2}}-a \sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{\sqrt{\frac{a^2+c^2}{a^2}} a+b}} \sqrt{b+a \sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right) a+\sqrt{\frac{a^2+c^2}{a^2}} a}{a \sqrt{\frac{a^2+c^2}{a^2}}-b}}}-\frac{\frac{2 a \left(b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)\right)}{a^2+c^2}-\frac{c \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1}}}{\sqrt{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}}\right) a^2}{c \left(a^2-b^2+c^2\right) e (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}+\frac{\sec ^{\frac{3}{2}}(d+e x) (b+a \cos (d+e x)+c \sin (d+e x))^2 \left(\frac{2 \left(\sin (d+e x) a^2+b c+c^2 \sin (d+e x)\right)}{a \left(a^2-b^2+c^2\right) (b+a \cos (d+e x)+c \sin (d+e x))}-\frac{2 \left(a^2+c^2\right)}{a c \left(a^2-b^2+c^2\right)}\right)}{e (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac{c \sec ^{\frac{3}{2}}(d+e x) (b+a \cos (d+e x)+c \sin (d+e x))^{3/2} \left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(1-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}\right)},-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}-1\right)}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \sqrt{\frac{a \sqrt{\frac{a^2+c^2}{a^2}}-a \sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{\sqrt{\frac{a^2+c^2}{a^2}} a+b}} \sqrt{b+a \sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{a^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right) a+\sqrt{\frac{a^2+c^2}{a^2}} a}{a \sqrt{\frac{a^2+c^2}{a^2}}-b}}}-\frac{\frac{2 a \left(b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)\right)}{a^2+c^2}-\frac{c \sin \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1}}}{\sqrt{b+a \sqrt{\frac{c^2}{a^2}+1} \cos \left(d+e x-\tan ^{-1}\left(\frac{c}{a}\right)\right)}}\right)}{\left(a^2-b^2+c^2\right) e (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac{2 b F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(1-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}\right) c},-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}-1\right) c}\right) \sec ^{\frac{3}{2}}(d+e x) \sec \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right) (b+a \cos (d+e x)+c \sin (d+e x))^{3/2} \sqrt{\frac{c \sqrt{\frac{a^2+c^2}{c^2}}-c \sqrt{\frac{a^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)}{b+c \sqrt{\frac{a^2+c^2}{c^2}}}} \sqrt{b+c \sqrt{\frac{a^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{c^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{a}{c}\right)\right) c+\sqrt{\frac{a^2+c^2}{c^2}} c}{c \sqrt{\frac{a^2+c^2}{c^2}}-b}}}{\sqrt{\frac{a^2}{c^2}+1} c \left(a^2-b^2+c^2\right) e (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}","-\frac{2 \sec ^{\frac{3}{2}}(d+e x) (a \cos (d+e x)+b+c \sin (d+e x))^2 E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \left(a^2-b^2+c^2\right) \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac{2 \sec ^{\frac{3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+b+c \sin (d+e x))}{e \left(a^2-b^2+c^2\right) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}",1,"(Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2*((-2*(a^2 + c^2))/(a*c*(a^2 - b^2 + c^2)) + (2*(b*c + a^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(a*(a^2 - b^2 + c^2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))))/(e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)) - (2*b*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sec[d + e*x]^(3/2)*Sec[d + e*x + ArcTan[a/c]]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])/(Sqrt[1 + a^2/c^2]*c*(a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)) - (a^2*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]]))/(c*(a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)) - (c*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]]))/((a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))","C",0
452,1,2708,492,6.5531279,"\int \frac{\sec ^{\frac{5}{2}}(d+e x)}{(a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx","Integrate[Sec[d + e*x]^(5/2)/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \sec ^{\frac{5}{2}}(d+e x) \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} (a \cos (d+e x)+b+c \sin (d+e x))^2 F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \left(a^2-b^2+c^2\right) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}+\frac{8 b \sec ^{\frac{5}{2}}(d+e x) (a \cos (d+e x)+b+c \sin (d+e x))^3 E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \left(a^2-b^2+c^2\right)^2 \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}+\frac{8 \sec ^{\frac{5}{2}}(d+e x) (b c \cos (d+e x)-a b \sin (d+e x)) (a \cos (d+e x)+b+c \sin (d+e x))^2}{3 e \left(a^2-b^2+c^2\right)^2 (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}-\frac{2 \sec ^{\frac{5}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+b+c \sin (d+e x))}{3 e \left(a^2-b^2+c^2\right) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}",1,"(Sec[d + e*x]^(5/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^3*((8*b*(a^2 + c^2))/(3*a*c*(-a^2 + b^2 - c^2)^2) + (2*(b*c + a^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(3*a*(a^2 - b^2 + c^2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2) - (2*(a^2*c + 3*b^2*c + c^3 + 4*a^2*b*Sin[d + e*x] + 4*b*c^2*Sin[d + e*x]))/(3*a*(a^2 - b^2 + c^2)^2*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))))/(e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (2*a^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sec[d + e*x]^(5/2)*Sec[d + e*x + ArcTan[a/c]]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])/(3*Sqrt[1 + a^2/c^2]*c*(a^2 - b^2 + c^2)^2*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (2*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sec[d + e*x]^(5/2)*Sec[d + e*x + ArcTan[a/c]]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])/(Sqrt[1 + a^2/c^2]*c*(a^2 - b^2 + c^2)^2*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (2*c*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Sin[d + e*x + ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sec[d + e*x]^(5/2)*Sec[d + e*x + ArcTan[a/c]]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])/(3*Sqrt[1 + a^2/c^2]*(a^2 - b^2 + c^2)^2*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (4*a^2*b*Sec[d + e*x]^(5/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]]))/(3*c*(a^2 - b^2 + c^2)^2*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (4*b*c*Sec[d + e*x]^(5/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Cos[d + e*x - ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])) - ((2*a*(b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]))/(a^2 + c^2) - (c*Sin[d + e*x - ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]))/Sqrt[b + a*Sqrt[1 + c^2/a^2]*Cos[d + e*x - ArcTan[c/a]]]))/(3*(a^2 - b^2 + c^2)^2*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2))","C",0
453,0,0,371,150.8129989,"\int \cos ^{\frac{3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} \, dx","Integrate[Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2),x]","\int \cos ^{\frac{3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} \, dx","\frac{2 \left(a^2-b^2+c^2\right) \cos ^{\frac{3}{2}}(d+e x) \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e (a \cos (d+e x)+b+c \sin (d+e x))^2}+\frac{8 b \cos ^{\frac{3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e (a \cos (d+e x)+b+c \sin (d+e x)) \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}}}-\frac{2 \cos ^{\frac{3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e (a \cos (d+e x)+b+c \sin (d+e x))}",1,"Integrate[Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2), x]","F",-1
454,0,0,118,21.2211215,"\int \sqrt{\cos (d+e x)} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)} \, dx","Integrate[Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]],x]","\int \sqrt{\cos (d+e x)} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)} \, dx","\frac{2 \sqrt{\cos (d+e x)} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}}}",1,"Integrate[Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]], x]","F",-1
455,1,506,118,2.9014963,"\int \frac{1}{\sqrt{\cos (d+e x)} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}} \, dx","Integrate[1/(Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]]),x]","\frac{4 \left(\sqrt{a^2-b^2+c^2}+i a-i b+c\right) (\cos (d+e x)+i \sin (d+e x)) \sqrt{-\frac{i \left(\sqrt{a^2-b^2+c^2}+(a-b) \tan \left(\frac{1}{2} (d+e x)\right)-c\right)}{\left(\sqrt{a^2-b^2+c^2}-i a+i b-c\right) \left(\tan \left(\frac{1}{2} (d+e x)\right)-i\right)}} \sqrt{-\frac{i \left(\sqrt{a^2-b^2+c^2}+(b-a) \tan \left(\frac{1}{2} (d+e x)\right)+c\right)}{\left(\sqrt{a^2-b^2+c^2}+i a-i b+c\right) \left(\tan \left(\frac{1}{2} (d+e x)\right)-i\right)}} \sqrt{\frac{\left(\sqrt{a^2-b^2+c^2}-i a+i b+c\right) (-\cos (d+e x)+i \sin (d+e x))}{\sqrt{a^2-b^2+c^2}+i a-i b+c}} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(-i a+i b+c+\sqrt{a^2-b^2+c^2}\right) (i \sin (d+e x)-\cos (d+e x))}{i a-i b+c+\sqrt{a^2-b^2+c^2}}}\right)|\frac{b+i \sqrt{a^2-b^2+c^2}}{b-i \sqrt{a^2-b^2+c^2}}\right)}{e \left(a+i \left(\sqrt{a^2-b^2+c^2}+i b+c\right)\right) \sqrt{\cos (d+e x)} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}","\frac{2 \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \sqrt{\cos (d+e x)} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}",1,"(4*(I*a - I*b + c + Sqrt[a^2 - b^2 + c^2])*EllipticF[ArcSin[Sqrt[(((-I)*a + I*b + c + Sqrt[a^2 - b^2 + c^2])*(-Cos[d + e*x] + I*Sin[d + e*x]))/(I*a - I*b + c + Sqrt[a^2 - b^2 + c^2])]], (b + I*Sqrt[a^2 - b^2 + c^2])/(b - I*Sqrt[a^2 - b^2 + c^2])]*Sqrt[(((-I)*a + I*b + c + Sqrt[a^2 - b^2 + c^2])*(-Cos[d + e*x] + I*Sin[d + e*x]))/(I*a - I*b + c + Sqrt[a^2 - b^2 + c^2])]*(Cos[d + e*x] + I*Sin[d + e*x])*Sqrt[((-I)*(-c + Sqrt[a^2 - b^2 + c^2] + (a - b)*Tan[(d + e*x)/2]))/(((-I)*a + I*b - c + Sqrt[a^2 - b^2 + c^2])*(-I + Tan[(d + e*x)/2]))]*Sqrt[((-I)*(c + Sqrt[a^2 - b^2 + c^2] + (-a + b)*Tan[(d + e*x)/2]))/((I*a - I*b + c + Sqrt[a^2 - b^2 + c^2])*(-I + Tan[(d + e*x)/2]))])/((a + I*(I*b + c + Sqrt[a^2 - b^2 + c^2]))*e*Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])","C",0
456,0,0,240,23.9934453,"\int \frac{1}{\cos ^{\frac{3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx","Integrate[1/(Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)),x]","\int \frac{1}{\cos ^{\frac{3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx","-\frac{2 (a \cos (d+e x)+b+c \sin (d+e x))^2 E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \left(a^2-b^2+c^2\right) \cos ^{\frac{3}{2}}(d+e x) \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}-\frac{2 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+b+c \sin (d+e x))}{e \left(a^2-b^2+c^2\right) \cos ^{\frac{3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}",1,"Integrate[1/(Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)), x]","F",-1
457,0,0,492,27.7442773,"\int \frac{1}{\cos ^{\frac{5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx","Integrate[1/(Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)),x]","\int \frac{1}{\cos ^{\frac{5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx","\frac{2 \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} (a \cos (d+e x)+b+c \sin (d+e x))^2 F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \left(a^2-b^2+c^2\right) \cos ^{\frac{5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}+\frac{8 b (a \cos (d+e x)+b+c \sin (d+e x))^3 E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(a,c)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \left(a^2-b^2+c^2\right)^2 \cos ^{\frac{5}{2}}(d+e x) \sqrt{\frac{a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}+\frac{8 (b c \cos (d+e x)-a b \sin (d+e x)) (a \cos (d+e x)+b+c \sin (d+e x))^2}{3 e \left(a^2-b^2+c^2\right)^2 \cos ^{\frac{5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}-\frac{2 (c \cos (d+e x)-a \sin (d+e x)) (a \cos (d+e x)+b+c \sin (d+e x))}{3 e \left(a^2-b^2+c^2\right) \cos ^{\frac{5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}}",1,"Integrate[1/(Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)), x]","F",-1
458,1,80,98,0.2241718,"\int \frac{1}{a+b \cot (x)+c \csc (x)} \, dx","Integrate[(a + b*Cot[x] + c*Csc[x])^(-1),x]","\frac{\frac{2 a c \tanh ^{-1}\left(\frac{a+(c-b) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\sqrt{a^2+b^2-c^2}}-b \log (a \sin (x)+b \cos (x)+c)+a x}{a^2+b^2}","\frac{2 a c \tanh ^{-1}\left(\frac{a-(b-c) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\left(a^2+b^2\right) \sqrt{a^2+b^2-c^2}}-\frac{b \log (a \sin (x)+b \cos (x)+c)}{a^2+b^2}+\frac{a x}{a^2+b^2}",1,"(a*x + (2*a*c*ArcTanh[(a + (-b + c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/Sqrt[a^2 + b^2 - c^2] - b*Log[c + b*Cos[x] + a*Sin[x]])/(a^2 + b^2)","A",1
459,1,50,51,0.0446711,"\int \frac{\csc (x)}{a+b \cot (x)+c \csc (x)} \, dx","Integrate[Csc[x]/(a + b*Cot[x] + c*Csc[x]),x]","-\frac{2 \tanh ^{-1}\left(\frac{a+(c-b) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\sqrt{a^2+b^2-c^2}}","-\frac{2 \tanh ^{-1}\left(\frac{a-(b-c) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\sqrt{a^2+b^2-c^2}}",1,"(-2*ArcTanh[(a + (-b + c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/Sqrt[a^2 + b^2 - c^2]","A",1
460,1,104,120,0.2817028,"\int \frac{\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx","Integrate[Csc[x]^2/(a + b*Cot[x] + c*Csc[x]),x]","\frac{\frac{2 a c \tanh ^{-1}\left(\frac{a+(c-b) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\sqrt{a^2+b^2-c^2}}+b \log (a \sin (x)+b \cos (x)+c)+(c-b) \log \left(\sin \left(\frac{x}{2}\right)\right)-(b+c) \log \left(\cos \left(\frac{x}{2}\right)\right)}{(c-b) (b+c)}","-\frac{2 a c \tanh ^{-1}\left(\frac{a-(b-c) \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2-c^2}}\right)}{\left(b^2-c^2\right) \sqrt{a^2+b^2-c^2}}-\frac{b \log \left(2 a \tan \left(\frac{x}{2}\right)-\left((b-c) \tan ^2\left(\frac{x}{2}\right)\right)+b+c\right)}{b^2-c^2}+\frac{\log \left(\tan \left(\frac{x}{2}\right)\right)}{b+c}",1,"((2*a*c*ArcTanh[(a + (-b + c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/Sqrt[a^2 + b^2 - c^2] - (b + c)*Log[Cos[x/2]] + (-b + c)*Log[Sin[x/2]] + b*Log[c + b*Cos[x] + a*Sin[x]])/((-b + c)*(b + c))","A",1
461,1,51,21,0.0232902,"\int \frac{\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx","Integrate[Csc[x]/(2 + 2*Cot[x] + 3*Csc[x]),x]","\tan ^{-1}\left(\sec \left(\frac{x}{2}\right) \left(\sin \left(\frac{x}{2}\right)+2 \cos \left(\frac{x}{2}\right)\right)\right)-\tan ^{-1}\left(\frac{\cos \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+2 \cos \left(\frac{x}{2}\right)}\right)","x+2 \tan ^{-1}\left(\frac{\cos (x)-\sin (x)}{\sin (x)+\cos (x)+2}\right)",1,"-ArcTan[Cos[x/2]/(2*Cos[x/2] + Sin[x/2])] + ArcTan[Sec[x/2]*(2*Cos[x/2] + Sin[x/2])]","B",1
462,1,2490,371,6.4403707,"\int \frac{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2}}{\csc ^{\frac{3}{2}}(d+e x)} \, dx","Integrate[(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)/Csc[d + e*x]^(3/2),x]","\text{Result too large to show}","\frac{2 \left(a^2-b^2+c^2\right) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \csc ^{\frac{3}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^2}+\frac{8 b (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \csc ^{\frac{3}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x)) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}}}-\frac{2 (a \cos (d+e x)-c \sin (d+e x)) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}{3 e \csc ^{\frac{3}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))}",1,"((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*((8*b*c)/(3*a) - (2*a*Cos[d + e*x])/3 + (2*c*Sin[d + e*x])/3))/(e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])) + (4*a*b*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]]))/(3*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)) + (4*b*c^2*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]]))/(3*a*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)) + (2*a*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sec[d + e*x + ArcTan[c/a]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])/(3*Sqrt[1 + c^2/a^2]*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)) + (2*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sec[d + e*x + ArcTan[c/a]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])/(a*Sqrt[1 + c^2/a^2]*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)) + (2*c^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sec[d + e*x + ArcTan[c/a]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])/(3*a*Sqrt[1 + c^2/a^2]*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2))","C",0
463,1,1580,118,6.2420645,"\int \frac{\sqrt{a+c \cot (d+e x)+b \csc (d+e x)}}{\sqrt{\csc (d+e x)}} \, dx","Integrate[Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]/Sqrt[Csc[d + e*x]],x]","\frac{\sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \left(-\frac{a F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(1-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}\right) c},-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}-1\right) c}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} c \sqrt{\frac{c \sqrt{\frac{a^2+c^2}{c^2}}-c \sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{b+c \sqrt{\frac{a^2+c^2}{c^2}}}} \sqrt{b+c \sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right) c+\sqrt{\frac{a^2+c^2}{c^2}} c}{c \sqrt{\frac{a^2+c^2}{c^2}}-b}}}-\frac{\frac{2 c \left(b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)\right)}{a^2+c^2}-\frac{a \sin \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} c}}{\sqrt{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}}\right) c^2}{a e \sqrt{\csc (d+e x)} \sqrt{b+c \cos (d+e x)+a \sin (d+e x)}}+\frac{2 \sqrt{a+c \cot (d+e x)+b \csc (d+e x)} c}{a e \sqrt{\csc (d+e x)}}+\frac{a \sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \left(-\frac{a F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(1-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}\right) c},-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}-1\right) c}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} c \sqrt{\frac{c \sqrt{\frac{a^2+c^2}{c^2}}-c \sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{b+c \sqrt{\frac{a^2+c^2}{c^2}}}} \sqrt{b+c \sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right) c+\sqrt{\frac{a^2+c^2}{c^2}} c}{c \sqrt{\frac{a^2+c^2}{c^2}}-b}}}-\frac{\frac{2 c \left(b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)\right)}{a^2+c^2}-\frac{a \sin \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} c}}{\sqrt{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}}\right)}{e \sqrt{\csc (d+e x)} \sqrt{b+c \cos (d+e x)+a \sin (d+e x)}}+\frac{2 b F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(1-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}\right)},-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}-1\right)}\right) \sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sec \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right) \sqrt{\frac{a \sqrt{\frac{a^2+c^2}{a^2}}-a \sqrt{\frac{a^2+c^2}{a^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)}{\sqrt{\frac{a^2+c^2}{a^2}} a+b}} \sqrt{b+a \sqrt{\frac{a^2+c^2}{a^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{a^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right) a+\sqrt{\frac{a^2+c^2}{a^2}} a}{a \sqrt{\frac{a^2+c^2}{a^2}}-b}}}{a \sqrt{\frac{c^2}{a^2}+1} e \sqrt{\csc (d+e x)} \sqrt{b+c \cos (d+e x)+a \sin (d+e x)}}","\frac{2 \sqrt{a+b \csc (d+e x)+c \cot (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \sqrt{\csc (d+e x)} \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}}}",1,"(2*c*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]])/(a*e*Sqrt[Csc[d + e*x]]) + (a*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]]))/(e*Sqrt[Csc[d + e*x]]*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]]) + (c^2*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]]))/(a*e*Sqrt[Csc[d + e*x]]*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]]) + (2*b*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sec[d + e*x + ArcTan[c/a]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])/(a*Sqrt[1 + c^2/a^2]*e*Sqrt[Csc[d + e*x]]*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]])","C",0
464,1,339,118,0.9073778,"\int \frac{\sqrt{\csc (d+e x)}}{\sqrt{a+c \cot (d+e x)+b \csc (d+e x)}} \, dx","Integrate[Sqrt[Csc[d + e*x]]/Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]],x]","\frac{2 \sqrt{\csc (d+e x)} \sec \left(\tan ^{-1}\left(\frac{c}{a}\right)+d+e x\right) \sqrt{-\frac{a \sqrt{\frac{c^2}{a^2}+1} \left(\sin \left(\tan ^{-1}\left(\frac{c}{a}\right)+d+e x\right)-1\right)}{a \sqrt{\frac{c^2}{a^2}+1}+b}} \sqrt{\frac{a \sqrt{\frac{c^2}{a^2}+1} \left(\sin \left(\tan ^{-1}\left(\frac{c}{a}\right)+d+e x\right)+1\right)}{a \sqrt{\frac{c^2}{a^2}+1}-b}} \sqrt{a \sqrt{\frac{c^2}{a^2}+1} \sin \left(\tan ^{-1}\left(\frac{c}{a}\right)+d+e x\right)+b} \sqrt{a \sin (d+e x)+b+c \cos (d+e x)} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)}{b-a \sqrt{\frac{c^2}{a^2}+1}},\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)}{\sqrt{\frac{c^2}{a^2}+1} a+b}\right)}{a e \sqrt{\frac{c^2}{a^2}+1} \sqrt{a+b \csc (d+e x)+c \cot (d+e x)}}","\frac{2 \sqrt{\csc (d+e x)} \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \sqrt{a+b \csc (d+e x)+c \cot (d+e x)}}",1,"(2*AppellF1[1/2, 1/2, 1/2, 3/2, (b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b - a*Sqrt[1 + c^2/a^2]), (b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b + a*Sqrt[1 + c^2/a^2])]*Sqrt[Csc[d + e*x]]*Sec[d + e*x + ArcTan[c/a]]*Sqrt[b + c*Cos[d + e*x] + a*Sin[d + e*x]]*Sqrt[-((a*Sqrt[1 + c^2/a^2]*(-1 + Sin[d + e*x + ArcTan[c/a]]))/(b + a*Sqrt[1 + c^2/a^2]))]*Sqrt[(a*Sqrt[1 + c^2/a^2]*(1 + Sin[d + e*x + ArcTan[c/a]]))/(-b + a*Sqrt[1 + c^2/a^2])]*Sqrt[b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]]])/(a*Sqrt[1 + c^2/a^2]*e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]])","C",0
465,1,1732,240,6.399703,"\int \frac{\csc ^{\frac{3}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2}} \, dx","Integrate[Csc[d + e*x]^(3/2)/(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2),x]","-\frac{\csc ^{\frac{3}{2}}(d+e x) (b+c \cos (d+e x)+a \sin (d+e x))^{3/2} \left(-\frac{a F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(1-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}\right) c},-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}-1\right) c}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} c \sqrt{\frac{c \sqrt{\frac{a^2+c^2}{c^2}}-c \sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{b+c \sqrt{\frac{a^2+c^2}{c^2}}}} \sqrt{b+c \sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right) c+\sqrt{\frac{a^2+c^2}{c^2}} c}{c \sqrt{\frac{a^2+c^2}{c^2}}-b}}}-\frac{\frac{2 c \left(b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)\right)}{a^2+c^2}-\frac{a \sin \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} c}}{\sqrt{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}}\right) c^2}{a \left(a^2-b^2+c^2\right) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2}}+\frac{\csc ^{\frac{3}{2}}(d+e x) (b+c \cos (d+e x)+a \sin (d+e x))^2 \left(\frac{2 \left(\sin (d+e x) a^2+b a+c^2 \sin (d+e x)\right)}{c \left(a^2-b^2+c^2\right) (b+c \cos (d+e x)+a \sin (d+e x))}-\frac{2 \left(a^2+c^2\right)}{a c \left(a^2-b^2+c^2\right)}\right)}{e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2}}-\frac{a \csc ^{\frac{3}{2}}(d+e x) (b+c \cos (d+e x)+a \sin (d+e x))^{3/2} \left(-\frac{a F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(1-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}\right) c},-\frac{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} \left(-\frac{b}{\sqrt{\frac{a^2}{c^2}+1} c}-1\right) c}\right) \sin \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} c \sqrt{\frac{c \sqrt{\frac{a^2+c^2}{c^2}}-c \sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{b+c \sqrt{\frac{a^2+c^2}{c^2}}}} \sqrt{b+c \sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{c^2}} \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right) c+\sqrt{\frac{a^2+c^2}{c^2}} c}{c \sqrt{\frac{a^2+c^2}{c^2}}-b}}}-\frac{\frac{2 c \left(b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)\right)}{a^2+c^2}-\frac{a \sin \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}{\sqrt{\frac{a^2}{c^2}+1} c}}{\sqrt{b+\sqrt{\frac{a^2}{c^2}+1} c \cos \left(d+e x-\tan ^{-1}\left(\frac{a}{c}\right)\right)}}\right)}{\left(a^2-b^2+c^2\right) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2}}-\frac{2 b F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(1-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}\right)},-\frac{b+a \sqrt{\frac{c^2}{a^2}+1} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)}{a \sqrt{\frac{c^2}{a^2}+1} \left(-\frac{b}{a \sqrt{\frac{c^2}{a^2}+1}}-1\right)}\right) \csc ^{\frac{3}{2}}(d+e x) \sec \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right) (b+c \cos (d+e x)+a \sin (d+e x))^{3/2} \sqrt{\frac{a \sqrt{\frac{a^2+c^2}{a^2}}-a \sqrt{\frac{a^2+c^2}{a^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)}{\sqrt{\frac{a^2+c^2}{a^2}} a+b}} \sqrt{b+a \sqrt{\frac{a^2+c^2}{a^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right)} \sqrt{\frac{\sqrt{\frac{a^2+c^2}{a^2}} \sin \left(d+e x+\tan ^{-1}\left(\frac{c}{a}\right)\right) a+\sqrt{\frac{a^2+c^2}{a^2}} a}{a \sqrt{\frac{a^2+c^2}{a^2}}-b}}}{a \left(a^2-b^2+c^2\right) \sqrt{\frac{c^2}{a^2}+1} e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2}}","-\frac{2 \csc ^{\frac{3}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^2 E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \left(a^2-b^2+c^2\right) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}-\frac{2 \csc ^{\frac{3}{2}}(d+e x) (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{e \left(a^2-b^2+c^2\right) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}",1,"(Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*((-2*(a^2 + c^2))/(a*c*(a^2 - b^2 + c^2)) + (2*(a*b + a^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(c*(a^2 - b^2 + c^2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x]))))/(e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)) - (a*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]]))/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)) - (c^2*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]]))/(a*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)) - (2*b*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Csc[d + e*x]^(3/2)*Sec[d + e*x + ArcTan[c/a]]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])/(a*(a^2 - b^2 + c^2)*Sqrt[1 + c^2/a^2]*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2))","C",0
466,1,2708,492,6.4938853,"\int \frac{\csc ^{\frac{5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx","Integrate[Csc[d + e*x]^(5/2)/(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \csc ^{\frac{5}{2}}(d+e x) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a \sin (d+e x)+b+c \cos (d+e x))^2 F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \left(a^2-b^2+c^2\right) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac{8 b \csc ^{\frac{5}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^3 E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \left(a^2-b^2+c^2\right)^2 \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac{8 \csc ^{\frac{5}{2}}(d+e x) (a b \cos (d+e x)-b c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))^2}{3 e \left(a^2-b^2+c^2\right)^2 (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}-\frac{2 \csc ^{\frac{5}{2}}(d+e x) (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{3 e \left(a^2-b^2+c^2\right) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}",1,"(Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^3*((8*b*(a^2 + c^2))/(3*a*c*(-a^2 + b^2 - c^2)^2) + (2*(a*b + a^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(3*c*(a^2 - b^2 + c^2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2) - (2*(a^3 + 3*a*b^2 + a*c^2 + 4*a^2*b*Sin[d + e*x] + 4*b*c^2*Sin[d + e*x]))/(3*c*(a^2 - b^2 + c^2)^2*(b + c*Cos[d + e*x] + a*Sin[d + e*x]))))/(e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) + (4*a*b*Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(5/2)*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]]))/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) + (4*b*c^2*Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(5/2)*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]]))/(3*a*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) + (2*a*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Csc[d + e*x]^(5/2)*Sec[d + e*x + ArcTan[c/a]]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(5/2)*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])/(3*(a^2 - b^2 + c^2)^2*Sqrt[1 + c^2/a^2]*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) + (2*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Csc[d + e*x]^(5/2)*Sec[d + e*x + ArcTan[c/a]]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(5/2)*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])/(a*(a^2 - b^2 + c^2)^2*Sqrt[1 + c^2/a^2]*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) + (2*c^2*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]])/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Csc[d + e*x]^(5/2)*Sec[d + e*x + ArcTan[c/a]]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(5/2)*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]]]*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])/(3*a*(a^2 - b^2 + c^2)^2*Sqrt[1 + c^2/a^2]*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2))","C",0
467,0,0,371,52.8556727,"\int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x) \, dx","Integrate[(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2),x]","\int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x) \, dx","\frac{2 \left(a^2-b^2+c^2\right) \sin ^{\frac{3}{2}}(d+e x) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e (a \sin (d+e x)+b+c \cos (d+e x))^2}+\frac{8 b \sin ^{\frac{3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e (a \sin (d+e x)+b+c \cos (d+e x)) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}}}-\frac{2 \sin ^{\frac{3}{2}}(d+e x) (a \cos (d+e x)-c \sin (d+e x)) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}{3 e (a \sin (d+e x)+b+c \cos (d+e x))}",1,"Integrate[(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2), x]","F",-1
468,0,0,118,12.3391063,"\int \sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sqrt{\sin (d+e x)} \, dx","Integrate[Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]],x]","\int \sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sqrt{\sin (d+e x)} \, dx","\frac{2 \sqrt{\sin (d+e x)} \sqrt{a+b \csc (d+e x)+c \cot (d+e x)} E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}}}",1,"Integrate[Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]], x]","F",-1
469,1,519,118,2.844959,"\int \frac{1}{\sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sqrt{\sin (d+e x)}} \, dx","Integrate[1/(Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]]),x]","\frac{4 \left(i \sqrt{a^2-b^2+c^2}-i a-b+c\right) (\cos (d+e x)+i \sin (d+e x)) \sqrt{-\frac{i \left(\sqrt{a^2-b^2+c^2}+a+(b-c) \tan \left(\frac{1}{2} (d+e x)\right)\right)}{\left(\sqrt{a^2-b^2+c^2}+a-i b+i c\right) \left(\tan \left(\frac{1}{2} (d+e x)\right)-i\right)}} \sqrt{-\frac{i \left(\sqrt{a^2-b^2+c^2}-a+(c-b) \tan \left(\frac{1}{2} (d+e x)\right)\right)}{\left(\sqrt{a^2-b^2+c^2}-a+i b-i c\right) \left(\tan \left(\frac{1}{2} (d+e x)\right)-i\right)}} \sqrt{\frac{\left(\sqrt{a^2-b^2+c^2}-a-i b+i c\right) (-\cos (d+e x)+i \sin (d+e x))}{\sqrt{a^2-b^2+c^2}-a+i b-i c}} F\left(\sin ^{-1}\left(\sqrt{\frac{\left(-a-i b+i c+\sqrt{a^2-b^2+c^2}\right) (i \sin (d+e x)-\cos (d+e x))}{-a+i b-i c+\sqrt{a^2-b^2+c^2}}}\right)|\frac{i b+\sqrt{a^2-b^2+c^2}}{i b-\sqrt{a^2-b^2+c^2}}\right)}{e \left(-\sqrt{a^2-b^2+c^2}+a+i b-i c\right) \sqrt{\sin (d+e x)} \sqrt{a+b \csc (d+e x)+c \cot (d+e x)}}","\frac{2 \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \sqrt{\sin (d+e x)} \sqrt{a+b \csc (d+e x)+c \cot (d+e x)}}",1,"(4*((-I)*a - b + c + I*Sqrt[a^2 - b^2 + c^2])*EllipticF[ArcSin[Sqrt[((-a - I*b + I*c + Sqrt[a^2 - b^2 + c^2])*(-Cos[d + e*x] + I*Sin[d + e*x]))/(-a + I*b - I*c + Sqrt[a^2 - b^2 + c^2])]], (I*b + Sqrt[a^2 - b^2 + c^2])/(I*b - Sqrt[a^2 - b^2 + c^2])]*Sqrt[((-a - I*b + I*c + Sqrt[a^2 - b^2 + c^2])*(-Cos[d + e*x] + I*Sin[d + e*x]))/(-a + I*b - I*c + Sqrt[a^2 - b^2 + c^2])]*(Cos[d + e*x] + I*Sin[d + e*x])*Sqrt[((-I)*(a + Sqrt[a^2 - b^2 + c^2] + (b - c)*Tan[(d + e*x)/2]))/((a - I*b + I*c + Sqrt[a^2 - b^2 + c^2])*(-I + Tan[(d + e*x)/2]))]*Sqrt[((-I)*(-a + Sqrt[a^2 - b^2 + c^2] + (-b + c)*Tan[(d + e*x)/2]))/((-a + I*b - I*c + Sqrt[a^2 - b^2 + c^2])*(-I + Tan[(d + e*x)/2]))])/((a + I*b - I*c - Sqrt[a^2 - b^2 + c^2])*e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]])","C",0
470,0,0,240,20.777725,"\int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)} \, dx","Integrate[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)),x]","\int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)} \, dx","-\frac{2 (a \sin (d+e x)+b+c \cos (d+e x))^2 E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{e \left(a^2-b^2+c^2\right) \sin ^{\frac{3}{2}}(d+e x) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}-\frac{2 (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{e \left(a^2-b^2+c^2\right) \sin ^{\frac{3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}",1,"Integrate[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)), x]","F",-1
471,0,0,492,25.2685077,"\int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)} \, dx","Integrate[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)),x]","\int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)} \, dx","\frac{2 \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a \sin (d+e x)+b+c \cos (d+e x))^2 F\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \left(a^2-b^2+c^2\right) \sin ^{\frac{5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac{8 b (a \sin (d+e x)+b+c \cos (d+e x))^3 E\left(\frac{1}{2} \left(d+e x-\tan ^{-1}(c,a)\right)|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right)}{3 e \left(a^2-b^2+c^2\right)^2 \sin ^{\frac{5}{2}}(d+e x) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac{8 (a b \cos (d+e x)-b c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))^2}{3 e \left(a^2-b^2+c^2\right)^2 \sin ^{\frac{5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}-\frac{2 (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{3 e \left(a^2-b^2+c^2\right) \sin ^{\frac{5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}",1,"Integrate[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)), x]","F",-1
472,1,1,1,0.0004324,"\int \frac{1}{\cos ^2(x)+\sin ^2(x)} \, dx","Integrate[(Cos[x]^2 + Sin[x]^2)^(-1),x]","x","x",1,"x","A",1
473,1,1,1,0.0003781,"\int \frac{1}{\left(\cos ^2(x)+\sin ^2(x)\right)^2} \, dx","Integrate[(Cos[x]^2 + Sin[x]^2)^(-2),x]","x","x",1,"x","A",1
474,1,1,1,0.0003305,"\int \frac{1}{\left(\cos ^2(x)+\sin ^2(x)\right)^3} \, dx","Integrate[(Cos[x]^2 + Sin[x]^2)^(-3),x]","x","x",1,"x","A",1
475,1,23,11,0.0069971,"\int \frac{1}{\cos ^2(x)-\sin ^2(x)} \, dx","Integrate[(Cos[x]^2 - Sin[x]^2)^(-1),x]","\frac{1}{2} \log (\sin (x)+\cos (x))-\frac{1}{2} \log (\cos (x)-\sin (x))","\frac{1}{2} \tanh ^{-1}(2 \sin (x) \cos (x))",1,"-1/2*Log[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x]]/2","B",1
476,1,8,13,0.0030688,"\int \frac{1}{\left(\cos ^2(x)-\sin ^2(x)\right)^2} \, dx","Integrate[(Cos[x]^2 - Sin[x]^2)^(-2),x]","\frac{1}{2} \tan (2 x)","\frac{\tan (x)}{1-\tan ^2(x)}",1,"Tan[2*x]/2","A",1
477,1,22,32,0.0063686,"\int \frac{1}{\left(\cos ^2(x)-\sin ^2(x)\right)^3} \, dx","Integrate[(Cos[x]^2 - Sin[x]^2)^(-3),x]","\frac{1}{4} \tanh ^{-1}(\sin (2 x))+\frac{1}{4} \tan (2 x) \sec (2 x)","\frac{\tan (x) \sec ^2(x)}{2 \left(1-\tan ^2(x)\right)^2}+\frac{1}{4} \tanh ^{-1}(2 \sin (x) \cos (x))",1,"ArcTanh[Sin[2*x]]/4 + (Sec[2*x]*Tan[2*x])/4","A",1
478,1,9,9,0.0317207,"\int \frac{1}{\cos ^2(x)+a^2 \sin ^2(x)} \, dx","Integrate[(Cos[x]^2 + a^2*Sin[x]^2)^(-1),x]","\frac{\tan ^{-1}(a \tan (x))}{a}","\frac{\tan ^{-1}(a \tan (x))}{a}",1,"ArcTan[a*Tan[x]]/a","A",1
479,1,11,11,0.0312357,"\int \frac{1}{b^2 \cos ^2(x)+\sin ^2(x)} \, dx","Integrate[(b^2*Cos[x]^2 + Sin[x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\tan (x)}{b}\right)}{b}","\frac{\tan ^{-1}\left(\frac{\tan (x)}{b}\right)}{b}",1,"ArcTan[Tan[x]/b]/b","A",1
480,1,15,15,0.0418657,"\int \frac{1}{b^2 \cos ^2(x)+a^2 \sin ^2(x)} \, dx","Integrate[(b^2*Cos[x]^2 + a^2*Sin[x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{a \tan (x)}{b}\right)}{a b}","\frac{\tan ^{-1}\left(\frac{a \tan (x)}{b}\right)}{a b}",1,"ArcTan[(a*Tan[x])/b]/(a*b)","A",1
481,1,25,53,0.0399055,"\int \frac{1}{4 \cos ^2(1+2 x)+3 \sin ^2(1+2 x)} \, dx","Integrate[(4*Cos[1 + 2*x]^2 + 3*Sin[1 + 2*x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{1}{2} \sqrt{3} \tan (2 x+1)\right)}{4 \sqrt{3}}","\frac{x}{2 \sqrt{3}}-\frac{\tan ^{-1}\left(\frac{\sin (2 x+1) \cos (2 x+1)}{\cos ^2(2 x+1)+2 \sqrt{3}+3}\right)}{4 \sqrt{3}}",1,"ArcTan[(Sqrt[3]*Tan[1 + 2*x])/2]/(4*Sqrt[3])","A",1
482,1,36,43,0.094965,"\int \frac{\sin ^2(x)}{a \cos ^2(x)+b \sin ^2(x)} \, dx","Integrate[Sin[x]^2/(a*Cos[x]^2 + b*Sin[x]^2),x]","\frac{x-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{b}}}{b-a}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{b} (a-b)}-\frac{x}{a-b}",1,"(x - (Sqrt[a]*ArcTan[(Sqrt[b]*Tan[x])/Sqrt[a]])/Sqrt[b])/(-a + b)","A",1
483,1,36,43,0.0549727,"\int \frac{\cos ^2(x)}{a \cos ^2(x)+b \sin ^2(x)} \, dx","Integrate[Cos[x]^2/(a*Cos[x]^2 + b*Sin[x]^2),x]","\frac{x-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a}}}{a-b}","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{a} (a-b)}",1,"(x - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[x])/Sqrt[a]])/Sqrt[a])/(a - b)","A",1
484,1,19,36,0.0457289,"\int \frac{1}{\sec ^2(x)+\tan ^2(x)} \, dx","Integrate[(Sec[x]^2 + Tan[x]^2)^(-1),x]","\sqrt{2} \tan ^{-1}\left(\sqrt{2} \tan (x)\right)-x","\sqrt{2} x-x+\sqrt{2} \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)",1,"-x + Sqrt[2]*ArcTan[Sqrt[2]*Tan[x]]","A",1
485,1,42,49,0.1368561,"\int \frac{1}{\left(\sec ^2(x)+\tan ^2(x)\right)^2} \, dx","Integrate[(Sec[x]^2 + Tan[x]^2)^(-2),x]","\frac{-3 x-\sin (2 x)+x \cos (2 x)}{\cos (2 x)-3}-\frac{\tan ^{-1}\left(\sqrt{2} \tan (x)\right)}{\sqrt{2}}","-\frac{x}{\sqrt{2}}+x+\frac{\tan (x)}{2 \tan ^2(x)+1}-\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)}{\sqrt{2}}",1,"-(ArcTan[Sqrt[2]*Tan[x]]/Sqrt[2]) + (-3*x + x*Cos[2*x] - Sin[2*x])/(-3 + Cos[2*x])","A",1
486,1,79,74,0.1797542,"\int \frac{1}{\left(\sec ^2(x)+\tan ^2(x)\right)^3} \, dx","Integrate[(Sec[x]^2 + Tan[x]^2)^(-3),x]","-\frac{(\cos (2 x)-3) \sec ^6(x) \left(-76 x-2 \sin (2 x)+3 \sin (4 x)+48 x \cos (2 x)-4 x \cos (4 x)+7 \sqrt{2} (\cos (2 x)-3)^2 \tan ^{-1}\left(\sqrt{2} \tan (x)\right)\right)}{64 \left(\tan ^2(x)+\sec ^2(x)\right)^3}","\frac{7 x}{4 \sqrt{2}}-x-\frac{\tan (x)}{4 \left(2 \tan ^2(x)+1\right)}+\frac{\tan (x)}{2 \left(2 \tan ^2(x)+1\right)^2}+\frac{7 \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)}{4 \sqrt{2}}",1,"-1/64*((-3 + Cos[2*x])*Sec[x]^6*(-76*x + 7*Sqrt[2]*ArcTan[Sqrt[2]*Tan[x]]*(-3 + Cos[2*x])^2 + 48*x*Cos[2*x] - 4*x*Cos[4*x] - 2*Sin[2*x] + 3*Sin[4*x]))/(Sec[x]^2 + Tan[x]^2)^3","A",1
487,1,1,1,0.0006227,"\int \frac{1}{\sec ^2(x)-\tan ^2(x)} \, dx","Integrate[(Sec[x]^2 - Tan[x]^2)^(-1),x]","x","x",1,"x","A",1
488,1,1,1,0.0005219,"\int \frac{1}{\left(\sec ^2(x)-\tan ^2(x)\right)^2} \, dx","Integrate[(Sec[x]^2 - Tan[x]^2)^(-2),x]","x","x",1,"x","A",1
489,1,1,1,0.0004601,"\int \frac{1}{\left(\sec ^2(x)-\tan ^2(x)\right)^3} \, dx","Integrate[(Sec[x]^2 - Tan[x]^2)^(-3),x]","x","x",1,"x","A",1
490,1,19,37,0.0423753,"\int \frac{1}{\cot ^2(x)+\csc ^2(x)} \, dx","Integrate[(Cot[x]^2 + Csc[x]^2)^(-1),x]","\sqrt{2} \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)-x","\sqrt{2} x-x-\sqrt{2} \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)",1,"-x + Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]]","A",1
491,1,64,47,0.1052129,"\int \frac{1}{\left(\cot ^2(x)+\csc ^2(x)\right)^2} \, dx","Integrate[(Cot[x]^2 + Csc[x]^2)^(-2),x]","\frac{(\cos (2 x)+3) \csc ^4(x) \left(6 x-2 \sin (2 x)+2 x \cos (2 x)-\sqrt{2} (\cos (2 x)+3) \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)\right)}{8 \left(\cot ^2(x)+\csc ^2(x)\right)^2}","-\frac{x}{\sqrt{2}}+x-\frac{\tan (x)}{\tan ^2(x)+2}+\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{\sqrt{2}}",1,"((3 + Cos[2*x])*Csc[x]^4*(6*x + 2*x*Cos[2*x] - Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]]*(3 + Cos[2*x]) - 2*Sin[2*x]))/(8*(Cot[x]^2 + Csc[x]^2)^2)","A",1
492,1,66,72,0.1560882,"\int \frac{1}{\left(\cot ^2(x)+\csc ^2(x)\right)^3} \, dx","Integrate[(Cot[x]^2 + Csc[x]^2)^(-3),x]","\frac{-76 x+2 \sin (2 x)+3 \sin (4 x)-48 x \cos (2 x)-4 x \cos (4 x)+7 \sqrt{2} (\cos (2 x)+3)^2 \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)}{8 (\cos (2 x)+3)^2}","\frac{7 x}{4 \sqrt{2}}-x+\frac{\tan (x)}{4 \left(\tan ^2(x)+2\right)}-\frac{\tan ^3(x)}{2 \left(\tan ^2(x)+2\right)^2}-\frac{7 \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{4 \sqrt{2}}",1,"(-76*x - 48*x*Cos[2*x] + 7*Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]]*(3 + Cos[2*x])^2 - 4*x*Cos[4*x] + 2*Sin[2*x] + 3*Sin[4*x])/(8*(3 + Cos[2*x])^2)","A",1
493,1,3,3,0.0004999,"\int \frac{1}{\cot ^2(x)-\csc ^2(x)} \, dx","Integrate[(Cot[x]^2 - Csc[x]^2)^(-1),x]","-x","-x",1,"-x","A",1
494,1,1,1,0.0004884,"\int \frac{1}{\left(\cot ^2(x)-\csc ^2(x)\right)^2} \, dx","Integrate[(Cot[x]^2 - Csc[x]^2)^(-2),x]","x","x",1,"x","A",1
495,1,3,3,0.0004619,"\int \frac{1}{\left(\cot ^2(x)-\csc ^2(x)\right)^3} \, dx","Integrate[(Cot[x]^2 - Csc[x]^2)^(-3),x]","-x","-x",1,"-x","A",1
496,1,33,33,0.0601985,"\int \frac{1}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx","Integrate[(a + b*Cos[x]^2 + c*Sin[x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a+b} \sqrt{a+c}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a+b} \sqrt{a+c}}",1,"ArcTan[(Sqrt[a + c]*Tan[x])/Sqrt[a + b]]/(Sqrt[a + b]*Sqrt[a + c])","A",1
497,1,507,239,3.0194148,"\int \frac{x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx","Integrate[x/(a + b*Cos[x]^2 + c*Sin[x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right) \left(2 x+\frac{i \left(-\text{Li}_2\left(\frac{\sqrt{a+b}-i \sqrt{a+c} \tan (x)}{\sqrt{a+b}-\sqrt{a+c}}\right)+\text{Li}_2\left(\frac{\sqrt{a+b}-i \sqrt{a+c} \tan (x)}{\sqrt{a+b}+\sqrt{a+c}}\right)-\text{Li}_2\left(\frac{i \sqrt{a+c} \tan (x)+\sqrt{a+b}}{\sqrt{a+b}-\sqrt{a+c}}\right)+\text{Li}_2\left(\frac{i \sqrt{a+c} \tan (x)+\sqrt{a+b}}{\sqrt{a+b}+\sqrt{a+c}}\right)+\log \left(\frac{\sqrt{a+c} (1+i \tan (x))}{\sqrt{a+b}+\sqrt{a+c}}\right) \log \left(1-\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right)-\log \left(\frac{i \sqrt{a+c} (\tan (x)+i)}{\sqrt{a+b}-\sqrt{a+c}}\right) \log \left(1-\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right)+\log \left(\frac{\sqrt{a+c} (1-i \tan (x))}{\sqrt{a+b}+\sqrt{a+c}}\right) \log \left(1+\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right)-\log \left(\frac{\sqrt{a+c} (1+i \tan (x))}{\sqrt{a+c}-\sqrt{a+b}}\right) \log \left(1+\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right)\right)}{\log \left(1-\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right)-\log \left(1+\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right)}\right)}{2 \sqrt{a+b} \sqrt{a+c}}","-\frac{\text{Li}_2\left(-\frac{(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt{a+b} \sqrt{a+c}}\right)}{4 \sqrt{a+b} \sqrt{a+c}}+\frac{\text{Li}_2\left(-\frac{(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt{a+b} \sqrt{a+c}}\right)}{4 \sqrt{a+b} \sqrt{a+c}}-\frac{i x \log \left(1+\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right)}{2 \sqrt{a+b} \sqrt{a+c}}+\frac{i x \log \left(1+\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right)}{2 \sqrt{a+b} \sqrt{a+c}}",1,"(ArcTan[(Sqrt[a + c]*Tan[x])/Sqrt[a + b]]*(2*x + (I*(Log[(Sqrt[a + c]*(1 + I*Tan[x]))/(Sqrt[a + b] + Sqrt[a + c])]*Log[1 - (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] - Log[(I*Sqrt[a + c]*(I + Tan[x]))/(Sqrt[a + b] - Sqrt[a + c])]*Log[1 - (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] + Log[(Sqrt[a + c]*(1 - I*Tan[x]))/(Sqrt[a + b] + Sqrt[a + c])]*Log[1 + (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] - Log[(Sqrt[a + c]*(1 + I*Tan[x]))/(-Sqrt[a + b] + Sqrt[a + c])]*Log[1 + (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] - PolyLog[2, (Sqrt[a + b] - I*Sqrt[a + c]*Tan[x])/(Sqrt[a + b] - Sqrt[a + c])] + PolyLog[2, (Sqrt[a + b] - I*Sqrt[a + c]*Tan[x])/(Sqrt[a + b] + Sqrt[a + c])] - PolyLog[2, (Sqrt[a + b] + I*Sqrt[a + c]*Tan[x])/(Sqrt[a + b] - Sqrt[a + c])] + PolyLog[2, (Sqrt[a + b] + I*Sqrt[a + c]*Tan[x])/(Sqrt[a + b] + Sqrt[a + c])]))/(Log[1 - (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] - Log[1 + (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]])))/(2*Sqrt[a + b]*Sqrt[a + c])","B",0
498,1,258,365,3.7908965,"\int \frac{x^2}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx","Integrate[x^2/(a + b*Cos[x]^2 + c*Sin[x]^2),x]","-\frac{i \left(-2 i x \text{Li}_2\left(\frac{(c-b) e^{2 i x}}{2 a+b+c-2 \sqrt{(a+b) (a+c)}}\right)+2 i x \text{Li}_2\left(\frac{(c-b) e^{2 i x}}{2 a+b+c+2 \sqrt{(a+b) (a+c)}}\right)+\text{Li}_3\left(\frac{(c-b) e^{2 i x}}{2 a+b+c-2 \sqrt{(a+b) (a+c)}}\right)-\text{Li}_3\left(\frac{(c-b) e^{2 i x}}{2 a+b+c+2 \sqrt{(a+b) (a+c)}}\right)+2 x^2 \log \left(1+\frac{e^{2 i x} (b-c)}{-2 \sqrt{(a+b) (a+c)}+2 a+b+c}\right)-2 x^2 \log \left(1+\frac{e^{2 i x} (b-c)}{2 \sqrt{(a+b) (a+c)}+2 a+b+c}\right)\right)}{4 \sqrt{(a+b) (a+c)}}","-\frac{x \text{Li}_2\left(-\frac{(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt{a+b} \sqrt{a+c}}\right)}{2 \sqrt{a+b} \sqrt{a+c}}+\frac{x \text{Li}_2\left(-\frac{(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt{a+b} \sqrt{a+c}}\right)}{2 \sqrt{a+b} \sqrt{a+c}}-\frac{i \text{Li}_3\left(-\frac{(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt{a+b} \sqrt{a+c}}\right)}{4 \sqrt{a+b} \sqrt{a+c}}+\frac{i \text{Li}_3\left(-\frac{(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt{a+b} \sqrt{a+c}}\right)}{4 \sqrt{a+b} \sqrt{a+c}}-\frac{i x^2 \log \left(1+\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right)}{2 \sqrt{a+b} \sqrt{a+c}}+\frac{i x^2 \log \left(1+\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right)}{2 \sqrt{a+b} \sqrt{a+c}}",1,"((-1/4*I)*(2*x^2*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[(a + b)*(a + c)])] - 2*x^2*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[(a + b)*(a + c)])] - (2*I)*x*PolyLog[2, ((-b + c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[(a + b)*(a + c)])] + (2*I)*x*PolyLog[2, ((-b + c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[(a + b)*(a + c)])] + PolyLog[3, ((-b + c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[(a + b)*(a + c)])] - PolyLog[3, ((-b + c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[(a + b)*(a + c)])]))/Sqrt[(a + b)*(a + c)]","A",1
499,1,149,195,0.9019578,"\int (a+b \sin (d+e x)) \left(b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right)^2 \, dx","Integrate[(a + b*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^2,x]","\frac{a \left(10 \left(7 a^3 b+8 a b^3\right) \cos (3 (d+e x))-2 a^3 b \cos (5 (d+e x))+5 \left(a^4+4 a^2 b^2\right) \sin (4 (d+e x))+60 \left(a^4+12 a^2 b^2+8 b^4\right) (d+e x)-40 \left(a^4+10 a^2 b^2+4 b^4\right) \sin (2 (d+e x))\right)-20 b \left(29 a^4+68 a^2 b^2+8 b^4\right) \cos (d+e x)}{160 e}","-\frac{\left(5 a^2+4 b^2\right) \cos (d+e x) (a \sin (d+e x)+b)^3}{20 e}-\frac{b \left(17 a^2+4 b^2\right) \cos (d+e x) (a \sin (d+e x)+b)^2}{20 e}-\frac{b \left(32 a^4+69 a^2 b^2+4 b^4\right) \cos (d+e x)}{10 e}-\frac{a \left(15 a^4+82 a^2 b^2+8 b^4\right) \sin (d+e x) \cos (d+e x)}{40 e}+\frac{3}{8} a x \left(a^4+12 a^2 b^2+8 b^4\right)-\frac{b \cos (d+e x) (a \sin (d+e x)+b)^4}{5 e}",1,"(-20*b*(29*a^4 + 68*a^2*b^2 + 8*b^4)*Cos[d + e*x] + a*(60*(a^4 + 12*a^2*b^2 + 8*b^4)*(d + e*x) + 10*(7*a^3*b + 8*a*b^3)*Cos[3*(d + e*x)] - 2*a^3*b*Cos[5*(d + e*x)] - 40*(a^4 + 10*a^2*b^2 + 4*b^4)*Sin[2*(d + e*x)] + 5*(a^4 + 4*a^2*b^2)*Sin[4*(d + e*x)]))/(160*e)","A",1
500,1,77,109,0.2855081,"\int (a+b \sin (d+e x)) \left(b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right) \, dx","Integrate[(a + b*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2),x]","\frac{a \left(6 \left(a^2+4 b^2\right) (d+e x)-3 \left(a^2+2 b^2\right) \sin (2 (d+e x))+a b \cos (3 (d+e x))\right)-3 b \left(11 a^2+4 b^2\right) \cos (d+e x)}{12 e}","\frac{a \left(a^2-6 b^2\right) \sin (d+e x) \cos (d+e x)}{6 e}+\frac{1}{2} a x \left(a^2+4 b^2\right)-\frac{a^2 \cos (d+e x) (a+b \sin (d+e x))^2}{3 b e}+\frac{\left(a^4-8 a^2 b^2-3 b^4\right) \cos (d+e x)}{3 b e}",1,"(-3*b*(11*a^2 + 4*b^2)*Cos[d + e*x] + a*(6*(a^2 + 4*b^2)*(d + e*x) + a*b*Cos[3*(d + e*x)] - 3*(a^2 + 2*b^2)*Sin[2*(d + e*x)]))/(12*e)","A",1
501,1,23,23,0.0583561,"\int \frac{a+b \sin (d+e x)}{b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)} \, dx","Integrate[(a + b*Sin[d + e*x])/(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2),x]","-\frac{\cos (d+e x)}{e (a \sin (d+e x)+b)}","-\frac{\cos (d+e x)}{e (a \sin (d+e x)+b)}",1,"-(Cos[d + e*x]/(e*(b + a*Sin[d + e*x])))","A",1
502,1,140,157,0.9613121,"\int \frac{a+b \sin (d+e x)}{\left(b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right)^2} \, dx","Integrate[(a + b*Sin[d + e*x])/(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^2,x]","-\frac{\frac{6 a b \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{\cos (d+e x) \left(a^4+a^2 \left(2 a^2+b^2\right) \sin ^2(d+e x)+3 a b \left(a^2+b^2\right) \sin (d+e x)-a^2 b^2+3 b^4\right)}{(a-b)^2 (a+b)^2 (a \sin (d+e x)+b)^3}}{3 e}","-\frac{\left(2 a^2+b^2\right) \cos (d+e x)}{3 e \left(a^2-b^2\right)^2 (a \sin (d+e x)+b)}+\frac{b \cos (d+e x)}{3 e \left(a^2-b^2\right) (a \sin (d+e x)+b)^2}+\frac{2 a b \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a^2-b^2}}\right)}{e \left(a^2-b^2\right)^{5/2}}-\frac{\cos (d+e x)}{3 e (a \sin (d+e x)+b)^3}",1,"-1/3*((6*a*b*ArcTan[(a + b*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (Cos[d + e*x]*(a^4 - a^2*b^2 + 3*b^4 + 3*a*b*(a^2 + b^2)*Sin[d + e*x] + a^2*(2*a^2 + b^2)*Sin[d + e*x]^2))/((a - b)^2*(a + b)^2*(b + a*Sin[d + e*x])^3))/e","A",1
503,1,286,242,0.7653205,"\int \frac{d+e \sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Integrate[(d + e*Sin[x])/(a + b*Sin[x] + c*Sin[x]^2),x]","\frac{\frac{\left(e \left(\sqrt{4 a c-b^2}+i b\right)-2 i c d\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b-i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{-i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}+\frac{\left(e \left(\sqrt{4 a c-b^2}-i b\right)+2 i c d\right) \tan ^{-1}\left(\frac{2 c+\tan \left(\frac{x}{2}\right) \left(b+i \sqrt{4 a c-b^2}\right)}{\sqrt{2} \sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}\right)}{\sqrt{i b \sqrt{4 a c-b^2}-2 c (a+c)+b^2}}}{\sqrt{2 a c-\frac{b^2}{2}}}","\frac{\sqrt{2} \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(b-\sqrt{b^2-4 a c}\right)+2 c}{\sqrt{2} \sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{-b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}+\frac{\sqrt{2} \left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b\right)+2 c}{\sqrt{2} \sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}\right)}{\sqrt{b \sqrt{b^2-4 a c}-2 c (a+c)+b^2}}",1,"((((-2*I)*c*d + (I*b + Sqrt[-b^2 + 4*a*c])*e)*ArcTan[(2*c + (b - I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) - I*b*Sqrt[-b^2 + 4*a*c]] + (((2*I)*c*d + ((-I)*b + Sqrt[-b^2 + 4*a*c])*e)*ArcTan[(2*c + (b + I*Sqrt[-b^2 + 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) + I*b*Sqrt[-b^2 + 4*a*c]])/Sqrt[-1/2*b^2 + 2*a*c]","C",1
504,1,140,331,0.8379471,"\int (a+b \sin (d+e x)) \left(b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right)^{3/2} \, dx","Integrate[(a + b*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2),x]","\frac{\sqrt{(a \sin (d+e x)+b)^2} \left(8 a \left(a^3+3 a b^2\right) \cos (3 (d+e x))+3 a b \left(20 \left(3 a^2+4 b^2\right) (d+e x)-8 \left(4 a^2+3 b^2\right) \sin (2 (d+e x))+a^2 \sin (4 (d+e x))\right)-24 \left(3 a^4+21 a^2 b^2+4 b^4\right) \cos (d+e x)\right)}{96 e (a \sin (d+e x)+b)}","-\frac{b \cos (d+e x) \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}{4 e}-\frac{\left(4 a^2+3 b^2\right) \cos (d+e x) \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}{12 e (a \sin (d+e x)+b)}+\frac{5 a^4 b x \left(3 a^2+4 b^2\right) \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}{8 \left(a^2 \sin (d+e x)+a b\right)^3}-\frac{a^4 b \left(29 a^2+6 b^2\right) \sin (d+e x) \cos (d+e x) \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}{24 e \left(a^2 \sin (d+e x)+a b\right)^3}-\frac{\left(4 a^4+28 a^2 b^2+3 b^4\right) \cos (d+e x) \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}{6 e (a \sin (d+e x)+b)^3}",1,"(Sqrt[(b + a*Sin[d + e*x])^2]*(-24*(3*a^4 + 21*a^2*b^2 + 4*b^4)*Cos[d + e*x] + 8*a*(a^3 + 3*a*b^2)*Cos[3*(d + e*x)] + 3*a*b*(20*(3*a^2 + 4*b^2)*(d + e*x) - 8*(4*a^2 + 3*b^2)*Sin[2*(d + e*x)] + a^2*Sin[4*(d + e*x)])))/(96*e*(b + a*Sin[d + e*x]))","A",1
505,1,70,185,0.1851201,"\int (a+b \sin (d+e x)) \sqrt{b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)} \, dx","Integrate[(a + b*Sin[d + e*x])*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2],x]","-\frac{\sqrt{(a \sin (d+e x)+b)^2} \left(4 \left(a^2+b^2\right) \cos (d+e x)+a b (\sin (2 (d+e x))-6 (d+e x))\right)}{4 e (a \sin (d+e x)+b)}","\frac{3 a^2 b x \sqrt{a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}{2 \left(a^2 \sin (d+e x)+a b\right)}-\frac{a^2 b \sin (d+e x) \cos (d+e x) \sqrt{a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}{2 e \left(a^2 \sin (d+e x)+a b\right)}-\frac{\left(a^2+b^2\right) \cos (d+e x) \sqrt{a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}{e (a \sin (d+e x)+b)}",1,"-1/4*(Sqrt[(b + a*Sin[d + e*x])^2]*(4*(a^2 + b^2)*Cos[d + e*x] + a*b*(-6*(d + e*x) + Sin[2*(d + e*x)])))/(e*(b + a*Sin[d + e*x]))","A",1
506,1,85,137,0.164816,"\int \frac{a+b \sin (d+e x)}{\sqrt{b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)}} \, dx","Integrate[(a + b*Sin[d + e*x])/Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2],x]","\frac{(a \sin (d+e x)+b) \left(b (d+e x)-2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)\right)}{a e \sqrt{(a \sin (d+e x)+b)^2}}","\frac{b x (a \sin (d+e x)+b)}{a \sqrt{a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}-\frac{2 \sqrt{a^2-b^2} (a \sin (d+e x)+b) \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a^2-b^2}}\right)}{a e \sqrt{a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}",1,"((b*(d + e*x) - 2*Sqrt[-a^2 + b^2]*ArcTan[(a + b*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])*(b + a*Sin[d + e*x]))/(a*e*Sqrt[(b + a*Sin[d + e*x])^2])","A",1
507,1,144,239,0.3472725,"\int \frac{a+b \sin (d+e x)}{\left(b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right)^{3/2}} \, dx","Integrate[(a + b*Sin[d + e*x])/(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2),x]","\frac{\sqrt{b^2-a^2} \cos (d+e x) \left(a^2-a b \sin (d+e x)-2 b^2\right)-2 a (a \sin (d+e x)+b)^2 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)}{2 e (b-a) (a+b) \sqrt{b^2-a^2} (a \sin (d+e x)+b) \sqrt{(a \sin (d+e x)+b)^2}}","-\frac{\cos (d+e x) (a \sin (d+e x)+b)}{2 e \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}-\frac{\left(a^2 \sin (d+e x)+a b\right)^3 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a^2-b^2}}\right)}{a^2 e \left(a^2-b^2\right)^{3/2} \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}+\frac{b \cos (d+e x) \left(a^2 \sin (d+e x)+a b\right)^3}{2 e \left(a^2-b^2\right) \left(a^4 \sin (d+e x)+a^3 b\right) \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}",1,"(-2*a*ArcTan[(a + b*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]]*(b + a*Sin[d + e*x])^2 + Sqrt[-a^2 + b^2]*Cos[d + e*x]*(a^2 - 2*b^2 - a*b*Sin[d + e*x]))/(2*(-a + b)*(a + b)*Sqrt[-a^2 + b^2]*e*(b + a*Sin[d + e*x])*Sqrt[(b + a*Sin[d + e*x])^2])","A",1
508,1,11,11,0.0501276,"\int \frac{a+b \cos (x)}{b^2+2 a b \cos (x)+a^2 \cos ^2(x)} \, dx","Integrate[(a + b*Cos[x])/(b^2 + 2*a*b*Cos[x] + a^2*Cos[x]^2),x]","\frac{\sin (x)}{a \cos (x)+b}","\frac{\sin (x)}{a \cos (x)+b}",1,"Sin[x]/(b + a*Cos[x])","A",1
509,1,241,246,0.5573397,"\int \frac{d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Integrate[(d + e*Cos[x])/(a + b*Cos[x] + c*Cos[x]^2),x]","\frac{\sqrt{2} \left(\frac{\left(e \left(\sqrt{b^2-4 a c}-b\right)+2 c d\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}-b+2 c\right)}{\sqrt{2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}-\frac{\left(e \left(\sqrt{b^2-4 a c}+b\right)-2 c d\right) \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \left(\sqrt{b^2-4 a c}+b-2 c\right)}{\sqrt{-2 b \sqrt{b^2-4 a c}+4 c (a+c)-2 b^2}}\right)}{\sqrt{-b \sqrt{b^2-4 a c}+2 c (a+c)-b^2}}\right)}{\sqrt{b^2-4 a c}}","\frac{2 \left(\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{-\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{-\sqrt{b^2-4 a c}+b-2 c} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}+\frac{2 \left(e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right) \tan ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right) \sqrt{\sqrt{b^2-4 a c}+b-2 c}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right)}{\sqrt{\sqrt{b^2-4 a c}+b-2 c} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}",1,"(Sqrt[2]*(-(((-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[((b - 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) - 2*b*Sqrt[b^2 - 4*a*c]]])/Sqrt[-b^2 + 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]]) + ((2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[((-b + 2*c + Sqrt[b^2 - 4*a*c])*Tan[x/2])/Sqrt[-2*b^2 + 4*c*(a + c) + 2*b*Sqrt[b^2 - 4*a*c]]])/Sqrt[-b^2 + 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]))/Sqrt[b^2 - 4*a*c]","A",1
510,1,153,144,2.1003995,"\int (a+b \tan (d+e x)) \left(b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right)^2 \, dx","Integrate[(a + b*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^2,x]","\frac{3 a^4 b \tan ^4(d+e x)+18 a^2 b \left(a^2+2 b^2\right) \tan ^2(d+e x)+6 \left(a^2+b^2\right) \left(i (a+i b)^3 \log (\tan (d+e x)+i)-i (a-i b)^3 \log (-\tan (d+e x)+i)\right)-12 a \left(a^4-2 a^2 b^2-4 b^4\right) \tan (d+e x)+4 a^3 \left(a^2+4 b^2\right) \tan ^3(d+e x)}{12 e}","-\frac{a \left(a^4-b^4\right) \tan (d+e x)}{e}+\frac{\left(a^2+b^2\right) (a \tan (d+e x)+b)^3}{3 e}+\frac{b \left(a^2+b^2\right) (a \tan (d+e x)+b)^2}{2 e}+\frac{b \left(3 a^2-b^2\right) \left(a^2+b^2\right) \log (\cos (d+e x))}{e}+a x \left(a^2-3 b^2\right) \left(a^2+b^2\right)+\frac{b (a \tan (d+e x)+b)^4}{4 e}",1,"(6*(a^2 + b^2)*((-I)*(a - I*b)^3*Log[I - Tan[d + e*x]] + I*(a + I*b)^3*Log[I + Tan[d + e*x]]) - 12*a*(a^4 - 2*a^2*b^2 - 4*b^4)*Tan[d + e*x] + 18*a^2*b*(a^2 + 2*b^2)*Tan[d + e*x]^2 + 4*a^3*(a^2 + 4*b^2)*Tan[d + e*x]^3 + 3*a^4*b*Tan[d + e*x]^4)/(12*e)","C",1
511,1,88,72,0.3260503,"\int (a+b \tan (d+e x)) \left(b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right) \, dx","Integrate[(a + b*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2),x]","\frac{2 a \left(a^2+2 b^2\right) \tan (d+e x)+\left(a^2+b^2\right) ((b+i a) \log (-\tan (d+e x)+i)+(b-i a) \log (\tan (d+e x)+i))+a^2 b \tan ^2(d+e x)}{2 e}","-\frac{b \left(a^2+b^2\right) \log (\cos (d+e x))}{e}-a x \left(a^2+b^2\right)+\frac{a^2 (a+b \tan (d+e x))^2}{2 b e}+\frac{2 a b^2 \tan (d+e x)}{e}",1,"((a^2 + b^2)*((I*a + b)*Log[I - Tan[d + e*x]] + ((-I)*a + b)*Log[I + Tan[d + e*x]]) + 2*a*(a^2 + 2*b^2)*Tan[d + e*x] + a^2*b*Tan[d + e*x]^2)/(2*e)","C",1
512,1,187,101,2.3383724,"\int \frac{a+b \tan (d+e x)}{b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx","Integrate[(a + b*Tan[d + e*x])/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2),x]","\frac{\frac{b (-((a+i b) \log (-\tan (d+e x)+i))-(a-i b) \log (\tan (d+e x)+i)+2 a \log (a \tan (d+e x)+b))}{a^2+b^2}+(a-b) (a+b) \left(\frac{2 a \left(2 b \log (a \tan (d+e x)+b)-\frac{a^2+b^2}{a \tan (d+e x)+b}\right)}{\left(a^2+b^2\right)^2}+\frac{i \log (-\tan (d+e x)+i)}{(a-i b)^2}-\frac{i \log (\tan (d+e x)+i)}{(a+i b)^2}\right)}{2 a e}","-\frac{a^2-b^2}{e \left(a^2+b^2\right) (a \tan (d+e x)+b)}+\frac{b \left(3 a^2-b^2\right) \log (a \sin (d+e x)+b \cos (d+e x))}{e \left(a^2+b^2\right)^2}-\frac{a x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^2}",1,"((b*(-((a + I*b)*Log[I - Tan[d + e*x]]) - (a - I*b)*Log[I + Tan[d + e*x]] + 2*a*Log[b + a*Tan[d + e*x]]))/(a^2 + b^2) + (a - b)*(a + b)*((I*Log[I - Tan[d + e*x]])/(a - I*b)^2 - (I*Log[I + Tan[d + e*x]])/(a + I*b)^2 + (2*a*(2*b*Log[b + a*Tan[d + e*x]] - (a^2 + b^2)/(b + a*Tan[d + e*x])))/(a^2 + b^2)^2))/(2*a*e)","C",1
513,1,308,197,5.4094266,"\int \frac{a+b \tan (d+e x)}{\left(b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right)^2} \, dx","Integrate[(a + b*Tan[d + e*x])/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^2,x]","\frac{3 b \left(\frac{a \left(-\frac{\left(a^2+b^2\right) \left(a^2+4 a b \tan (d+e x)+5 b^2\right)}{(a \tan (d+e x)+b)^2}-2 \left(a^2-3 b^2\right) \log (a \tan (d+e x)+b)\right)}{\left(a^2+b^2\right)^3}+\frac{\log (-\tan (d+e x)+i)}{(a-i b)^3}+\frac{\log (\tan (d+e x)+i)}{(a+i b)^3}\right)-(a-b) (a+b) \left(-\frac{6 a \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^3 (a \tan (d+e x)+b)}+\frac{6 a b}{\left(a^2+b^2\right)^2 (a \tan (d+e x)+b)^2}+\frac{2 a}{\left(a^2+b^2\right) (a \tan (d+e x)+b)^3}+\frac{24 a b (a-b) (a+b) \log (a \tan (d+e x)+b)}{\left(a^2+b^2\right)^4}+\frac{3 i \log (-\tan (d+e x)+i)}{(a-i b)^4}-\frac{3 i \log (\tan (d+e x)+i)}{(a+i b)^4}\right)}{6 a e}","-\frac{a^2-b^2}{3 e \left(a^2+b^2\right) (a \tan (d+e x)+b)^3}-\frac{b \left(3 a^2-b^2\right)}{2 e \left(a^2+b^2\right)^2 (a \tan (d+e x)+b)^2}+\frac{a^4-6 a^2 b^2+b^4}{e \left(a^2+b^2\right)^3 (a \tan (d+e x)+b)}-\frac{b \left(5 a^4-10 a^2 b^2+b^4\right) \log (a \sin (d+e x)+b \cos (d+e x))}{e \left(a^2+b^2\right)^4}+\frac{a x \left(a^4-10 a^2 b^2+5 b^4\right)}{\left(a^2+b^2\right)^4}",1,"(-((a - b)*(a + b)*(((3*I)*Log[I - Tan[d + e*x]])/(a - I*b)^4 - ((3*I)*Log[I + Tan[d + e*x]])/(a + I*b)^4 + (24*a*(a - b)*b*(a + b)*Log[b + a*Tan[d + e*x]])/(a^2 + b^2)^4 + (2*a)/((a^2 + b^2)*(b + a*Tan[d + e*x])^3) + (6*a*b)/((a^2 + b^2)^2*(b + a*Tan[d + e*x])^2) - (6*a*(a^2 - 3*b^2))/((a^2 + b^2)^3*(b + a*Tan[d + e*x])))) + 3*b*(Log[I - Tan[d + e*x]]/(a - I*b)^3 + Log[I + Tan[d + e*x]]/(a + I*b)^3 + (a*(-2*(a^2 - 3*b^2)*Log[b + a*Tan[d + e*x]] - ((a^2 + b^2)*(a^2 + 5*b^2 + 4*a*b*Tan[d + e*x]))/(b + a*Tan[d + e*x])^2))/(a^2 + b^2)^3))/(6*a*e)","C",1
514,1,147,284,1.3323502,"\int (a+b \tan (d+e x)) \left(b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right)^{3/2} \, dx","Integrate[(a + b*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2),x]","\frac{\sqrt{(a \tan (d+e x)+b)^2} \left(2 a^3 b \tan ^3(d+e x)+3 a^2 \left(a^2+3 b^2\right) \tan ^2(d+e x)+6 a b \left(2 a^2+3 b^2\right) \tan (d+e x)-3 \left(a^2+b^2\right) \left((a-i b)^2 \log (-\tan (d+e x)+i)+(a+i b)^2 \log (\tan (d+e x)+i)\right)\right)}{6 e (a \tan (d+e x)+b)}","\frac{b \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}{3 e}+\frac{\left(a^2+b^2\right) \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}{2 e (a \tan (d+e x)+b)}-\frac{2 a^4 b x \left(a^2+b^2\right) \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}{\left(a^2 \tan (d+e x)+a b\right)^3}+\frac{a^4 b \left(a^2+b^2\right) \tan (d+e x) \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}{e \left(a^2 \tan (d+e x)+a b\right)^3}+\frac{\left(a^4-b^4\right) \log (\cos (d+e x)) \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}{e (a \tan (d+e x)+b)^3}",1,"(Sqrt[(b + a*Tan[d + e*x])^2]*(-3*(a^2 + b^2)*((a - I*b)^2*Log[I - Tan[d + e*x]] + (a + I*b)^2*Log[I + Tan[d + e*x]]) + 6*a*b*(2*a^2 + 3*b^2)*Tan[d + e*x] + 3*a^2*(a^2 + 3*b^2)*Tan[d + e*x]^2 + 2*a^3*b*Tan[d + e*x]^3))/(6*e*(b + a*Tan[d + e*x]))","C",1
515,1,58,122,0.3019985,"\int (a+b \tan (d+e x)) \sqrt{b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx","Integrate[(a + b*Tan[d + e*x])*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2],x]","\frac{\sqrt{(a \tan (d+e x)+b)^2} \left(a b \tan (d+e x)-\left(a^2+b^2\right) \log (\cos (d+e x))\right)}{e (a \tan (d+e x)+b)}","\frac{a^2 b \tan (d+e x) \sqrt{a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}}{e \left(a^2 \tan (d+e x)+a b\right)}-\frac{\left(a^2+b^2\right) \log (\cos (d+e x)) \sqrt{a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}}{e (a \tan (d+e x)+b)}",1,"(Sqrt[(b + a*Tan[d + e*x])^2]*(-((a^2 + b^2)*Log[Cos[d + e*x]]) + a*b*Tan[d + e*x]))/(e*(b + a*Tan[d + e*x]))","A",1
516,1,88,138,0.6843222,"\int \frac{a+b \tan (d+e x)}{\sqrt{b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx","Integrate[(a + b*Tan[d + e*x])/Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2],x]","\frac{(a \tan (d+e x)+b) \left(4 a b \tan ^{-1}(\tan (d+e x))-\left(a^2-b^2\right) \left(\log \left(\sec ^2(d+e x)\right)-2 \log (a \tan (d+e x)+b)\right)\right)}{2 e \left(a^2+b^2\right) \sqrt{(a \tan (d+e x)+b)^2}}","\frac{2 b x \left(a^2 \tan (d+e x)+a b\right)}{\left(a^2+b^2\right) \sqrt{a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}}+\frac{\left(a^2-b^2\right) (a \tan (d+e x)+b) \log (a \sin (d+e x)+b \cos (d+e x))}{e \left(a^2+b^2\right) \sqrt{a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}}",1,"((4*a*b*ArcTan[Tan[d + e*x]] - (a^2 - b^2)*(Log[Sec[d + e*x]^2] - 2*Log[b + a*Tan[d + e*x]]))*(b + a*Tan[d + e*x]))/(2*(a^2 + b^2)*e*Sqrt[(b + a*Tan[d + e*x])^2])","A",1
517,1,268,316,3.4409738,"\int \frac{a+b \tan (d+e x)}{\left(b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right)^{3/2}} \, dx","Integrate[(a + b*Tan[d + e*x])/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2),x]","\frac{(a \tan (d+e x)+b)^3 \left(b \left(\frac{2 a \left(2 b \log (a \tan (d+e x)+b)-\frac{a^2+b^2}{a \tan (d+e x)+b}\right)}{\left(a^2+b^2\right)^2}+\frac{i \log (-\tan (d+e x)+i)}{(a-i b)^2}-\frac{i \log (\tan (d+e x)+i)}{(a+i b)^2}\right)+(a-b) (a+b) \left(\frac{a \left(-\frac{\left(a^2+b^2\right) \left(a^2+4 a b \tan (d+e x)+5 b^2\right)}{(a \tan (d+e x)+b)^2}-2 \left(a^2-3 b^2\right) \log (a \tan (d+e x)+b)\right)}{\left(a^2+b^2\right)^3}+\frac{\log (-\tan (d+e x)+i)}{(a-i b)^3}+\frac{\log (\tan (d+e x)+i)}{(a+i b)^3}\right)\right)}{2 a e \left((a \tan (d+e x)+b)^2\right)^{3/2}}","-\frac{\left(a^2-b^2\right) (a \tan (d+e x)+b)}{2 e \left(a^2+b^2\right) \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}-\frac{4 b x \left(a^2-b^2\right) \left(a^2 \tan (d+e x)+a b\right)^3}{a^2 \left(a^2+b^2\right)^3 \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}-\frac{\left(a^4-6 a^2 b^2+b^4\right) (a \tan (d+e x)+b)^3 \log (a \sin (d+e x)+b \cos (d+e x))}{e \left(a^2+b^2\right)^3 \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}-\frac{b \left(3 a^2-b^2\right) \left(a^2 \tan (d+e x)+a b\right)^3}{e \left(a^2+b^2\right)^2 \left(a^4 \tan (d+e x)+a^3 b\right) \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}",1,"((b + a*Tan[d + e*x])^3*(b*((I*Log[I - Tan[d + e*x]])/(a - I*b)^2 - (I*Log[I + Tan[d + e*x]])/(a + I*b)^2 + (2*a*(2*b*Log[b + a*Tan[d + e*x]] - (a^2 + b^2)/(b + a*Tan[d + e*x])))/(a^2 + b^2)^2) + (a - b)*(a + b)*(Log[I - Tan[d + e*x]]/(a - I*b)^3 + Log[I + Tan[d + e*x]]/(a + I*b)^3 + (a*(-2*(a^2 - 3*b^2)*Log[b + a*Tan[d + e*x]] - ((a^2 + b^2)*(a^2 + 5*b^2 + 4*a*b*Tan[d + e*x]))/(b + a*Tan[d + e*x])^2))/(a^2 + b^2)^3)))/(2*a*e*((b + a*Tan[d + e*x])^2)^(3/2))","C",1
518,1,130,184,0.8744157,"\int (a+b \sec (d+e x)) \left(b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right)^2 \, dx","Integrate[(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^2,x]","\frac{3 b \left(19 a^4+56 a^2 b^2+8 b^4\right) \tanh ^{-1}(\sin (d+e x))+8 a^3 \left(a^2+4 b^2\right) \tan ^3(d+e x)+3 a \tan (d+e x) \left(2 a^3 b \sec ^3(d+e x)+a b \left(19 a^2+24 b^2\right) \sec (d+e x)+8 \left(a^4+10 a^2 b^2+4 b^4\right)\right)+24 a b^4 e x}{24 e}","\frac{a^2 b \left(41 a^2+26 b^2\right) \tan (d+e x) \sec (d+e x)}{24 e}+\frac{\left(4 a^2+7 b^2\right) \tan (d+e x) \left(a^2 \sec (d+e x)+a b\right)^2}{12 a e}+\frac{b \tan (d+e x) \left(a^2 \sec (d+e x)+a b\right)^3}{4 a^2 e}+\frac{a \left(4 a^4+50 a^2 b^2+19 b^4\right) \tan (d+e x)}{6 e}+\frac{b \left(19 a^4+56 a^2 b^2+8 b^4\right) \tanh ^{-1}(\sin (d+e x))}{8 e}+a b^4 x",1,"(24*a*b^4*e*x + 3*b*(19*a^4 + 56*a^2*b^2 + 8*b^4)*ArcTanh[Sin[d + e*x]] + 3*a*(8*(a^4 + 10*a^2*b^2 + 4*b^4) + a*b*(19*a^2 + 24*b^2)*Sec[d + e*x] + 2*a^3*b*Sec[d + e*x]^3)*Tan[d + e*x] + 8*a^3*(a^2 + 4*b^2)*Tan[d + e*x]^3)/(24*e)","A",1
519,1,64,76,0.2838077,"\int (a+b \sec (d+e x)) \left(b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right) \, dx","Integrate[(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2),x]","\frac{b \left(5 a^2+2 b^2\right) \tanh ^{-1}(\sin (d+e x))+a \tan (d+e x) \left(2 a^2+a b \sec (d+e x)+4 b^2\right)+2 a b^2 e x}{2 e}","\frac{a \left(a^2+2 b^2\right) \tan (d+e x)}{e}+\frac{b \left(5 a^2+2 b^2\right) \tanh ^{-1}(\sin (d+e x))}{2 e}+\frac{a^2 b \tan (d+e x) \sec (d+e x)}{2 e}+a b^2 x",1,"(2*a*b^2*e*x + b*(5*a^2 + 2*b^2)*ArcTanh[Sin[d + e*x]] + a*(2*a^2 + 4*b^2 + a*b*Sec[d + e*x])*Tan[d + e*x])/(2*e)","A",1
520,1,97,92,0.387364,"\int \frac{a+b \sec (d+e x)}{b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)} \, dx","Integrate[(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2),x]","\frac{2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)+\frac{a (a d+a e x-b \sin (d+e x)+b (d+e x) \cos (d+e x))}{a+b \cos (d+e x)}}{b^2 e}","-\frac{a^2 \tan (d+e x)}{b e \left(a^2 \sec (d+e x)+a b\right)}-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{b^2 e}+\frac{a x}{b^2}",1,"(2*Sqrt[-a^2 + b^2]*ArcTanh[((-a + b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]] + (a*(a*d + a*e*x + b*(d + e*x)*Cos[d + e*x] - b*Sin[d + e*x]))/(a + b*Cos[d + e*x]))/(b^2*e)","A",1
521,1,276,230,1.5287189,"\int \frac{a+b \sec (d+e x)}{\left(b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right)^2} \, dx","Integrate[(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^2,x]","\frac{\sec ^3(d+e x) (a+b \cos (d+e x)) (a+b \sec (d+e x)) \left(-2 a^3 b \sin (d+e x)+\frac{a^2 b \left(7 a^2-9 b^2\right) \sin (d+e x) (a+b \cos (d+e x))}{(a-b) (a+b)}-\frac{a b \left(11 a^4-23 a^2 b^2+18 b^4\right) \sin (d+e x) (a+b \cos (d+e x))^2}{(a-b)^2 (a+b)^2}+\frac{6 \left(-2 a^6+5 a^4 b^2-3 a^2 b^4+2 b^6\right) (a+b \cos (d+e x))^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+6 a (d+e x) (a+b \cos (d+e x))^3\right)}{6 b^4 e (a \cos (d+e x)+b) (a \sec (d+e x)+b)^4}","-\frac{a \left(3 a^2-5 b^2\right) \tan (d+e x)}{6 b^2 e \left(a^2-b^2\right) (a \sec (d+e x)+b)^2}-\frac{\left(a^2-2 b^2\right) \left(2 a^4-a^2 b^2+b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{b^4 e (a-b)^{5/2} (a+b)^{5/2}}-\frac{a \left(6 a^4-11 a^2 b^2+11 b^4\right) \tan (d+e x)}{6 b^3 e \left(a^2-b^2\right)^2 (a \sec (d+e x)+b)}-\frac{a^4 \tan (d+e x)}{3 b e \left(a^2 \sec (d+e x)+a b\right)^3}+\frac{a x}{b^4}",1,"((a + b*Cos[d + e*x])*Sec[d + e*x]^3*(a + b*Sec[d + e*x])*(6*a*(d + e*x)*(a + b*Cos[d + e*x])^3 + (6*(-2*a^6 + 5*a^4*b^2 - 3*a^2*b^4 + 2*b^6)*ArcTanh[((-a + b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]]*(a + b*Cos[d + e*x])^3)/(-a^2 + b^2)^(5/2) - 2*a^3*b*Sin[d + e*x] + (a^2*b*(7*a^2 - 9*b^2)*(a + b*Cos[d + e*x])*Sin[d + e*x])/((a - b)*(a + b)) - (a*b*(11*a^4 - 23*a^2*b^2 + 18*b^4)*(a + b*Cos[d + e*x])^2*Sin[d + e*x])/((a - b)^2*(a + b)^2)))/(6*b^4*e*(b + a*Cos[d + e*x])*(b + a*Sec[d + e*x])^4)","A",1
522,1,128,359,0.8179641,"\int (a+b \sec (d+e x)) \left(b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right)^{3/2} \, dx","Integrate[(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2),x]","\frac{\cos (d+e x) \sqrt{(a \sec (d+e x)+b)^2} \left(2 a^3 b \tan ^3(d+e x)+3 a \tan (d+e x) \left(a \left(a^2+3 b^2\right) \sec (d+e x)+8 a^2 b+6 b^3\right)+3 \left(a^4+9 a^2 b^2+2 b^4\right) \tanh ^{-1}(\sin (d+e x))+6 a b^3 e x\right)}{6 e (a+b \cos (d+e x))}","\frac{a^5 \left(3 a^2+5 b^2\right) \tan (d+e x) \sec (d+e x) \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}{6 e \left(a^2 \sec (d+e x)+a b\right)^3}+\frac{a^4 b \left(11 a^2+8 b^2\right) \tan (d+e x) \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}{3 e \left(a^2 \sec (d+e x)+a b\right)^3}+\frac{\left(a^4+9 a^2 b^2+2 b^4\right) \tanh ^{-1}(\sin (d+e x)) \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}{2 e (a \sec (d+e x)+b)^3}+\frac{a^4 b^3 x \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}{\left(a^2 \sec (d+e x)+a b\right)^3}+\frac{b \tan (d+e x) \left(a^3 \sec (d+e x)+a^2 b\right)^2 \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}{3 e \left(a^2 \sec (d+e x)+a b\right)^3}",1,"(Cos[d + e*x]*Sqrt[(b + a*Sec[d + e*x])^2]*(6*a*b^3*e*x + 3*(a^4 + 9*a^2*b^2 + 2*b^4)*ArcTanh[Sin[d + e*x]] + 3*a*(8*a^2*b + 6*b^3 + a*(a^2 + 3*b^2)*Sec[d + e*x])*Tan[d + e*x] + 2*a^3*b*Tan[d + e*x]^3))/(6*e*(a + b*Cos[d + e*x]))","A",1
523,1,67,173,0.2588514,"\int (a+b \sec (d+e x)) \sqrt{b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)} \, dx","Integrate[(a + b*Sec[d + e*x])*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2],x]","\frac{\cos (d+e x) \sqrt{(a \sec (d+e x)+b)^2} \left(\left(a^2+b^2\right) \tanh ^{-1}(\sin (d+e x))+a b (\tan (d+e x)+e x)\right)}{e (a+b \cos (d+e x))}","\frac{a^2 b x \sqrt{a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2}}{a^2 \sec (d+e x)+a b}+\frac{a^2 b \tan (d+e x) \sqrt{a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2}}{e \left(a^2 \sec (d+e x)+a b\right)}+\frac{\left(a^2+b^2\right) \tanh ^{-1}(\sin (d+e x)) \sqrt{a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2}}{e (a \sec (d+e x)+b)}",1,"(Cos[d + e*x]*Sqrt[(b + a*Sec[d + e*x])^2]*((a^2 + b^2)*ArcTanh[Sin[d + e*x]] + a*b*(e*x + Tan[d + e*x])))/(e*(a + b*Cos[d + e*x]))","A",1
524,1,92,142,0.3677916,"\int \frac{a+b \sec (d+e x)}{\sqrt{b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)}} \, dx","Integrate[(a + b*Sec[d + e*x])/Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2],x]","\frac{\sec (d+e x) (a+b \cos (d+e x)) \left(2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)+a (d+e x)\right)}{b e \sqrt{(a \sec (d+e x)+b)^2}}","\frac{x \left(a^2 \sec (d+e x)+a b\right)}{b \sqrt{a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2}}-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right) (a \sec (d+e x)+b)}{b e \sqrt{a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2}}",1,"((a*(d + e*x) + 2*Sqrt[-a^2 + b^2]*ArcTanh[((-a + b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])*(a + b*Cos[d + e*x])*Sec[d + e*x])/(b*e*Sqrt[(b + a*Sec[d + e*x])^2])","A",1
525,1,216,330,1.003586,"\int \frac{a+b \sec (d+e x)}{\left(b^2+2 a b \sec (d+e x)+a^2 \sec ^2(d+e x)\right)^{3/2}} \, dx","Integrate[(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2),x]","\frac{\sec ^2(d+e x) (a+b \cos (d+e x)) (a+b \sec (d+e x)) \left(\frac{a b \left(3 a^2-4 b^2\right) \sin (d+e x) (a+b \cos (d+e x))}{(b-a) (a+b)}+a^2 b \sin (d+e x)+\frac{2 \left(2 a^4-3 a^2 b^2+2 b^4\right) (a+b \cos (d+e x))^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+2 a (d+e x) (a+b \cos (d+e x))^2\right)}{2 b^3 e (a \cos (d+e x)+b) \left((a \sec (d+e x)+b)^2\right)^{3/2}}","-\frac{\tan (d+e x) \left(a^2 \sec (d+e x)+a b\right)}{2 b e \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}+\frac{x \left(a^2 \sec (d+e x)+a b\right)^3}{a^2 b^3 \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}-\frac{\left(2 a^4-3 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right) (a \sec (d+e x)+b)^3}{b^3 e (a-b)^{3/2} (a+b)^{3/2} \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}-\frac{\left(2 a^2-3 b^2\right) \tan (d+e x) \left(a^2 \sec (d+e x)+a b\right)^3}{2 b^2 e \left(a^2-b^2\right) \left(a^3 \sec (d+e x)+a^2 b\right) \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}",1,"((a + b*Cos[d + e*x])*Sec[d + e*x]^2*(a + b*Sec[d + e*x])*(2*a*(d + e*x)*(a + b*Cos[d + e*x])^2 + (2*(2*a^4 - 3*a^2*b^2 + 2*b^4)*ArcTanh[((-a + b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]]*(a + b*Cos[d + e*x])^2)/(-a^2 + b^2)^(3/2) + a^2*b*Sin[d + e*x] + (a*b*(3*a^2 - 4*b^2)*(a + b*Cos[d + e*x])*Sin[d + e*x])/((-a + b)*(a + b))))/(2*b^3*e*(b + a*Cos[d + e*x])*((b + a*Sec[d + e*x])^2)^(3/2))","A",1
526,1,19,17,0.0053106,"\int \frac{\cos (x)-i \sin (x)}{\cos (x)+i \sin (x)} \, dx","Integrate[(Cos[x] - I*Sin[x])/(Cos[x] + I*Sin[x]),x]","\frac{1}{2} \sin (2 x)+\frac{1}{2} i \cos (2 x)","\frac{1}{2} i (\cos (x)-i \sin (x))^2",1,"(I/2)*Cos[2*x] + Sin[2*x]/2","A",1
527,1,19,17,0.0057988,"\int \frac{\cos (x)+i \sin (x)}{\cos (x)-i \sin (x)} \, dx","Integrate[(Cos[x] + I*Sin[x])/(Cos[x] - I*Sin[x]),x]","\frac{1}{2} \sin (2 x)-\frac{1}{2} i \cos (2 x)","-\frac{i}{2 (\cos (x)-i \sin (x))^2}",1,"(-1/2*I)*Cos[2*x] + Sin[2*x]/2","A",1
528,1,6,6,0.0255322,"\int \frac{\cos (x)-\sin (x)}{\cos (x)+\sin (x)} \, dx","Integrate[(Cos[x] - Sin[x])/(Cos[x] + Sin[x]),x]","\log (\sin (x)+\cos (x))","\log (\sin (x)+\cos (x))",1,"Log[Cos[x] + Sin[x]]","A",1
529,1,39,47,0.1421895,"\int \frac{B \cos (x)+C \sin (x)}{b \cos (x)+c \sin (x)} \, dx","Integrate[(B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x]),x]","\frac{x (b B+c C)+(B c-b C) \log (b \cos (x)+c \sin (x))}{b^2+c^2}","\frac{x (b B+c C)}{b^2+c^2}+\frac{(B c-b C) \log (b \cos (x)+c \sin (x))}{b^2+c^2}",1,"((b*B + c*C)*x + (B*c - b*C)*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",1
530,1,75,74,0.240547,"\int \frac{B \cos (x)+C \sin (x)}{(b \cos (x)+c \sin (x))^2} \, dx","Integrate[(B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^2,x]","\frac{b C-B c}{\left(b^2+c^2\right) (b \cos (x)+c \sin (x))}+\frac{2 (b B+c C) \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)}{\left(b^2+c^2\right)^{3/2}}","-\frac{B c-b C}{\left(b^2+c^2\right) (b \cos (x)+c \sin (x))}-\frac{(b B+c C) \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{\left(b^2+c^2\right)^{3/2}}",1,"(2*(b*B + c*C)*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2) + (-(B*c) + b*C)/((b^2 + c^2)*(b*Cos[x] + c*Sin[x]))","A",1
531,1,64,66,0.1892068,"\int \frac{B \cos (x)+C \sin (x)}{(b \cos (x)+c \sin (x))^3} \, dx","Integrate[(B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^3,x]","\frac{C \left(b^2+c^2\right)+b \sin (2 x) (b B+c C)-c \cos (2 x) (b B+c C)}{2 b \left(b^2+c^2\right) (b \cos (x)+c \sin (x))^2}","\frac{\sin (x) (b B+c C)}{b \left(b^2+c^2\right) (b \cos (x)+c \sin (x))}-\frac{B c-b C}{2 \left(b^2+c^2\right) (b \cos (x)+c \sin (x))^2}",1,"((b^2 + c^2)*C - c*(b*B + c*C)*Cos[2*x] + b*(b*B + c*C)*Sin[2*x])/(2*b*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2)","A",1
532,1,78,84,0.2526593,"\int \frac{A+B \cos (x)+C \sin (x)}{b \cos (x)+c \sin (x)} \, dx","Integrate[(A + B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x]),x]","\frac{2 A \sqrt{b^2+c^2} \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)+x (b B+c C)+(B c-b C) \log (b \cos (x)+c \sin (x))}{b^2+c^2}","-\frac{A \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{\sqrt{b^2+c^2}}+\frac{x (b B+c C)}{b^2+c^2}+\frac{(B c-b C) \log (b \cos (x)+c \sin (x))}{b^2+c^2}",1,"((b*B + c*C)*x + 2*A*Sqrt[b^2 + c^2]*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]] + (B*c - b*C)*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",1
533,1,92,85,0.2753145,"\int \frac{A+B \cos (x)+C \sin (x)}{(b \cos (x)+c \sin (x))^2} \, dx","Integrate[(A + B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^2,x]","\frac{A \left(b^2+c^2\right) \sin (x)+b (b C-B c)}{b \left(b^2+c^2\right) (b \cos (x)+c \sin (x))}+\frac{2 (b B+c C) \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)}{\left(b^2+c^2\right)^{3/2}}","-\frac{-A b \sin (x)+A c \cos (x)-b C+B c}{\left(b^2+c^2\right) (b \cos (x)+c \sin (x))}-\frac{(b B+c C) \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{\left(b^2+c^2\right)^{3/2}}",1,"(2*(b*B + c*C)*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2) + (b*(-(B*c) + b*C) + A*(b^2 + c^2)*Sin[x])/(b*(b^2 + c^2)*(b*Cos[x] + c*Sin[x]))","A",1
534,1,122,129,0.6609713,"\int \frac{A+B \cos (x)+C \sin (x)}{(b \cos (x)+c \sin (x))^3} \, dx","Integrate[(A + B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^3,x]","\frac{A b^2 \sin (x)-A b c \cos (x)+b^2 B \sin (2 x)+b^2 C-c \cos (2 x) (b B+c C)+b c C \sin (2 x)+c^2 C}{2 b \left(b^2+c^2\right) (b \cos (x)+c \sin (x))^2}+\frac{A \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-c}{\sqrt{b^2+c^2}}\right)}{\left(b^2+c^2\right)^{3/2}}","-\frac{-A b \sin (x)+A c \cos (x)-b C+B c}{2 \left(b^2+c^2\right) (b \cos (x)+c \sin (x))^2}-\frac{A \tanh ^{-1}\left(\frac{c \cos (x)-b \sin (x)}{\sqrt{b^2+c^2}}\right)}{2 \left(b^2+c^2\right)^{3/2}}-\frac{c \cos (x) (b B+c C)-b \sin (x) (b B+c C)}{\left(b^2+c^2\right)^2 (b \cos (x)+c \sin (x))}",1,"(A*ArcTanh[(-c + b*Tan[x/2])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2) + (b^2*C + c^2*C - A*b*c*Cos[x] - c*(b*B + c*C)*Cos[2*x] + A*b^2*Sin[x] + b^2*B*Sin[2*x] + b*c*C*Sin[2*x])/(2*b*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2)","A",1
535,1,95,115,0.2864845,"\int \frac{A+B \cos (x)}{a+b \cos (x)+c \sin (x)} \, dx","Integrate[(A + B*Cos[x])/(a + b*Cos[x] + c*Sin[x]),x]","\frac{B (c \log (a+b \cos (x)+c \sin (x))+b x)-\frac{2 \left(A \left(b^2+c^2\right)-a b B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\sqrt{-a^2+b^2+c^2}}}{b^2+c^2}","-\frac{2 \left(a b B-A \left(b^2+c^2\right)\right) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(b^2+c^2\right) \sqrt{a^2-b^2-c^2}}+\frac{B c \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}+\frac{b B x}{b^2+c^2}",1,"((-2*(-(a*b*B) + A*(b^2 + c^2))*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/Sqrt[-a^2 + b^2 + c^2] + B*(b*x + c*Log[a + b*Cos[x] + c*Sin[x]]))/(b^2 + c^2)","A",1
536,1,118,113,0.3258625,"\int \frac{A+B \cos (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Integrate[(A + B*Cos[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","\frac{\sin (x) \left(A \left(b^2+c^2\right)-a b B\right)+c (a A-b B)}{b \left(-a^2+b^2+c^2\right) (a+b \cos (x)+c \sin (x))}+\frac{2 (a A-b B) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{3/2}}","\frac{2 (a A-b B) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(a^2-b^2-c^2\right)^{3/2}}+\frac{-\sin (x) (A b-a B)+A c \cos (x)+B c}{\left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))}",1,"(2*(a*A - b*B)*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(3/2) + ((a*A - b*B)*c + (-(a*b*B) + A*(b^2 + c^2))*Sin[x])/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[x] + c*Sin[x]))","A",1
537,1,326,200,0.9370984,"\int \frac{A+B \cos (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx","Integrate[(A + B*Cos[x])/(a + b*Cos[x] + c*Sin[x])^3,x]","\frac{-6 a^3 A c+4 a^3 b B \sin (x)-2 b c \cos (x) \left(2 a^2 A-3 a b B+A \left(b^2+c^2\right)\right)+c \cos (2 x) \left(a^2 (-b) B+3 a A \left(b^2+c^2\right)-2 b B \left(b^2+c^2\right)\right)-8 a^2 A b^2 \sin (x)-12 a^2 A c^2 \sin (x)+a^2 b^2 B \sin (2 x)+9 a^2 b B c-3 a A b^3 \sin (2 x)-3 a A b^2 c-3 a A b c^2 \sin (2 x)-3 a A c^3+2 a b^3 B \sin (x)+8 a b B c^2 \sin (x)+2 A b^4 \sin (x)+2 A b^2 c^2 \sin (x)+2 b^4 B \sin (2 x)+2 b^2 B c^2 \sin (2 x)}{4 b \left(-a^2+b^2+c^2\right)^2 (a+b \cos (x)+c \sin (x))^2}-\frac{\left(2 a^2 A-3 a b B+A \left(b^2+c^2\right)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{5/2}}","\frac{\left(2 a^2 A-3 a b B+A \left(b^2+c^2\right)\right) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(a^2-b^2-c^2\right)^{5/2}}+\frac{-\sin (x) \left(a^2 (-B)+3 a A b-2 b^2 B\right)+c \cos (x) (3 a A-2 b B)+a B c}{2 \left(a^2-b^2-c^2\right)^2 (a+b \cos (x)+c \sin (x))}+\frac{-\sin (x) (A b-a B)+A c \cos (x)+B c}{2 \left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))^2}",1,"-(((2*a^2*A - 3*a*b*B + A*(b^2 + c^2))*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(5/2)) + (-6*a^3*A*c - 3*a*A*b^2*c + 9*a^2*b*B*c - 3*a*A*c^3 - 2*b*c*(2*a^2*A - 3*a*b*B + A*(b^2 + c^2))*Cos[x] + c*(-(a^2*b*B) + 3*a*A*(b^2 + c^2) - 2*b*B*(b^2 + c^2))*Cos[2*x] - 8*a^2*A*b^2*Sin[x] + 2*A*b^4*Sin[x] + 4*a^3*b*B*Sin[x] + 2*a*b^3*B*Sin[x] - 12*a^2*A*c^2*Sin[x] + 2*A*b^2*c^2*Sin[x] + 8*a*b*B*c^2*Sin[x] - 3*a*A*b^3*Sin[2*x] + a^2*b^2*B*Sin[2*x] + 2*b^4*B*Sin[2*x] - 3*a*A*b*c^2*Sin[2*x] + 2*b^2*B*c^2*Sin[2*x])/(4*b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[x] + c*Sin[x])^2)","A",1
538,1,147,84,0.2215134,"\int \frac{A+B \cos (x)}{a+b \cos (x)+i b \sin (x)} \, dx","Integrate[(A + B*Cos[x])/(a + b*Cos[x] + I*b*Sin[x]),x]","\frac{2 \left(a^2 B-2 a A b+b^2 B\right) \tan ^{-1}\left(\frac{(a+b) \cot \left(\frac{x}{2}\right)}{a-b}\right)+2 i a A b \log \left(a^2+2 a b \cos (x)+b^2\right)-i a^2 B \log \left(a^2+2 a b \cos (x)+b^2\right)-i b^2 B \log \left(a^2+2 a b \cos (x)+b^2\right)+a^2 B x+2 a A b x+2 a b B \sin (x)+2 i a b B \cos (x)-b^2 B x}{4 a^2 b}","\frac{i \left(a^2 (-B)+2 a A b-b^2 B\right) \log (a+i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac{x (2 a A-b B)}{2 a^2}+\frac{B \sin (x)}{2 a}+\frac{i B \cos (x)}{2 a}",1,"(2*a*A*b*x + a^2*B*x - b^2*B*x + 2*(-2*a*A*b + a^2*B + b^2*B)*ArcTan[((a + b)*Cot[x/2])/(a - b)] + (2*I)*a*b*B*Cos[x] + (2*I)*a*A*b*Log[a^2 + b^2 + 2*a*b*Cos[x]] - I*a^2*B*Log[a^2 + b^2 + 2*a*b*Cos[x]] - I*b^2*B*Log[a^2 + b^2 + 2*a*b*Cos[x]] + 2*a*b*B*Sin[x])/(4*a^2*b)","A",1
539,1,147,84,0.1910455,"\int \frac{A+B \cos (x)}{a+b \cos (x)-i b \sin (x)} \, dx","Integrate[(A + B*Cos[x])/(a + b*Cos[x] - I*b*Sin[x]),x]","\frac{2 \left(a^2 B-2 a A b+b^2 B\right) \tan ^{-1}\left(\frac{(a+b) \cot \left(\frac{x}{2}\right)}{a-b}\right)-2 i a A b \log \left(a^2+2 a b \cos (x)+b^2\right)+i a^2 B \log \left(a^2+2 a b \cos (x)+b^2\right)+i b^2 B \log \left(a^2+2 a b \cos (x)+b^2\right)+a^2 B x+2 a A b x+2 a b B \sin (x)-2 i a b B \cos (x)-b^2 B x}{4 a^2 b}","-\frac{i \left(a^2 (-B)+2 a A b-b^2 B\right) \log (a-i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac{x (2 a A-b B)}{2 a^2}+\frac{B \sin (x)}{2 a}-\frac{i B \cos (x)}{2 a}",1,"(2*a*A*b*x + a^2*B*x - b^2*B*x + 2*(-2*a*A*b + a^2*B + b^2*B)*ArcTan[((a + b)*Cot[x/2])/(a - b)] - (2*I)*a*b*B*Cos[x] - (2*I)*a*A*b*Log[a^2 + b^2 + 2*a*b*Cos[x]] + I*a^2*B*Log[a^2 + b^2 + 2*a*b*Cos[x]] + I*b^2*B*Log[a^2 + b^2 + 2*a*b*Cos[x]] + 2*a*b*B*Sin[x])/(4*a^2*b)","A",1
540,1,96,116,0.2878026,"\int \frac{A+C \sin (x)}{a+b \cos (x)+c \sin (x)} \, dx","Integrate[(A + C*Sin[x])/(a + b*Cos[x] + c*Sin[x]),x]","\frac{C (c x-b \log (a+b \cos (x)+c \sin (x)))-\frac{2 \left(A \left(b^2+c^2\right)-a c C\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\sqrt{-a^2+b^2+c^2}}}{b^2+c^2}","\frac{2 \left(A \left(b^2+c^2\right)-a c C\right) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(b^2+c^2\right) \sqrt{a^2-b^2-c^2}}-\frac{b C \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}+\frac{c C x}{b^2+c^2}",1,"((-2*(A*(b^2 + c^2) - a*c*C)*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/Sqrt[-a^2 + b^2 + c^2] + C*(c*x - b*Log[a + b*Cos[x] + c*Sin[x]]))/(b^2 + c^2)","A",1
541,1,123,114,0.3859869,"\int \frac{A+C \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Integrate[(A + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","\frac{a^2 (-C)+\sin (x) \left(A \left(b^2+c^2\right)-a c C\right)+a A c+b^2 C}{b \left(-a^2+b^2+c^2\right) (a+b \cos (x)+c \sin (x))}+\frac{2 (a A-c C) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{3/2}}","\frac{2 (a A-c C) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(a^2-b^2-c^2\right)^{3/2}}-\frac{-\cos (x) (A c-a C)+A b \sin (x)+b C}{\left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))}",1,"(2*(a*A - c*C)*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(3/2) + (a*A*c - a^2*C + b^2*C + (A*(b^2 + c^2) - a*c*C)*Sin[x])/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[x] + c*Sin[x]))","A",1
542,1,361,200,0.9252039,"\int \frac{A+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx","Integrate[(A + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^3,x]","\frac{2 a^4 C-6 a^3 A c+4 a^3 c C \sin (x)-2 b c \cos (x) \left(2 a^2 A-3 a c C+A \left(b^2+c^2\right)\right)-c \cos (2 x) \left(a^2 c C-3 a A \left(b^2+c^2\right)+2 c C \left(b^2+c^2\right)\right)-8 a^2 A b^2 \sin (x)-12 a^2 A c^2 \sin (x)-4 a^2 b^2 C+a^2 b c C \sin (2 x)+5 a^2 c^2 C-3 a A b^3 \sin (2 x)-3 a A b^2 c-3 a A b c^2 \sin (2 x)-3 a A c^3+2 a b^2 c C \sin (x)+8 a c^3 C \sin (x)+2 A b^4 \sin (x)+2 A b^2 c^2 \sin (x)+2 b^4 C+2 b^3 c C \sin (2 x)+4 b^2 c^2 C+2 b c^3 C \sin (2 x)+2 c^4 C}{4 b \left(-a^2+b^2+c^2\right)^2 (a+b \cos (x)+c \sin (x))^2}-\frac{\left(2 a^2 A-3 a c C+A \left(b^2+c^2\right)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{5/2}}","\frac{\left(2 a^2 A-3 a c C+A \left(b^2+c^2\right)\right) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(a^2-b^2-c^2\right)^{5/2}}-\frac{-\cos (x) \left(a^2 (-C)+3 a A c-2 c^2 C\right)+b \sin (x) (3 a A-2 c C)+a b C}{2 \left(a^2-b^2-c^2\right)^2 (a+b \cos (x)+c \sin (x))}-\frac{-\cos (x) (A c-a C)+A b \sin (x)+b C}{2 \left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))^2}",1,"-(((2*a^2*A + A*(b^2 + c^2) - 3*a*c*C)*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(5/2)) + (-6*a^3*A*c - 3*a*A*b^2*c - 3*a*A*c^3 + 2*a^4*C - 4*a^2*b^2*C + 2*b^4*C + 5*a^2*c^2*C + 4*b^2*c^2*C + 2*c^4*C - 2*b*c*(2*a^2*A + A*(b^2 + c^2) - 3*a*c*C)*Cos[x] - c*(-3*a*A*(b^2 + c^2) + a^2*c*C + 2*c*(b^2 + c^2)*C)*Cos[2*x] - 8*a^2*A*b^2*Sin[x] + 2*A*b^4*Sin[x] - 12*a^2*A*c^2*Sin[x] + 2*A*b^2*c^2*Sin[x] + 4*a^3*c*C*Sin[x] + 2*a*b^2*c*C*Sin[x] + 8*a*c^3*C*Sin[x] - 3*a*A*b^3*Sin[2*x] - 3*a*A*b*c^2*Sin[2*x] + a^2*b*c*C*Sin[2*x] + 2*b^3*c*C*Sin[2*x] + 2*b*c^3*C*Sin[2*x])/(4*b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[x] + c*Sin[x])^2)","A",1
543,1,152,85,0.2925647,"\int \frac{A+C \sin (x)}{a+b \cos (x)+i b \sin (x)} \, dx","Integrate[(A + C*Sin[x])/(a + b*Cos[x] + I*b*Sin[x]),x]","\frac{\left(-2 i a^2 C-4 a A b+2 i b^2 C\right) \tan ^{-1}\left(\frac{(a+b) \cot \left(\frac{x}{2}\right)}{a-b}\right)+2 i a A b \log \left(a^2+2 a b \cos (x)+b^2\right)-a^2 C \log \left(a^2+2 a b \cos (x)+b^2\right)+b^2 C \log \left(a^2+2 a b \cos (x)+b^2\right)-i a^2 C x+2 a A b x+2 i a b C \sin (x)-2 a b C \cos (x)-i b^2 C x}{4 a^2 b}","\frac{\left(a^2 (-C)+2 i a A b+b^2 C\right) \log (a+i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac{x (2 a A-i b C)}{2 a^2}+\frac{i C \sin (x)}{2 a}-\frac{C \cos (x)}{2 a}",1,"(2*a*A*b*x - I*a^2*C*x - I*b^2*C*x + (-4*a*A*b - (2*I)*a^2*C + (2*I)*b^2*C)*ArcTan[((a + b)*Cot[x/2])/(a - b)] - 2*a*b*C*Cos[x] + (2*I)*a*A*b*Log[a^2 + b^2 + 2*a*b*Cos[x]] - a^2*C*Log[a^2 + b^2 + 2*a*b*Cos[x]] + b^2*C*Log[a^2 + b^2 + 2*a*b*Cos[x]] + (2*I)*a*b*C*Sin[x])/(4*a^2*b)","A",1
544,1,152,85,0.2580312,"\int \frac{A+C \sin (x)}{a+b \cos (x)-i b \sin (x)} \, dx","Integrate[(A + C*Sin[x])/(a + b*Cos[x] - I*b*Sin[x]),x]","\frac{2 i \left(a^2 C+2 i a A b-b^2 C\right) \tan ^{-1}\left(\frac{(a+b) \cot \left(\frac{x}{2}\right)}{a-b}\right)-2 i a A b \log \left(a^2+2 a b \cos (x)+b^2\right)-a^2 C \log \left(a^2+2 a b \cos (x)+b^2\right)+b^2 C \log \left(a^2+2 a b \cos (x)+b^2\right)+i a^2 C x+2 a A b x-2 i a b C \sin (x)-2 a b C \cos (x)+i b^2 C x}{4 a^2 b}","-\frac{\left(a^2 C+2 i a A b-b^2 C\right) \log (a-i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac{x (2 a A+i b C)}{2 a^2}-\frac{i C \sin (x)}{2 a}-\frac{C \cos (x)}{2 a}",1,"(2*a*A*b*x + I*a^2*C*x + I*b^2*C*x + (2*I)*((2*I)*a*A*b + a^2*C - b^2*C)*ArcTan[((a + b)*Cot[x/2])/(a - b)] - 2*a*b*C*Cos[x] - (2*I)*a*A*b*Log[a^2 + b^2 + 2*a*b*Cos[x]] - a^2*C*Log[a^2 + b^2 + 2*a*b*Cos[x]] + b^2*C*Log[a^2 + b^2 + 2*a*b*Cos[x]] - (2*I)*a*b*C*Sin[x])/(4*a^2*b)","A",1
545,1,98,119,0.3849383,"\int \frac{B \cos (x)+C \sin (x)}{a+b \cos (x)+c \sin (x)} \, dx","Integrate[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x]),x]","\frac{\frac{2 a (b B+c C) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\sqrt{-a^2+b^2+c^2}}+(B c-b C) \log (a+b \cos (x)+c \sin (x))+x (b B+c C)}{b^2+c^2}","-\frac{2 a (b B+c C) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(b^2+c^2\right) \sqrt{a^2-b^2-c^2}}+\frac{(B c-b C) \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}+\frac{x (b B+c C)}{b^2+c^2}",1,"((b*B + c*C)*x + (2*a*(b*B + c*C)*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/Sqrt[-a^2 + b^2 + c^2] + (B*c - b*C)*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",1
546,1,116,110,0.4057914,"\int \frac{B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Integrate[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","-\frac{a^2 C+a \sin (x) (b B+c C)-b^2 C+b B c}{b \left(-a^2+b^2+c^2\right) (a+b \cos (x)+c \sin (x))}-\frac{2 (b B+c C) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{3/2}}","\frac{a B \sin (x)-a C \cos (x)-b C+B c}{\left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))}-\frac{2 (b B+c C) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(a^2-b^2-c^2\right)^{3/2}}",1,"(-2*(b*B + c*C)*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(3/2) - (b*B*c + a^2*C - b^2*C + a*(b*B + c*C)*Sin[x])/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[x] + c*Sin[x]))","A",1
547,1,311,197,0.9047109,"\int \frac{B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx","Integrate[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^3,x]","\frac{3 a (b B+c C) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{5/2}}+\frac{2 a^4 C+4 a^3 b B \sin (x)+4 a^3 c C \sin (x)-c \cos (2 x) \left(a^2+2 \left(b^2+c^2\right)\right) (b B+c C)+a^2 b^2 B \sin (2 x)-4 a^2 b^2 C+9 a^2 b B c+a^2 b c C \sin (2 x)+5 a^2 c^2 C+2 a b^3 B \sin (x)+2 a b^2 c C \sin (x)+8 a b B c^2 \sin (x)+6 a b c \cos (x) (b B+c C)+8 a c^3 C \sin (x)+2 b^4 B \sin (2 x)+2 b^4 C+2 b^3 c C \sin (2 x)+2 b^2 B c^2 \sin (2 x)+4 b^2 c^2 C+2 b c^3 C \sin (2 x)+2 c^4 C}{4 b \left(-a^2+b^2+c^2\right)^2 (a+b \cos (x)+c \sin (x))^2}","-\frac{3 a (b B+c C) \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right)}{\left(a^2-b^2-c^2\right)^{5/2}}+\frac{-\cos (x) \left(C \left(a^2+2 c^2\right)+2 b B c\right)+\sin (x) \left(a^2 B+2 b (b B+c C)\right)+a (B c-b C)}{2 \left(a^2-b^2-c^2\right)^2 (a+b \cos (x)+c \sin (x))}+\frac{a B \sin (x)-a C \cos (x)-b C+B c}{2 \left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))^2}",1,"(3*a*(b*B + c*C)*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(5/2) + (9*a^2*b*B*c + 2*a^4*C - 4*a^2*b^2*C + 2*b^4*C + 5*a^2*c^2*C + 4*b^2*c^2*C + 2*c^4*C + 6*a*b*c*(b*B + c*C)*Cos[x] - c*(a^2 + 2*(b^2 + c^2))*(b*B + c*C)*Cos[2*x] + 4*a^3*b*B*Sin[x] + 2*a*b^3*B*Sin[x] + 8*a*b*B*c^2*Sin[x] + 4*a^3*c*C*Sin[x] + 2*a*b^2*c*C*Sin[x] + 8*a*c^3*C*Sin[x] + a^2*b^2*B*Sin[2*x] + 2*b^4*B*Sin[2*x] + 2*b^2*B*c^2*Sin[2*x] + a^2*b*c*C*Sin[2*x] + 2*b^3*c*C*Sin[2*x] + 2*b*c^3*C*Sin[2*x])/(4*b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[x] + c*Sin[x])^2)","A",1
548,1,195,92,0.3237734,"\int \frac{B \cos (x)+C \sin (x)}{a+b \cos (x)+i b \sin (x)} \, dx","Integrate[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + I*b*Sin[x]),x]","\frac{x \left(a^2 B-i a^2 C-b^2 B-i b^2 C\right)}{4 a^2 b}-\frac{i \left(a^2 B-i a^2 C+b^2 B+i b^2 C\right) \log \left(a^2+2 a b \cos (x)+b^2\right)}{4 a^2 b}-\frac{\left(a^2 B-i a^2 C+b^2 B+i b^2 C\right) \tan ^{-1}\left(\frac{(a+b) \cos \left(\frac{x}{2}\right)}{b \sin \left(\frac{x}{2}\right)-a \sin \left(\frac{x}{2}\right)}\right)}{2 a^2 b}+\frac{(B+i C) \sin (x)}{2 a}+\frac{i (B+i C) \cos (x)}{2 a}","-\frac{\left(a^2 (C+i B)+i b^2 (B+i C)\right) \log (a+i b \sin (x)+b \cos (x))}{2 a^2 b}-\frac{b x (B+i C)}{2 a^2}+\frac{(-C+i B) (\cos (x)-i \sin (x))}{2 a}",1,"((a^2*B - b^2*B - I*a^2*C - I*b^2*C)*x)/(4*a^2*b) - ((a^2*B + b^2*B - I*a^2*C + I*b^2*C)*ArcTan[((a + b)*Cos[x/2])/(-(a*Sin[x/2]) + b*Sin[x/2])])/(2*a^2*b) + ((I/2)*(B + I*C)*Cos[x])/a - ((I/4)*(a^2*B + b^2*B - I*a^2*C + I*b^2*C)*Log[a^2 + b^2 + 2*a*b*Cos[x]])/(a^2*b) + ((B + I*C)*Sin[x])/(2*a)","B",1
549,1,195,90,0.293893,"\int \frac{B \cos (x)+C \sin (x)}{a+b \cos (x)-i b \sin (x)} \, dx","Integrate[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] - I*b*Sin[x]),x]","\frac{x \left(a^2 B+i a^2 C-b^2 B+i b^2 C\right)}{4 a^2 b}+\frac{i \left(a^2 B+i a^2 C+b^2 B-i b^2 C\right) \log \left(a^2+2 a b \cos (x)+b^2\right)}{4 a^2 b}+\frac{\left(a^2 B+i a^2 C+b^2 B-i b^2 C\right) \tan ^{-1}\left(\frac{(a+b) \cos \left(\frac{x}{2}\right)}{a \sin \left(\frac{x}{2}\right)-b \sin \left(\frac{x}{2}\right)}\right)}{2 a^2 b}+\frac{(B-i C) \sin (x)}{2 a}-\frac{i (B-i C) \cos (x)}{2 a}","\frac{\left(i a^2 (B+i C)+b^2 (C+i B)\right) \log (a-i b \sin (x)+b \cos (x))}{2 a^2 b}-\frac{b x (B-i C)}{2 a^2}-\frac{(C+i B) (\cos (x)+i \sin (x))}{2 a}",1,"((a^2*B - b^2*B + I*a^2*C + I*b^2*C)*x)/(4*a^2*b) + ((a^2*B + b^2*B + I*a^2*C - I*b^2*C)*ArcTan[((a + b)*Cos[x/2])/(a*Sin[x/2] - b*Sin[x/2])])/(2*a^2*b) - ((I/2)*(B - I*C)*Cos[x])/a + ((I/4)*(a^2*B + b^2*B + I*a^2*C - I*b^2*C)*Log[a^2 + b^2 + 2*a*b*Cos[x]])/(a^2*b) + ((B - I*C)*Sin[x])/(2*a)","B",1
550,1,110,131,0.385691,"\int \frac{A+B \cos (x)+C \sin (x)}{a+b \cos (x)+c \sin (x)} \, dx","Integrate[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x]),x]","\frac{\frac{2 \left(a (b B+c C)-A \left(b^2+c^2\right)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\sqrt{-a^2+b^2+c^2}}+(B c-b C) \log (a+b \cos (x)+c \sin (x))+x (b B+c C)}{b^2+c^2}","\frac{2 \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right) \left(A \left(b^2+c^2\right)-a (b B+c C)\right)}{\left(b^2+c^2\right) \sqrt{a^2-b^2-c^2}}+\frac{(B c-b C) \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}+\frac{x (b B+c C)}{b^2+c^2}",1,"((b*B + c*C)*x + (2*(-(A*(b^2 + c^2)) + a*(b*B + c*C))*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/Sqrt[-a^2 + b^2 + c^2] + (B*c - b*C)*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",1
551,1,137,127,0.5042053,"\int \frac{A+B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Integrate[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","\frac{a^2 (-C)+\sin (x) \left(A \left(b^2+c^2\right)-a (b B+c C)\right)+a A c+b (b C-B c)}{b \left(-a^2+b^2+c^2\right) (a+b \cos (x)+c \sin (x))}+\frac{2 (a A-b B-c C) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{3/2}}","\frac{2 \tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right) (a A-b B-c C)}{\left(a^2-b^2-c^2\right)^{3/2}}+\frac{-\sin (x) (A b-a B)+\cos (x) (A c-a C)-b C+B c}{\left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))}",1,"(2*(a*A - b*B - c*C)*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(3/2) + (a*A*c - a^2*C + b*(-(B*c) + b*C) + (A*(b^2 + c^2) - a*(b*B + c*C))*Sin[x])/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[x] + c*Sin[x]))","A",1
552,1,452,237,1.2807712,"\int \frac{A+B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx","Integrate[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^3,x]","\frac{2 a^4 C-6 a^3 A c+4 a^3 b B \sin (x)+4 a^3 c C \sin (x)-2 b c \cos (x) \left(2 a^2 A-3 a (b B+c C)+A \left(b^2+c^2\right)\right)-c \cos (2 x) \left(a^2 (b B+c C)-3 a A \left(b^2+c^2\right)+2 \left(b^2+c^2\right) (b B+c C)\right)-8 a^2 A b^2 \sin (x)-12 a^2 A c^2 \sin (x)+a^2 b^2 B \sin (2 x)-4 a^2 b^2 C+9 a^2 b B c+a^2 b c C \sin (2 x)+5 a^2 c^2 C-3 a A b^3 \sin (2 x)-3 a A b^2 c-3 a A b c^2 \sin (2 x)-3 a A c^3+2 a b^3 B \sin (x)+2 a b^2 c C \sin (x)+8 a b B c^2 \sin (x)+8 a c^3 C \sin (x)+2 A b^4 \sin (x)+2 A b^2 c^2 \sin (x)+2 b^4 B \sin (2 x)+2 b^4 C+2 b^3 c C \sin (2 x)+2 b^2 B c^2 \sin (2 x)+4 b^2 c^2 C+2 b c^3 C \sin (2 x)+2 c^4 C}{4 b \left(-a^2+b^2+c^2\right)^2 (a+b \cos (x)+c \sin (x))^2}-\frac{\left(2 a^2 A-3 a (b B+c C)+A \left(b^2+c^2\right)\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{-a^2+b^2+c^2}}\right)}{\left(-a^2+b^2+c^2\right)^{5/2}}","\frac{\tan ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)+c}{\sqrt{a^2-b^2-c^2}}\right) \left(2 a^2 A-3 a (b B+c C)+A \left(b^2+c^2\right)\right)}{\left(a^2-b^2-c^2\right)^{5/2}}+\frac{-\sin (x) \left(a^2 (-B)+3 a A b-2 b (b B+c C)\right)+\cos (x) \left(a^2 (-C)+3 a A c-2 c (b B+c C)\right)+a (B c-b C)}{2 \left(a^2-b^2-c^2\right)^2 (a+b \cos (x)+c \sin (x))}+\frac{-\sin (x) (A b-a B)+\cos (x) (A c-a C)-b C+B c}{2 \left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))^2}",1,"-(((2*a^2*A + A*(b^2 + c^2) - 3*a*(b*B + c*C))*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(5/2)) + (-6*a^3*A*c - 3*a*A*b^2*c + 9*a^2*b*B*c - 3*a*A*c^3 + 2*a^4*C - 4*a^2*b^2*C + 2*b^4*C + 5*a^2*c^2*C + 4*b^2*c^2*C + 2*c^4*C - 2*b*c*(2*a^2*A + A*(b^2 + c^2) - 3*a*(b*B + c*C))*Cos[x] - c*(-3*a*A*(b^2 + c^2) + a^2*(b*B + c*C) + 2*(b^2 + c^2)*(b*B + c*C))*Cos[2*x] - 8*a^2*A*b^2*Sin[x] + 2*A*b^4*Sin[x] + 4*a^3*b*B*Sin[x] + 2*a*b^3*B*Sin[x] - 12*a^2*A*c^2*Sin[x] + 2*A*b^2*c^2*Sin[x] + 8*a*b*B*c^2*Sin[x] + 4*a^3*c*C*Sin[x] + 2*a*b^2*c*C*Sin[x] + 8*a*c^3*C*Sin[x] - 3*a*A*b^3*Sin[2*x] + a^2*b^2*B*Sin[2*x] + 2*b^4*B*Sin[2*x] - 3*a*A*b*c^2*Sin[2*x] + 2*b^2*B*c^2*Sin[2*x] + a^2*b*c*C*Sin[2*x] + 2*b^3*c*C*Sin[2*x] + 2*b*c^3*C*Sin[2*x])/(4*b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[x] + c*Sin[x])^2)","A",1
553,1,165,105,0.4445504,"\int \frac{A+B \cos (x)+C \sin (x)}{a+b \cos (x)+i b \sin (x)} \, dx","Integrate[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + I*b*Sin[x]),x]","\frac{x \left(a^2 (B-i C)+2 a A b-b^2 (B+i C)\right)+\left(a^2 (-C-i B)+2 i a A b+b^2 (C-i B)\right) \log \left(a^2+2 a b \cos (x)+b^2\right)+2 \left(a^2 (B-i C)-2 a A b+b^2 (B+i C)\right) \tan ^{-1}\left(\frac{(a+b) \cot \left(\frac{x}{2}\right)}{a-b}\right)+2 a b (B+i C) \sin (x)+2 i a b (B+i C) \cos (x)}{4 a^2 b}","\frac{i \left(-\left(a^2 (B-i C)\right)+2 a A b-b^2 (B+i C)\right) \log (a+i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac{x (2 a A-b (B+i C))}{2 a^2}+\frac{(-C+i B) (\cos (x)-i \sin (x))}{2 a}",1,"((2*a*A*b + a^2*(B - I*C) - b^2*(B + I*C))*x + 2*(-2*a*A*b + a^2*(B - I*C) + b^2*(B + I*C))*ArcTan[((a + b)*Cot[x/2])/(a - b)] + (2*I)*a*b*(B + I*C)*Cos[x] + ((2*I)*a*A*b + a^2*((-I)*B - C) + b^2*((-I)*B + C))*Log[a^2 + b^2 + 2*a*b*Cos[x]] + 2*a*b*(B + I*C)*Sin[x])/(4*a^2*b)","A",1
554,1,167,103,0.4509328,"\int \frac{A+B \cos (x)+C \sin (x)}{a+b \cos (x)-i b \sin (x)} \, dx","Integrate[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] - I*b*Sin[x]),x]","\frac{\frac{\left(i a^2 (B+i C)-2 i a A b+b^2 (C+i B)\right) \log \left(a^2+2 a b \cos (x)+b^2\right)}{b}+\frac{2 \left(a^2 (B+i C)-2 a A b+b^2 (B-i C)\right) \tan ^{-1}\left(\frac{(a+b) \cot \left(\frac{x}{2}\right)}{a-b}\right)}{b}+x \left(\frac{a^2 (B+i C)}{b}+2 a A-b (B-i C)\right)+2 a (B-i C) \sin (x)-2 i a (B-i C) \cos (x)}{4 a^2}","-\frac{i \left(-\left(a^2 (B+i C)\right)+2 a A b-b^2 (B-i C)\right) \log (a-i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac{x (2 a A-b B+i b C)}{2 a^2}-\frac{(C+i B) (\cos (x)+i \sin (x))}{2 a}",1,"((2*a*A - b*(B - I*C) + (a^2*(B + I*C))/b)*x + (2*(-2*a*A*b + b^2*(B - I*C) + a^2*(B + I*C))*ArcTan[((a + b)*Cot[x/2])/(a - b)])/b - (2*I)*a*(B - I*C)*Cos[x] + (((-2*I)*a*A*b + I*a^2*(B + I*C) + b^2*(I*B + C))*Log[a^2 + b^2 + 2*a*b*Cos[x]])/b + 2*a*(B - I*C)*Sin[x])/(4*a^2)","A",1
555,1,32,24,0.0938213,"\int \frac{b^2+c^2+a b \cos (x)+a c \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Integrate[(b^2 + c^2 + a*b*Cos[x] + a*c*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","\frac{a c+b^2 \sin (x)+c^2 \sin (x)}{b (a+b \cos (x)+c \sin (x))}","-\frac{c \cos (x)-b \sin (x)}{a+b \cos (x)+c \sin (x)}",1,"(a*c + b^2*Sin[x] + c^2*Sin[x])/(b*(a + b*Cos[x] + c*Sin[x]))","A",1
556,1,7823,390,6.9408373,"\int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx","Integrate[(a + b*Cos[x] + c*Sin[x])^(5/2)*(d + b*e*Cos[x] + c*e*Sin[x]),x]","\text{Result too large to show}","-\frac{2 \left(a^2-b^2-c^2\right) \left(15 a^2 e+56 a d+25 e \left(b^2+c^2\right)\right) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{105 \sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 \left(15 a^3 e+161 a^2 d+145 a e \left(b^2+c^2\right)+63 d \left(b^2+c^2\right)\right) \sqrt{a+b \cos (x)+c \sin (x)} E\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{105 \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}-\frac{2}{105} \sqrt{a+b \cos (x)+c \sin (x)} \left(c \cos (x) \left(15 a^2 e+56 a d+25 e \left(b^2+c^2\right)\right)-b \sin (x) \left(15 a^2 e+56 a d+25 e \left(b^2+c^2\right)\right)\right)-\frac{2}{35} (a+b \cos (x)+c \sin (x))^{3/2} (c \cos (x) (5 a e+7 d)-b \sin (x) (5 a e+7 d))-\frac{2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))",1,"Result too large to show","C",0
557,1,5218,294,6.5844717,"\int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx","Integrate[(a + b*Cos[x] + c*Sin[x])^(3/2)*(d + b*e*Cos[x] + c*e*Sin[x]),x]","\text{Result too large to show}","-\frac{2 \left(a^2-b^2-c^2\right) (3 a e+5 d) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{15 \sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 \left(3 a^2 e+20 a d+9 e \left(b^2+c^2\right)\right) \sqrt{a+b \cos (x)+c \sin (x)} E\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{15 \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}-\frac{2}{15} \sqrt{a+b \cos (x)+c \sin (x)} (c \cos (x) (3 a e+5 d)-b \sin (x) (3 a e+5 d))-\frac{2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))",1,"Result too large to show","C",0
558,1,3006,229,6.3698133,"\int \sqrt{a+b \cos (x)+c \sin (x)} (d+b e \cos (x)+c e \sin (x)) \, dx","Integrate[Sqrt[a + b*Cos[x] + c*Sin[x]]*(d + b*e*Cos[x] + c*e*Sin[x]),x]","\text{Result too large to show}","-\frac{2 e \left(a^2-b^2-c^2\right) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{3 \sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 (a e+3 d) \sqrt{a+b \cos (x)+c \sin (x)} E\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{3 \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}-\frac{2}{3} \sqrt{a+b \cos (x)+c \sin (x)} (c e \cos (x)-b e \sin (x))",1,"Sqrt[a + b*Cos[x] + c*Sin[x]]*((2*b*(3*d + a*e))/(3*c) - (2*c*e*Cos[x])/3 + (2*b*e*Sin[x])/3) + (2*a*d*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c) + (2*b^2*e*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(3*Sqrt[1 + b^2/c^2]*c) + (2*c*e*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(3*Sqrt[1 + b^2/c^2]) + (b^2*d*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/c + c*d*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]) + (a*b^2*e*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/(3*c) + (a*c*e*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/3","C",0
559,1,1319,180,6.3052161,"\int \frac{d+b e \cos (x)+c e \sin (x)}{\sqrt{a+b \cos (x)+c \sin (x)}} \, dx","Integrate[(d + b*e*Cos[x] + c*e*Sin[x])/Sqrt[a + b*Cos[x] + c*Sin[x]],x]","\frac{e \left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(1-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}\right)},-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}-1\right)}\right) \sin \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \sqrt{\frac{b \sqrt{\frac{b^2+c^2}{b^2}}-b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{a+b \sqrt{\frac{b^2+c^2}{b^2}}}} \sqrt{a+b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)} \sqrt{\frac{\sqrt{\frac{b^2+c^2}{b^2}} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right) b+\sqrt{\frac{b^2+c^2}{b^2}} b}{b \sqrt{\frac{b^2+c^2}{b^2}}-a}}}-\frac{\frac{2 b \left(a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)\right)}{b^2+c^2}-\frac{c \sin \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1}}}{\sqrt{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}}\right) b^2}{c}+\frac{2 e \sqrt{a+b \cos (x)+c \sin (x)} b}{c}+c e \left(-\frac{c F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(1-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}\right)},-\frac{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \left(-\frac{a}{b \sqrt{\frac{c^2}{b^2}+1}}-1\right)}\right) \sin \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1} \sqrt{\frac{b \sqrt{\frac{b^2+c^2}{b^2}}-b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{a+b \sqrt{\frac{b^2+c^2}{b^2}}}} \sqrt{a+b \sqrt{\frac{b^2+c^2}{b^2}} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)} \sqrt{\frac{\sqrt{\frac{b^2+c^2}{b^2}} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right) b+\sqrt{\frac{b^2+c^2}{b^2}} b}{b \sqrt{\frac{b^2+c^2}{b^2}}-a}}}-\frac{\frac{2 b \left(a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)\right)}{b^2+c^2}-\frac{c \sin \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}{b \sqrt{\frac{c^2}{b^2}+1}}}{\sqrt{a+b \sqrt{\frac{c^2}{b^2}+1} \cos \left(x-\tan ^{-1}\left(\frac{c}{b}\right)\right)}}\right)+\frac{2 d F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};-\frac{a+\sqrt{\frac{b^2}{c^2}+1} c \sin \left(x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{\sqrt{\frac{b^2}{c^2}+1} \left(1-\frac{a}{\sqrt{\frac{b^2}{c^2}+1} c}\right) c},-\frac{a+\sqrt{\frac{b^2}{c^2}+1} c \sin \left(x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{\sqrt{\frac{b^2}{c^2}+1} \left(-\frac{a}{\sqrt{\frac{b^2}{c^2}+1} c}-1\right) c}\right) \sec \left(x+\tan ^{-1}\left(\frac{b}{c}\right)\right) \sqrt{\frac{c \sqrt{\frac{b^2+c^2}{c^2}}-c \sqrt{\frac{b^2+c^2}{c^2}} \sin \left(x+\tan ^{-1}\left(\frac{b}{c}\right)\right)}{a+c \sqrt{\frac{b^2+c^2}{c^2}}}} \sqrt{a+c \sqrt{\frac{b^2+c^2}{c^2}} \sin \left(x+\tan ^{-1}\left(\frac{b}{c}\right)\right)} \sqrt{\frac{\sqrt{\frac{b^2+c^2}{c^2}} \sin \left(x+\tan ^{-1}\left(\frac{b}{c}\right)\right) c+\sqrt{\frac{b^2+c^2}{c^2}} c}{c \sqrt{\frac{b^2+c^2}{c^2}}-a}}}{\sqrt{\frac{b^2}{c^2}+1} c}","\frac{2 (d-a e) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{\sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 e \sqrt{a+b \cos (x)+c \sin (x)} E\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{\sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}",1,"(2*b*e*Sqrt[a + b*Cos[x] + c*Sin[x]])/c + (2*d*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c) + (b^2*e*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/c + c*e*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]])","C",0
560,1,3176,250,6.5712695,"\int \frac{d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx","Integrate[(d + b*e*Cos[x] + c*e*Sin[x])/(a + b*Cos[x] + c*Sin[x])^(3/2),x]","\text{Result too large to show}","\frac{2 (d-a e) \sqrt{a+b \cos (x)+c \sin (x)} E\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{\left(a^2-b^2-c^2\right) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 e \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{\sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{\left(a^2-b^2-c^2\right) \sqrt{a+b \cos (x)+c \sin (x)}}",1,"Sqrt[a + b*Cos[x] + c*Sin[x]]*((2*(b^2 + c^2)*(-d + a*e))/(b*c*(-a^2 + b^2 + c^2)) - (2*(-(a*c*d) + a^2*c*e - b^2*d*Sin[x] - c^2*d*Sin[x] + a*b^2*e*Sin[x] + a*c^2*e*Sin[x]))/(b*(-a^2 + b^2 + c^2)*(a + b*Cos[x] + c*Sin[x]))) - (2*a*d*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)) + (2*b^2*e*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*(-a^2 + b^2 + c^2)) + (2*c*e*AppellF1[1/2, 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[x + ArcTan[b/c]])/(-a + c*Sqrt[(b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*(-a^2 + b^2 + c^2)) - (b^2*d*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/(c*(-a^2 + b^2 + c^2)) - (c*d*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/(-a^2 + b^2 + c^2) + (a*b^2*e*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/(c*(-a^2 + b^2 + c^2)) + (a*c*e*(-((c*AppellF1[-1/2, -1/2, -1/2, 1/2, -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(1 - a/(b*Sqrt[1 + c^2/b^2])))), -((a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*(-1 - a/(b*Sqrt[1 + c^2/b^2]))))]*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] - b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(a + b*Sqrt[(b^2 + c^2)/b^2])]*Sqrt[a + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]]]*Sqrt[(b*Sqrt[(b^2 + c^2)/b^2] + b*Sqrt[(b^2 + c^2)/b^2]*Cos[x - ArcTan[c/b]])/(-a + b*Sqrt[(b^2 + c^2)/b^2])])) - ((2*b*(a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]))/(b^2 + c^2) - (c*Sin[x - ArcTan[c/b]])/(b*Sqrt[1 + c^2/b^2]))/Sqrt[a + b*Sqrt[1 + c^2/b^2]*Cos[x - ArcTan[c/b]]]))/(-a^2 + b^2 + c^2)","C",0
561,1,5554,378,6.9518546,"\int \frac{d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx","Integrate[(d + b*e*Cos[x] + c*e*Sin[x])/(a + b*Cos[x] + c*Sin[x])^(5/2),x]","\text{Result too large to show}","-\frac{2 (d-a e) \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}} F\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{3 \left(a^2-b^2-c^2\right) \sqrt{a+b \cos (x)+c \sin (x)}}+\frac{2 \left(a^2 (-e)+4 a d-3 e \left(b^2+c^2\right)\right) \sqrt{a+b \cos (x)+c \sin (x)} E\left(\frac{1}{2} \left(x-\tan ^{-1}(b,c)\right)|\frac{2 \sqrt{b^2+c^2}}{a+\sqrt{b^2+c^2}}\right)}{3 \left(a^2-b^2-c^2\right)^2 \sqrt{\frac{a+b \cos (x)+c \sin (x)}{a+\sqrt{b^2+c^2}}}}+\frac{2 (c \cos (x) (d-a e)-b \sin (x) (d-a e))}{3 \left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))^{3/2}}+\frac{2 \left(c \cos (x) \left(a^2 (-e)+4 a d-3 e \left(b^2+c^2\right)\right)-b \sin (x) \left(a^2 (-e)+4 a d-3 e \left(b^2+c^2\right)\right)\right)}{3 \left(a^2-b^2-c^2\right)^2 \sqrt{a+b \cos (x)+c \sin (x)}}",1,"Result too large to show","C",0
562,1,80,84,0.2542549,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{a+c \sin (d+e x)} \, dx","Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x]),x]","\frac{\frac{2 (A c-a C) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-c^2}}\right)}{\sqrt{a^2-c^2}}+B \log (a+c \sin (d+e x))+C (d+e x)}{c e}","\frac{2 (A c-a C) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-c^2}}\right)}{c e \sqrt{a^2-c^2}}+\frac{B \log (a+c \sin (d+e x))}{c e}+\frac{C x}{c}",1,"(C*(d + e*x) + (2*(A*c - a*C)*ArcTan[(c + a*Tan[(d + e*x)/2])/Sqrt[a^2 - c^2]])/Sqrt[a^2 - c^2] + B*Log[a + c*Sin[d + e*x]])/(c*e)","A",1
563,1,114,118,0.4328886,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^2} \, dx","Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^2,x]","\frac{\frac{B \left(a^2-c^2\right)-c (A c-a C) \cos (d+e x)}{c (c-a) (a+c) (a+c \sin (d+e x))}+\frac{2 (a A-c C) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-c^2}}\right)}{\left(a^2-c^2\right)^{3/2}}}{e}","\frac{2 (a A-c C) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-c^2}}\right)}{e \left(a^2-c^2\right)^{3/2}}+\frac{(A c-a C) \cos (d+e x)}{e \left(a^2-c^2\right) (a+c \sin (d+e x))}-\frac{B}{c e (a+c \sin (d+e x))}",1,"((2*(a*A - c*C)*ArcTan[(c + a*Tan[(d + e*x)/2])/Sqrt[a^2 - c^2]])/(a^2 - c^2)^(3/2) + (B*(a^2 - c^2) - c*(A*c - a*C)*Cos[d + e*x])/(c*(-a + c)*(a + c)*(a + c*Sin[d + e*x])))/e","A",1
564,1,174,185,0.9139422,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^3} \, dx","Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^3,x]","\frac{\frac{B \left(c^2-a^2\right)+c (A c-a C) \cos (d+e x)}{c (a-c) (a+c) (a+c \sin (d+e x))^2}+\frac{2 \left(2 a^2 A-3 a c C+A c^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-c^2}}\right)}{\left(a^2-c^2\right)^{5/2}}-\frac{\left(a^2 C-3 a A c+2 c^2 C\right) \cos (d+e x)}{(a-c)^2 (a+c)^2 (a+c \sin (d+e x))}}{2 e}","\frac{\left(2 a^2 A-3 a c C+A c^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-c^2}}\right)}{e \left(a^2-c^2\right)^{5/2}}+\frac{\left(a^2 (-C)+3 a A c-2 c^2 C\right) \cos (d+e x)}{2 e \left(a^2-c^2\right)^2 (a+c \sin (d+e x))}+\frac{(A c-a C) \cos (d+e x)}{2 e \left(a^2-c^2\right) (a+c \sin (d+e x))^2}-\frac{B}{2 c e (a+c \sin (d+e x))^2}",1,"((2*(2*a^2*A + A*c^2 - 3*a*c*C)*ArcTan[(c + a*Tan[(d + e*x)/2])/Sqrt[a^2 - c^2]])/(a^2 - c^2)^(5/2) + (B*(-a^2 + c^2) + c*(A*c - a*C)*Cos[d + e*x])/((a - c)*c*(a + c)*(a + c*Sin[d + e*x])^2) - ((-3*a*A*c + a^2*C + 2*c^2*C)*Cos[d + e*x])/((a - c)^2*(a + c)^2*(a + c*Sin[d + e*x])))/(2*e)","A",1
565,1,244,258,2.6372032,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4} \, dx","Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^4,x]","\frac{\frac{2 B \left(c^2-a^2\right)+2 c (A c-a C) \cos (d+e x)}{c (a-c) (a+c) (a+c \sin (d+e x))^3}+\frac{\left(-2 a^2 C+5 a A c-3 c^2 C\right) \cos (d+e x)}{(a-c)^2 (a+c)^2 (a+c \sin (d+e x))^2}+\frac{6 \left(2 a^3 A-4 a^2 c C+3 a A c^2-c^3 C\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-c^2}}\right)}{\left(a^2-c^2\right)^{7/2}}+\frac{\left(-2 a^3 C+11 a^2 A c-13 a c^2 C+4 A c^3\right) \cos (d+e x)}{(a-c)^3 (a+c)^3 (a+c \sin (d+e x))}}{6 e}","\frac{\left(-2 a^2 C+5 a A c-3 c^2 C\right) \cos (d+e x)}{6 e \left(a^2-c^2\right)^2 (a+c \sin (d+e x))^2}+\frac{(A c-a C) \cos (d+e x)}{3 e \left(a^2-c^2\right) (a+c \sin (d+e x))^3}+\frac{\left(2 a^3 A-4 a^2 c C+3 a A c^2-c^3 C\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (d+e x)\right)+c}{\sqrt{a^2-c^2}}\right)}{e \left(a^2-c^2\right)^{7/2}}+\frac{\left(-2 a^3 C+11 a^2 A c-13 a c^2 C+4 A c^3\right) \cos (d+e x)}{6 e \left(a^2-c^2\right)^3 (a+c \sin (d+e x))}-\frac{B}{3 c e (a+c \sin (d+e x))^3}",1,"((6*(2*a^3*A + 3*a*A*c^2 - 4*a^2*c*C - c^3*C)*ArcTan[(c + a*Tan[(d + e*x)/2])/Sqrt[a^2 - c^2]])/(a^2 - c^2)^(7/2) + (2*B*(-a^2 + c^2) + 2*c*(A*c - a*C)*Cos[d + e*x])/((a - c)*c*(a + c)*(a + c*Sin[d + e*x])^3) + ((5*a*A*c - 2*a^2*C - 3*c^2*C)*Cos[d + e*x])/((a - c)^2*(a + c)^2*(a + c*Sin[d + e*x])^2) + ((11*a^2*A*c + 4*A*c^3 - 2*a^3*C - 13*a*c^2*C)*Cos[d + e*x])/((a - c)^3*(a + c)^3*(a + c*Sin[d + e*x])))/(6*e)","A",1
566,1,145,131,0.5892174,"\int (a+b \cos (c+d x) \sin (c+d x))^m \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^m,x]","\frac{\sec (2 (c+d x)) \sqrt{-\frac{b (\sin (2 (c+d x))-1)}{2 a+b}} \sqrt{\frac{b (\sin (2 (c+d x))+1)}{b-2 a}} \left(a+\frac{1}{2} b \sin (2 (c+d x))\right)^{m+1} F_1\left(m+1;\frac{1}{2},\frac{1}{2};m+2;\frac{2 a+b \sin (2 (c+d x))}{2 a-b},\frac{2 a+b \sin (2 (c+d x))}{2 a+b}\right)}{b d (m+1)}","-\frac{\cos (2 c+2 d x) \left(a+\frac{1}{2} b \sin (2 c+2 d x)\right)^m \left(\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sin (2 c+2 d x)),\frac{b (1-\sin (2 c+2 d x))}{2 a+b}\right)}{\sqrt{2} d \sqrt{\sin (2 c+2 d x)+1}}",1,"(AppellF1[1 + m, 1/2, 1/2, 2 + m, (2*a + b*Sin[2*(c + d*x)])/(2*a - b), (2*a + b*Sin[2*(c + d*x)])/(2*a + b)]*Sec[2*(c + d*x)]*Sqrt[-((b*(-1 + Sin[2*(c + d*x)]))/(2*a + b))]*Sqrt[(b*(1 + Sin[2*(c + d*x)]))/(-2*a + b)]*(a + (b*Sin[2*(c + d*x)])/2)^(1 + m))/(b*d*(1 + m))","A",0
567,1,75,107,0.2717218,"\int (a+b \cos (c+d x) \sin (c+d x))^3 \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^3,x]","\frac{-9 \left(16 a^2 b+b^3\right) \cos (2 (c+d x))+6 a \left(4 \left(8 a^2+3 b^2\right) (c+d x)-3 b^2 \sin (4 (c+d x))\right)+b^3 \cos (6 (c+d x))}{192 d}","-\frac{b \left(16 a^2+b^2\right) \cos (2 c+2 d x)}{24 d}+\frac{1}{8} a x \left(8 a^2+3 b^2\right)-\frac{5 a b^2 \sin (2 c+2 d x) \cos (2 c+2 d x)}{48 d}-\frac{b \cos (2 c+2 d x) (2 a+b \sin (2 c+2 d x))^2}{48 d}",1,"(-9*(16*a^2*b + b^3)*Cos[2*(c + d*x)] + b^3*Cos[6*(c + d*x)] + 6*a*(4*(8*a^2 + 3*b^2)*(c + d*x) - 3*b^2*Sin[4*(c + d*x)]))/(192*d)","A",1
568,1,48,61,0.1545291,"\int (a+b \cos (c+d x) \sin (c+d x))^2 \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^2,x]","-\frac{-4 \left(8 a^2+b^2\right) (c+d x)+16 a b \cos (2 (c+d x))+b^2 \sin (4 (c+d x))}{32 d}","\frac{1}{8} x \left(8 a^2+b^2\right)-\frac{a b \cos (2 c+2 d x)}{2 d}-\frac{b^2 \sin (2 c+2 d x) \cos (2 c+2 d x)}{16 d}",1,"-1/32*(-4*(8*a^2 + b^2)*(c + d*x) + 16*a*b*Cos[2*(c + d*x)] + b^2*Sin[4*(c + d*x)])/d","A",1
569,1,38,20,0.0071562,"\int (a+b \cos (c+d x) \sin (c+d x)) \, dx","Integrate[a + b*Cos[c + d*x]*Sin[c + d*x],x]","a x+\frac{b \sin (2 c) \sin (2 d x)}{4 d}-\frac{b \cos (2 c) \cos (2 d x)}{4 d}","a x+\frac{b \sin ^2(c+d x)}{2 d}",1,"a*x - (b*Cos[2*c]*Cos[2*d*x])/(4*d) + (b*Sin[2*c]*Sin[2*d*x])/(4*d)","A",1
570,1,48,48,0.0756261,"\int \frac{1}{a+b \cos (c+d x) \sin (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^(-1),x]","\frac{2 \tan ^{-1}\left(\frac{2 a \tan (c+d x)+b}{\sqrt{4 a^2-b^2}}\right)}{d \sqrt{4 a^2-b^2}}","\frac{2 \tan ^{-1}\left(\frac{2 a \tan (c+d x)+b}{\sqrt{4 a^2-b^2}}\right)}{d \sqrt{4 a^2-b^2}}",1,"(2*ArcTan[(b + 2*a*Tan[c + d*x])/Sqrt[4*a^2 - b^2]])/(Sqrt[4*a^2 - b^2]*d)","A",1
571,1,94,95,0.4080174,"\int \frac{1}{(a+b \cos (c+d x) \sin (c+d x))^2} \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^(-2),x]","\frac{2 \left(\frac{4 a \tan ^{-1}\left(\frac{2 a \tan (c+d x)+b}{\sqrt{4 a^2-b^2}}\right)}{\left(4 a^2-b^2\right)^{3/2}}+\frac{b \cos (2 (c+d x))}{(2 a-b) (2 a+b) (2 a+b \sin (2 (c+d x)))}\right)}{d}","\frac{8 a \tan ^{-1}\left(\frac{2 a \tan (c+d x)+b}{\sqrt{4 a^2-b^2}}\right)}{d \left(4 a^2-b^2\right)^{3/2}}+\frac{2 b \cos (2 c+2 d x)}{d \left(4 a^2-b^2\right) (2 a+b \sin (2 c+2 d x))}",1,"(2*((4*a*ArcTan[(b + 2*a*Tan[c + d*x])/Sqrt[4*a^2 - b^2]])/(4*a^2 - b^2)^(3/2) + (b*Cos[2*(c + d*x)])/((2*a - b)*(2*a + b)*(2*a + b*Sin[2*(c + d*x)]))))/d","A",1
572,1,120,149,0.9297808,"\int \frac{1}{(a+b \cos (c+d x) \sin (c+d x))^3} \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^(-3),x]","\frac{2 \left(\frac{2 \left(8 a^2+b^2\right) \tan ^{-1}\left(\frac{2 a \tan (c+d x)+b}{\sqrt{4 a^2-b^2}}\right)}{\left(4 a^2-b^2\right)^{5/2}}+\frac{b \cos (2 (c+d x)) \left(16 a^2+6 a b \sin (2 (c+d x))-b^2\right)}{\left(b^2-4 a^2\right)^2 (2 a+b \sin (2 (c+d x)))^2}\right)}{d}","\frac{4 \left(8 a^2+b^2\right) \tan ^{-1}\left(\frac{2 a \tan (c+d x)+b}{\sqrt{4 a^2-b^2}}\right)}{d \left(4 a^2-b^2\right)^{5/2}}+\frac{12 a b \cos (2 c+2 d x)}{d \left(4 a^2-b^2\right)^2 (2 a+b \sin (2 c+2 d x))}+\frac{2 b \cos (2 c+2 d x)}{d \left(4 a^2-b^2\right) (2 a+b \sin (2 c+2 d x))^2}",1,"(2*((2*(8*a^2 + b^2)*ArcTan[(b + 2*a*Tan[c + d*x])/Sqrt[4*a^2 - b^2]])/(4*a^2 - b^2)^(5/2) + (b*Cos[2*(c + d*x)]*(16*a^2 - b^2 + 6*a*b*Sin[2*(c + d*x)]))/((-4*a^2 + b^2)^2*(2*a + b*Sin[2*(c + d*x)])^2)))/d","A",1
573,1,202,265,1.8476088,"\int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^(5/2),x]","\frac{-32 a \left(4 a^2-b^2\right) \sqrt{\frac{2 a+b \sin (2 (c+d x))}{2 a+b}} F\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)-b \left(88 a^2 \cos (2 (c+d x))+b \sin (4 (c+d x)) (28 a+3 b \sin (2 (c+d x)))\right)+2 \left(184 a^3+92 a^2 b+18 a b^2+9 b^3\right) \sqrt{\frac{2 a+b \sin (2 (c+d x))}{2 a+b}} E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{120 d \sqrt{4 a+2 b \sin (2 (c+d x))}}","-\frac{2 \sqrt{2} a \left(4 a^2-b^2\right) \sqrt{\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}} F\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{15 d \sqrt{2 a+b \sin (2 c+2 d x)}}+\frac{\left(92 a^2+9 b^2\right) \sqrt{2 a+b \sin (2 c+2 d x)} E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{60 \sqrt{2} d \sqrt{\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}}}-\frac{b \cos (2 c+2 d x) (2 a+b \sin (2 c+2 d x))^{3/2}}{20 \sqrt{2} d}-\frac{2 \sqrt{2} a b \cos (2 c+2 d x) \sqrt{2 a+b \sin (2 c+2 d x)}}{15 d}",1,"(2*(184*a^3 + 92*a^2*b + 18*a*b^2 + 9*b^3)*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*(c + d*x)])/(2*a + b)] - 32*a*(4*a^2 - b^2)*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*(c + d*x)])/(2*a + b)] - b*(88*a^2*Cos[2*(c + d*x)] + b*(28*a + 3*b*Sin[2*(c + d*x)])*Sin[4*(c + d*x)]))/(120*d*Sqrt[4*a + 2*b*Sin[2*(c + d*x)]])","A",1
574,1,167,212,1.4563591,"\int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^(3/2),x]","\frac{-\left(4 a^2-b^2\right) \sqrt{\frac{2 a+b \sin (2 (c+d x))}{2 a+b}} F\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)-b \cos (2 (c+d x)) (2 a+b \sin (2 (c+d x)))+8 a (2 a+b) \sqrt{\frac{2 a+b \sin (2 (c+d x))}{2 a+b}} E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{6 d \sqrt{4 a+2 b \sin (2 (c+d x))}}","-\frac{\left(4 a^2-b^2\right) \sqrt{\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}} F\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{6 \sqrt{2} d \sqrt{2 a+b \sin (2 c+2 d x)}}-\frac{b \cos (2 c+2 d x) \sqrt{2 a+b \sin (2 c+2 d x)}}{6 \sqrt{2} d}+\frac{2 \sqrt{2} a \sqrt{2 a+b \sin (2 c+2 d x)} E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{3 d \sqrt{\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}}}",1,"(-(b*Cos[2*(c + d*x)]*(2*a + b*Sin[2*(c + d*x)])) + 8*a*(2*a + b)*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*(c + d*x)])/(2*a + b)] - (4*a^2 - b^2)*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*(c + d*x)])/(2*a + b)])/(6*d*Sqrt[4*a + 2*b*Sin[2*(c + d*x)]])","A",1
575,1,75,76,0.1081448,"\int \sqrt{a+b \cos (c+d x) \sin (c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]*Sin[c + d*x]],x]","\frac{(2 a+b) \sqrt{\frac{2 a+b \sin (2 (c+d x))}{2 a+b}} E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{d \sqrt{4 a+2 b \sin (2 (c+d x))}}","\frac{\sqrt{2 a+b \sin (2 c+2 d x)} E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{\sqrt{2} d \sqrt{\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}}}",1,"((2*a + b)*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*(c + d*x)])/(2*a + b)])/(d*Sqrt[4*a + 2*b*Sin[2*(c + d*x)]])","A",1
576,1,70,76,0.1337262,"\int \frac{1}{\sqrt{a+b \cos (c+d x) \sin (c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Cos[c + d*x]*Sin[c + d*x]],x]","\frac{\sqrt{\frac{2 a+b \sin (2 (c+d x))}{2 a+b}} F\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{d \sqrt{a+\frac{1}{2} b \sin (2 (c+d x))}}","\frac{\sqrt{2} \sqrt{\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}} F\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{d \sqrt{2 a+b \sin (2 c+2 d x)}}",1,"(EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*(c + d*x)])/(2*a + b)])/(d*Sqrt[a + (b*Sin[2*(c + d*x)])/2])","A",1
577,1,101,143,0.4193655,"\int \frac{1}{(a+b \cos (c+d x) \sin (c+d x))^{3/2}} \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^(-3/2),x]","\frac{2 \left((2 a+b) \sqrt{\frac{2 a+b \sin (2 (c+d x))}{2 a+b}} E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)+b \cos (2 (c+d x))\right)}{d \left(4 a^2-b^2\right) \sqrt{a+\frac{1}{2} b \sin (2 (c+d x))}}","\frac{2 \sqrt{2} b \cos (2 c+2 d x)}{d \left(4 a^2-b^2\right) \sqrt{2 a+b \sin (2 c+2 d x)}}+\frac{2 \sqrt{2} \sqrt{2 a+b \sin (2 c+2 d x)} E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{d \left(4 a^2-b^2\right) \sqrt{\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}}}",1,"(2*(b*Cos[2*(c + d*x)] + (2*a + b)*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*(c + d*x)])/(2*a + b)]))/((4*a^2 - b^2)*d*Sqrt[a + (b*Sin[2*(c + d*x)])/2])","A",1
578,1,201,295,1.5157407,"\int \frac{1}{(a+b \cos (c+d x) \sin (c+d x))^{5/2}} \, dx","Integrate[(a + b*Cos[c + d*x]*Sin[c + d*x])^(-5/2),x]","-\frac{4 \sqrt{2} \left(b \cos (2 (c+d x)) \left(-20 a^2-8 a b \sin (2 (c+d x))+b^2\right)+(2 a-b) (2 a+b)^2 \left(\frac{2 a+b \sin (2 (c+d x))}{2 a+b}\right)^{3/2} F\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)-\frac{8 a (2 a+b \sin (2 (c+d x)))^2 E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{\sqrt{\frac{2 a+b \sin (2 (c+d x))}{2 a+b}}}\right)}{3 d \left(b^2-4 a^2\right)^2 (2 a+b \sin (2 (c+d x)))^{3/2}}","\frac{32 \sqrt{2} a b \cos (2 c+2 d x)}{3 d \left(4 a^2-b^2\right)^2 \sqrt{2 a+b \sin (2 c+2 d x)}}+\frac{4 \sqrt{2} b \cos (2 c+2 d x)}{3 d \left(4 a^2-b^2\right) (2 a+b \sin (2 c+2 d x))^{3/2}}-\frac{4 \sqrt{2} \sqrt{\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}} F\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{3 d \left(4 a^2-b^2\right) \sqrt{2 a+b \sin (2 c+2 d x)}}+\frac{32 \sqrt{2} a \sqrt{2 a+b \sin (2 c+2 d x)} E\left(c+d x-\frac{\pi }{4}|\frac{2 b}{2 a+b}\right)}{3 d \left(4 a^2-b^2\right)^2 \sqrt{\frac{2 a+b \sin (2 c+2 d x)}{2 a+b}}}",1,"(-4*Sqrt[2]*((-8*a*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*(2*a + b*Sin[2*(c + d*x)])^2)/Sqrt[(2*a + b*Sin[2*(c + d*x)])/(2*a + b)] + (2*a - b)*(2*a + b)^2*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*((2*a + b*Sin[2*(c + d*x)])/(2*a + b))^(3/2) + b*Cos[2*(c + d*x)]*(-20*a^2 + b^2 - 8*a*b*Sin[2*(c + d*x)])))/(3*(-4*a^2 + b^2)^2*d*(2*a + b*Sin[2*(c + d*x)])^(3/2))","A",1
579,1,340,461,0.8714975,"\int \frac{x^3}{a+b \cos (x) \sin (x)} \, dx","Integrate[x^3/(a + b*Cos[x]*Sin[x]),x]","\frac{-6 x^2 \text{Li}_2\left(-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right)+6 x^2 \text{Li}_2\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)-6 i x \text{Li}_3\left(-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right)+6 i x \text{Li}_3\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)+3 \text{Li}_4\left(-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right)-3 \text{Li}_4\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)-4 i x^3 \log \left(1+\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right)+4 i x^3 \log \left(1-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{4 \sqrt{4 a^2-b^2}}","-\frac{3 x^2 \text{Li}_2\left(\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right)}{2 \sqrt{4 a^2-b^2}}+\frac{3 x^2 \text{Li}_2\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)}{2 \sqrt{4 a^2-b^2}}-\frac{3 i x \text{Li}_3\left(\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right)}{2 \sqrt{4 a^2-b^2}}+\frac{3 i x \text{Li}_3\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)}{2 \sqrt{4 a^2-b^2}}+\frac{3 \text{Li}_4\left(\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right)}{4 \sqrt{4 a^2-b^2}}-\frac{3 \text{Li}_4\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)}{4 \sqrt{4 a^2-b^2}}-\frac{i x^3 \log \left(1-\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right)}{\sqrt{4 a^2-b^2}}+\frac{i x^3 \log \left(1-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{\sqrt{4 a^2-b^2}}",1,"((-4*I)*x^3*Log[1 + (I*b*E^((2*I)*x))/(-2*a + Sqrt[4*a^2 - b^2])] + (4*I)*x^3*Log[1 - (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])] - 6*x^2*PolyLog[2, ((-I)*b*E^((2*I)*x))/(-2*a + Sqrt[4*a^2 - b^2])] + 6*x^2*PolyLog[2, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])] - (6*I)*x*PolyLog[3, ((-I)*b*E^((2*I)*x))/(-2*a + Sqrt[4*a^2 - b^2])] + (6*I)*x*PolyLog[3, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])] + 3*PolyLog[4, ((-I)*b*E^((2*I)*x))/(-2*a + Sqrt[4*a^2 - b^2])] - 3*PolyLog[4, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])])/(4*Sqrt[4*a^2 - b^2])","A",1
580,1,256,340,0.7202547,"\int \frac{x^2}{a+b \cos (x) \sin (x)} \, dx","Integrate[x^2/(a + b*Cos[x]*Sin[x]),x]","-\frac{i \left(-2 i x \text{Li}_2\left(-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right)+2 i x \text{Li}_2\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)+\text{Li}_3\left(-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right)-\text{Li}_3\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)+2 x^2 \log \left(1+\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}-2 a}\right)-2 x^2 \log \left(1-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)\right)}{2 \sqrt{4 a^2-b^2}}","-\frac{x \text{Li}_2\left(\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right)}{\sqrt{4 a^2-b^2}}+\frac{x \text{Li}_2\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)}{\sqrt{4 a^2-b^2}}-\frac{i \text{Li}_3\left(\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right)}{2 \sqrt{4 a^2-b^2}}+\frac{i \text{Li}_3\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)}{2 \sqrt{4 a^2-b^2}}-\frac{i x^2 \log \left(1-\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right)}{\sqrt{4 a^2-b^2}}+\frac{i x^2 \log \left(1-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{\sqrt{4 a^2-b^2}}",1,"((-1/2*I)*(2*x^2*Log[1 + (I*b*E^((2*I)*x))/(-2*a + Sqrt[4*a^2 - b^2])] - 2*x^2*Log[1 - (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])] - (2*I)*x*PolyLog[2, ((-I)*b*E^((2*I)*x))/(-2*a + Sqrt[4*a^2 - b^2])] + (2*I)*x*PolyLog[2, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])] + PolyLog[3, ((-I)*b*E^((2*I)*x))/(-2*a + Sqrt[4*a^2 - b^2])] - PolyLog[3, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])]))/Sqrt[4*a^2 - b^2]","A",1
581,1,788,225,1.4064009,"\int \frac{x}{a+b \cos (x) \sin (x)} \, dx","Integrate[x/(a + b*Cos[x]*Sin[x]),x]","\frac{1}{2} \left(\frac{\pi  \tan ^{-1}\left(\frac{2 a \tan (x)+b}{\sqrt{4 a^2-b^2}}\right)}{\sqrt{4 a^2-b^2}}+\frac{i \left(\text{Li}_2\left(\frac{\left(2 a-i \sqrt{b^2-4 a^2}\right) \left(2 a+b-\sqrt{b^2-4 a^2} \cot \left(x+\frac{\pi }{4}\right)\right)}{b \left(2 a+b+\sqrt{b^2-4 a^2} \cot \left(x+\frac{\pi }{4}\right)\right)}\right)-\text{Li}_2\left(\frac{\left(2 a+i \sqrt{b^2-4 a^2}\right) \left(2 a+b-\sqrt{b^2-4 a^2} \cot \left(x+\frac{\pi }{4}\right)\right)}{b \left(2 a+b+\sqrt{b^2-4 a^2} \cot \left(x+\frac{\pi }{4}\right)\right)}\right)\right)+(\pi -4 x) \tanh ^{-1}\left(\frac{(2 a+b) \tan \left(x+\frac{\pi }{4}\right)}{\sqrt{b^2-4 a^2}}\right)+2 \cos ^{-1}\left(-\frac{2 a}{b}\right) \tanh ^{-1}\left(\frac{(2 a-b) \cot \left(x+\frac{\pi }{4}\right)}{\sqrt{b^2-4 a^2}}\right)-\log \left(\frac{(2 a+b) \left(-i \sqrt{b^2-4 a^2}-2 a+b\right) \left(1+i \cot \left(x+\frac{\pi }{4}\right)\right)}{b \left(\sqrt{b^2-4 a^2} \cot \left(x+\frac{\pi }{4}\right)+2 a+b\right)}\right) \left(\cos ^{-1}\left(-\frac{2 a}{b}\right)+2 i \tanh ^{-1}\left(\frac{(2 a-b) \cot \left(x+\frac{\pi }{4}\right)}{\sqrt{b^2-4 a^2}}\right)\right)-\log \left(\frac{(2 a+b) \left(\sqrt{b^2-4 a^2}+2 i a-i b\right) \left(\cot \left(x+\frac{\pi }{4}\right)+i\right)}{b \left(\sqrt{b^2-4 a^2} \cot \left(x+\frac{\pi }{4}\right)+2 a+b\right)}\right) \left(\cos ^{-1}\left(-\frac{2 a}{b}\right)-2 i \tanh ^{-1}\left(\frac{(2 a-b) \cot \left(x+\frac{\pi }{4}\right)}{\sqrt{b^2-4 a^2}}\right)\right)+\log \left(\frac{\sqrt[4]{-1} e^{-i x} \sqrt{b^2-4 a^2}}{2 \sqrt{b} \sqrt{a+b \sin (x) \cos (x)}}\right) \left(\cos ^{-1}\left(-\frac{2 a}{b}\right)+2 i \left(\tanh ^{-1}\left(\frac{(2 a+b) \tan \left(x+\frac{\pi }{4}\right)}{\sqrt{b^2-4 a^2}}\right)+\tanh ^{-1}\left(\frac{(2 a-b) \cot \left(x+\frac{\pi }{4}\right)}{\sqrt{b^2-4 a^2}}\right)\right)\right)+\log \left(-\frac{(-1)^{3/4} e^{i x} \sqrt{b^2-4 a^2}}{2 \sqrt{b} \sqrt{a+b \sin (x) \cos (x)}}\right) \left(-2 i \tanh ^{-1}\left(\frac{(2 a+b) \tan \left(x+\frac{\pi }{4}\right)}{\sqrt{b^2-4 a^2}}\right)-2 i \tanh ^{-1}\left(\frac{(2 a-b) \cot \left(x+\frac{\pi }{4}\right)}{\sqrt{b^2-4 a^2}}\right)+\cos ^{-1}\left(-\frac{2 a}{b}\right)\right)}{\sqrt{b^2-4 a^2}}\right)","-\frac{\text{Li}_2\left(\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right)}{2 \sqrt{4 a^2-b^2}}+\frac{\text{Li}_2\left(\frac{i b e^{2 i x}}{2 a+\sqrt{4 a^2-b^2}}\right)}{2 \sqrt{4 a^2-b^2}}-\frac{i x \log \left(1-\frac{i b e^{2 i x}}{2 a-\sqrt{4 a^2-b^2}}\right)}{\sqrt{4 a^2-b^2}}+\frac{i x \log \left(1-\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{\sqrt{4 a^2-b^2}}",1,"((Pi*ArcTan[(b + 2*a*Tan[x])/Sqrt[4*a^2 - b^2]])/Sqrt[4*a^2 - b^2] + (2*ArcCos[(-2*a)/b]*ArcTanh[((2*a - b)*Cot[Pi/4 + x])/Sqrt[-4*a^2 + b^2]] + (Pi - 4*x)*ArcTanh[((2*a + b)*Tan[Pi/4 + x])/Sqrt[-4*a^2 + b^2]] - (ArcCos[(-2*a)/b] + (2*I)*ArcTanh[((2*a - b)*Cot[Pi/4 + x])/Sqrt[-4*a^2 + b^2]])*Log[((2*a + b)*(-2*a + b - I*Sqrt[-4*a^2 + b^2])*(1 + I*Cot[Pi/4 + x]))/(b*(2*a + b + Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))] - (ArcCos[(-2*a)/b] - (2*I)*ArcTanh[((2*a - b)*Cot[Pi/4 + x])/Sqrt[-4*a^2 + b^2]])*Log[((2*a + b)*((2*I)*a - I*b + Sqrt[-4*a^2 + b^2])*(I + Cot[Pi/4 + x]))/(b*(2*a + b + Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))] + (ArcCos[(-2*a)/b] + (2*I)*(ArcTanh[((2*a - b)*Cot[Pi/4 + x])/Sqrt[-4*a^2 + b^2]] + ArcTanh[((2*a + b)*Tan[Pi/4 + x])/Sqrt[-4*a^2 + b^2]]))*Log[((-1)^(1/4)*Sqrt[-4*a^2 + b^2])/(2*Sqrt[b]*E^(I*x)*Sqrt[a + b*Cos[x]*Sin[x]])] + (ArcCos[(-2*a)/b] - (2*I)*ArcTanh[((2*a - b)*Cot[Pi/4 + x])/Sqrt[-4*a^2 + b^2]] - (2*I)*ArcTanh[((2*a + b)*Tan[Pi/4 + x])/Sqrt[-4*a^2 + b^2]])*Log[-1/2*((-1)^(3/4)*Sqrt[-4*a^2 + b^2]*E^(I*x))/(Sqrt[b]*Sqrt[a + b*Cos[x]*Sin[x]])] + I*(PolyLog[2, ((2*a - I*Sqrt[-4*a^2 + b^2])*(2*a + b - Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))/(b*(2*a + b + Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))] - PolyLog[2, ((2*a + I*Sqrt[-4*a^2 + b^2])*(2*a + b - Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))/(b*(2*a + b + Sqrt[-4*a^2 + b^2]*Cot[Pi/4 + x]))]))/Sqrt[-4*a^2 + b^2])/2","B",1
582,0,0,20,1.580984,"\int \frac{1}{x (a+b \cos (x) \sin (x))} \, dx","Integrate[1/(x*(a + b*Cos[x]*Sin[x])),x]","\int \frac{1}{x (a+b \cos (x) \sin (x))} \, dx","\text{Int}\left(\frac{1}{x \left(a+\frac{1}{2} b \sin (2 x)\right)},x\right)",0,"Integrate[1/(x*(a + b*Cos[x]*Sin[x])), x]","A",-1
583,0,0,79,5.4648004,"\int \frac{(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx","Integrate[((b*x)^(2 - n)*Sin[a*x]^n)/(a*c*x*Cos[a*x] - c*Sin[a*x])^2,x]","\int \frac{(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx","\frac{b^2 (1-n) \text{Int}\left((b x)^{-n} \sin ^{n-2}(a x),x\right)}{a^2 c^2}+\frac{b (b x)^{1-n} \sin ^{n-1}(a x)}{a^2 \left(a c^2 x \cos (a x)-c^2 \sin (a x)\right)}",0,"Integrate[((b*x)^(2 - n)*Sin[a*x]^n)/(a*c*x*Cos[a*x] - c*Sin[a*x])^2, x]","A",-1
584,0,0,79,4.9678346,"\int \frac{(b x)^{2-n} \cos ^n(a x)}{(c \cos (a x)+a c x \sin (a x))^2} \, dx","Integrate[((b*x)^(2 - n)*Cos[a*x]^n)/(c*Cos[a*x] + a*c*x*Sin[a*x])^2,x]","\int \frac{(b x)^{2-n} \cos ^n(a x)}{(c \cos (a x)+a c x \sin (a x))^2} \, dx","\frac{b^2 (1-n) \text{Int}\left((b x)^{-n} \cos ^{n-2}(a x),x\right)}{a^2 c^2}-\frac{b (b x)^{1-n} \cos ^{n-1}(a x)}{a^2 \left(a c^2 x \sin (a x)+c^2 \cos (a x)\right)}",0,"Integrate[((b*x)^(2 - n)*Cos[a*x]^n)/(c*Cos[a*x] + a*c*x*Sin[a*x])^2, x]","A",-1
585,1,198,175,1.446722,"\int \frac{\sin ^6(a x)}{x^4 (a x \cos (a x)-\sin (a x))^2} \, dx","Integrate[Sin[a*x]^6/(x^4*(a*x*Cos[a*x] - Sin[a*x])^2),x]","\frac{-32 a^3 x^3 \text{Si}(2 a x) (a x \cos (a x)-\sin (a x))+256 a^3 x^3 \text{Si}(4 a x) (a x \cos (a x)-\sin (a x))-8 a^3 x^3 \cos (a x)+24 a^3 x^3 \cos (3 a x)+32 a^3 x^3 \cos (5 a x)-12 a^2 x^2 \sin (a x)+44 a^2 x^2 \sin (3 a x)-24 a^2 x^2 \sin (5 a x)+10 \sin (a x)-5 \sin (3 a x)+\sin (5 a x)+8 a x \cos (a x)-12 a x \cos (3 a x)+4 a x \cos (5 a x)}{48 x^3 (a x \cos (a x)-\sin (a x))}","-\frac{2}{3} a^3 \text{Si}(2 a x)+\frac{16}{3} a^3 \text{Si}(4 a x)+\frac{\sin ^4(a x)}{a^2 x^5}+\frac{\sin ^5(a x)}{a^2 x^5 (a x \cos (a x)-\sin (a x))}+\frac{a^2}{x}+\frac{32 a^2 \sin ^4(a x)}{3 x}-\frac{10 a^2 \sin ^2(a x)}{x}+\frac{\sin ^3(a x) \cos (a x)}{a x^4}-\frac{4 \sin ^4(a x)}{3 x^3}+\frac{\sin ^2(a x)}{x^3}-\frac{8 a \sin ^3(a x) \cos (a x)}{3 x^2}+\frac{a \sin (a x) \cos (a x)}{x^2}",1,"(8*a*x*Cos[a*x] - 8*a^3*x^3*Cos[a*x] - 12*a*x*Cos[3*a*x] + 24*a^3*x^3*Cos[3*a*x] + 4*a*x*Cos[5*a*x] + 32*a^3*x^3*Cos[5*a*x] + 10*Sin[a*x] - 12*a^2*x^2*Sin[a*x] - 5*Sin[3*a*x] + 44*a^2*x^2*Sin[3*a*x] + Sin[5*a*x] - 24*a^2*x^2*Sin[5*a*x] - 32*a^3*x^3*(a*x*Cos[a*x] - Sin[a*x])*SinIntegral[2*a*x] + 256*a^3*x^3*(a*x*Cos[a*x] - Sin[a*x])*SinIntegral[4*a*x])/(48*x^3*(a*x*Cos[a*x] - Sin[a*x]))","A",1
586,1,142,131,0.970445,"\int \frac{\sin ^5(a x)}{x^3 (a x \cos (a x)-\sin (a x))^2} \, dx","Integrate[Sin[a*x]^5/(x^3*(a*x*Cos[a*x] - Sin[a*x])^2),x]","\frac{-2 a^2 x^2 \text{Si}(a x) (a x \cos (a x)-\sin (a x))+54 a^2 x^2 \text{Si}(3 a x) (a x \cos (a x)-\sin (a x))-a^2 x^2+8 a^2 x^2 \cos (2 a x)+9 a^2 x^2 \cos (4 a x)+12 a x \sin (2 a x)-6 a x \sin (4 a x)-4 \cos (2 a x)+\cos (4 a x)+3}{16 x^2 (a x \cos (a x)-\sin (a x))}","-\frac{1}{8} a^2 \text{Si}(a x)+\frac{27}{8} a^2 \text{Si}(3 a x)+\frac{\sin ^3(a x)}{a^2 x^4}+\frac{\sin ^4(a x)}{a^2 x^4 (a x \cos (a x)-\sin (a x))}+\frac{\sin ^2(a x) \cos (a x)}{a x^3}-\frac{3 \sin ^3(a x)}{2 x^2}+\frac{\sin (a x)}{x^2}+\frac{a \cos (a x)}{x}-\frac{9 a \sin ^2(a x) \cos (a x)}{2 x}",1,"(3 - a^2*x^2 - 4*Cos[2*a*x] + 8*a^2*x^2*Cos[2*a*x] + Cos[4*a*x] + 9*a^2*x^2*Cos[4*a*x] + 12*a*x*Sin[2*a*x] - 6*a*x*Sin[4*a*x] - 2*a^2*x^2*(a*x*Cos[a*x] - Sin[a*x])*SinIntegral[a*x] + 54*a^2*x^2*(a*x*Cos[a*x] - Sin[a*x])*SinIntegral[3*a*x])/(16*x^2*(a*x*Cos[a*x] - Sin[a*x]))","A",1
587,1,77,80,0.8207099,"\int \frac{\sin ^4(a x)}{x^2 (a x \cos (a x)-\sin (a x))^2} \, dx","Integrate[Sin[a*x]^4/(x^2*(a*x*Cos[a*x] - Sin[a*x])^2),x]","\frac{8 a x \text{Si}(2 a x) (a x \cos (a x)-\sin (a x))+3 \sin (a x)-\sin (3 a x)+2 a x \cos (a x)+2 a x \cos (3 a x)}{4 x (a x \cos (a x)-\sin (a x))}","\frac{\sin ^2(a x)}{a^2 x^3}+\frac{\sin ^3(a x)}{a^2 x^3 (a x \cos (a x)-\sin (a x))}+2 a \text{Si}(2 a x)+\frac{\sin (a x) \cos (a x)}{a x^2}-\frac{2 \sin ^2(a x)}{x}+\frac{1}{x}",1,"(2*a*x*Cos[a*x] + 2*a*x*Cos[3*a*x] + 3*Sin[a*x] - Sin[3*a*x] + 8*a*x*(a*x*Cos[a*x] - Sin[a*x])*SinIntegral[2*a*x])/(4*x*(a*x*Cos[a*x] - Sin[a*x]))","A",1
588,1,242,56,7.4188976,"\int \frac{\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx","Integrate[Sin[a*x]^3/(x*(a*x*Cos[a*x] - Sin[a*x])^2),x]","\frac{-i e \text{Ci}(i-a x) (a x \cos (a x)-\sin (a x))+i e \text{Ci}(a x+i) (a x \cos (a x)-\sin (a x))-i e \text{Ei}(-i a x-1) \sin (a x)+i e \text{Ei}(i a x-1) \sin (a x)+i e a x \text{Ei}(-i a x-1) \cos (a x)-i e a x \text{Ei}(i a x-1) \cos (a x)-2 \text{Si}(a x) \sin (a x)-e \text{Si}(i-a x) \sin (a x)+e \text{Si}(a x+i) \sin (a x)+2 a x \text{Si}(a x) \cos (a x)+e a x \text{Si}(i-a x) \cos (a x)-e a x \text{Si}(a x+i) \cos (a x)+\cos (2 a x)+1}{2 a x \cos (a x)-2 \sin (a x)}","\frac{\sin (a x)}{a^2 x^2}+\frac{\sin ^2(a x)}{a^2 x^2 (a x \cos (a x)-\sin (a x))}+\text{Si}(a x)+\frac{\cos (a x)}{a x}",1,"(1 + Cos[2*a*x] + I*a*E*x*Cos[a*x]*ExpIntegralEi[-1 - I*a*x] - I*a*E*x*Cos[a*x]*ExpIntegralEi[-1 + I*a*x] - I*E*CosIntegral[I - a*x]*(a*x*Cos[a*x] - Sin[a*x]) + I*E*CosIntegral[I + a*x]*(a*x*Cos[a*x] - Sin[a*x]) - I*E*ExpIntegralEi[-1 - I*a*x]*Sin[a*x] + I*E*ExpIntegralEi[-1 + I*a*x]*Sin[a*x] + 2*a*x*Cos[a*x]*SinIntegral[a*x] - 2*Sin[a*x]*SinIntegral[a*x] + a*E*x*Cos[a*x]*SinIntegral[I - a*x] - E*Sin[a*x]*SinIntegral[I - a*x] - a*E*x*Cos[a*x]*SinIntegral[I + a*x] + E*Sin[a*x]*SinIntegral[I + a*x])/(2*a*x*Cos[a*x] - 2*Sin[a*x])","C",0
589,1,24,35,0.2766087,"\int \frac{\sin ^2(a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx","Integrate[Sin[a*x]^2/(a*x*Cos[a*x] - Sin[a*x])^2,x]","\frac{\cos (a x)}{a^2 x \cos (a x)-a \sin (a x)}","\frac{1}{a^2 x}+\frac{\sin (a x)}{a^2 x (a x \cos (a x)-\sin (a x))}",1,"Cos[a*x]/(a^2*x*Cos[a*x] - a*Sin[a*x])","A",1
590,1,20,20,0.0301568,"\int \frac{x \sin (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx","Integrate[(x*Sin[a*x])/(a*x*Cos[a*x] - Sin[a*x])^2,x]","-\frac{1}{a^2 (\sin (a x)-a x \cos (a x))}","\frac{1}{a^2 (a x \cos (a x)-\sin (a x))}",1,"-(1/(a^2*(-(a*x*Cos[a*x]) + Sin[a*x])))","A",1
591,1,32,35,0.4558613,"\int \frac{x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx","Integrate[x^2/(a*x*Cos[a*x] - Sin[a*x])^2,x]","\frac{a x \sin (a x)+\cos (a x)}{a^3 (a x \cos (a x)-\sin (a x))}","\frac{x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\cot (a x)}{a^3}",1,"(Cos[a*x] + a*x*Sin[a*x])/(a^3*(a*x*Cos[a*x] - Sin[a*x]))","A",1
592,1,157,104,1.003233,"\int \frac{x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx","Integrate[(x^3*Csc[a*x])/(a*x*Cos[a*x] - Sin[a*x])^2,x]","\frac{a^2 x^2 \csc (a x)+a^2 x^2 \log \left(1-e^{i a x}\right) \cot (a x)-a^2 x^2 \log \left(1+e^{i a x}\right) \cot (a x)+i \text{Li}_2\left(-e^{i a x}\right) (a x \cot (a x)-1)-i \text{Li}_2\left(e^{i a x}\right) (a x \cot (a x)-1)-a x \log \left(1-e^{i a x}\right)+a x \log \left(1+e^{i a x}\right)+\csc (a x)}{a^4 (a x \cot (a x)-1)}","\frac{i \text{Li}_2\left(-e^{i a x}\right)}{a^4}-\frac{i \text{Li}_2\left(e^{i a x}\right)}{a^4}-\frac{\csc (a x)}{a^4}-\frac{2 x \tanh ^{-1}\left(e^{i a x}\right)}{a^3}-\frac{x \cot (a x) \csc (a x)}{a^3}+\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}",1,"(Csc[a*x] + a^2*x^2*Csc[a*x] - a*x*Log[1 - E^(I*a*x)] + a^2*x^2*Cot[a*x]*Log[1 - E^(I*a*x)] + a*x*Log[1 + E^(I*a*x)] - a^2*x^2*Cot[a*x]*Log[1 + E^(I*a*x)] + I*(-1 + a*x*Cot[a*x])*PolyLog[2, -E^(I*a*x)] - I*(-1 + a*x*Cot[a*x])*PolyLog[2, E^(I*a*x)])/(a^4*(-1 + a*x*Cot[a*x]))","A",1
593,1,102,127,1.0493609,"\int \frac{x^4 \csc ^2(a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx","Integrate[(x^4*Csc[a*x]^2)/(a*x*Cos[a*x] - Sin[a*x])^2,x]","\frac{a^3 \left(-x^2\right) \cot (a x)-2 i a \left(a^2 x^2+\text{Li}_2\left(e^{2 i a x}\right)\right)+\frac{\left(a^2 x^2+1\right)^2 \sin (a x)}{x (a x \cos (a x)-\sin (a x))}+a^2 x+4 a^2 x \log \left(1-e^{2 i a x}\right)+\frac{1}{x}}{a^6}","-\frac{2 i \text{Li}_2\left(e^{2 i a x}\right)}{a^5}-\frac{\cot (a x)}{a^5}+\frac{4 x \log \left(1-e^{2 i a x}\right)}{a^4}-\frac{x \csc ^2(a x)}{a^4}-\frac{2 i x^2}{a^3}-\frac{2 x^2 \cot (a x)}{a^3}-\frac{x^2 \cot (a x) \csc ^2(a x)}{a^3}+\frac{x^3 \csc ^3(a x)}{a^2 (a x \cos (a x)-\sin (a x))}",1,"(x^(-1) + a^2*x - a^3*x^2*Cot[a*x] + 4*a^2*x*Log[1 - E^((2*I)*a*x)] - (2*I)*a*(a^2*x^2 + PolyLog[2, E^((2*I)*a*x)]) + ((1 + a^2*x^2)^2*Sin[a*x])/(x*(a*x*Cos[a*x] - Sin[a*x])))/a^6","A",1
594,1,194,176,1.2231994,"\int \frac{\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx","Integrate[Cos[a*x]^6/(x^4*(Cos[a*x] + a*x*Sin[a*x])^2),x]","\frac{32 a^3 x^3 \text{Si}(2 a x) (a x \sin (a x)+\cos (a x))+256 a^3 x^3 \text{Si}(4 a x) (a x \sin (a x)+\cos (a x))-8 a^3 x^3 \sin (a x)-24 a^3 x^3 \sin (3 a x)+32 a^3 x^3 \sin (5 a x)+12 a^2 x^2 \cos (a x)+44 a^2 x^2 \cos (3 a x)+24 a^2 x^2 \cos (5 a x)+8 a x \sin (a x)+12 a x \sin (3 a x)+4 a x \sin (5 a x)-10 \cos (a x)-5 \cos (3 a x)-\cos (5 a x)}{48 x^3 (a x \sin (a x)+\cos (a x))}","\frac{2}{3} a^3 \text{Si}(2 a x)+\frac{16}{3} a^3 \text{Si}(4 a x)+\frac{\cos ^4(a x)}{a^2 x^5}-\frac{\cos ^5(a x)}{a^2 x^5 (a x \sin (a x)+\cos (a x))}+\frac{a^2}{x}+\frac{32 a^2 \cos ^4(a x)}{3 x}-\frac{10 a^2 \cos ^2(a x)}{x}-\frac{\sin (a x) \cos ^3(a x)}{a x^4}-\frac{4 \cos ^4(a x)}{3 x^3}+\frac{\cos ^2(a x)}{x^3}+\frac{8 a \sin (a x) \cos ^3(a x)}{3 x^2}-\frac{a \sin (a x) \cos (a x)}{x^2}",1,"(-10*Cos[a*x] + 12*a^2*x^2*Cos[a*x] - 5*Cos[3*a*x] + 44*a^2*x^2*Cos[3*a*x] - Cos[5*a*x] + 24*a^2*x^2*Cos[5*a*x] + 8*a*x*Sin[a*x] - 8*a^3*x^3*Sin[a*x] + 12*a*x*Sin[3*a*x] - 24*a^3*x^3*Sin[3*a*x] + 4*a*x*Sin[5*a*x] + 32*a^3*x^3*Sin[5*a*x] + 32*a^3*x^3*(Cos[a*x] + a*x*Sin[a*x])*SinIntegral[2*a*x] + 256*a^3*x^3*(Cos[a*x] + a*x*Sin[a*x])*SinIntegral[4*a*x])/(48*x^3*(Cos[a*x] + a*x*Sin[a*x]))","A",1
595,1,136,132,0.7951793,"\int \frac{\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx","Integrate[Cos[a*x]^5/(x^3*(Cos[a*x] + a*x*Sin[a*x])^2),x]","-\frac{2 a^2 x^2 \text{Ci}(a x) (a x \sin (a x)+\cos (a x))+54 a^2 x^2 \text{Ci}(3 a x) (a x \sin (a x)+\cos (a x))-a^2 x^2-8 a^2 x^2 \cos (2 a x)+9 a^2 x^2 \cos (4 a x)-12 a x \sin (2 a x)-6 a x \sin (4 a x)+4 \cos (2 a x)+\cos (4 a x)+3}{16 x^2 (a x \sin (a x)+\cos (a x))}","-\frac{1}{8} a^2 \text{Ci}(a x)-\frac{27}{8} a^2 \text{Ci}(3 a x)+\frac{\cos ^3(a x)}{a^2 x^4}-\frac{\cos ^4(a x)}{a^2 x^4 (a x \sin (a x)+\cos (a x))}-\frac{\sin (a x) \cos ^2(a x)}{a x^3}-\frac{3 \cos ^3(a x)}{2 x^2}+\frac{\cos (a x)}{x^2}-\frac{a \sin (a x)}{x}+\frac{9 a \sin (a x) \cos ^2(a x)}{2 x}",1,"-1/16*(3 - a^2*x^2 + 4*Cos[2*a*x] - 8*a^2*x^2*Cos[2*a*x] + Cos[4*a*x] + 9*a^2*x^2*Cos[4*a*x] + 2*a^2*x^2*CosIntegral[a*x]*(Cos[a*x] + a*x*Sin[a*x]) + 54*a^2*x^2*CosIntegral[3*a*x]*(Cos[a*x] + a*x*Sin[a*x]) - 12*a*x*Sin[2*a*x] - 6*a*x*Sin[4*a*x])/(x^2*(Cos[a*x] + a*x*Sin[a*x]))","A",1
596,1,71,80,0.6772288,"\int \frac{\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx","Integrate[Cos[a*x]^4/(x^2*(Cos[a*x] + a*x*Sin[a*x])^2),x]","-\frac{8 a x \text{Si}(2 a x) (a x \sin (a x)+\cos (a x))-2 a x \sin (a x)+2 a x \sin (3 a x)+3 \cos (a x)+\cos (3 a x)}{4 x (a x \sin (a x)+\cos (a x))}","\frac{\cos ^2(a x)}{a^2 x^3}-\frac{\cos ^3(a x)}{a^2 x^3 (a x \sin (a x)+\cos (a x))}-2 a \text{Si}(2 a x)-\frac{\sin (a x) \cos (a x)}{a x^2}-\frac{2 \cos ^2(a x)}{x}+\frac{1}{x}",1,"-1/4*(3*Cos[a*x] + Cos[3*a*x] - 2*a*x*Sin[a*x] + 2*a*x*Sin[3*a*x] + 8*a*x*(Cos[a*x] + a*x*Sin[a*x])*SinIntegral[2*a*x])/(x*(Cos[a*x] + a*x*Sin[a*x]))","A",1
597,1,237,56,7.4026096,"\int \frac{\cos ^3(a x)}{x (\cos (a x)+a x \sin (a x))^2} \, dx","Integrate[Cos[a*x]^3/(x*(Cos[a*x] + a*x*Sin[a*x])^2),x]","\frac{-e a x \text{Ci}(a x+i) \sin (a x)-e \text{Ci}(a x+i) \cos (a x)+2 \text{Ci}(a x) (a x \sin (a x)+\cos (a x))-e \text{Ci}(i-a x) (a x \sin (a x)+\cos (a x))+e a x \text{Ei}(-i a x-1) \sin (a x)+e a x \text{Ei}(i a x-1) \sin (a x)+e \text{Ei}(-i a x-1) \cos (a x)+e \text{Ei}(i a x-1) \cos (a x)-i e a x \text{Si}(i-a x) \sin (a x)-i e a x \text{Si}(a x+i) \sin (a x)-i e \text{Si}(i-a x) \cos (a x)-i e \text{Si}(a x+i) \cos (a x)+\cos (2 a x)-1}{2 (a x \sin (a x)+\cos (a x))}","\frac{\cos (a x)}{a^2 x^2}-\frac{\cos ^2(a x)}{a^2 x^2 (a x \sin (a x)+\cos (a x))}+\text{Ci}(a x)-\frac{\sin (a x)}{a x}",1,"(-1 + Cos[2*a*x] - E*Cos[a*x]*CosIntegral[I + a*x] + E*Cos[a*x]*ExpIntegralEi[-1 - I*a*x] + E*Cos[a*x]*ExpIntegralEi[-1 + I*a*x] - a*E*x*CosIntegral[I + a*x]*Sin[a*x] + a*E*x*ExpIntegralEi[-1 - I*a*x]*Sin[a*x] + a*E*x*ExpIntegralEi[-1 + I*a*x]*Sin[a*x] + 2*CosIntegral[a*x]*(Cos[a*x] + a*x*Sin[a*x]) - E*CosIntegral[I - a*x]*(Cos[a*x] + a*x*Sin[a*x]) - I*E*Cos[a*x]*SinIntegral[I - a*x] - I*a*E*x*Sin[a*x]*SinIntegral[I - a*x] - I*E*Cos[a*x]*SinIntegral[I + a*x] - I*a*E*x*Sin[a*x]*SinIntegral[I + a*x])/(2*(Cos[a*x] + a*x*Sin[a*x]))","C",0
598,1,22,34,0.2141359,"\int \frac{\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx","Integrate[Cos[a*x]^2/(Cos[a*x] + a*x*Sin[a*x])^2,x]","\frac{\sin (a x)}{a (a x \sin (a x)+\cos (a x))}","\frac{1}{a^2 x}-\frac{\cos (a x)}{a^2 x (a x \sin (a x)+\cos (a x))}",1,"Sin[a*x]/(a*(Cos[a*x] + a*x*Sin[a*x]))","A",1
599,1,19,19,0.0186365,"\int \frac{x \cos (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx","Integrate[(x*Cos[a*x])/(Cos[a*x] + a*x*Sin[a*x])^2,x]","-\frac{1}{a^2 (a x \sin (a x)+\cos (a x))}","-\frac{1}{a^2 (a x \sin (a x)+\cos (a x))}",1,"-(1/(a^2*(Cos[a*x] + a*x*Sin[a*x])))","A",1
600,1,31,33,0.4056348,"\int \frac{x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx","Integrate[x^2/(Cos[a*x] + a*x*Sin[a*x])^2,x]","\frac{\sin (a x)-a x \cos (a x)}{a^3 (a x \sin (a x)+\cos (a x))}","\frac{\tan (a x)}{a^3}-\frac{x \sec (a x)}{a^2 (a x \sin (a x)+\cos (a x))}",1,"(-(a*x*Cos[a*x]) + Sin[a*x])/(a^3*(Cos[a*x] + a*x*Sin[a*x]))","A",1
601,1,176,110,1.1118478,"\int \frac{x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx","Integrate[(x^3*Sec[a*x])/(Cos[a*x] + a*x*Sin[a*x])^2,x]","-\frac{a^2 x^2 \sec (a x)-a^2 x^2 \log \left(1-i e^{i a x}\right) \tan (a x)+a^2 x^2 \log \left(1+i e^{i a x}\right) \tan (a x)-i \text{Li}_2\left(-i e^{i a x}\right) (a x \tan (a x)+1)+i \text{Li}_2\left(i e^{i a x}\right) (a x \tan (a x)+1)-a x \log \left(1-i e^{i a x}\right)+a x \log \left(1+i e^{i a x}\right)+\sec (a x)}{a^4 (a x \tan (a x)+1)}","\frac{i \text{Li}_2\left(-i e^{i a x}\right)}{a^4}-\frac{i \text{Li}_2\left(i e^{i a x}\right)}{a^4}-\frac{\sec (a x)}{a^4}-\frac{2 i x \tan ^{-1}\left(e^{i a x}\right)}{a^3}+\frac{x \tan (a x) \sec (a x)}{a^3}-\frac{x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}",1,"-((-(a*x*Log[1 - I*E^(I*a*x)]) + a*x*Log[1 + I*E^(I*a*x)] + Sec[a*x] + a^2*x^2*Sec[a*x] - a^2*x^2*Log[1 - I*E^(I*a*x)]*Tan[a*x] + a^2*x^2*Log[1 + I*E^(I*a*x)]*Tan[a*x] - I*PolyLog[2, (-I)*E^(I*a*x)]*(1 + a*x*Tan[a*x]) + I*PolyLog[2, I*E^(I*a*x)]*(1 + a*x*Tan[a*x]))/(a^4*(1 + a*x*Tan[a*x])))","A",1
602,1,130,124,1.0739138,"\int \frac{x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx","Integrate[(x^4*Sec[a*x]^2)/(Cos[a*x] + a*x*Sin[a*x])^2,x]","\frac{a^3 x^3 \tan ^2(a x)-a x \left(a^2 x^2+2 i a x-4 \log \left(1+e^{2 i a x}\right)+1\right)+\left(-2 i a^3 x^3+2 a^2 x^2+4 a^2 x^2 \log \left(1+e^{2 i a x}\right)+1\right) \tan (a x)-2 i \text{Li}_2\left(-e^{2 i a x}\right) (a x \tan (a x)+1)}{a^5 (a x \tan (a x)+1)}","-\frac{2 i \text{Li}_2\left(-e^{2 i a x}\right)}{a^5}+\frac{\tan (a x)}{a^5}+\frac{4 x \log \left(1+e^{2 i a x}\right)}{a^4}-\frac{x \sec ^2(a x)}{a^4}-\frac{2 i x^2}{a^3}+\frac{2 x^2 \tan (a x)}{a^3}+\frac{x^2 \tan (a x) \sec ^2(a x)}{a^3}-\frac{x^3 \sec ^3(a x)}{a^2 (a x \sin (a x)+\cos (a x))}",1,"(-(a*x*(1 + (2*I)*a*x + a^2*x^2 - 4*Log[1 + E^((2*I)*a*x)])) + (1 + 2*a^2*x^2 - (2*I)*a^3*x^3 + 4*a^2*x^2*Log[1 + E^((2*I)*a*x)])*Tan[a*x] + a^3*x^3*Tan[a*x]^2 - (2*I)*PolyLog[2, -E^((2*I)*a*x)]*(1 + a*x*Tan[a*x]))/(a^5*(1 + a*x*Tan[a*x]))","A",1
603,1,64,157,0.2242769,"\int \sec ^4(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Integrate[Sec[2*(a + b*x)]^4*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","-\frac{(7 \cos (3 (a+b x))+2 \cos (7 (a+b x))) \csc (a+b x) \sec ^3(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}{35 b}","\frac{c \tan (2 a+2 b x) \sec ^3(2 a+2 b x)}{7 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{6 \tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{35 b c}-\frac{4 \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{35 b}-\frac{2 c \tan (2 a+2 b x)}{5 b \sqrt{c \sec (2 a+2 b x)-c}}",1,"-1/35*((7*Cos[3*(a + b*x)] + 2*Cos[7*(a + b*x)])*Csc[a + b*x]*Sec[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/b","A",1
604,1,62,110,0.1777766,"\int \sec ^3(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Integrate[Sec[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{(5 \cos (a+b x)+2 \cos (5 (a+b x))) \csc (a+b x) \sec ^2(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}{15 b}","\frac{\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b c}+\frac{2 \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{15 b}+\frac{7 c \tan (2 a+2 b x)}{15 b \sqrt{c \sec (2 a+2 b x)-c}}",1,"((5*Cos[a + b*x] + 2*Cos[5*(a + b*x)])*Csc[a + b*x]*Sec[2*(a + b*x)]^2*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(15*b)","A",1
605,1,44,72,0.1764799,"\int \sec ^2(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Integrate[Sec[2*(a + b*x)]^2*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\sqrt{c \tan (a+b x) \tan (2 (a+b x))} (\tan (2 (a+b x))-\cot (a+b x))}{3 b}","\frac{\tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{3 b}-\frac{c \tan (2 a+2 b x)}{3 b \sqrt{c \sec (2 a+2 b x)-c}}",1,"(Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]]*(-Cot[a + b*x] + Tan[2*(a + b*x)]))/(3*b)","A",1
606,1,30,33,0.0817831,"\int \sec (2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Integrate[Sec[2*(a + b*x)]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\cot (a+b x) \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}{b}","\frac{c \tan (2 a+2 b x)}{b \sqrt{c \sec (2 a+2 b x)-c}}",1,"(Cot[a + b*x]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/b","A",1
607,1,73,45,0.128462,"\int \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Integrate[Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","-\frac{\sqrt{\cos (2 (a+b x))} \csc (a+b x) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \tanh ^{-1}\left(\frac{\sqrt{2} \cos (a+b x)}{\sqrt{\cos (2 (a+b x))}}\right)}{\sqrt{2} b}","-\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{b}",1,"-((ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2*(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]]*Csc[a + b*x]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(Sqrt[2]*b))","A",1
608,1,92,84,0.2403025,"\int \cos (2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Integrate[Cos[2*(a + b*x)]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","-\frac{\csc (a+b x) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(\cos (a+b x)+\cos (3 (a+b x))-\sqrt{2} \sqrt{\cos (2 (a+b x))} \tanh ^{-1}\left(\frac{\sqrt{2} \cos (a+b x)}{\sqrt{\cos (2 (a+b x))}}\right)\right)}{4 b}","\frac{\sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{2 b}-\frac{c \sin (2 a+2 b x)}{2 b \sqrt{c \sec (2 a+2 b x)-c}}",1,"-1/4*((Cos[a + b*x] - Sqrt[2]*ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2*(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]] + Cos[3*(a + b*x)])*Csc[a + b*x]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/b","A",1
609,1,105,129,0.2493352,"\int \cos ^2(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Integrate[Cos[2*(a + b*x)]^2*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(2 (-\sin (2 (a+b x))+\sin (4 (a+b x))+\cot (a+b x))-3 \sqrt{2} \sqrt{\cos (2 (a+b x))} \csc (a+b x) \tanh ^{-1}\left(\frac{\sqrt{2} \cos (a+b x)}{\sqrt{\cos (2 (a+b x))}}\right)\right)}{16 b}","\frac{3 c \sin (2 a+2 b x)}{8 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{c \sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{3 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{8 b}",1,"((-3*Sqrt[2]*ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2*(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]]*Csc[a + b*x] + 2*(Cot[a + b*x] - Sin[2*(a + b*x)] + Sin[4*(a + b*x)]))*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(16*b)","A",1
610,1,116,176,0.2936945,"\int \cos ^3(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Integrate[Cos[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(30 \sin (2 (a+b x))-2 \sin (4 (a+b x))+4 \sin (6 (a+b x))-26 \cot (a+b x)+15 \sqrt{2} \sqrt{\cos (2 (a+b x))} \csc (a+b x) \tanh ^{-1}\left(\frac{\sqrt{2} \cos (a+b x)}{\sqrt{\cos (2 (a+b x))}}\right)\right)}{96 b}","-\frac{5 c \sin (2 a+2 b x)}{16 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{c \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{5 c \sin (2 a+2 b x) \cos (2 a+2 b x)}{24 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{5 \sqrt{c} \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{16 b}",1,"((-26*Cot[a + b*x] + 15*Sqrt[2]*ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2*(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]]*Csc[a + b*x] + 30*Sin[2*(a + b*x)] - 2*Sin[4*(a + b*x)] + 4*Sin[6*(a + b*x)])*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(96*b)","A",1
611,1,85,208,0.3474944,"\int \sec ^4(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Integrate[Sec[2*(a + b*x)]^4*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{\cot (a+b x) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \left(188 \cot (a+b x) \cot (2 (a+b x))+35 \sec ^3(2 (a+b x))-50 \sec ^2(2 (a+b x))+52 \sec (2 (a+b x))-84\right)}{315 b}","\frac{c^2 \tan (2 a+2 b x) \sec ^4(2 a+2 b x)}{9 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{17 c^2 \tan (2 a+2 b x) \sec ^3(2 a+2 b x)}{63 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{34 c^2 \tan (2 a+2 b x)}{45 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{34 \tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{105 b}+\frac{68 c \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{315 b}",1,"(Cot[a + b*x]*(-84 + 188*Cot[a + b*x]*Cot[2*(a + b*x)] + 52*Sec[2*(a + b*x)] - 50*Sec[2*(a + b*x)]^2 + 35*Sec[2*(a + b*x)]^3)*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))/(315*b)","A",1
612,1,73,148,0.2141286,"\int \sec ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Integrate[Sec[2*(a + b*x)]^3*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\frac{\cot (a+b x) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \left(76 \cot (a+b x) \cot (2 (a+b x))-15 \sec ^2(2 (a+b x))+24 \sec (2 (a+b x))-28\right)}{105 b}","-\frac{76 c^2 \tan (2 a+2 b x)}{105 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{5/2}}{7 b c}+\frac{2 \tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{35 b}+\frac{19 c \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{105 b}",1,"-1/105*(Cot[a + b*x]*(-28 + 76*Cot[a + b*x]*Cot[2*(a + b*x)] + 24*Sec[2*(a + b*x)] - 15*Sec[2*(a + b*x)]^2)*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))/b","A",1
613,1,59,110,0.225715,"\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Integrate[Sec[2*(a + b*x)]^2*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{\cot (a+b x) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} (4 \cot (a+b x) \cot (2 (a+b x))+\sec (2 (a+b x))-2)}{5 b}","\frac{4 c^2 \tan (2 a+2 b x)}{5 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{c \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{5 b}+\frac{\tan (2 a+2 b x) (c \sec (2 a+2 b x)-c)^{3/2}}{5 b}",1,"(Cot[a + b*x]*(-2 + 4*Cot[a + b*x]*Cot[2*(a + b*x)] + Sec[2*(a + b*x)])*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))/(5*b)","A",1
614,1,51,75,0.158729,"\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Integrate[Sec[2*(a + b*x)]*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\frac{\cot (a+b x) (4 \cot (a+b x) \cot (2 (a+b x))-1) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}{3 b}","\frac{c \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{3 b}-\frac{4 c^2 \tan (2 a+2 b x)}{3 b \sqrt{c \sec (2 a+2 b x)-c}}",1,"-1/3*(Cot[a + b*x]*(-1 + 4*Cot[a + b*x]*Cot[2*(a + b*x)])*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))/b","A",1
615,1,86,80,0.1534736,"\int (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Integrate[(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{c \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(2 \cot (a+b x)+\sqrt{2} \sqrt{\cos (2 (a+b x))} \csc (a+b x) \tanh ^{-1}\left(\frac{\sqrt{2} \cos (a+b x)}{\sqrt{\cos (2 (a+b x))}}\right)\right)}{2 b}","\frac{c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{b}+\frac{c^2 \tan (2 a+2 b x)}{b \sqrt{c \sec (2 a+2 b x)-c}}",1,"(c*(2*Cot[a + b*x] + Sqrt[2]*ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2*(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]]*Csc[a + b*x])*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(2*b)","A",1
616,1,93,86,0.2443815,"\int \cos (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Integrate[Cos[2*(a + b*x)]*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{c \csc (a+b x) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(\cos (a+b x)+\cos (3 (a+b x))-3 \sqrt{2} \sqrt{\cos (2 (a+b x))} \tanh ^{-1}\left(\frac{\sqrt{2} \cos (a+b x)}{\sqrt{\cos (2 (a+b x))}}\right)\right)}{4 b}","\frac{c^2 \sin (2 a+2 b x)}{2 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{3 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{2 b}",1,"(c*(Cos[a + b*x] - 3*Sqrt[2]*ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2*(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]] + Cos[3*(a + b*x)])*Csc[a + b*x]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(4*b)","A",1
617,1,105,133,0.2713102,"\int \cos ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Integrate[Cos[2*(a + b*x)]^2*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{c \csc (a+b x) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(-5 \cos (a+b x)-6 \cos (3 (a+b x))+\cos (5 (a+b x))+7 \sqrt{2} \sqrt{\cos (2 (a+b x))} \tanh ^{-1}\left(\frac{\sqrt{2} \cos (a+b x)}{\sqrt{\cos (2 (a+b x))}}\right)\right)}{16 b}","\frac{7 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{8 b}-\frac{7 c^2 \sin (2 a+2 b x)}{8 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{c^2 \sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b \sqrt{c \sec (2 a+2 b x)-c}}",1,"(c*(-5*Cos[a + b*x] + 7*Sqrt[2]*ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2*(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]] - 6*Cos[3*(a + b*x)] + Cos[5*(a + b*x)])*Csc[a + b*x]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(16*b)","A",1
618,1,117,182,0.229881,"\int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Integrate[Cos[2*(a + b*x)]^3*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{c \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(-42 \sin (2 (a+b x))+14 \sin (4 (a+b x))-4 \sin (6 (a+b x))+38 \cot (a+b x)-33 \sqrt{2} \sqrt{\cos (2 (a+b x))} \csc (a+b x) \tanh ^{-1}\left(\frac{\sqrt{2} \cos (a+b x)}{\sqrt{\cos (2 (a+b x))}}\right)\right)}{96 b}","-\frac{11 c^{3/2} \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{16 b}+\frac{11 c^2 \sin (2 a+2 b x)}{16 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{c^2 \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{11 c^2 \sin (2 a+2 b x) \cos (2 a+2 b x)}{24 b \sqrt{c \sec (2 a+2 b x)-c}}",1,"(c*(38*Cot[a + b*x] - 33*Sqrt[2]*ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2*(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]]*Csc[a + b*x] - 42*Sin[2*(a + b*x)] + 14*Sin[4*(a + b*x)] - 4*Sin[6*(a + b*x)])*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(96*b)","A",1
619,1,112,175,0.6727454,"\int \frac{\sec ^4(2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Integrate[Sec[2*(a + b*x)]^4/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\sin (a+b x) \cos (a+b x) \sec ^3(2 (a+b x)) \left(4 \cos (2 (a+b x))+26 \cos (4 (a+b x))+30 \cos ^2(2 (a+b x)) \tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \sqrt{\tan ^2(a+b x)-1}+38\right)}{30 b \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}","\frac{\tan (2 a+2 b x) \sec ^2(2 a+2 b x)}{5 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{\tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{15 b c}+\frac{14 \tan (2 a+2 b x)}{15 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{\sqrt{2} b \sqrt{c}}",1,"(Cos[a + b*x]*Sec[2*(a + b*x)]^3*Sin[a + b*x]*(38 + 4*Cos[2*(a + b*x)] + 26*Cos[4*(a + b*x)] + 30*ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Cos[2*(a + b*x)]^2*Sqrt[-1 + Tan[a + b*x]^2]))/(30*b*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])","A",1
620,1,89,129,0.3910559,"\int \frac{\sec ^3(2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Integrate[Sec[2*(a + b*x)]^3/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\cos ^2(a+b x) \csc (2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(3 \sqrt{\tan ^2(a+b x)-1} \tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right)+2 \sec (2 (a+b x))+2\right)}{3 b c}","\frac{\tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{3 b c}+\frac{2 \tan (2 a+2 b x)}{3 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{\sqrt{2} b \sqrt{c}}",1,"(Cos[a + b*x]^2*Csc[2*(a + b*x)]*(2 + 2*Sec[2*(a + b*x)] + 3*ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sqrt[-1 + Tan[a + b*x]^2])*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(3*b*c)","A",1
621,1,67,88,0.2427215,"\int \frac{\sec ^2(2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Integrate[Sec[2*(a + b*x)]^2/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\left(\sqrt{\tan ^2(a+b x)-1} \tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right)+2\right) \tan (2 (a+b x))}{2 b \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}","\frac{\tan (2 a+2 b x)}{b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{\sqrt{2} b \sqrt{c}}",1,"((2 + ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sqrt[-1 + Tan[a + b*x]^2])*Tan[2*(a + b*x)])/(2*b*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])","A",1
622,1,64,55,0.1485627,"\int \frac{\sec (2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Integrate[Sec[2*(a + b*x)]/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \sqrt{\tan ^2(a+b x)-1} \tan (2 (a+b x))}{2 b \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{\sqrt{2} b \sqrt{c}}",1,"(ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sqrt[-1 + Tan[a + b*x]^2]*Tan[2*(a + b*x)])/(2*b*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])","A",1
623,1,94,100,0.3129081,"\int \frac{1}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Integrate[1/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\tan (a+b x) \left(2 \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-2 \tan ^2(a+b x)}\right)-\sqrt{2} \tanh ^{-1}\left(\sqrt{1-\tan ^2(a+b x)}\right)\right)}{b \sqrt{2-2 \tan ^2(a+b x)} \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{b \sqrt{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{\sqrt{2} b \sqrt{c}}",1,"((2*ArcTanh[Sqrt[2 - 2*Tan[a + b*x]^2]/2] - Sqrt[2]*ArcTanh[Sqrt[1 - Tan[a + b*x]^2]])*Tan[a + b*x])/(b*Sqrt[2 - 2*Tan[a + b*x]^2]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])","A",1
624,1,166,138,2.3211805,"\int \frac{\cos (2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Integrate[Cos[2*(a + b*x)]/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\tan (a+b x) \left(\sqrt{2} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-2 \tan ^2(a+b x)}\right)-\tanh ^{-1}\left(\sqrt{1-\tan ^2(a+b x)}\right)+\sqrt{2} \cos ^2(a+b x) \sqrt{\frac{1}{\sec (2 (a+b x))+1}} \left(\tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \sqrt{\tan ^2(a+b x)-1} \sec (2 (a+b x))+2\right)\right)}{2 b \sqrt{1-\tan ^2(a+b x)} \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}","\frac{\sin (2 a+2 b x)}{2 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{2 b \sqrt{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{\sqrt{2} b \sqrt{c}}",1,"(Tan[a + b*x]*(Sqrt[2]*ArcTanh[Sqrt[2 - 2*Tan[a + b*x]^2]/2] - ArcTanh[Sqrt[1 - Tan[a + b*x]^2]] + Sqrt[2]*Cos[a + b*x]^2*Sqrt[(1 + Sec[2*(a + b*x)])^(-1)]*(2 + ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sec[2*(a + b*x)]*Sqrt[-1 + Tan[a + b*x]^2])))/(2*b*Sqrt[1 - Tan[a + b*x]^2]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])","A",1
625,1,186,182,2.9018541,"\int \frac{\cos ^2(2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Integrate[Cos[2*(a + b*x)]^2/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{\tan (a+b x) \left(7 \sqrt{2} \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-2 \tan ^2(a+b x)}\right)-7 \tanh ^{-1}\left(\sqrt{1-\tan ^2(a+b x)}\right)+\sqrt{2} \cos ^2(a+b x) \sec (2 (a+b x)) \sqrt{\frac{1}{\sec (2 (a+b x))+1}} \left(2 (\cos (2 (a+b x))+\cos (4 (a+b x))+1)+\sqrt{\tan ^2(a+b x)-1} \tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right)\right)\right)}{8 b \sqrt{1-\tan ^2(a+b x)} \sqrt{c \tan (a+b x) \tan (2 (a+b x))}}","\frac{\sin (2 a+2 b x)}{8 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{\sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{7 \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{8 b \sqrt{c}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{\sqrt{2} b \sqrt{c}}",1,"(Tan[a + b*x]*(7*Sqrt[2]*ArcTanh[Sqrt[2 - 2*Tan[a + b*x]^2]/2] - 7*ArcTanh[Sqrt[1 - Tan[a + b*x]^2]] + Sqrt[2]*Cos[a + b*x]^2*Sec[2*(a + b*x)]*Sqrt[(1 + Sec[2*(a + b*x)])^(-1)]*(2*(1 + Cos[2*(a + b*x)] + Cos[4*(a + b*x)]) + ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sqrt[-1 + Tan[a + b*x]^2])))/(8*b*Sqrt[1 - Tan[a + b*x]^2]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])","A",1
626,1,100,180,1.304336,"\int \frac{\sec ^4(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Integrate[Sec[2*(a + b*x)]^4/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\frac{\cot (a+b x) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(\csc ^2(a+b x) ((19 \cos (4 (a+b x))+11) \sec (2 (a+b x))-24)-66 \tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \sqrt{\tan ^2(a+b x)-1}\right)}{48 b c^2}","-\frac{11 \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{4 \sqrt{2} b c^{3/2}}+\frac{7 \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{12 b c^2}-\frac{\tan (2 a+2 b x) \sec ^2(2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}+\frac{13 \tan (2 a+2 b x)}{6 b c \sqrt{c \sec (2 a+2 b x)-c}}",1,"-1/48*(Cot[a + b*x]*(Csc[a + b*x]^2*(-24 + (11 + 19*Cos[4*(a + b*x)])*Sec[2*(a + b*x)]) - 66*ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sqrt[-1 + Tan[a + b*x]^2])*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(b*c^2)","A",1
627,1,94,128,0.6030914,"\int \frac{\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Integrate[Sec[2*(a + b*x)]^3/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{\tan (2 (a+b x)) \left(4 \sec (2 (a+b x))+7 \sin ^2(a+b x) \tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \sqrt{\tan ^2(a+b x)-1} \sec (2 (a+b x))-5\right)}{4 b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}","-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{4 \sqrt{2} b c^{3/2}}+\frac{\tan (2 a+2 b x)}{b c \sqrt{c \sec (2 a+2 b x)-c}}-\frac{\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}",1,"((-5 + 4*Sec[2*(a + b*x)] + 7*ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sec[2*(a + b*x)]*Sin[a + b*x]^2*Sqrt[-1 + Tan[a + b*x]^2])*Tan[2*(a + b*x)])/(4*b*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))","A",1
628,1,84,93,0.5884618,"\int \frac{\sec ^2(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Integrate[Sec[2*(a + b*x)]^2/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{\tan (2 (a+b x)) \left(3 \sin ^2(a+b x) \tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \sqrt{\tan ^2(a+b x)-1} \sec (2 (a+b x))-1\right)}{4 b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{4 \sqrt{2} b c^{3/2}}-\frac{\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}",1,"((-1 + 3*ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sec[2*(a + b*x)]*Sin[a + b*x]^2*Sqrt[-1 + Tan[a + b*x]^2])*Tan[2*(a + b*x)])/(4*b*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))","A",1
629,1,83,93,0.6233801,"\int \frac{\sec (2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Integrate[Sec[2*(a + b*x)]/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\frac{\tan (2 (a+b x)) \left(\sin ^2(a+b x) \tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \sqrt{\tan ^2(a+b x)-1} \sec (2 (a+b x))+1\right)}{4 b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{4 \sqrt{2} b c^{3/2}}-\frac{\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}",1,"-1/4*((1 + ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sec[2*(a + b*x)]*Sin[a + b*x]^2*Sqrt[-1 + Tan[a + b*x]^2])*Tan[2*(a + b*x)])/(b*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))","A",1
630,1,196,138,3.700386,"\int \frac{1}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Integrate[(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(-3/2),x]","-\frac{\cot (a+b x) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left(\tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \sqrt{-\left(\tan ^2(a+b x)-1\right)^2}+\cot ^2(a+b x) \left(\cos (2 (a+b x)) \sec ^2(a+b x)\right)^{3/2}+4 \sqrt{2} \cos (2 (a+b x)) \sec ^2(a+b x) \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-2 \tan ^2(a+b x)}\right)-4 \cos (2 (a+b x)) \sec ^2(a+b x) \tanh ^{-1}\left(\sqrt{1-\tan ^2(a+b x)}\right)\right)}{8 b c^2 \sqrt{1-\tan ^2(a+b x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{b c^{3/2}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{4 \sqrt{2} b c^{3/2}}-\frac{\tan (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}",1,"-1/8*(Cot[a + b*x]*(4*Sqrt[2]*ArcTanh[Sqrt[2 - 2*Tan[a + b*x]^2]/2]*Cos[2*(a + b*x)]*Sec[a + b*x]^2 - 4*ArcTanh[Sqrt[1 - Tan[a + b*x]^2]]*Cos[2*(a + b*x)]*Sec[a + b*x]^2 + Cot[a + b*x]^2*(Cos[2*(a + b*x)]*Sec[a + b*x]^2)^(3/2) + ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Sqrt[-(-1 + Tan[a + b*x]^2)^2])*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]])/(b*c^2*Sqrt[1 - Tan[a + b*x]^2])","A",0
631,1,342,178,6.2020506,"\int \frac{\cos (2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Integrate[Cos[2*(a + b*x)]/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{\tan ^2(a+b x) \tan ^2(2 (a+b x)) \left(\frac{1}{2} \sin (2 (a+b x))-\frac{1}{4} \cot (a+b x)-\frac{1}{8} \cot (a+b x) \csc ^2(a+b x)\right)}{b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}-\frac{3 \tan ^{\frac{3}{2}}(a+b x) \tan ^{\frac{3}{2}}(2 (a+b x)) \left(\frac{\tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \tan ^{\frac{3}{2}}(a+b x) \sqrt{\tan ^2(a+b x)-1} \sqrt{\tan (2 (a+b x))} \csc ^2(a+b x) \sec ^2(a+b x)}{\left(\tan ^2(a+b x)+1\right)^2}+\frac{\sqrt{2} \cos (2 (a+b x)) \tan ^{\frac{3}{2}}(a+b x) \sqrt{\tan (2 (a+b x))} \csc ^2(a+b x) \sec ^2(a+b x) \left(2 \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-2 \tan ^2(a+b x)}\right)-\sqrt{2} \tanh ^{-1}\left(\sqrt{1-\tan ^2(a+b x)}\right)\right)}{\sqrt{1-\tan ^2(a+b x)} \left(\tan ^2(a+b x)+1\right)}\right)}{8 b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}","-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{2 b c^{3/2}}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{4 \sqrt{2} b c^{3/2}}-\frac{3 \sin (2 a+2 b x)}{4 b c \sqrt{c \sec (2 a+2 b x)-c}}-\frac{\sin (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}",1,"(-3*Tan[a + b*x]^(3/2)*((ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Csc[a + b*x]^2*Sec[a + b*x]^2*Tan[a + b*x]^(3/2)*Sqrt[-1 + Tan[a + b*x]^2]*Sqrt[Tan[2*(a + b*x)]])/(1 + Tan[a + b*x]^2)^2 + (Sqrt[2]*(2*ArcTanh[Sqrt[2 - 2*Tan[a + b*x]^2]/2] - Sqrt[2]*ArcTanh[Sqrt[1 - Tan[a + b*x]^2]])*Cos[2*(a + b*x)]*Csc[a + b*x]^2*Sec[a + b*x]^2*Tan[a + b*x]^(3/2)*Sqrt[Tan[2*(a + b*x)]])/(Sqrt[1 - Tan[a + b*x]^2]*(1 + Tan[a + b*x]^2)))*Tan[2*(a + b*x)]^(3/2))/(8*b*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2)) + ((-1/4*Cot[a + b*x] - (Cot[a + b*x]*Csc[a + b*x]^2)/8 + Sin[2*(a + b*x)]/2)*Tan[a + b*x]^2*Tan[2*(a + b*x)]^2)/(b*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))","A",1
632,1,356,234,6.2067308,"\int \frac{\cos ^2(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Integrate[Cos[2*(a + b*x)]^2/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{\tan ^2(a+b x) \tan ^2(2 (a+b x)) \left(\frac{7}{8} \sin (2 (a+b x))+\frac{1}{8} \sin (4 (a+b x))-\frac{5}{8} \cot (a+b x)-\frac{1}{8} \cot (a+b x) \csc ^2(a+b x)\right)}{b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}+\frac{\tan ^{\frac{3}{2}}(a+b x) \tan ^{\frac{3}{2}}(2 (a+b x)) \left(-\frac{7 \tan ^{-1}\left(\sqrt{\tan ^2(a+b x)-1}\right) \tan ^{\frac{3}{2}}(a+b x) \sqrt{\tan ^2(a+b x)-1} \sqrt{\tan (2 (a+b x))} \csc ^2(a+b x) \sec ^2(a+b x)}{\left(\tan ^2(a+b x)+1\right)^2}-\frac{19 \cos (2 (a+b x)) \tan ^{\frac{3}{2}}(a+b x) \sqrt{\tan (2 (a+b x))} \csc ^2(a+b x) \sec ^2(a+b x) \left(2 \tanh ^{-1}\left(\frac{1}{2} \sqrt{2-2 \tan ^2(a+b x)}\right)-\sqrt{2} \tanh ^{-1}\left(\sqrt{1-\tan ^2(a+b x)}\right)\right)}{\sqrt{2} \sqrt{1-\tan ^2(a+b x)} \left(\tan ^2(a+b x)+1\right)}\right)}{16 b (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}","-\frac{19 \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right)}{8 b c^{3/2}}+\frac{13 \tanh ^{-1}\left(\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{2} \sqrt{c \sec (2 a+2 b x)-c}}\right)}{4 \sqrt{2} b c^{3/2}}-\frac{7 \sin (2 a+2 b x)}{8 b c \sqrt{c \sec (2 a+2 b x)-c}}-\frac{\sin (2 a+2 b x) \cos (2 a+2 b x)}{2 b c \sqrt{c \sec (2 a+2 b x)-c}}-\frac{\sin (2 a+2 b x) \cos (2 a+2 b x)}{4 b (c \sec (2 a+2 b x)-c)^{3/2}}",1,"(Tan[a + b*x]^(3/2)*((-7*ArcTan[Sqrt[-1 + Tan[a + b*x]^2]]*Csc[a + b*x]^2*Sec[a + b*x]^2*Tan[a + b*x]^(3/2)*Sqrt[-1 + Tan[a + b*x]^2]*Sqrt[Tan[2*(a + b*x)]])/(1 + Tan[a + b*x]^2)^2 - (19*(2*ArcTanh[Sqrt[2 - 2*Tan[a + b*x]^2]/2] - Sqrt[2]*ArcTanh[Sqrt[1 - Tan[a + b*x]^2]])*Cos[2*(a + b*x)]*Csc[a + b*x]^2*Sec[a + b*x]^2*Tan[a + b*x]^(3/2)*Sqrt[Tan[2*(a + b*x)]])/(Sqrt[2]*Sqrt[1 - Tan[a + b*x]^2]*(1 + Tan[a + b*x]^2)))*Tan[2*(a + b*x)]^(3/2))/(16*b*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2)) + (((-5*Cot[a + b*x])/8 - (Cot[a + b*x]*Csc[a + b*x]^2)/8 + (7*Sin[2*(a + b*x)])/8 + Sin[4*(a + b*x)]/8)*Tan[a + b*x]^2*Tan[2*(a + b*x)]^2)/(b*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))","A",1
633,1,16,16,0.0313641,"\int \frac{\cot (x) \csc (x)}{\sqrt{\sin (2 x)}} \, dx","Integrate[(Cot[x]*Csc[x])/Sqrt[Sin[2*x]],x]","-\frac{1}{3} \sqrt{\sin (2 x)} \cot (x) \csc (x)","-\frac{2 \cos (x) \cot (x)}{3 \sqrt{\sin (2 x)}}",1,"-1/3*(Cot[x]*Csc[x]*Sqrt[Sin[2*x]])","A",1
634,1,119,69,5.9486144,"\int \frac{\csc ^2(x) \sec (x)}{\sqrt{\sin (2 x)} (-2+\tan (x))} \, dx","Integrate[(Csc[x]^2*Sec[x])/(Sqrt[Sin[2*x]]*(-2 + Tan[x])),x]","\frac{1}{4} \sqrt{\sin (2 x)} \left(\left(\frac{2 \cot (x)}{3}+1\right) \csc (x)+5 \sqrt{\frac{\cos (x)}{2 \cos (x)-2}} \sqrt{\tan \left(\frac{x}{2}\right)} \sec (x) \left(F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{2}\right)}}\right)\right|-1\right)-\Pi \left(-\frac{2}{-1+\sqrt{5}};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{2}\right)}}\right)\right|-1\right)-\Pi \left(\frac{1}{2} \left(-1+\sqrt{5}\right);\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{2}\right)}}\right)\right|-1\right)\right)\right)","\frac{\cos (x)}{2 \sqrt{\sin (2 x)}}-\frac{5 \sin (x) \tanh ^{-1}\left(\frac{\sqrt{\tan (x)}}{\sqrt{2}}\right)}{2 \sqrt{2} \sqrt{\sin (2 x)} \sqrt{\tan (x)}}+\frac{\cos (x) \cot (x)}{3 \sqrt{\sin (2 x)}}",1,"(Sqrt[Sin[2*x]]*((1 + (2*Cot[x])/3)*Csc[x] + 5*Sqrt[Cos[x]/(-2 + 2*Cos[x])]*(EllipticF[ArcSin[1/Sqrt[Tan[x/2]]], -1] - EllipticPi[-2/(-1 + Sqrt[5]), ArcSin[1/Sqrt[Tan[x/2]]], -1] - EllipticPi[(-1 + Sqrt[5])/2, ArcSin[1/Sqrt[Tan[x/2]]], -1])*Sec[x]*Sqrt[Tan[x/2]]))/4","C",1
635,1,183,79,4.955293,"\int \frac{\cos ^2(x) \sin (x)}{\left(\sin ^2(x)-\sin (2 x)\right) \sin ^{\frac{5}{2}}(2 x)} \, dx","Integrate[(Cos[x]^2*Sin[x])/((Sin[x]^2 - Sin[2*x])*Sin[2*x]^(5/2)),x]","\frac{1}{96} \sqrt{\sin (2 x)} \sec (x) \left(2 \cot ^2(x)+6 \cot (x)+2 \csc ^2(x)+15 \sqrt{2} \sqrt{\frac{\cos (x)}{\cos (x)-1}} \sqrt{\tan \left(\frac{x}{2}\right)} F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{2}\right)}}\right)\right|-1\right)-15 \sqrt{2} \sqrt{\frac{\cos (x)}{\cos (x)-1}} \sqrt{\tan \left(\frac{x}{2}\right)} \Pi \left(-\frac{2}{-1+\sqrt{5}};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{2}\right)}}\right)\right|-1\right)-15 \sqrt{2} \sqrt{\frac{\cos (x)}{\cos (x)-1}} \sqrt{\tan \left(\frac{x}{2}\right)} \Pi \left(\frac{1}{2} \left(-1+\sqrt{5}\right);\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{2}\right)}}\right)\right|-1\right)-2\right)","\frac{\sin (x) \cos ^4(x)}{3 \sin ^{\frac{5}{2}}(2 x)}+\frac{\sin ^2(x) \cos ^3(x)}{2 \sin ^{\frac{5}{2}}(2 x)}-\frac{5 \sin ^5(x) \tanh ^{-1}\left(\frac{\sqrt{\tan (x)}}{\sqrt{2}}\right)}{2 \sqrt{2} \sin ^{\frac{5}{2}}(2 x) \tan ^{\frac{5}{2}}(x)}",1,"(Sec[x]*Sqrt[Sin[2*x]]*(-2 + 6*Cot[x] + 2*Cot[x]^2 + 2*Csc[x]^2 + 15*Sqrt[2]*Sqrt[Cos[x]/(-1 + Cos[x])]*EllipticF[ArcSin[1/Sqrt[Tan[x/2]]], -1]*Sqrt[Tan[x/2]] - 15*Sqrt[2]*Sqrt[Cos[x]/(-1 + Cos[x])]*EllipticPi[-2/(-1 + Sqrt[5]), ArcSin[1/Sqrt[Tan[x/2]]], -1]*Sqrt[Tan[x/2]] - 15*Sqrt[2]*Sqrt[Cos[x]/(-1 + Cos[x])]*EllipticPi[(-1 + Sqrt[5])/2, ArcSin[1/Sqrt[Tan[x/2]]], -1]*Sqrt[Tan[x/2]]))/96","C",1
636,1,184,95,15.721013,"\int \frac{\cos ^3(x) \cos (2 x)}{\left(\sin ^2(x)-\sin (2 x)\right) \sin ^{\frac{5}{2}}(2 x)} \, dx","Integrate[(Cos[x]^3*Cos[2*x])/((Sin[x]^2 - Sin[2*x])*Sin[2*x]^(5/2)),x]","\frac{1}{960} \sqrt{\sin (2 x)} \sec (x) \left(20 \cot ^2(x)-114 \cot (x)+24 \cot (x) \csc ^2(x)-45 \sqrt{2} \sqrt{\frac{\cos (x)}{\cos (x)-1}} \sqrt{\tan \left(\frac{x}{2}\right)} F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{2}\right)}}\right)\right|-1\right)+45 \sqrt{2} \sqrt{\frac{\cos (x)}{\cos (x)-1}} \sqrt{\tan \left(\frac{x}{2}\right)} \Pi \left(-\frac{2}{-1+\sqrt{5}};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{2}\right)}}\right)\right|-1\right)+45 \sqrt{2} \sqrt{\frac{\cos (x)}{\cos (x)-1}} \sqrt{\tan \left(\frac{x}{2}\right)} \Pi \left(\frac{1}{2} \left(-1+\sqrt{5}\right);\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{x}{2}\right)}}\right)\right|-1\right)\right)","\frac{\cos ^5(x)}{5 \sin ^{\frac{5}{2}}(2 x)}+\frac{\sin (x) \cos ^4(x)}{6 \sin ^{\frac{5}{2}}(2 x)}-\frac{3 \sin ^2(x) \cos ^3(x)}{4 \sin ^{\frac{5}{2}}(2 x)}+\frac{3 \sin ^5(x) \tanh ^{-1}\left(\frac{\sqrt{\tan (x)}}{\sqrt{2}}\right)}{4 \sqrt{2} \sin ^{\frac{5}{2}}(2 x) \tan ^{\frac{5}{2}}(x)}",1,"(Sec[x]*Sqrt[Sin[2*x]]*(-114*Cot[x] + 20*Cot[x]^2 + 24*Cot[x]*Csc[x]^2 - 45*Sqrt[2]*Sqrt[Cos[x]/(-1 + Cos[x])]*EllipticF[ArcSin[1/Sqrt[Tan[x/2]]], -1]*Sqrt[Tan[x/2]] + 45*Sqrt[2]*Sqrt[Cos[x]/(-1 + Cos[x])]*EllipticPi[-2/(-1 + Sqrt[5]), ArcSin[1/Sqrt[Tan[x/2]]], -1]*Sqrt[Tan[x/2]] + 45*Sqrt[2]*Sqrt[Cos[x]/(-1 + Cos[x])]*EllipticPi[(-1 + Sqrt[5])/2, ArcSin[1/Sqrt[Tan[x/2]]], -1]*Sqrt[Tan[x/2]]))/960","C",1
637,1,51,30,1.2180979,"\int (b \sec (c+d x)+a \sin (c+d x))^n (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx","Integrate[(b*Sec[c + d*x] + a*Sin[c + d*x])^n*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]),x]","\frac{\sec (c+d x) (a \sin (2 (c+d x))+2 b) (a \sin (c+d x)+b \sec (c+d x))^n}{2 d (n+1)}","\frac{(a \sin (c+d x)+b \sec (c+d x))^{n+1}}{d (n+1)}",1,"(Sec[c + d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^n*(2*b + a*Sin[2*(c + d*x)]))/(2*d*(1 + n))","A",1
638,1,938,26,6.5379134,"\int (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx","Integrate[(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]),x]","\frac{a^4 \cos (4 c) \cos (4 d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \cos ^5(c+d x)}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}-\frac{4 a^3 \cos (2 d x) (a \cos (2 c)+4 b \sin (2 c)) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \cos ^5(c+d x)}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac{4 a^3 (a \sin (2 c)-4 b \cos (2 c)) \sin (2 d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \cos ^5(c+d x)}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}-\frac{a^4 \sin (4 c) \sin (4 d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \cos ^5(c+d x)}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac{32 a^3 b \sec (c) \sin (d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \cos ^4(c+d x)}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac{16 a b^2 \sec (c) (3 a \cos (c)+2 b \sin (c)) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \cos ^3(c+d x)}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac{32 a b^3 \sec (c) \sin (d x) (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \cos ^2(c+d x)}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}+\frac{8 b^4 (b \sec (c+d x)+a \sin (c+d x))^3 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \cos (c+d x)}{d (3 a \cos (c+d x)+a \cos (3 c+3 d x)+4 b \sin (c+d x)) (2 b+a \sin (2 c+2 d x))^3}","\frac{(a \sin (c+d x)+b \sec (c+d x))^4}{4 d}",1,"(8*b^4*Cos[c + d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) + (a^4*Cos[4*c]*Cos[4*d*x]*Cos[c + d*x]^5*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) + (16*a*b^2*Cos[c + d*x]^3*Sec[c]*(3*a*Cos[c] + 2*b*Sin[c])*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) - (4*a^3*Cos[2*d*x]*Cos[c + d*x]^5*(a*Cos[2*c] + 4*b*Sin[2*c])*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) + (32*a*b^3*Cos[c + d*x]^2*Sec[c]*Sin[d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) + (32*a^3*b*Cos[c + d*x]^4*Sec[c]*Sin[d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) + (4*a^3*Cos[c + d*x]^5*(-4*b*Cos[2*c] + a*Sin[2*c])*Sin[2*d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3) - (a^4*Cos[c + d*x]^5*Sin[4*c]*Sin[4*d*x]*(b*Sec[c + d*x] + a*Sin[c + d*x])^3*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]))/(d*(3*a*Cos[c + d*x] + a*Cos[3*c + 3*d*x] + 4*b*Sin[c + d*x])*(2*b + a*Sin[2*c + 2*d*x])^3)","B",1
639,1,31,26,1.2238327,"\int (b \sec (c+d x)+a \sin (c+d x))^2 (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx","Integrate[(b*Sec[c + d*x] + a*Sin[c + d*x])^2*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]),x]","\frac{\sec ^3(c+d x) (a \sin (2 (c+d x))+2 b)^3}{24 d}","\frac{(a \sin (c+d x)+b \sec (c+d x))^3}{3 d}",1,"(Sec[c + d*x]^3*(2*b + a*Sin[2*(c + d*x)])^3)/(24*d)","A",1
640,1,67,26,0.0373861,"\int (b \sec (c+d x)+a \sin (c+d x)) (a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)) \, dx","Integrate[(b*Sec[c + d*x] + a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x]),x]","-\frac{a^2 \cos ^2(c+d x)}{2 d}-\frac{a b \tan ^{-1}(\tan (c+d x))}{d}+\frac{a b \tan (c+d x)}{d}+a b x+\frac{b^2 \sec ^2(c+d x)}{2 d}","\frac{(a \sin (c+d x)+b \sec (c+d x))^2}{2 d}",1,"a*b*x - (a*b*ArcTan[Tan[c + d*x]])/d - (a^2*Cos[c + d*x]^2)/(2*d) + (b^2*Sec[c + d*x]^2)/(2*d) + (a*b*Tan[c + d*x])/d","B",1
641,1,29,22,0.4627857,"\int \frac{a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{b \sec (c+d x)+a \sin (c+d x)} \, dx","Integrate[(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x])/(b*Sec[c + d*x] + a*Sin[c + d*x]),x]","\frac{\log (a \sin (2 (c+d x))+2 b)-\log (\cos (c+d x))}{d}","\frac{\log (a \sin (c+d x)+b \sec (c+d x))}{d}",1,"(-Log[Cos[c + d*x]] + Log[2*b + a*Sin[2*(c + d*x)]])/d","A",1
642,1,27,24,0.3046978,"\int \frac{a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{(b \sec (c+d x)+a \sin (c+d x))^2} \, dx","Integrate[(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x])/(b*Sec[c + d*x] + a*Sin[c + d*x])^2,x]","-\frac{2 \cos (c+d x)}{d (a \sin (2 (c+d x))+2 b)}","-\frac{1}{d (a \sin (c+d x)+b \sec (c+d x))}",1,"(-2*Cos[c + d*x])/(d*(2*b + a*Sin[2*(c + d*x)]))","A",1
643,1,29,26,0.7100307,"\int \frac{a \cos (c+d x)+b \sec (c+d x) \tan (c+d x)}{(b \sec (c+d x)+a \sin (c+d x))^3} \, dx","Integrate[(a*Cos[c + d*x] + b*Sec[c + d*x]*Tan[c + d*x])/(b*Sec[c + d*x] + a*Sin[c + d*x])^3,x]","-\frac{2 \cos ^2(c+d x)}{d (a \sin (2 (c+d x))+2 b)^2}","-\frac{1}{2 d (a \sin (c+d x)+b \sec (c+d x))^2}",1,"(-2*Cos[c + d*x]^2)/(d*(2*b + a*Sin[2*(c + d*x)])^2)","A",1
644,0,0,21,0.0451701,"\int F(c,d,\cos (a+b x),r,s) \sin (a+b x) \, dx","Integrate[F[c, d, Cos[a + b*x], r, s]*Sin[a + b*x],x]","\int F(c,d,\cos (a+b x),r,s) \sin (a+b x) \, dx","\text{Int}(\sin (a+b x) F(c,d,\cos (a+b x),r,s),x)",0,"Integrate[F[c, d, Cos[a + b*x], r, s]*Sin[a + b*x], x]","A",-1
645,0,0,21,0.0348951,"\int \cos (a+b x) F(c,d,\sin (a+b x),r,s) \, dx","Integrate[Cos[a + b*x]*F[c, d, Sin[a + b*x], r, s],x]","\int \cos (a+b x) F(c,d,\sin (a+b x),r,s) \, dx","\text{Int}(\cos (a+b x) F(c,d,\sin (a+b x),r,s),x)",0,"Integrate[Cos[a + b*x]*F[c, d, Sin[a + b*x], r, s], x]","A",-1
646,0,0,23,0.0808462,"\int F(c,d,\tan (a+b x),r,s) \sec ^2(a+b x) \, dx","Integrate[F[c, d, Tan[a + b*x], r, s]*Sec[a + b*x]^2,x]","\int F(c,d,\tan (a+b x),r,s) \sec ^2(a+b x) \, dx","\text{Int}\left(\sec ^2(a+b x) F(c,d,\tan (a+b x),r,s),x\right)",0,"Integrate[F[c, d, Tan[a + b*x], r, s]*Sec[a + b*x]^2, x]","A",-1
647,0,0,23,0.0730988,"\int \csc ^2(a+b x) F(c,d,\cot (a+b x),r,s) \, dx","Integrate[Csc[a + b*x]^2*F[c, d, Cot[a + b*x], r, s],x]","\int \csc ^2(a+b x) F(c,d,\cot (a+b x),r,s) \, dx","\text{Int}\left(\csc ^2(a+b x) F(c,d,\cot (a+b x),r,s),x\right)",0,"Integrate[Csc[a + b*x]^2*F[c, d, Cot[a + b*x], r, s], x]","A",-1
648,1,12,12,0.0175749,"\int \frac{\sin (x)}{a+b \cos (x)} \, dx","Integrate[Sin[x]/(a + b*Cos[x]),x]","-\frac{\log (a+b \cos (x))}{b}","-\frac{\log (a+b \cos (x))}{b}",1,"-(Log[a + b*Cos[x]]/b)","A",1
649,1,19,20,0.0314719,"\int (a+b \cos (x))^n \sin (x) \, dx","Integrate[(a + b*Cos[x])^n*Sin[x],x]","-\frac{(a+b \cos (x))^{n+1}}{b n+b}","-\frac{(a+b \cos (x))^{n+1}}{b (n+1)}",1,"-((a + b*Cos[x])^(1 + n)/(b + b*n))","A",1
650,1,5,5,0.02088,"\int \frac{\sin (x)}{\sqrt{1+\cos ^2(x)}} \, dx","Integrate[Sin[x]/Sqrt[1 + Cos[x]^2],x]","-\sinh ^{-1}(\cos (x))","-\sinh ^{-1}(\cos (x))",1,"-ArcSinh[Cos[x]]","A",1
651,1,5,5,2.5662171,"\int \cos (\cos (x)) \sin (x) \, dx","Integrate[Cos[Cos[x]]*Sin[x],x]","-\sin (\cos (x))","-\sin (\cos (x))",1,"-Sin[Cos[x]]","A",1
652,1,21,28,1.4760018,"\int \cos (x) \cos (\cos (x)) \sin (x) \sin (\cos (x)) \, dx","Integrate[Cos[x]*Cos[Cos[x]]*Sin[x]*Sin[Cos[x]],x]","\frac{1}{4} \cos (x) \cos (2 \cos (x))-\frac{1}{8} \sin (2 \cos (x))","\frac{\cos (x)}{4}-\frac{1}{2} \cos (x) \sin ^2(\cos (x))-\frac{1}{4} \cos (\cos (x)) \sin (\cos (x))",1,"(Cos[x]*Cos[2*Cos[x]])/4 - Sin[2*Cos[x]]/8","A",1
653,1,26,26,4.5034086,"\int \cos (\cos (x)) \sin (x) \sin ^2(6 \cos (x)) \, dx","Integrate[Cos[Cos[x]]*Sin[x]*Sin[6*Cos[x]]^2,x]","-\frac{1}{2} \sin (\cos (x))+\frac{1}{44} \sin (11 \cos (x))+\frac{1}{52} \sin (13 \cos (x))","-\frac{1}{2} \sin (\cos (x))+\frac{1}{44} \sin (11 \cos (x))+\frac{1}{52} \sin (13 \cos (x))",1,"-1/2*Sin[Cos[x]] + Sin[11*Cos[x]]/44 + Sin[13*Cos[x]]/52","A",1
654,1,137,36,0.2808985,"\int \cos ^3(x) \left(a+b \cos ^2(x)\right)^3 \sin (x) \, dx","Integrate[Cos[x]^3*(a + b*Cos[x]^2)^3*Sin[x],x]","\frac{1}{32} \left(-4 a^3 \cos (2 x)-a^3 \cos (4 x)-12 a^2 b \cos ^4(x)-4 a^2 b \cos (3 x) \cos ^3(x)-8 a b^2 \cos ^6(x)-\frac{1}{32} a b^2 (48 \cos (2 x)+36 \cos (4 x)+16 \cos (6 x)+3 \cos (8 x))-2 b^3 \cos ^8(x)-\frac{1}{320} b^3 (140 \cos (2 x)+100 \cos (4 x)+50 \cos (6 x)+15 \cos (8 x)+2 \cos (10 x))\right)","\frac{a \left(a+b \cos ^2(x)\right)^4}{8 b^2}-\frac{\left(a+b \cos ^2(x)\right)^5}{10 b^2}",1,"(-12*a^2*b*Cos[x]^4 - 8*a*b^2*Cos[x]^6 - 2*b^3*Cos[x]^8 - 4*a^3*Cos[2*x] - 4*a^2*b*Cos[x]^3*Cos[3*x] - a^3*Cos[4*x] - (a*b^2*(48*Cos[2*x] + 36*Cos[4*x] + 16*Cos[6*x] + 3*Cos[8*x]))/32 - (b^3*(140*Cos[2*x] + 100*Cos[4*x] + 50*Cos[6*x] + 15*Cos[8*x] + 2*Cos[10*x]))/320)/32","B",1
655,1,9,9,2.6782289,"\int \sin (3 x) \sin (\cos (3 x)) \, dx","Integrate[Sin[3*x]*Sin[Cos[3*x]],x]","\frac{1}{3} \cos (\cos (3 x))","\frac{1}{3} \cos (\cos (3 x))",1,"Cos[Cos[3*x]]/3","A",1
656,1,24,31,0.1215478,"\int e^{\cos (1+3 x)} \cos (1+3 x) \sin (1+3 x) \, dx","Integrate[E^Cos[1 + 3*x]*Cos[1 + 3*x]*Sin[1 + 3*x],x]","\frac{2}{3} \sin ^2\left(\frac{1}{2} (3 x+1)\right) e^{\cos (3 x+1)}","\frac{1}{3} e^{\cos (3 x+1)}-\frac{1}{3} e^{\cos (3 x+1)} \cos (3 x+1)",1,"(2*E^Cos[1 + 3*x]*Sin[(1 + 3*x)/2]^2)/3","A",1
657,1,162,9,2.2165726,"\int \frac{\cos ^2(x) \sin (x)}{\sqrt{1-\cos ^6(x)}} \, dx","Integrate[(Cos[x]^2*Sin[x])/Sqrt[1 - Cos[x]^6],x]","-\frac{i \sin (x) \cos ^2(x) \sqrt{1-\frac{2 i \tan ^2(x)}{\sqrt{3}-3 i}} \sqrt{1+\frac{2 i \tan ^2(x)}{\sqrt{3}+3 i}} \Pi \left(\frac{3}{2}+\frac{i \sqrt{3}}{2};i \sinh ^{-1}\left(\sqrt{-\frac{2 i}{-3 i+\sqrt{3}}} \tan (x)\right)|\frac{3 i-\sqrt{3}}{3 i+\sqrt{3}}\right)}{\sqrt{2} \sqrt{-\frac{i}{\sqrt{3}-3 i}} \sqrt{1-\cos ^6(x)}}","-\frac{1}{3} \sin ^{-1}\left(\cos ^3(x)\right)",1,"((-I)*Cos[x]^2*EllipticPi[3/2 + (I/2)*Sqrt[3], I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[3])]*Tan[x]], (3*I - Sqrt[3])/(3*I + Sqrt[3])]*Sin[x]*Sqrt[1 - ((2*I)*Tan[x]^2)/(-3*I + Sqrt[3])]*Sqrt[1 + ((2*I)*Tan[x]^2)/(3*I + Sqrt[3])])/(Sqrt[2]*Sqrt[(-I)/(-3*I + Sqrt[3])]*Sqrt[1 - Cos[x]^6])","C",1
658,1,59,71,0.1602755,"\int \frac{\sin ^5(x)}{\sqrt{1-5 \cos (x)}} \, dx","Integrate[Sin[x]^5/Sqrt[1 - 5*Cos[x]],x]","\frac{180607 \left(\sqrt{1-5 \cos (x)}-1\right)}{562500}+\sqrt{1-5 \cos (x)} \left(-\frac{6772 \cos (x)}{196875}-\frac{2227 \cos (2 x)}{39375}+\frac{4 \cos (3 x)}{1575}+\frac{1}{180} \cos (4 x)\right)","\frac{2 (1-5 \cos (x))^{9/2}}{28125}-\frac{8 (1-5 \cos (x))^{7/2}}{21875}-\frac{88 (1-5 \cos (x))^{5/2}}{15625}+\frac{64 (1-5 \cos (x))^{3/2}}{3125}+\frac{1152 \sqrt{1-5 \cos (x)}}{3125}",1,"(180607*(-1 + Sqrt[1 - 5*Cos[x]]))/562500 + Sqrt[1 - 5*Cos[x]]*((-6772*Cos[x])/196875 - (2227*Cos[2*x])/39375 + (4*Cos[3*x])/1575 + Cos[4*x]/180)","A",1
659,1,18,18,0.0483637,"\int e^{n \cos (a+b x)} \sin (a+b x) \, dx","Integrate[E^(n*Cos[a + b*x])*Sin[a + b*x],x]","-\frac{e^{n \cos (a+b x)}}{b n}","-\frac{e^{n \cos (a+b x)}}{b n}",1,"-(E^(n*Cos[a + b*x])/(b*n))","A",1
660,1,23,23,0.2346123,"\int e^{n \cos (a c+b c x)} \sin (c (a+b x)) \, dx","Integrate[E^(n*Cos[a*c + b*c*x])*Sin[c*(a + b*x)],x]","-\frac{e^{n \cos (c (a+b x))}}{b c n}","-\frac{e^{n \cos (c (a+b x))}}{b c n}",1,"-(E^(n*Cos[c*(a + b*x)])/(b*c*n))","A",1
661,1,23,24,0.0423181,"\int e^{n \cos (c (a+b x))} \sin (a c+b c x) \, dx","Integrate[E^(n*Cos[c*(a + b*x)])*Sin[a*c + b*c*x],x]","-\frac{e^{n \cos (c (a+b x))}}{b c n}","-\frac{e^{n \cos (a c+b c x)}}{b c n}",1,"-(E^(n*Cos[c*(a + b*x)])/(b*c*n))","A",1
662,1,14,14,0.0398833,"\int e^{n \cos (a+b x)} \tan (a+b x) \, dx","Integrate[E^(n*Cos[a + b*x])*Tan[a + b*x],x]","-\frac{\text{Ei}(n \cos (a+b x))}{b}","-\frac{\text{Ei}(n \cos (a+b x))}{b}",1,"-(ExpIntegralEi[n*Cos[a + b*x]]/b)","A",1
663,1,19,19,0.0608756,"\int e^{n \cos (a c+b c x)} \tan (c (a+b x)) \, dx","Integrate[E^(n*Cos[a*c + b*c*x])*Tan[c*(a + b*x)],x]","-\frac{\text{Ei}(n \cos (c (a+b x)))}{b c}","-\frac{\text{Ei}(n \cos (c (a+b x)))}{b c}",1,"-(ExpIntegralEi[n*Cos[c*(a + b*x)]]/(b*c))","A",1
664,1,19,20,0.0573791,"\int e^{n \cos (c (a+b x))} \tan (a c+b c x) \, dx","Integrate[E^(n*Cos[c*(a + b*x)])*Tan[a*c + b*c*x],x]","-\frac{\text{Ei}(n \cos (c (a+b x)))}{b c}","-\frac{\text{Ei}(n \cos (a c+b x c))}{b c}",1,"-(ExpIntegralEi[n*Cos[c*(a + b*x)]]/(b*c))","A",1
665,1,11,11,0.0061461,"\int \frac{\cos (x)}{a+b \sin (x)} \, dx","Integrate[Cos[x]/(a + b*Sin[x]),x]","\frac{\log (a+b \sin (x))}{b}","\frac{\log (a+b \sin (x))}{b}",1,"Log[a + b*Sin[x]]/b","A",1
666,1,18,19,0.0202786,"\int \cos (x) (a+b \sin (x))^n \, dx","Integrate[Cos[x]*(a + b*Sin[x])^n,x]","\frac{(a+b \sin (x))^{n+1}}{b n+b}","\frac{(a+b \sin (x))^{n+1}}{b (n+1)}",1,"(a + b*Sin[x])^(1 + n)/(b + b*n)","A",1
667,1,3,3,0.0077359,"\int \frac{\cos (x)}{\sqrt{1+\sin ^2(x)}} \, dx","Integrate[Cos[x]/Sqrt[1 + Sin[x]^2],x]","\sinh ^{-1}(\sin (x))","\sinh ^{-1}(\sin (x))",1,"ArcSinh[Sin[x]]","A",1
668,1,7,7,0.0083603,"\int \frac{\cos (x)}{\sqrt{4-\sin ^2(x)}} \, dx","Integrate[Cos[x]/Sqrt[4 - Sin[x]^2],x]","\sin ^{-1}\left(\frac{\sin (x)}{2}\right)","\sin ^{-1}\left(\frac{\sin (x)}{2}\right)",1,"ArcSin[Sin[x]/2]","A",1
669,1,13,13,0.0282248,"\int \frac{\cos (3 x)}{\sqrt{4-\sin ^2(3 x)}} \, dx","Integrate[Cos[3*x]/Sqrt[4 - Sin[3*x]^2],x]","\frac{1}{3} \sin ^{-1}\left(\frac{1}{2} \sin (3 x)\right)","\frac{1}{3} \sin ^{-1}\left(\frac{1}{2} \sin (3 x)\right)",1,"ArcSin[Sin[3*x]/2]/3","A",1
670,1,21,21,0.0145379,"\int \cos (x) \sqrt{1+\csc (x)} \, dx","Integrate[Cos[x]*Sqrt[1 + Csc[x]],x]","\sin (x) \sqrt{\csc (x)+1}+\tanh ^{-1}\left(\sqrt{\csc (x)+1}\right)","\sin (x) \sqrt{\csc (x)+1}+\tanh ^{-1}\left(\sqrt{\csc (x)+1}\right)",1,"ArcTanh[Sqrt[1 + Csc[x]]] + Sqrt[1 + Csc[x]]*Sin[x]","A",1
671,1,28,28,0.0183806,"\int \cos (x) \sqrt{4-\sin ^2(x)} \, dx","Integrate[Cos[x]*Sqrt[4 - Sin[x]^2],x]","2 \sin ^{-1}\left(\frac{\sin (x)}{2}\right)+\frac{1}{2} \sin (x) \sqrt{4-\sin ^2(x)}","2 \sin ^{-1}\left(\frac{\sin (x)}{2}\right)+\frac{1}{2} \sin (x) \sqrt{4-\sin ^2(x)}",1,"2*ArcSin[Sin[x]/2] + (Sin[x]*Sqrt[4 - Sin[x]^2])/2","A",1
672,1,14,14,0.006938,"\int \cos (x) \sin (x) \sqrt{1+\sin ^2(x)} \, dx","Integrate[Cos[x]*Sin[x]*Sqrt[1 + Sin[x]^2],x]","\frac{1}{3} \left(\sin ^2(x)+1\right)^{3/2}","\frac{1}{3} \left(\sin ^2(x)+1\right)^{3/2}",1,"(1 + Sin[x]^2)^(3/2)/3","A",1
673,1,40,19,0.0182296,"\int \frac{\cos (x)}{\sqrt{2 \sin (x)+\sin ^2(x)}} \, dx","Integrate[Cos[x]/Sqrt[2*Sin[x] + Sin[x]^2],x]","\frac{2 \sqrt{\sin (x)} \sqrt{\sin (x)+2} \sinh ^{-1}\left(\frac{\sqrt{\sin (x)}}{\sqrt{2}}\right)}{\sqrt{\sin (x) (\sin (x)+2)}}","2 \tanh ^{-1}\left(\frac{\sin (x)}{\sqrt{\sin ^2(x)+2 \sin (x)}}\right)",1,"(2*ArcSinh[Sqrt[Sin[x]]/Sqrt[2]]*Sqrt[Sin[x]]*Sqrt[2 + Sin[x]])/Sqrt[Sin[x]*(2 + Sin[x])]","B",1
674,1,3,3,1.4658056,"\int \cos (x) \cos (\sin (x)) \, dx","Integrate[Cos[x]*Cos[Sin[x]],x]","\sin (\sin (x))","\sin (\sin (x))",1,"Sin[Sin[x]]","A",1
675,1,4,4,8.9588888,"\int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx","Integrate[Cos[x]*Cos[Sin[x]]*Cos[Sin[Sin[x]]],x]","\sin (\sin (\sin (x)))","\sin (\sin (\sin (x)))",1,"Sin[Sin[Sin[x]]]","A",1
676,1,4,4,0.0050285,"\int \cos (x) \sec (\sin (x)) \, dx","Integrate[Cos[x]*Sec[Sin[x]],x]","\tanh ^{-1}(\sin (\sin (x)))","\tanh ^{-1}(\sin (\sin (x)))",1,"ArcTanh[Sin[Sin[x]]]","A",1
677,1,128,36,0.3488046,"\int \cos (x) \sin ^3(x) \left(a+b \sin ^2(x)\right)^3 \, dx","Integrate[Cos[x]*Sin[x]^3*(a + b*Sin[x]^2)^3,x]","\frac{-20 \left(64 a^3+24 a b^2+7 b^3\right) \cos (2 x)+20 \left(16 a^3+18 a b^2+5 b^3\right) \cos (4 x)+b \left(3840 a^2 \sin ^4(x)-1280 a^2 \sin (3 x) \sin ^3(x)+2560 a b \sin ^6(x)-10 b (16 a+5 b) \cos (6 x)+15 b (2 a+b) \cos (8 x)+640 b^2 \sin ^8(x)-2 b^2 \cos (10 x)\right)}{10240}","\frac{\left(a+b \sin ^2(x)\right)^5}{10 b^2}-\frac{a \left(a+b \sin ^2(x)\right)^4}{8 b^2}",1,"(-20*(64*a^3 + 24*a*b^2 + 7*b^3)*Cos[2*x] + 20*(16*a^3 + 18*a*b^2 + 5*b^3)*Cos[4*x] + b*(-10*b*(16*a + 5*b)*Cos[6*x] + 15*b*(2*a + b)*Cos[8*x] - 2*b^2*Cos[10*x] + 3840*a^2*Sin[x]^4 + 2560*a*b*Sin[x]^6 + 640*b^2*Sin[x]^8 - 1280*a^2*Sin[x]^3*Sin[3*x]))/10240","B",1
678,1,9,14,0.0098068,"\int e^{\sin (x)} \cos (x) \sin (x) \, dx","Integrate[E^Sin[x]*Cos[x]*Sin[x],x]","e^{\sin (x)} (\sin (x)-1)","e^{\sin (x)} \sin (x)-e^{\sin (x)}",1,"E^Sin[x]*(-1 + Sin[x])","A",1
679,1,20,25,0.0199736,"\int \frac{\cos ^3(x)}{\sqrt{\sin ^3(x)}} \, dx","Integrate[Cos[x]^3/Sqrt[Sin[x]^3],x]","\frac{\sin (x) (\cos (2 x)-7)}{3 \sqrt{\sin ^3(x)}}","-\frac{2 \sin (x)}{\sqrt{\sin ^3(x)}}-\frac{2}{3} \sqrt{\sin ^3(x)}",1,"((-7 + Cos[2*x])*Sin[x])/(3*Sqrt[Sin[x]^3])","A",1
680,1,10,10,0.0104207,"\int \frac{e^{\sqrt{\sin (x)}} \cos (x)}{\sqrt{\sin (x)}} \, dx","Integrate[(E^Sqrt[Sin[x]]*Cos[x])/Sqrt[Sin[x]],x]","2 e^{\sqrt{\sin (x)}}","2 e^{\sqrt{\sin (x)}}",1,"2*E^Sqrt[Sin[x]]","A",1
681,1,6,6,0.0091131,"\int e^{4+\sin (x)} \cos (x) \, dx","Integrate[E^(4 + Sin[x])*Cos[x],x]","e^{\sin (x)+4}","e^{\sin (x)+4}",1,"E^(4 + Sin[x])","A",1
682,1,7,10,0.030446,"\int e^{\cos (x) \sin (x)} \cos (2 x) \, dx","Integrate[E^(Cos[x]*Sin[x])*Cos[2*x],x]","e^{\sin (x) \cos (x)}","e^{\frac{1}{2} \sin (2 x)}",1,"E^(Cos[x]*Sin[x])","A",1
683,1,10,10,0.0091441,"\int e^{\cos \left(\frac{x}{2}\right) \sin \left(\frac{x}{2}\right)} \cos (x) \, dx","Integrate[E^(Cos[x/2]*Sin[x/2])*Cos[x],x]","2 e^{\frac{\sin (x)}{2}}","2 e^{\frac{\sin (x)}{2}}",1,"2*E^(Sin[x]/2)","A",1
684,1,17,17,0.0165414,"\int e^{n \sin (a+b x)} \cos (a+b x) \, dx","Integrate[E^(n*Sin[a + b*x])*Cos[a + b*x],x]","\frac{e^{n \sin (a+b x)}}{b n}","\frac{e^{n \sin (a+b x)}}{b n}",1,"E^(n*Sin[a + b*x])/(b*n)","A",1
685,1,23,22,0.1361741,"\int e^{n \sin (a c+b c x)} \cos (c (a+b x)) \, dx","Integrate[E^(n*Sin[a*c + b*c*x])*Cos[c*(a + b*x)],x]","\frac{e^{n \sin (a c+b c x)}}{b c n}","\frac{e^{n \sin (c (a+b x))}}{b c n}",1,"E^(n*Sin[a*c + b*c*x])/(b*c*n)","A",1
686,1,23,23,0.0425373,"\int e^{n \sin (c (a+b x))} \cos (a c+b c x) \, dx","Integrate[E^(n*Sin[c*(a + b*x)])*Cos[a*c + b*c*x],x]","\frac{e^{n \sin (a c+b c x)}}{b c n}","\frac{e^{n \sin (a c+b c x)}}{b c n}",1,"E^(n*Sin[a*c + b*c*x])/(b*c*n)","A",1
687,1,13,13,0.0387303,"\int e^{n \sin (a+b x)} \cot (a+b x) \, dx","Integrate[E^(n*Sin[a + b*x])*Cot[a + b*x],x]","\frac{\text{Ei}(n \sin (a+b x))}{b}","\frac{\text{Ei}(n \sin (a+b x))}{b}",1,"ExpIntegralEi[n*Sin[a + b*x]]/b","A",1
688,1,18,18,0.0658281,"\int e^{n \sin (a c+b c x)} \cot (c (a+b x)) \, dx","Integrate[E^(n*Sin[a*c + b*c*x])*Cot[c*(a + b*x)],x]","\frac{\text{Ei}(n \sin (c (a+b x)))}{b c}","\frac{\text{Ei}(n \sin (c (a+b x)))}{b c}",1,"ExpIntegralEi[n*Sin[c*(a + b*x)]]/(b*c)","A",1
689,1,18,19,0.0604002,"\int e^{n \sin (c (a+b x))} \cot (a c+b c x) \, dx","Integrate[E^(n*Sin[c*(a + b*x)])*Cot[a*c + b*c*x],x]","\frac{\text{Ei}(n \sin (c (a+b x)))}{b c}","\frac{\text{Ei}(n \sin (a c+b x c))}{b c}",1,"ExpIntegralEi[n*Sin[c*(a + b*x)]]/(b*c)","A",1
690,1,20,11,0.0563255,"\int \frac{\sec ^2(x)}{a+b \tan (x)} \, dx","Integrate[Sec[x]^2/(a + b*Tan[x]),x]","\frac{\log (a \cos (x)+b \sin (x))-\log (\cos (x))}{b}","\frac{\log (a+b \tan (x))}{b}",1,"(-Log[Cos[x]] + Log[a*Cos[x] + b*Sin[x]])/b","A",1
691,1,23,11,0.0059434,"\int \frac{\sec ^2(x)}{1-\tan ^2(x)} \, dx","Integrate[Sec[x]^2/(1 - Tan[x]^2),x]","\frac{1}{2} \log (\sin (x)+\cos (x))-\frac{1}{2} \log (\cos (x)-\sin (x))","\frac{1}{2} \tanh ^{-1}(2 \sin (x) \cos (x))",1,"-1/2*Log[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x]]/2","B",1
692,1,9,27,0.0238518,"\int \frac{\sec ^2(x)}{9+\tan ^2(x)} \, dx","Integrate[Sec[x]^2/(9 + Tan[x]^2),x]","-\frac{1}{3} \tan ^{-1}(3 \cot (x))","\frac{x}{3}-\frac{1}{3} \tan ^{-1}\left(\frac{2 \sin (x) \cos (x)}{2 \cos ^2(x)+1}\right)",1,"-1/3*ArcTan[3*Cot[x]]","A",1
693,1,18,19,0.1887794,"\int \sec ^2(x) (a+b \tan (x))^n \, dx","Integrate[Sec[x]^2*(a + b*Tan[x])^n,x]","\frac{(a+b \tan (x))^{n+1}}{b n+b}","\frac{(a+b \tan (x))^{n+1}}{b (n+1)}",1,"(a + b*Tan[x])^(1 + n)/(b + b*n)","A",1
694,1,4,4,0.0052642,"\int \sec ^2(x) \left(1+\frac{1}{1+\tan ^2(x)}\right) \, dx","Integrate[Sec[x]^2*(1 + (1 + Tan[x]^2)^(-1)),x]","x+\tan (x)","x+\tan (x)",1,"x + Tan[x]","A",1
695,1,4,4,0.0027967,"\int \frac{\sec ^2(x) \left(2+\tan ^2(x)\right)}{1+\tan ^2(x)} \, dx","Integrate[(Sec[x]^2*(2 + Tan[x]^2))/(1 + Tan[x]^2),x]","x+\tan (x)","x+\tan (x)",1,"x + Tan[x]","A",1
696,1,31,33,0.0336341,"\int \frac{\sec ^2(x)}{2+2 \tan (x)+\tan ^2(x)} \, dx","Integrate[Sec[x]^2/(2 + 2*Tan[x] + Tan[x]^2),x]","2 \left(\frac{1}{4} \tan ^{-1}(\sec (x) (\sin (x)+\cos (x)))-\frac{1}{4} \tan ^{-1}\left(\frac{\cos (x)}{\sin (x)+\cos (x)}\right)\right)","x-\tan ^{-1}\left(\frac{-2 \cos ^2(x)+\sin (x) \cos (x)+1}{\cos ^2(x)+2 \sin (x) \cos (x)+2}\right)",1,"2*(-1/4*ArcTan[Cos[x]/(Cos[x] + Sin[x])] + ArcTan[Sec[x]*(Cos[x] + Sin[x])]/4)","A",1
697,1,16,10,0.0353552,"\int \frac{\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx","Integrate[Sec[x]^2/(Tan[x]^2 + Tan[x]^3),x]","-\cot (x)-\log (\sin (x))+\log (\sin (x)+\cos (x))","\log (\cot (x)+1)-\cot (x)",1,"-Cot[x] - Log[Sin[x]] + Log[Cos[x] + Sin[x]]","A",1
698,1,16,10,0.0344052,"\int \frac{\sec ^2(x)}{-\tan ^2(x)+\tan ^3(x)} \, dx","Integrate[Sec[x]^2/(-Tan[x]^2 + Tan[x]^3),x]","\cot (x)-\log (\sin (x))+\log (\cos (x)-\sin (x))","\cot (x)+\log (1-\cot (x))",1,"Cot[x] + Log[Cos[x] - Sin[x]] - Log[Sin[x]]","A",1
699,1,74,176,0.1150192,"\int \frac{\sec ^2(x)}{3-4 \tan ^3(x)} \, dx","Integrate[Sec[x]^2/(3 - 4*Tan[x]^3),x]","\frac{2 \sqrt{3} \tan ^{-1}\left(\frac{2\ 6^{2/3} \tan (x)+3}{3 \sqrt{3}}\right)+\log \left(2 \sqrt[3]{6} \tan ^2(x)+6^{2/3} \tan (x)+3\right)-2 \log \left(3-6^{2/3} \tan (x)\right)}{6\ 6^{2/3}}","\frac{x}{3\ 2^{2/3} \sqrt[6]{3}}+\frac{\log \left(2 \sqrt[3]{2} \tan ^2(x)+2^{2/3} \sqrt[3]{3} \tan (x)+3^{2/3}\right)}{6\ 6^{2/3}}-\frac{\log \left(\sqrt[3]{3}-2^{2/3} \tan (x)\right)}{3\ 6^{2/3}}-\frac{\tan ^{-1}\left(\frac{-2\ 6^{2/3} \cos ^2(x)+2 \left(3-2 \sqrt[3]{6}\right) \sin (x) \cos (x)+6^{2/3}}{\left(6-4 \sqrt[3]{6}\right) \cos ^2(x)+2\ 6^{2/3} \sin (x) \cos (x)+4 \sqrt[3]{6}+3\ 2^{2/3} \sqrt[6]{3}}\right)}{3\ 2^{2/3} \sqrt[6]{3}}",1,"(2*Sqrt[3]*ArcTan[(3 + 2*6^(2/3)*Tan[x])/(3*Sqrt[3])] - 2*Log[3 - 6^(2/3)*Tan[x]] + Log[3 + 6^(2/3)*Tan[x] + 2*6^(1/3)*Tan[x]^2])/(6*6^(2/3))","A",1
700,1,22,53,0.0540018,"\int \frac{\sec ^2(x)}{11-5 \tan (x)+5 \tan ^2(x)} \, dx","Integrate[Sec[x]^2/(11 - 5*Tan[x] + 5*Tan[x]^2),x]","-\frac{2 \tan ^{-1}\left(\sqrt{\frac{5}{39}} (1-2 \tan (x))\right)}{\sqrt{195}}","\frac{2 x}{\sqrt{195}}-\frac{2 \tan ^{-1}\left(\frac{10 \cos ^2(x)+12 \sin (x) \cos (x)-5}{12 \cos ^2(x)-10 \sin (x) \cos (x)+\sqrt{195}+10}\right)}{\sqrt{195}}",1,"(-2*ArcTan[Sqrt[5/39]*(1 - 2*Tan[x])])/Sqrt[195]","A",1
701,1,54,28,0.3835727,"\int \frac{\sec ^2(x) (a+b \tan (x))}{c+d \tan (x)} \, dx","Integrate[(Sec[x]^2*(a + b*Tan[x]))/(c + d*Tan[x]),x]","\frac{\cos (x) (a+b \tan (x)) ((b c-a d) (\log (\cos (x))-\log (c \cos (x)+d \sin (x)))+b d \tan (x))}{d^2 (a \cos (x)+b \sin (x))}","\frac{b \tan (x)}{d}-\frac{(b c-a d) \log (c+d \tan (x))}{d^2}",1,"(Cos[x]*(a + b*Tan[x])*((b*c - a*d)*(Log[Cos[x]] - Log[c*Cos[x] + d*Sin[x]]) + b*d*Tan[x]))/(d^2*(a*Cos[x] + b*Sin[x]))","A",1
702,1,62,53,0.5920382,"\int \frac{\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx","Integrate[(Sec[x]^2*(a + b*Tan[x])^2)/(c + d*Tan[x]),x]","\frac{b^2 d^2 \sec ^2(x)-2 \left(b d \tan (x) (b c-2 a d)+(b c-a d)^2 (\log (\cos (x))-\log (c \cos (x)+d \sin (x)))\right)}{2 d^3}","\frac{(b c-a d)^2 \log (c+d \tan (x))}{d^3}-\frac{b \tan (x) (b c-a d)}{d^2}+\frac{(a+b \tan (x))^2}{2 d}",1,"(b^2*d^2*Sec[x]^2 - 2*((b*c - a*d)^2*(Log[Cos[x]] - Log[c*Cos[x] + d*Sin[x]]) + b*d*(b*c - 2*a*d)*Tan[x]))/(2*d^3)","A",1
703,1,133,78,0.960213,"\int \frac{\sec ^2(x) (a+b \tan (x))^3}{c+d \tan (x)} \, dx","Integrate[(Sec[x]^2*(a + b*Tan[x])^3)/(c + d*Tan[x]),x]","\frac{(a+b \tan (x))^3 (c \cos (x)+d \sin (x)) \left(b d^2 (9 a \sin (2 x) (a d-b c)+b (9 a d-3 b c+2 b d \tan (x)))+6 \cos ^2(x) (b c-a d)^3 (\log (\cos (x))-\log (c \cos (x)+d \sin (x)))+b^3 (-d) \left(d^2-3 c^2\right) \sin (2 x)\right)}{6 d^4 (c+d \tan (x)) (a \cos (x)+b \sin (x))^3}","-\frac{(b c-a d)^3 \log (c+d \tan (x))}{d^4}+\frac{b \tan (x) (b c-a d)^2}{d^3}-\frac{(b c-a d) (a+b \tan (x))^2}{2 d^2}+\frac{(a+b \tan (x))^3}{3 d}",1,"((c*Cos[x] + d*Sin[x])*(a + b*Tan[x])^3*(6*(b*c - a*d)^3*Cos[x]^2*(Log[Cos[x]] - Log[c*Cos[x] + d*Sin[x]]) - b^3*d*(-3*c^2 + d^2)*Sin[2*x] + b*d^2*(9*a*(-(b*c) + a*d)*Sin[2*x] + b*(-3*b*c + 9*a*d + 2*b*d*Tan[x]))))/(6*d^4*(a*Cos[x] + b*Sin[x])^3*(c + d*Tan[x]))","A",1
704,1,12,12,0.03459,"\int \frac{\sec ^2(x) \tan ^2(x)}{\left(2+\tan ^3(x)\right)^2} \, dx","Integrate[(Sec[x]^2*Tan[x]^2)/(2 + Tan[x]^3)^2,x]","-\frac{1}{3 \left(\tan ^3(x)+2\right)}","-\frac{1}{3 \left(\tan ^3(x)+2\right)}",1,"-1/3*1/(2 + Tan[x]^3)","A",1
705,1,67,33,0.0248202,"\int \sec ^2(x) \tan ^6(x) \left(1+\tan ^2(x)\right)^3 \, dx","Integrate[Sec[x]^2*Tan[x]^6*(1 + Tan[x]^2)^3,x]","-\frac{16 \tan (x)}{3003}+\frac{1}{13} \tan (x) \sec ^{12}(x)-\frac{27}{143} \tan (x) \sec ^{10}(x)+\frac{53}{429} \tan (x) \sec ^8(x)-\frac{5 \tan (x) \sec ^6(x)}{3003}-\frac{2 \tan (x) \sec ^4(x)}{1001}-\frac{8 \tan (x) \sec ^2(x)}{3003}","\frac{\tan ^{13}(x)}{13}+\frac{3 \tan ^{11}(x)}{11}+\frac{\tan ^9(x)}{3}+\frac{\tan ^7(x)}{7}",1,"(-16*Tan[x])/3003 - (8*Sec[x]^2*Tan[x])/3003 - (2*Sec[x]^4*Tan[x])/1001 - (5*Sec[x]^6*Tan[x])/3003 + (53*Sec[x]^8*Tan[x])/429 - (27*Sec[x]^10*Tan[x])/143 + (Sec[x]^12*Tan[x])/13","B",1
706,1,32,46,0.2213132,"\int \frac{\sec ^2(x) \left(2+\tan ^2(x)\right)}{1+\tan ^3(x)} \, dx","Integrate[(Sec[x]^2*(2 + Tan[x]^2))/(1 + Tan[x]^3),x]","-\frac{2 \tan ^{-1}\left(\frac{1-2 \tan (x)}{\sqrt{3}}\right)}{\sqrt{3}}-\log (\cos (x))+\log (\sin (x)+\cos (x))","\frac{2 x}{\sqrt{3}}+\log (\tan (x)+1)+\frac{2 \tan ^{-1}\left(\frac{1-2 \cos ^2(x)}{-2 \sin (x) \cos (x)+\sqrt{3}+2}\right)}{\sqrt{3}}",1,"(-2*ArcTan[(1 - 2*Tan[x])/Sqrt[3]])/Sqrt[3] - Log[Cos[x]] + Log[Cos[x] + Sin[x]]","A",1
707,1,4,4,0.0023553,"\int \left(1+\cos ^2(x)\right) \sec ^2(x) \, dx","Integrate[(1 + Cos[x]^2)*Sec[x]^2,x]","x+\tan (x)","x+\tan (x)",1,"x + Tan[x]","A",1
708,1,29,21,0.0295279,"\int \frac{\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx","Integrate[Sec[x]^2/(1 + Sec[x]^2 - 3*Tan[x]),x]","2 \left(\frac{1}{2} \log (2 \cos (x)-\sin (x))-\frac{1}{2} \log (\cos (x)-\sin (x))\right)","\log (2 \cos (x)-\sin (x))-\log (\cos (x)-\sin (x))",1,"2*(-1/2*Log[Cos[x] - Sin[x]] + Log[2*Cos[x] - Sin[x]]/2)","A",1
709,1,43,9,0.0424197,"\int \frac{\sec ^2(x)}{\sqrt{4-\sec ^2(x)}} \, dx","Integrate[Sec[x]^2/Sqrt[4 - Sec[x]^2],x]","\frac{\sqrt{2 \cos (2 x)+1} \sec (x) \tan ^{-1}\left(\frac{\sin (x)}{\sqrt{3-4 \sin ^2(x)}}\right)}{\sqrt{4-\sec ^2(x)}}","\sin ^{-1}\left(\frac{\tan (x)}{\sqrt{3}}\right)",1,"(ArcTan[Sin[x]/Sqrt[3 - 4*Sin[x]^2]]*Sqrt[1 + 2*Cos[2*x]]*Sec[x])/Sqrt[4 - Sec[x]^2]","B",1
710,1,47,9,0.0602003,"\int \frac{\sec ^2(x)}{\sqrt{1-4 \tan ^2(x)}} \, dx","Integrate[Sec[x]^2/Sqrt[1 - 4*Tan[x]^2],x]","\frac{\sqrt{5 \cos (2 x)-3} \sec (x) \tan ^{-1}\left(\frac{2 \sin (x)}{\sqrt{1-5 \sin ^2(x)}}\right)}{2 \sqrt{2-8 \tan ^2(x)}}","\frac{1}{2} \sin ^{-1}(2 \tan (x))",1,"(ArcTan[(2*Sin[x])/Sqrt[1 - 5*Sin[x]^2]]*Sqrt[-3 + 5*Cos[2*x]]*Sec[x])/(2*Sqrt[2 - 8*Tan[x]^2])","B",1
711,1,46,14,0.046986,"\int \frac{\sec ^2(x)}{\sqrt{-4+\tan ^2(x)}} \, dx","Integrate[Sec[x]^2/Sqrt[-4 + Tan[x]^2],x]","\frac{\sqrt{5 \cos (2 x)+3} \sec (x) \tan ^{-1}\left(\frac{\sin (x)}{\sqrt{4-5 \sin ^2(x)}}\right)}{\sqrt{2} \sqrt{\tan ^2(x)-4}}","\tanh ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)-4}}\right)",1,"(ArcTan[Sin[x]/Sqrt[4 - 5*Sin[x]^2]]*Sqrt[3 + 5*Cos[2*x]]*Sec[x])/(Sqrt[2]*Sqrt[-4 + Tan[x]^2])","B",1
712,1,52,19,0.4934531,"\int \sqrt{1-\cot ^2(x)} \sec ^2(x) \, dx","Integrate[Sqrt[1 - Cot[x]^2]*Sec[x]^2,x]","\tan (x) \sqrt{1-\cot ^2(x)} \sec (2 x) \left(\cos (2 x)-\cos (x) \sqrt{-\cos (2 x)} \tan ^{-1}\left(\frac{\cos (x)}{\sqrt{-\cos (2 x)}}\right)\right)","\tan (x) \sqrt{1-\cot ^2(x)}+\sin ^{-1}(\cot (x))",1,"(-(ArcTan[Cos[x]/Sqrt[-Cos[2*x]]]*Cos[x]*Sqrt[-Cos[2*x]]) + Cos[2*x])*Sqrt[1 - Cot[x]^2]*Sec[2*x]*Tan[x]","B",1
713,1,63,26,0.1151167,"\int \sec ^2(x) \sqrt{1-\tan ^2(x)} \, dx","Integrate[Sec[x]^2*Sqrt[1 - Tan[x]^2],x]","\frac{\cos (2 x) \tan (x)+\sqrt{\cos ^2(x)} \cos (x) \sqrt{1-\tan ^2(x)} \sin ^{-1}\left(\frac{\sin (x)}{\sqrt{\cos ^2(x)}}\right)}{2 \sqrt{\cos ^2(x)} \sqrt{\cos (2 x)}}","\frac{1}{2} \tan (x) \sqrt{1-\tan ^2(x)}+\frac{1}{2} \sin ^{-1}(\tan (x))",1,"(Cos[2*x]*Tan[x] + ArcSin[Sin[x]/Sqrt[Cos[x]^2]]*Cos[x]*Sqrt[Cos[x]^2]*Sqrt[1 - Tan[x]^2])/(2*Sqrt[Cos[x]^2]*Sqrt[Cos[2*x]])","B",1
714,1,4,4,0.0592686,"\int e^{\tan (x)} \sec ^2(x) \, dx","Integrate[E^Tan[x]*Sec[x]^2,x]","e^{\tan (x)}","e^{\tan (x)}",1,"E^Tan[x]","A",1
715,1,25,17,0.0167416,"\int \sec ^4(x) \left(-1+\sec ^2(x)\right)^2 \tan (x) \, dx","Integrate[Sec[x]^4*(-1 + Sec[x]^2)^2*Tan[x],x]","\frac{\sec ^8(x)}{8}-\frac{\sec ^6(x)}{3}+\frac{\sec ^4(x)}{4}","\frac{\tan ^8(x)}{8}+\frac{\tan ^6(x)}{6}",1,"Sec[x]^4/4 - Sec[x]^6/3 + Sec[x]^8/8","A",1
716,1,20,12,0.0590873,"\int \frac{\csc ^2(x)}{a+b \cot (x)} \, dx","Integrate[Csc[x]^2/(a + b*Cot[x]),x]","\frac{\log (\sin (x))-\log (a \sin (x)+b \cos (x))}{b}","-\frac{\log (a+b \cot (x))}{b}",1,"(Log[Sin[x]] - Log[b*Cos[x] + a*Sin[x]])/b","A",1
717,1,19,20,0.2047386,"\int (a+b \cot (x))^n \csc ^2(x) \, dx","Integrate[(a + b*Cot[x])^n*Csc[x]^2,x]","-\frac{(a+b \cot (x))^{n+1}}{b n+b}","-\frac{(a+b \cot (x))^{n+1}}{b (n+1)}",1,"-((a + b*Cot[x])^(1 + n)/(b + b*n))","A",1
718,1,6,6,0.0024683,"\int \csc ^2(x) \left(1+\sin ^2(x)\right) \, dx","Integrate[Csc[x]^2*(1 + Sin[x]^2),x]","x-\cot (x)","x-\cot (x)",1,"x - Cot[x]","A",1
719,1,6,6,0.005209,"\int \left(1+\frac{1}{1+\cot ^2(x)}\right) \csc ^2(x) \, dx","Integrate[(1 + (1 + Cot[x]^2)^(-1))*Csc[x]^2,x]","x-\cot (x)","x-\cot (x)",1,"x - Cot[x]","A",1
720,1,56,28,0.3773244,"\int \frac{(a+b \cot (x)) \csc ^2(x)}{c+d \cot (x)} \, dx","Integrate[((a + b*Cot[x])*Csc[x]^2)/(c + d*Cot[x]),x]","\frac{\sin (x) (a+b \cot (x)) (-(b c-a d) (\log (\sin (x))-\log (c \sin (x)+d \cos (x)))-b d \cot (x))}{d^2 (a \sin (x)+b \cos (x))}","\frac{(b c-a d) \log (c+d \cot (x))}{d^2}-\frac{b \cot (x)}{d}",1,"((a + b*Cot[x])*(-(b*d*Cot[x]) - (b*c - a*d)*(Log[Sin[x]] - Log[d*Cos[x] + c*Sin[x]]))*Sin[x])/(d^2*(b*Cos[x] + a*Sin[x]))","A",1
721,1,62,53,0.5614344,"\int \frac{(a+b \cot (x))^2 \csc ^2(x)}{c+d \cot (x)} \, dx","Integrate[((a + b*Cot[x])^2*Csc[x]^2)/(c + d*Cot[x]),x]","\frac{2 b d \cot (x) (b c-2 a d)+2 (b c-a d)^2 (\log (\sin (x))-\log (c \sin (x)+d \cos (x)))-b^2 d^2 \csc ^2(x)}{2 d^3}","-\frac{(b c-a d)^2 \log (c+d \cot (x))}{d^3}+\frac{b \cot (x) (b c-a d)}{d^2}-\frac{(a+b \cot (x))^2}{2 d}",1,"(2*b*d*(b*c - 2*a*d)*Cot[x] - b^2*d^2*Csc[x]^2 + 2*(b*c - a*d)^2*(Log[Sin[x]] - Log[d*Cos[x] + c*Sin[x]]))/(2*d^3)","A",1
722,1,135,78,1.3044305,"\int \frac{(a+b \cot (x))^3 \csc ^2(x)}{c+d \cot (x)} \, dx","Integrate[((a + b*Cot[x])^3*Csc[x]^2)/(c + d*Cot[x]),x]","\frac{(a+b \cot (x))^3 (c \sin (x)+d \cos (x)) \left(b d \left(\sin (2 x) \left(-9 a^2 d^2+9 a b c d+b^2 \left(d^2-3 c^2\right)\right)+3 b d (b c-3 a d)\right)-6 \sin ^2(x) (b c-a d)^3 (\log (\sin (x))-\log (c \sin (x)+d \cos (x)))-2 b^3 d^3 \cot (x)\right)}{6 d^4 (c+d \cot (x)) (a \sin (x)+b \cos (x))^3}","\frac{(b c-a d)^3 \log (c+d \cot (x))}{d^4}-\frac{b \cot (x) (b c-a d)^2}{d^3}+\frac{(b c-a d) (a+b \cot (x))^2}{2 d^2}-\frac{(a+b \cot (x))^3}{3 d}",1,"((a + b*Cot[x])^3*(d*Cos[x] + c*Sin[x])*(-2*b^3*d^3*Cot[x] - 6*(b*c - a*d)^3*(Log[Sin[x]] - Log[d*Cos[x] + c*Sin[x]])*Sin[x]^2 + b*d*(3*b*d*(b*c - 3*a*d) + (9*a*b*c*d - 9*a^2*d^2 + b^2*(-3*c^2 + d^2))*Sin[2*x])))/(6*d^4*(c + d*Cot[x])*(b*Cos[x] + a*Sin[x])^3)","A",1
723,1,6,6,0.0702772,"\int e^{-\cot (x)} \csc ^2(x) \, dx","Integrate[Csc[x]^2/E^Cot[x],x]","e^{-\cot (x)}","e^{-\cot (x)}",1,"E^(-Cot[x])","A",1
724,1,20,11,0.0156963,"\int \frac{\sec (x) \tan (x)}{a+b \sec (x)} \, dx","Integrate[(Sec[x]*Tan[x])/(a + b*Sec[x]),x]","\frac{\log (a \cos (x)+b)}{b}-\frac{\log (\cos (x))}{b}","\frac{\log (a+b \sec (x))}{b}",1,"-(Log[Cos[x]]/b) + Log[b + a*Cos[x]]/b","A",1
725,1,5,5,0.0218098,"\int \frac{\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx","Integrate[(Sec[x]*Tan[x])/(1 + Sec[x]^2),x]","-\tan ^{-1}(\cos (x))","-\tan ^{-1}(\cos (x))",1,"-ArcTan[Cos[x]]","A",1
726,1,11,11,0.025864,"\int \frac{\sec (x) \tan (x)}{9+4 \sec ^2(x)} \, dx","Integrate[(Sec[x]*Tan[x])/(9 + 4*Sec[x]^2),x]","-\frac{1}{6} \tan ^{-1}\left(\frac{3 \cos (x)}{2}\right)","-\frac{1}{6} \tan ^{-1}\left(\frac{3 \cos (x)}{2}\right)",1,"-1/6*ArcTan[(3*Cos[x])/2]","A",1
727,1,9,7,0.0049712,"\int \frac{\sec (x) \tan (x)}{\sec (x)+\sec ^2(x)} \, dx","Integrate[(Sec[x]*Tan[x])/(Sec[x] + Sec[x]^2),x]","-2 \log \left(\cos \left(\frac{x}{2}\right)\right)","-\log (\cos (x)+1)",1,"-2*Log[Cos[x/2]]","A",1
728,1,38,5,0.0282209,"\int \frac{\sec (x) \tan (x)}{\sqrt{4+\sec ^2(x)}} \, dx","Integrate[(Sec[x]*Tan[x])/Sqrt[4 + Sec[x]^2],x]","\frac{\sqrt{2 \cos (2 x)+3} \sec (x) \tanh ^{-1}\left(\sqrt{4 \cos ^2(x)+1}\right)}{\sqrt{\sec ^2(x)+4}}","\text{csch}^{-1}(2 \cos (x))",1,"(ArcTanh[Sqrt[1 + 4*Cos[x]^2]]*Sqrt[3 + 2*Cos[2*x]]*Sec[x])/Sqrt[4 + Sec[x]^2]","B",1
729,1,13,13,0.0167306,"\int \frac{\sec (x) \tan (x)}{\sqrt{1+\cos ^2(x)}} \, dx","Integrate[(Sec[x]*Tan[x])/Sqrt[1 + Cos[x]^2],x]","\sqrt{\cos ^2(x)+1} \sec (x)","\sqrt{\cos ^2(x)+1} \sec (x)",1,"Sqrt[1 + Cos[x]^2]*Sec[x]","A",1
730,1,4,4,0.0080481,"\int e^{\sec (x)} \sec (x) \tan (x) \, dx","Integrate[E^Sec[x]*Sec[x]*Tan[x],x]","e^{\sec (x)}","e^{\sec (x)}",1,"E^Sec[x]","A",1
731,1,9,9,0.0069749,"\int 2^{\sec (x)} \sec (x) \tan (x) \, dx","Integrate[2^Sec[x]*Sec[x]*Tan[x],x]","\frac{2^{\sec (x)}}{\log (2)}","\frac{2^{\sec (x)}}{\log (2)}",1,"2^Sec[x]/Log[2]","A",1
732,1,20,12,0.0749785,"\int \frac{\sec (2 x) \tan (2 x)}{(1+\sec (2 x))^{3/2}} \, dx","Integrate[(Sec[2*x]*Tan[2*x])/(1 + Sec[2*x])^(3/2),x]","-\frac{2 \cos ^2(x) \sec (2 x)}{(\sec (2 x)+1)^{3/2}}","-\frac{1}{\sqrt{\sec (2 x)+1}}",1,"(-2*Cos[x]^2*Sec[2*x])/(1 + Sec[2*x])^(3/2)","A",1
733,1,43,43,0.0518927,"\int \sqrt{1+5 \cos ^2(3 x)} \sec (3 x) \tan (3 x) \, dx","Integrate[Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x]*Tan[3*x],x]","\frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x)-\frac{1}{3} \sqrt{5} \sinh ^{-1}\left(\sqrt{5} \cos (3 x)\right)","\frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x)-\frac{1}{3} \sqrt{5} \sinh ^{-1}\left(\sqrt{5} \cos (3 x)\right)",1,"-1/3*(Sqrt[5]*ArcSinh[Sqrt[5]*Cos[3*x]]) + (Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3","A",1
734,1,22,22,0.0254137,"\int \frac{\sec (3 x) \tan (3 x)}{\sqrt{1+5 \cos ^2(3 x)}} \, dx","Integrate[(Sec[3*x]*Tan[3*x])/Sqrt[1 + 5*Cos[3*x]^2],x]","\frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x)","\frac{1}{3} \sqrt{5 \cos ^2(3 x)+1} \sec (3 x)",1,"(Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3","A",1
735,1,20,12,0.0162659,"\int \frac{\cot (x) \csc (x)}{a+b \csc (x)} \, dx","Integrate[(Cot[x]*Csc[x])/(a + b*Csc[x]),x]","\frac{\log (\sin (x))}{b}-\frac{\log (a \sin (x)+b)}{b}","-\frac{\log (a+b \csc (x))}{b}",1,"Log[Sin[x]]/b - Log[b + a*Sin[x]]/b","A",1
736,1,14,14,0.0175737,"\int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx","Integrate[5^Csc[3*x]*Cot[3*x]*Csc[3*x],x]","-\frac{5^{\csc (3 x)}}{3 \log (5)}","-\frac{5^{\csc (3 x)}}{3 \log (5)}",1,"-1/3*5^Csc[3*x]/Log[5]","A",1
737,1,3,3,0.0119922,"\int \frac{\cot (x) \csc (x)}{1+\csc ^2(x)} \, dx","Integrate[(Cot[x]*Csc[x])/(1 + Csc[x]^2),x]","\tan ^{-1}(\sin (x))","\tan ^{-1}(\sin (x))",1,"ArcTan[Sin[x]]","A",1
738,1,41,43,0.6586448,"\int \frac{\cot (6 x) \csc (6 x)}{\left(5-11 \csc ^2(6 x)\right)^2} \, dx","Integrate[(Cot[6*x]*Csc[6*x])/(5 - 11*Csc[6*x]^2)^2,x]","\frac{\sin (6 x)}{30 (5 \cos (12 x)+17)}-\frac{\tanh ^{-1}\left(\sqrt{\frac{5}{11}} \sin (6 x)\right)}{60 \sqrt{55}}","\frac{\sin (6 x)}{60 \left(11-5 \sin ^2(6 x)\right)}-\frac{\tanh ^{-1}\left(\sqrt{\frac{5}{11}} \sin (6 x)\right)}{60 \sqrt{55}}",1,"-1/60*ArcTanh[Sqrt[5/11]*Sin[6*x]]/Sqrt[55] + Sin[6*x]/(30*(17 + 5*Cos[12*x]))","A",1
739,1,14,14,0.0179076,"\int \frac{\cot (x) \csc (x)}{\sqrt{1+\sin ^2(x)}} \, dx","Integrate[(Cot[x]*Csc[x])/Sqrt[1 + Sin[x]^2],x]","\sqrt{\sin ^2(x)+1} (-\csc (x))","\sqrt{\sin ^2(x)+1} (-\csc (x))",1,"-(Csc[x]*Sqrt[1 + Sin[x]^2])","A",1
740,1,28,43,0.0549124,"\int \frac{\cot (5 x) \csc ^3(5 x)}{\sqrt{1+\sin ^2(5 x)}} \, dx","Integrate[(Cot[5*x]*Csc[5*x]^3)/Sqrt[1 + Sin[5*x]^2],x]","-\frac{1}{15} \sqrt{\sin ^2(5 x)+1} \csc (5 x) \left(\csc ^2(5 x)-2\right)","\frac{2}{15} \sqrt{\sin ^2(5 x)+1} \csc (5 x)-\frac{1}{15} \sqrt{\sin ^2(5 x)+1} \csc ^3(5 x)",1,"-1/15*(Csc[5*x]*(-2 + Csc[5*x]^2)*Sqrt[1 + Sin[5*x]^2])","A",1
741,1,28,43,0.0577596,"\int e^{n \sin (a+b x)} \sin (2 a+2 b x) \, dx","Integrate[E^(n*Sin[a + b*x])*Sin[2*a + 2*b*x],x]","\frac{2 e^{n \sin (a+b x)} (n \sin (a+b x)-1)}{b n^2}","\frac{2 \sin (a+b x) e^{n \sin (a+b x)}}{b n}-\frac{2 e^{n \sin (a+b x)}}{b n^2}",1,"(2*E^(n*Sin[a + b*x])*(-1 + n*Sin[a + b*x]))/(b*n^2)","A",1
742,1,28,43,0.0280522,"\int e^{n \sin (a+b x)} \sin (2 (a+b x)) \, dx","Integrate[E^(n*Sin[a + b*x])*Sin[2*(a + b*x)],x]","\frac{2 e^{n \sin (a+b x)} (n \sin (a+b x)-1)}{b n^2}","\frac{2 \sin (a+b x) e^{n \sin (a+b x)}}{b n}-\frac{2 e^{n \sin (a+b x)}}{b n^2}",1,"(2*E^(n*Sin[a + b*x])*(-1 + n*Sin[a + b*x]))/(b*n^2)","A",1
743,1,36,64,0.064465,"\int e^{n \sin \left(\frac{a}{2}+\frac{b x}{2}\right)} \sin (a+b x) \, dx","Integrate[E^(n*Sin[a/2 + (b*x)/2])*Sin[a + b*x],x]","\frac{4 e^{n \sin \left(\frac{1}{2} (a+b x)\right)} \left(n \sin \left(\frac{1}{2} (a+b x)\right)-1\right)}{b n^2}","\frac{4 \sin \left(\frac{a}{2}+\frac{b x}{2}\right) e^{n \sin \left(\frac{a}{2}+\frac{b x}{2}\right)}}{b n}-\frac{4 e^{n \sin \left(\frac{a}{2}+\frac{b x}{2}\right)}}{b n^2}",1,"(4*E^(n*Sin[(a + b*x)/2])*(-1 + n*Sin[(a + b*x)/2]))/(b*n^2)","A",1
744,1,36,64,0.0288982,"\int e^{n \sin \left(\frac{1}{2} (a+b x)\right)} \sin (a+b x) \, dx","Integrate[E^(n*Sin[(a + b*x)/2])*Sin[a + b*x],x]","\frac{4 e^{n \sin \left(\frac{1}{2} (a+b x)\right)} \left(n \sin \left(\frac{1}{2} (a+b x)\right)-1\right)}{b n^2}","\frac{4 \sin \left(\frac{a}{2}+\frac{b x}{2}\right) e^{n \sin \left(\frac{a}{2}+\frac{b x}{2}\right)}}{b n}-\frac{4 e^{n \sin \left(\frac{a}{2}+\frac{b x}{2}\right)}}{b n^2}",1,"(4*E^(n*Sin[(a + b*x)/2])*(-1 + n*Sin[(a + b*x)/2]))/(b*n^2)","A",1
745,1,28,43,0.1408423,"\int e^{n \cos (a+b x)} \sin (2 a+2 b x) \, dx","Integrate[E^(n*Cos[a + b*x])*Sin[2*a + 2*b*x],x]","-\frac{2 e^{n \cos (a+b x)} (n \cos (a+b x)-1)}{b n^2}","\frac{2 e^{n \cos (a+b x)}}{b n^2}-\frac{2 \cos (a+b x) e^{n \cos (a+b x)}}{b n}",1,"(-2*E^(n*Cos[a + b*x])*(-1 + n*Cos[a + b*x]))/(b*n^2)","A",1
746,1,28,43,0.0338926,"\int e^{n \cos (a+b x)} \sin (2 (a+b x)) \, dx","Integrate[E^(n*Cos[a + b*x])*Sin[2*(a + b*x)],x]","-\frac{2 e^{n \cos (a+b x)} (n \cos (a+b x)-1)}{b n^2}","\frac{2 e^{n \cos (a+b x)}}{b n^2}-\frac{2 \cos (a+b x) e^{n \cos (a+b x)}}{b n}",1,"(-2*E^(n*Cos[a + b*x])*(-1 + n*Cos[a + b*x]))/(b*n^2)","A",1
747,1,36,64,0.1732639,"\int e^{n \cos \left(\frac{a}{2}+\frac{b x}{2}\right)} \sin (a+b x) \, dx","Integrate[E^(n*Cos[a/2 + (b*x)/2])*Sin[a + b*x],x]","-\frac{4 e^{n \cos \left(\frac{1}{2} (a+b x)\right)} \left(n \cos \left(\frac{1}{2} (a+b x)\right)-1\right)}{b n^2}","\frac{4 e^{n \cos \left(\frac{a}{2}+\frac{b x}{2}\right)}}{b n^2}-\frac{4 \cos \left(\frac{a}{2}+\frac{b x}{2}\right) e^{n \cos \left(\frac{a}{2}+\frac{b x}{2}\right)}}{b n}",1,"(-4*E^(n*Cos[(a + b*x)/2])*(-1 + n*Cos[(a + b*x)/2]))/(b*n^2)","A",1
748,1,36,64,0.0330538,"\int e^{n \cos \left(\frac{1}{2} (a+b x)\right)} \sin (a+b x) \, dx","Integrate[E^(n*Cos[(a + b*x)/2])*Sin[a + b*x],x]","-\frac{4 e^{n \cos \left(\frac{1}{2} (a+b x)\right)} \left(n \cos \left(\frac{1}{2} (a+b x)\right)-1\right)}{b n^2}","\frac{4 e^{n \cos \left(\frac{a}{2}+\frac{b x}{2}\right)}}{b n^2}-\frac{4 \cos \left(\frac{a}{2}+\frac{b x}{2}\right) e^{n \cos \left(\frac{a}{2}+\frac{b x}{2}\right)}}{b n}",1,"(-4*E^(n*Cos[(a + b*x)/2])*(-1 + n*Cos[(a + b*x)/2]))/(b*n^2)","A",1
749,1,9,9,0.005379,"\int \csc (x) \log (\tan (x)) \sec (x) \, dx","Integrate[Csc[x]*Log[Tan[x]]*Sec[x],x]","\frac{1}{2} \log ^2(\tan (x))","\frac{1}{2} \log ^2(\tan (x))",1,"Log[Tan[x]]^2/2","A",1
750,1,9,9,0.0099399,"\int \csc (2 x) \log (\tan (x)) \, dx","Integrate[Csc[2*x]*Log[Tan[x]],x]","\frac{1}{4} \log ^2(\tan (x))","\frac{1}{4} \log ^2(\tan (x))",1,"Log[Tan[x]]^2/4","A",1
751,1,3,3,0.0004774,"\int e^{\cos ^2(x)+\sin ^2(x)} \, dx","Integrate[E^(Cos[x]^2 + Sin[x]^2),x]","e x","e x",1,"E*x","A",1
752,1,8,8,0.0036606,"\int x \sec ^2(x) \, dx","Integrate[x*Sec[x]^2,x]","x \tan (x)+\log (\cos (x))","x \tan (x)+\log (\cos (x))",1,"Log[Cos[x]] + x*Tan[x]","A",1
753,1,28,34,0.0168785,"\int x \cos ^4\left(x^2\right) \, dx","Integrate[x*Cos[x^2]^4,x]","\frac{3 x^2}{16}+\frac{1}{8} \sin \left(2 x^2\right)+\frac{1}{64} \sin \left(4 x^2\right)","\frac{3 x^2}{16}+\frac{1}{8} \sin \left(x^2\right) \cos ^3\left(x^2\right)+\frac{3}{16} \sin \left(x^2\right) \cos \left(x^2\right)",1,"(3*x^2)/16 + Sin[2*x^2]/8 + Sin[4*x^2]/64","A",1
754,1,10,10,0.0026783,"\int \sqrt{\cos (x)} \sin (x) \, dx","Integrate[Sqrt[Cos[x]]*Sin[x],x]","-\frac{2}{3} \cos ^{\frac{3}{2}}(x)","-\frac{2}{3} \cos ^{\frac{3}{2}}(x)",1,"(-2*Cos[x]^(3/2))/3","A",1
755,1,11,11,0.0081672,"\int e^{-2 x} \tan \left(e^{-2 x}\right) \, dx","Integrate[Tan[E^(-2*x)]/E^(2*x),x]","\frac{1}{2} \log \left(\cos \left(e^{-2 x}\right)\right)","\frac{1}{2} \log \left(\cos \left(e^{-2 x}\right)\right)",1,"Log[Cos[E^(-2*x)]]/2","A",1
756,1,9,7,0.0055035,"\int \frac{\sec (x) \sin (2 x)}{1+\cos (x)} \, dx","Integrate[(Sec[x]*Sin[2*x])/(1 + Cos[x]),x]","-4 \log \left(\cos \left(\frac{x}{2}\right)\right)","-2 \log (\cos (x)+1)",1,"-4*Log[Cos[x/2]]","A",1
757,1,19,19,0.0085034,"\int x \sec ^2(3 x) \, dx","Integrate[x*Sec[3*x]^2,x]","\frac{1}{3} x \tan (3 x)+\frac{1}{9} \log (\cos (3 x))","\frac{1}{3} x \tan (3 x)+\frac{1}{9} \log (\cos (3 x))",1,"Log[Cos[3*x]]/9 + (x*Tan[3*x])/3","A",1
758,1,26,37,0.0281394,"\int e^{-2 \pi  x} \cos (2 \pi  x) \, dx","Integrate[Cos[2*Pi*x]/E^(2*Pi*x),x]","\frac{e^{-2 \pi  x} (\sin (2 \pi  x)-\cos (2 \pi  x))}{4 \pi }","\frac{e^{-2 \pi  x} \sin (2 \pi  x)}{4 \pi }-\frac{e^{-2 \pi  x} \cos (2 \pi  x)}{4 \pi }",1,"(-Cos[2*Pi*x] + Sin[2*Pi*x])/(4*E^(2*Pi*x)*Pi)","A",1
759,1,49,12,0.0267192,"\int \left(\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right) \, dx","Integrate[Cos[x]^12*Sin[x]^10 - Cos[x]^10*Sin[x]^12,x]","\frac{21 \sin (2 x)}{1048576}-\frac{15 \sin (6 x)}{1048576}+\frac{15 \sin (10 x)}{2097152}-\frac{5 \sin (14 x)}{2097152}+\frac{\sin (18 x)}{2097152}-\frac{\sin (22 x)}{23068672}","\frac{1}{11} \sin ^{11}(x) \cos ^{11}(x)",1,"(21*Sin[2*x])/1048576 - (15*Sin[6*x])/1048576 + (15*Sin[10*x])/2097152 - (5*Sin[14*x])/2097152 + Sin[18*x]/2097152 - Sin[22*x]/23068672","B",1
760,1,19,9,0.0092561,"\int x \cot \left(x^2\right) \, dx","Integrate[x*Cot[x^2],x]","\frac{1}{2} \log \left(\tan \left(x^2\right)\right)+\frac{1}{2} \log \left(\cos \left(x^2\right)\right)","\frac{1}{2} \log \left(\sin \left(x^2\right)\right)",1,"Log[Cos[x^2]]/2 + Log[Tan[x^2]]/2","B",1
761,1,8,8,0.0168897,"\int x \sec ^2\left(x^2\right) \, dx","Integrate[x*Sec[x^2]^2,x]","\frac{\tan \left(x^2\right)}{2}","\frac{\tan \left(x^2\right)}{2}",1,"Tan[x^2]/2","A",1
762,1,15,15,0.0154038,"\int \frac{\sin (8 x)}{9+\sin ^4(4 x)} \, dx","Integrate[Sin[8*x]/(9 + Sin[4*x]^4),x]","\frac{1}{12} \tan ^{-1}\left(\frac{1}{3} \sin ^2(4 x)\right)","\frac{1}{12} \tan ^{-1}\left(\frac{1}{3} \sin ^2(4 x)\right)",1,"ArcTan[Sin[4*x]^2/3]/12","A",1
763,1,20,23,0.01231,"\int \frac{\cos (2 x)}{8+\sin ^2(2 x)} \, dx","Integrate[Cos[2*x]/(8 + Sin[2*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sqrt{2}}\right)}{4 \sqrt{2}}","\frac{\tan ^{-1}\left(\frac{\sin (2 x)}{2 \sqrt{2}}\right)}{4 \sqrt{2}}",1,"ArcTan[(Cos[x]*Sin[x])/Sqrt[2]]/(4*Sqrt[2])","A",1
764,1,37,37,0.0226721,"\int x \left(\cos ^3\left(x^2\right)-\sin ^3\left(x^2\right)\right) \, dx","Integrate[x*(Cos[x^2]^3 - Sin[x^2]^3),x]","-\frac{1}{6} \sin ^3\left(x^2\right)+\frac{\sin \left(x^2\right)}{2}+\frac{3 \cos \left(x^2\right)}{8}-\frac{1}{24} \cos \left(3 x^2\right)","-\frac{1}{6} \sin ^3\left(x^2\right)+\frac{\sin \left(x^2\right)}{2}-\frac{1}{6} \cos ^3\left(x^2\right)+\frac{\cos \left(x^2\right)}{2}",1,"(3*Cos[x^2])/8 - Cos[3*x^2]/24 + Sin[x^2]/2 - Sin[x^2]^3/6","A",1
765,1,12,10,0.0185602,"\int \frac{\cos (x) \sin (x)}{1-\cos (x)} \, dx","Integrate[(Cos[x]*Sin[x])/(1 - Cos[x]),x]","\cos (x)+2 \log \left(\sin \left(\frac{x}{2}\right)\right)","\cos (x)+\log (1-\cos (x))",1,"Cos[x] + 2*Log[Sin[x/2]]","A",1
766,1,8,8,0.001423,"\int x \cos \left(x^2\right) \, dx","Integrate[x*Cos[x^2],x]","\frac{\sin \left(x^2\right)}{2}","\frac{\sin \left(x^2\right)}{2}",1,"Sin[x^2]/2","A",1
767,1,10,10,0.0027179,"\int x^2 \cos \left(4 x^3\right) \, dx","Integrate[x^2*Cos[4*x^3],x]","\frac{1}{12} \sin \left(4 x^3\right)","\frac{1}{12} \sin \left(4 x^3\right)",1,"Sin[4*x^3]/12","A",1
768,1,8,8,0.0016708,"\int x^3 \cos \left(x^4\right) \, dx","Integrate[x^3*Cos[x^4],x]","\frac{\sin \left(x^4\right)}{4}","\frac{\sin \left(x^4\right)}{4}",1,"Sin[x^4]/4","A",1
769,1,10,10,0.009898,"\int x \sin \left(\frac{x^2}{2}\right) \, dx","Integrate[x*Sin[x^2/2],x]","-\cos \left(\frac{x^2}{2}\right)","-\cos \left(\frac{x^2}{2}\right)",1,"-Cos[x^2/2]","A",1
770,1,8,8,0.005658,"\int x \sec \left(x^2\right) \tan \left(x^2\right) \, dx","Integrate[x*Sec[x^2]*Tan[x^2],x]","\frac{\sec \left(x^2\right)}{2}","\frac{\sec \left(x^2\right)}{2}",1,"Sec[x^2]/2","A",1
771,1,12,10,0.0228152,"\int \frac{\tan ^2\left(\frac{1}{x}\right)}{x^2} \, dx","Integrate[Tan[x^(-1)]^2/x^2,x]","\tan ^{-1}\left(\tan \left(\frac{1}{x}\right)\right)-\tan \left(\frac{1}{x}\right)","\frac{1}{x}-\tan \left(\frac{1}{x}\right)",1,"ArcTan[Tan[x^(-1)]] - Tan[x^(-1)]","A",1
772,1,11,11,0.0177397,"\int x \tan \left(1+x^2\right) \, dx","Integrate[x*Tan[1 + x^2],x]","-\frac{1}{2} \log \left(\cos \left(x^2+1\right)\right)","-\frac{1}{2} \log \left(\cos \left(x^2+1\right)\right)",1,"-1/2*Log[Cos[1 + x^2]]","A",1
773,1,12,12,0.0056838,"\int \sin (\pi  (1+2 x)) \, dx","Integrate[Sin[Pi*(1 + 2*x)],x]","\frac{\cos (2 \pi  x)}{2 \pi }","\frac{\cos (2 \pi  x)}{2 \pi }",1,"Cos[2*Pi*x]/(2*Pi)","A",1
774,1,25,21,0.0169722,"\int \frac{\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx","Integrate[(Cot[x] + Csc[x]^2)/(1 - Cos[x]^2),x]","-\frac{2 \cot (x)}{3}-\frac{\csc ^2(x)}{2}-\frac{1}{3} \cot (x) \csc ^2(x)","-\frac{1}{3} \cot ^3(x)-\frac{\cot ^2(x)}{2}-\cot (x)",1,"(-2*Cot[x])/3 - Csc[x]^2/2 - (Cot[x]*Csc[x]^2)/3","A",1
775,1,19,19,0.0089293,"\int x^2 \cos \left(4 x^3\right) \cos \left(5 x^3\right) \, dx","Integrate[x^2*Cos[4*x^3]*Cos[5*x^3],x]","\frac{\sin \left(x^3\right)}{6}+\frac{1}{54} \sin \left(9 x^3\right)","\frac{\sin \left(x^3\right)}{6}+\frac{1}{54} \sin \left(9 x^3\right)",1,"Sin[x^3]/6 + Sin[9*x^3]/54","A",1
776,1,35,47,0.0314846,"\int x^{14} \sin \left(x^3\right) \, dx","Integrate[x^14*Sin[x^3],x]","\frac{4}{3} x^3 \left(x^6-6\right) \sin \left(x^3\right)-\frac{1}{3} \left(x^{12}-12 x^6+24\right) \cos \left(x^3\right)","-8 x^3 \sin \left(x^3\right)-8 \cos \left(x^3\right)-\frac{1}{3} x^{12} \cos \left(x^3\right)+\frac{4}{3} x^9 \sin \left(x^3\right)+4 x^6 \cos \left(x^3\right)",1,"-1/3*((24 - 12*x^6 + x^12)*Cos[x^3]) + (4*x^3*(-6 + x^6)*Sin[x^3])/3","A",1
777,1,28,35,0.0438105,"\int e^{-3 x^3} x^2 \sin \left(2 x^3\right) \, dx","Integrate[(x^2*Sin[2*x^3])/E^(3*x^3),x]","-\frac{1}{39} e^{-3 x^3} \left(3 \sin \left(2 x^3\right)+2 \cos \left(2 x^3\right)\right)","-\frac{1}{13} e^{-3 x^3} \sin \left(2 x^3\right)-\frac{2}{39} e^{-3 x^3} \cos \left(2 x^3\right)",1,"-1/39*(2*Cos[2*x^3] + 3*Sin[2*x^3])/E^(3*x^3)","A",1
778,1,4,4,0.0014525,"\int 2 x \cos \left(x^2\right) \, dx","Integrate[2*x*Cos[x^2],x]","\sin \left(x^2\right)","\sin \left(x^2\right)",1,"Sin[x^2]","A",1
779,1,6,6,0.0031675,"\int 3 x^2 \cos \left(7+x^3\right) \, dx","Integrate[3*x^2*Cos[7 + x^3],x]","\sin \left(x^3+7\right)","\sin \left(x^3+7\right)",1,"Sin[7 + x^3]","A",1
780,1,7,7,0.0073753,"\int \left(\frac{1}{1+x^2}+\sin (x)\right) \, dx","Integrate[(1 + x^2)^(-1) + Sin[x],x]","\tan ^{-1}(x)-\cos (x)","\tan ^{-1}(x)-\cos (x)",1,"ArcTan[x] - Cos[x]","A",1
781,1,21,10,0.012587,"\int x \sin \left(1+x^2\right) \, dx","Integrate[x*Sin[1 + x^2],x]","\frac{1}{2} \sin (1) \sin \left(x^2\right)-\frac{1}{2} \cos (1) \cos \left(x^2\right)","-\frac{1}{2} \cos \left(x^2+1\right)",1,"-1/2*(Cos[1]*Cos[x^2]) + (Sin[1]*Sin[x^2])/2","B",1
782,1,10,10,0.0025923,"\int x \cos \left(1+x^2\right) \, dx","Integrate[x*Cos[1 + x^2],x]","\frac{1}{2} \sin \left(x^2+1\right)","\frac{1}{2} \sin \left(x^2+1\right)",1,"Sin[1 + x^2]/2","A",1
783,1,10,10,0.0026231,"\int \left(1+x^2 \cos \left(x^3\right)\right) \, dx","Integrate[1 + x^2*Cos[x^3],x]","\frac{\sin \left(x^3\right)}{3}+x","\frac{\sin \left(x^3\right)}{3}+x",1,"x + Sin[x^3]/3","A",1
784,1,21,10,0.0129409,"\int x^2 \sin \left(1+x^3\right) \, dx","Integrate[x^2*Sin[1 + x^3],x]","\frac{1}{3} \sin (1) \sin \left(x^3\right)-\frac{1}{3} \cos (1) \cos \left(x^3\right)","-\frac{1}{3} \cos \left(x^3+1\right)",1,"-1/3*(Cos[1]*Cos[x^3]) + (Sin[1]*Sin[x^3])/3","B",1
785,1,6,6,0.001651,"\int 12 x^2 \cos \left(x^3\right) \, dx","Integrate[12*x^2*Cos[x^3],x]","4 \sin \left(x^3\right)","4 \sin \left(x^3\right)",1,"4*Sin[x^3]","A",1
786,1,14,14,0.0279813,"\int (1+x) \sin (1+x) \, dx","Integrate[(1 + x)*Sin[1 + x],x]","\sin (x+1)-(x+1) \cos (x+1)","\sin (x+1)-(x+1) \cos (x+1)",1,"-((1 + x)*Cos[1 + x]) + Sin[1 + x]","A",1
787,1,20,20,0.0070044,"\int x^5 \cos \left(x^3\right) \, dx","Integrate[x^5*Cos[x^3],x]","\frac{1}{3} x^3 \sin \left(x^3\right)+\frac{\cos \left(x^3\right)}{3}","\frac{1}{3} x^3 \sin \left(x^3\right)+\frac{\cos \left(x^3\right)}{3}",1,"Cos[x^3]/3 + (x^3*Sin[x^3])/3","A",1
788,1,16,23,0.0128394,"\int e^{-3 x} \cos (x) \, dx","Integrate[Cos[x]/E^(3*x),x]","\frac{1}{10} e^{-3 x} (\sin (x)-3 \cos (x))","\frac{1}{10} e^{-3 x} \sin (x)-\frac{3}{10} e^{-3 x} \cos (x)",1,"(-3*Cos[x] + Sin[x])/(10*E^(3*x))","A",1
789,1,20,20,0.0019846,"\int x^3 \sin \left(x^2\right) \, dx","Integrate[x^3*Sin[x^2],x]","\frac{\sin \left(x^2\right)}{2}-\frac{1}{2} x^2 \cos \left(x^2\right)","\frac{\sin \left(x^2\right)}{2}-\frac{1}{2} x^2 \cos \left(x^2\right)",1,"-1/2*(x^2*Cos[x^2]) + Sin[x^2]/2","A",1
790,1,20,20,0.0064584,"\int x^3 \cos \left(x^2\right) \, dx","Integrate[x^3*Cos[x^2],x]","\frac{1}{2} x^2 \sin \left(x^2\right)+\frac{\cos \left(x^2\right)}{2}","\frac{1}{2} x^2 \sin \left(x^2\right)+\frac{\cos \left(x^2\right)}{2}",1,"Cos[x^2]/2 + (x^2*Sin[x^2])/2","A",1
791,1,9,9,1.3838466,"\int \cos (x) \cos (2 \sin (x)) \, dx","Integrate[Cos[x]*Cos[2*Sin[x]],x]","\frac{1}{2} \sin (2 \sin (x))","\frac{1}{2} \sin (2 \sin (x))",1,"Sin[2*Sin[x]]/2","A",1
792,1,11,11,0.0264655,"\int \frac{\cos (x) \sin (x)}{1+\cos ^2(x)} \, dx","Integrate[(Cos[x]*Sin[x])/(1 + Cos[x]^2),x]","-\frac{1}{2} \log (\cos (2 x)+3)","-\frac{1}{2} \log \left(\cos ^2(x)+1\right)",1,"-1/2*Log[3 + Cos[2*x]]","A",1
793,1,10,10,0.0179504,"\int (1+\cos (x)) (x+\sin (x))^3 \, dx","Integrate[(1 + Cos[x])*(x + Sin[x])^3,x]","\frac{1}{4} (x+\sin (x))^4","\frac{1}{4} (x+\sin (x))^4",1,"(x + Sin[x])^4/4","A",1
794,1,9,9,0.0036702,"\int (1+\cos (x)) \csc ^2(x) \, dx","Integrate[(1 + Cos[x])*Csc[x]^2,x]","-\cot (x)-\csc (x)","-\cot (x)-\csc (x)",1,"-Cot[x] - Csc[x]","A",1
795,1,5,5,0.0103641,"\int \sin (x) \tan ^2(x) \, dx","Integrate[Sin[x]*Tan[x]^2,x]","\cos (x)+\sec (x)","\cos (x)+\sec (x)",1,"Cos[x] + Sec[x]","A",1
796,1,13,13,0.2928573,"\int e^{\sin (x)} \sec ^2(x) \left(x \cos ^3(x)-\sin (x)\right) \, dx","Integrate[E^Sin[x]*Sec[x]^2*(x*Cos[x]^3 - Sin[x]),x]","e^{\sin (x)} (x \cos (x)-1) \sec (x)","e^{\sin (x)} (x \cos (x)-1) \sec (x)",1,"E^Sin[x]*(-1 + x*Cos[x])*Sec[x]","A",1
797,1,9,9,0.0163148,"\int x \csc ^2(x) \, dx","Integrate[x*Csc[x]^2,x]","\log (\sin (x))-x \cot (x)","\log (\sin (x))-x \cot (x)",1,"-(x*Cot[x]) + Log[Sin[x]]","A",1
798,1,20,20,0.0112653,"\int \cos (x) \sin \left(\frac{\pi }{6}+x\right) \, dx","Integrate[Cos[x]*Sin[Pi/6 + x],x]","\frac{x}{4}-\frac{1}{4} \cos \left(2 x+\frac{\pi }{6}\right)","\frac{x}{4}-\frac{1}{4} \cos \left(2 x+\frac{\pi }{6}\right)",1,"x/4 - Cos[Pi/6 + 2*x]/4","A",1
799,1,19,19,0.0098538,"\int x \sin ^3\left(x^2\right) \, dx","Integrate[x*Sin[x^2]^3,x]","\frac{1}{24} \cos \left(3 x^2\right)-\frac{3 \cos \left(x^2\right)}{8}","\frac{1}{6} \cos ^3\left(x^2\right)-\frac{\cos \left(x^2\right)}{2}",1,"(-3*Cos[x^2])/8 + Cos[3*x^2]/24","A",1
800,1,14,14,0.0049124,"\int \sin ^2(x) \tan (x) \, dx","Integrate[Sin[x]^2*Tan[x],x]","\frac{\cos ^2(x)}{2}-\log (\cos (x))","\frac{\cos ^2(x)}{2}-\log (\cos (x))",1,"Cos[x]^2/2 - Log[Cos[x]]","A",1
801,1,20,22,0.0230923,"\int \cos ^2(x) \cot ^3(x) \, dx","Integrate[Cos[x]^2*Cot[x]^3,x]","\frac{1}{2} \left(\sin ^2(x)-\csc ^2(x)-4 \log (\sin (x))\right)","\frac{\sin ^2(x)}{2}-\frac{1}{2} \csc ^2(x)-2 \log (\sin (x))",1,"(-Csc[x]^2 - 4*Log[Sin[x]] + Sin[x]^2)/2","A",1
802,1,36,5,0.0072297,"\int \sec (x) (1-\sin (x)) \, dx","Integrate[Sec[x]*(1 - Sin[x]),x]","\log (\cos (x))-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\log (\sin (x)+1)",1,"Log[Cos[x]] - Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]]","B",1
803,1,20,7,0.0059903,"\int (1+\cos (x)) \csc (x) \, dx","Integrate[(1 + Cos[x])*Csc[x],x]","\log \left(\sin \left(\frac{x}{2}\right)\right)+\log (\sin (x))-\log \left(\cos \left(\frac{x}{2}\right)\right)","\log (1-\cos (x))",1,"-Log[Cos[x/2]] + Log[Sin[x/2]] + Log[Sin[x]]","B",1
804,1,8,5,0.0018974,"\int \cos ^2(x) \left(1-\tan ^2(x)\right) \, dx","Integrate[Cos[x]^2*(1 - Tan[x]^2),x]","\frac{1}{2} \sin (2 x)","\sin (x) \cos (x)",1,"Sin[2*x]/2","A",1
805,1,61,15,0.0094989,"\int \csc (2 x) (\cos (x)+\sin (x)) \, dx","Integrate[Csc[2*x]*(Cos[x] + Sin[x]),x]","\frac{1}{2} \log \left(\sin \left(\frac{x}{2}\right)\right)-\frac{1}{2} \log \left(\cos \left(\frac{x}{2}\right)\right)-\frac{1}{2} \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\frac{1}{2} \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\frac{1}{2} \tanh ^{-1}(\sin (x))-\frac{1}{2} \tanh ^{-1}(\cos (x))",1,"-1/2*Log[Cos[x/2]] - Log[Cos[x/2] - Sin[x/2]]/2 + Log[Sin[x/2]]/2 + Log[Cos[x/2] + Sin[x/2]]/2","B",1
806,1,26,11,0.0923992,"\int \frac{\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx","Integrate[(Cos[x]*(-3 + 2*Sin[x]))/(2 - 3*Sin[x] + Sin[x]^2),x]","\log (2-\sin (x))+2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)","\log \left(\sin ^2(x)-3 \sin (x)+2\right)",1,"2*Log[Cos[x/2] - Sin[x/2]] + Log[2 - Sin[x]]","B",1
807,1,82,20,0.1724024,"\int \frac{\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx","Integrate[(Cos[x]^2*Sin[x])/(5 + Cos[x]^2),x]","\frac{1}{20} \left(-20 \cos (x)+21 \sqrt{5} \tan ^{-1}\left(\frac{1}{\sqrt{5}}-\sqrt{\frac{6}{5}} \tan \left(\frac{x}{2}\right)\right)+21 \sqrt{5} \tan ^{-1}\left(\sqrt{\frac{6}{5}} \tan \left(\frac{x}{2}\right)+\frac{1}{\sqrt{5}}\right)-\sqrt{5} \tan ^{-1}\left(\frac{\cos (x)}{\sqrt{5}}\right)\right)","\sqrt{5} \tan ^{-1}\left(\frac{\cos (x)}{\sqrt{5}}\right)-\cos (x)",1,"(-(Sqrt[5]*ArcTan[Cos[x]/Sqrt[5]]) + 21*Sqrt[5]*ArcTan[1/Sqrt[5] - Sqrt[6/5]*Tan[x/2]] + 21*Sqrt[5]*ArcTan[1/Sqrt[5] + Sqrt[6/5]*Tan[x/2]] - 20*Cos[x])/20","B",1
808,1,11,11,0.0080155,"\int \frac{\cos (x)}{\sin (x)+\sin ^2(x)} \, dx","Integrate[Cos[x]/(Sin[x] + Sin[x]^2),x]","\log (\sin (x))-\log (\sin (x)+1)","\log (\sin (x))-\log (\sin (x)+1)",1,"Log[Sin[x]] - Log[1 + Sin[x]]","A",1
809,1,26,26,0.0431605,"\int \frac{\cos (x)}{\sin (x)+\sin ^{\sqrt{2}}(x)} \, dx","Integrate[Cos[x]/(Sin[x] + Sin[x]^Sqrt[2]),x]","\log (\sin (x))-\left(1+\sqrt{2}\right) \log \left(\sin ^{\sqrt{2}-1}(x)+1\right)","\log (\sin (x))-\left(1+\sqrt{2}\right) \log \left(\sin ^{\sqrt{2}-1}(x)+1\right)",1,"Log[Sin[x]] - (1 + Sqrt[2])*Log[1 + Sin[x]^(-1 + Sqrt[2])]","A",1
810,1,39,24,0.0331802,"\int \frac{1}{2 \sin (x)+\sin (2 x)} \, dx","Integrate[(2*Sin[x] + Sin[2*x])^(-1),x]","\frac{1-2 \cos ^2\left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)}{4 (\cos (x)+1)}","\frac{1}{8} \tan ^2\left(\frac{x}{2}\right)+\frac{1}{4} \log \left(\tan \left(\frac{x}{2}\right)\right)",1,"(1 - 2*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]))/(4*(1 + Cos[x]))","A",1
811,1,29,40,0.0410933,"\int \left(-3+4 x+x^2\right) \sin (2 x) \, dx","Integrate[(-3 + 4*x + x^2)*Sin[2*x],x]","\frac{1}{4} \left(\left(-2 x^2-8 x+7\right) \cos (2 x)+2 (x+2) \sin (2 x)\right)","-\frac{1}{2} x^2 \cos (2 x)+\frac{1}{2} x \sin (2 x)+\sin (2 x)-2 x \cos (2 x)+\frac{7}{4} \cos (2 x)",1,"((7 - 8*x - 2*x^2)*Cos[2*x] + 2*(2 + x)*Sin[2*x])/4","A",1
812,1,22,27,0.0294767,"\int e^{-3 x} \cos (4 x) \, dx","Integrate[Cos[4*x]/E^(3*x),x]","\frac{1}{25} e^{-3 x} (4 \sin (4 x)-3 \cos (4 x))","\frac{4}{25} e^{-3 x} \sin (4 x)-\frac{3}{25} e^{-3 x} \cos (4 x)",1,"(-3*Cos[4*x] + 4*Sin[4*x])/(25*E^(3*x))","A",1
813,1,31,23,0.0248392,"\int \frac{\cos (x) \sin (x)}{\sqrt{1+\sin (x)}} \, dx","Integrate[(Cos[x]*Sin[x])/Sqrt[1 + Sin[x]],x]","\frac{2 (\sin (x)-2) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}{3 \sqrt{\sin (x)+1}}","\frac{2}{3} (\sin (x)+1)^{3/2}-2 \sqrt{\sin (x)+1}",1,"(2*(Cos[x/2] + Sin[x/2])^2*(-2 + Sin[x]))/(3*Sqrt[1 + Sin[x]])","A",1
814,1,46,30,0.01741,"\int \left(x+60 \cos ^5(x) \sin ^4(x)\right) \, dx","Integrate[x + 60*Cos[x]^5*Sin[x]^4,x]","\frac{x^2}{2}+\frac{45 \sin (x)}{32}-\frac{5}{16} \sin (3 x)-\frac{3}{16} \sin (5 x)+\frac{15}{448} \sin (7 x)+\frac{5}{192} \sin (9 x)","\frac{x^2}{2}+\frac{20 \sin ^9(x)}{3}-\frac{120 \sin ^7(x)}{7}+12 \sin ^5(x)",1,"x^2/2 + (45*Sin[x])/32 - (5*Sin[3*x])/16 - (3*Sin[5*x])/16 + (15*Sin[7*x])/448 + (5*Sin[9*x])/192","A",1
815,1,6,6,0.0014758,"\int \cos (x) (\sec (x)+\tan (x)) \, dx","Integrate[Cos[x]*(Sec[x] + Tan[x]),x]","x-\cos (x)","x-\cos (x)",1,"x - Cos[x]","A",1
816,1,7,7,0.0035495,"\int \cos (x) \left(\sec ^3(x)+\tan (x)\right) \, dx","Integrate[Cos[x]*(Sec[x]^3 + Tan[x]),x]","\tan (x)-\cos (x)","\tan (x)-\cos (x)",1,"-Cos[x] + Tan[x]","A",1
817,1,10,13,0.0035905,"\int \frac{1}{2} \left(-\cot (x) \csc (x)+\csc ^2(x)\right) \, dx","Integrate[(-(Cot[x]*Csc[x]) + Csc[x]^2)/2,x]","\frac{1}{2} \tan \left(\frac{x}{2}\right)","\frac{\csc (x)}{2}-\frac{\cot (x)}{2}",1,"Tan[x/2]/2","A",1
818,1,11,11,0.005779,"\int \left(-\csc ^2(x)+\sin (2 x)\right) \, dx","Integrate[-Csc[x]^2 + Sin[2*x],x]","\cot (x)-\frac{1}{2} \cos (2 x)","\cot (x)-\frac{1}{2} \cos (2 x)",1,"-1/2*Cos[2*x] + Cot[x]","A",1
819,1,10,10,0.0100066,"\int (2 \cot (2 x)-3 \sin (3 x)) \, dx","Integrate[2*Cot[2*x] - 3*Sin[3*x],x]","\cos (3 x)+\log (\sin (2 x))","\cos (3 x)+\log (\sin (2 x))",1,"Cos[3*x] + Log[Sin[2*x]]","A",1
820,1,10,10,0.0088942,"\int x \sin \left(2 x^2\right) \, dx","Integrate[x*Sin[2*x^2],x]","-\frac{1}{4} \cos \left(2 x^2\right)","-\frac{1}{4} \cos \left(2 x^2\right)",1,"-1/4*Cos[2*x^2]","A",1
821,1,18,18,0.0311137,"\int -\cos (1-x) \sin (1-x) \sqrt{1+\sin ^2(1-x)} \, dx","Integrate[-(Cos[1 - x]*Sin[1 - x]*Sqrt[1 + Sin[1 - x]^2]),x]","\frac{1}{3} \left(\sin ^2(1-x)+1\right)^{3/2}","\frac{1}{3} \left(\sin ^2(1-x)+1\right)^{3/2}",1,"(1 + Sin[1 - x]^2)^(3/2)/3","A",1
822,1,10,10,0.005643,"\int \frac{\cos \left(\frac{1}{x}\right) \sin \left(\frac{1}{x}\right)}{x^2} \, dx","Integrate[(Cos[x^(-1)]*Sin[x^(-1)])/x^2,x]","\frac{1}{2} \cos ^2\left(\frac{1}{x}\right)","-\frac{1}{2} \sin ^2\left(\frac{1}{x}\right)",1,"Cos[x^(-1)]^2/2","A",1
823,1,25,16,0.021286,"\int \cos \left(\frac{1}{2} (1+3 x)\right) \sin ^3\left(\frac{1}{2} (1+3 x)\right) \, dx","Integrate[Cos[(1 + 3*x)/2]*Sin[(1 + 3*x)/2]^3,x]","\frac{1}{2} \left(\frac{1}{24} \cos (6 x+2)-\frac{1}{6} \cos (3 x+1)\right)","\frac{1}{6} \sin ^4\left(\frac{3 x}{2}+\frac{1}{2}\right)",1,"(-1/6*Cos[1 + 3*x] + Cos[2 + 6*x]/24)/2","A",1
824,1,7,7,0.0059166,"\int 4 x \tan \left(x^2\right) \, dx","Integrate[4*x*Tan[x^2],x]","-2 \log \left(\cos \left(x^2\right)\right)","-2 \log \left(\cos \left(x^2\right)\right)",1,"-2*Log[Cos[x^2]]","A",1
825,1,13,13,0.0173955,"\int x \sec \left(5-x^2\right) \, dx","Integrate[x*Sec[5 - x^2],x]","-\frac{1}{2} \tanh ^{-1}\left(\sin \left(5-x^2\right)\right)","-\frac{1}{2} \tanh ^{-1}\left(\sin \left(5-x^2\right)\right)",1,"-1/2*ArcTanh[Sin[5 - x^2]]","A",1
826,1,21,5,0.0143999,"\int \frac{\csc \left(\frac{1}{x}\right)}{x^2} \, dx","Integrate[Csc[x^(-1)]/x^2,x]","\log \left(\cos \left(\frac{1}{2 x}\right)\right)-\log \left(\sin \left(\frac{1}{2 x}\right)\right)","\tanh ^{-1}\left(\cos \left(\frac{1}{x}\right)\right)",1,"Log[Cos[1/(2*x)]] - Log[Sin[1/(2*x)]]","B",1
827,1,7,7,0.0066651,"\int (\csc (x)-\sec (x)) (\cos (x)+\sin (x)) \, dx","Integrate[(Csc[x] - Sec[x])*(Cos[x] + Sin[x]),x]","\log (\sin (x))+\log (\cos (x))","\log (\sin (x))+\log (\cos (x))",1,"Log[Cos[x]] + Log[Sin[x]]","A",1
828,1,4,4,0.0014057,"\int (-\cos (3 x) \sin (2 x)+\cos (2 x) \sin (3 x)) \, dx","Integrate[-(Cos[3*x]*Sin[2*x]) + Cos[2*x]*Sin[3*x],x]","-\cos (x)","-\cos (x)",1,"-Cos[x]","A",1
829,1,21,13,0.0063704,"\int 4 x \sec ^2(2 x) \, dx","Integrate[4*x*Sec[2*x]^2,x]","4 \left(\frac{1}{2} x \tan (2 x)+\frac{1}{4} \log (\cos (2 x))\right)","2 x \tan (2 x)+\log (\cos (2 x))",1,"4*(Log[Cos[2*x]]/4 + (x*Tan[2*x])/2)","A",1
830,1,18,16,0.0242023,"\int 4 \sin ^2(x) \tan ^2(x) \, dx","Integrate[4*Sin[x]^2*Tan[x]^2,x]","4 \left(-\frac{3 x}{2}+\frac{1}{4} \sin (2 x)+\tan (x)\right)","-6 x+6 \tan (x)-2 \sin ^2(x) \tan (x)",1,"4*((-3*x)/2 + Sin[2*x]/4 + Tan[x])","A",1
831,1,26,32,0.0210337,"\int \cos ^4(x) \cot ^2(x) \, dx","Integrate[Cos[x]^4*Cot[x]^2,x]","-\frac{15 x}{8}-\frac{1}{2} \sin (2 x)-\frac{1}{32} \sin (4 x)-\cot (x)","-\frac{15 x}{8}-\frac{15 \cot (x)}{8}+\frac{1}{4} \cos ^4(x) \cot (x)+\frac{5}{8} \cos ^2(x) \cot (x)",1,"(-15*x)/8 - Cot[x] - Sin[2*x]/2 - Sin[4*x]/32","A",1
832,1,16,18,0.0070914,"\int 16 \cos ^2(x) \sin ^2(x) \, dx","Integrate[16*Cos[x]^2*Sin[x]^2,x]","4 \left(\frac{x}{2}-\frac{1}{8} \sin (4 x)\right)","2 x-4 \sin (x) \cos ^3(x)+2 \sin (x) \cos (x)",1,"4*(x/2 - Sin[4*x]/8)","A",1
833,1,32,34,0.0089123,"\int 8 \cos ^2(x) \sin ^4(x) \, dx","Integrate[8*Cos[x]^2*Sin[x]^4,x]","8 \left(\frac{x}{16}-\frac{1}{64} \sin (2 x)-\frac{1}{64} \sin (4 x)+\frac{1}{192} \sin (6 x)\right)","\frac{x}{2}-\frac{4}{3} \sin ^3(x) \cos ^3(x)-\sin (x) \cos ^3(x)+\frac{1}{2} \sin (x) \cos (x)",1,"8*(x/16 - Sin[2*x]/64 - Sin[4*x]/64 + Sin[6*x]/192)","A",1
834,1,33,13,0.009518,"\int 35 \cos ^3(x) \sin ^4(x) \, dx","Integrate[35*Cos[x]^3*Sin[x]^4,x]","35 \left(\frac{3 \sin (x)}{64}-\frac{1}{64} \sin (3 x)-\frac{1}{320} \sin (5 x)+\frac{1}{448} \sin (7 x)\right)","7 \sin ^5(x)-5 \sin ^7(x)",1,"35*((3*Sin[x])/64 - Sin[3*x]/64 - Sin[5*x]/320 + Sin[7*x]/448)","B",1
835,1,24,46,0.0071248,"\int 4 \cos ^4(x) \sin ^4(x) \, dx","Integrate[4*Cos[x]^4*Sin[x]^4,x]","4 \left(\frac{3 x}{128}-\frac{1}{128} \sin (4 x)+\frac{\sin (8 x)}{1024}\right)","\frac{3 x}{32}-\frac{1}{2} \sin ^3(x) \cos ^5(x)-\frac{1}{4} \sin (x) \cos ^5(x)+\frac{1}{16} \sin (x) \cos ^3(x)+\frac{3}{32} \sin (x) \cos (x)",1,"4*((3*x)/128 - Sin[4*x]/128 + Sin[8*x]/1024)","A",1
836,1,9,9,0.004787,"\int \frac{\cos (x)}{-\sin (x)+\sin ^3(x)} \, dx","Integrate[Cos[x]/(-Sin[x] + Sin[x]^3),x]","\log (\cos (x))-\log (\sin (x))","\log (\cos (x))-\log (\sin (x))",1,"Log[Cos[x]] - Log[Sin[x]]","A",1
837,1,17,14,0.0050115,"\int \left(-1+2 \cos ^2(x)+\cos (x) \sin (x)\right) \, dx","Integrate[-1 + 2*Cos[x]^2 + Cos[x]*Sin[x],x]","\frac{1}{2} \sin (2 x)-\frac{\cos ^2(x)}{2}","\frac{\sin ^2(x)}{2}+\sin (x) \cos (x)",1,"-1/2*Cos[x]^2 + Sin[2*x]/2","A",1
838,1,1,1,0.0003433,"\int \left(\cos ^2(x)+\sin ^2(x)\right) \, dx","Integrate[Cos[x]^2 + Sin[x]^2,x]","x","x",1,"x","A",1
839,1,8,6,0.0023438,"\int \left(-\cos ^2(x)+\sin ^2(x)\right) \, dx","Integrate[-Cos[x]^2 + Sin[x]^2,x]","-\frac{1}{2} \sin (2 x)","\sin (x) (-\cos (x))",1,"-1/2*Sin[2*x]","A",1
840,1,9,9,0.0062285,"\int 2^{\sin (x)} \cos (x) \, dx","Integrate[2^Sin[x]*Cos[x],x]","\frac{2^{\sin (x)}}{\log (2)}","\frac{2^{\sin (x)}}{\log (2)}",1,"2^Sin[x]/Log[2]","A",1
841,1,8,8,0.0044448,"\int \left(\tan ^3(x)+\tan ^5(x)\right) \, dx","Integrate[Tan[x]^3 + Tan[x]^5,x]","\frac{\tan ^4(x)}{4}","\frac{\tan ^4(x)}{4}",1,"Tan[x]^4/4","A",1
842,1,6,6,0.0236287,"\int x \sec (x) (2+x \tan (x)) \, dx","Integrate[x*Sec[x]*(2 + x*Tan[x]),x]","x^2 \sec (x)","x^2 \sec (x)",1,"x^2*Sec[x]","A",1
843,1,8,8,0.0177117,"\int \frac{\cot \left(\sqrt{x}\right) \csc \left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[(Cot[Sqrt[x]]*Csc[Sqrt[x]])/Sqrt[x],x]","-2 \csc \left(\sqrt{x}\right)","-2 \csc \left(\sqrt{x}\right)",1,"-2*Csc[Sqrt[x]]","A",1
844,1,12,8,0.0124045,"\int \frac{\cos \left(\sqrt{x}\right) \sin \left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[(Cos[Sqrt[x]]*Sin[Sqrt[x]])/Sqrt[x],x]","-\frac{1}{2} \cos \left(2 \sqrt{x}\right)","\sin ^2\left(\sqrt{x}\right)",1,"-1/2*Cos[2*Sqrt[x]]","A",1
845,1,8,8,0.016671,"\int \frac{\sec \left(\sqrt{x}\right) \tan \left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[(Sec[Sqrt[x]]*Tan[Sqrt[x]])/Sqrt[x],x]","2 \sec \left(\sqrt{x}\right)","2 \sec \left(\sqrt{x}\right)",1,"2*Sec[Sqrt[x]]","A",1
846,1,55,55,0.0784842,"\int \frac{\sin ^2(x)}{a+b \sin (2 x)} \, dx","Integrate[Sin[x]^2/(a + b*Sin[2*x]),x]","\frac{\tan ^{-1}\left(\frac{a \tan (x)+b}{\sqrt{a^2-b^2}}\right)}{2 \sqrt{a^2-b^2}}-\frac{\log (a+b \sin (2 x))}{4 b}","\frac{\tan ^{-1}\left(\frac{a \tan (x)+b}{\sqrt{a^2-b^2}}\right)}{2 \sqrt{a^2-b^2}}-\frac{\log (a+b \sin (2 x))}{4 b}",1,"ArcTan[(b + a*Tan[x])/Sqrt[a^2 - b^2]]/(2*Sqrt[a^2 - b^2]) - Log[a + b*Sin[2*x]]/(4*b)","A",1
847,1,54,55,0.0604952,"\int \frac{\cos ^2(x)}{a+b \sin (2 x)} \, dx","Integrate[Cos[x]^2/(a + b*Sin[2*x]),x]","\frac{1}{4} \left(\frac{2 \tan ^{-1}\left(\frac{a \tan (x)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{\log (a+b \sin (2 x))}{b}\right)","\frac{\tan ^{-1}\left(\frac{a \tan (x)+b}{\sqrt{a^2-b^2}}\right)}{2 \sqrt{a^2-b^2}}+\frac{\log (a+b \sin (2 x))}{4 b}",1,"((2*ArcTan[(b + a*Tan[x])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + Log[a + b*Sin[2*x]]/b)/4","A",1
848,1,48,52,0.0865196,"\int \frac{\sin ^2(x)}{a+b \cos (2 x)} \, dx","Integrate[Sin[x]^2/(a + b*Cos[2*x]),x]","-\frac{\frac{(a+b) \tanh ^{-1}\left(\frac{(a-b) \tan (x)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+x}{2 b}","\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (x)}{\sqrt{a+b}}\right)}{2 b \sqrt{a-b}}-\frac{x}{2 b}",1,"-1/2*(x + ((a + b)*ArcTanh[((a - b)*Tan[x])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2])/b","A",1
849,1,50,52,0.0540284,"\int \frac{\cos ^2(x)}{a+b \cos (2 x)} \, dx","Integrate[Cos[x]^2/(a + b*Cos[2*x]),x]","\frac{\frac{(a-b) \tanh ^{-1}\left(\frac{(a-b) \tan (x)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+x}{2 b}","\frac{x}{2 b}-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (x)}{\sqrt{a+b}}\right)}{2 b \sqrt{a+b}}",1,"(x + ((a - b)*ArcTanh[((a - b)*Tan[x])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2])/(2*b)","A",1
850,1,31,30,0.0384811,"\int \frac{\tan (c+d x)}{\sqrt{a \sin ^2(c+d x)}} \, dx","Integrate[Tan[c + d*x]/Sqrt[a*Sin[c + d*x]^2],x]","\frac{\sin (c+d x) \tanh ^{-1}(\sin (c+d x))}{d \sqrt{a \sin ^2(c+d x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin ^2(c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(ArcTanh[Sin[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a*Sin[c + d*x]^2])","A",1
851,1,49,31,0.0617737,"\int \frac{\cot (c+d x)}{\sqrt{a \cos ^2(c+d x)}} \, dx","Integrate[Cot[c + d*x]/Sqrt[a*Cos[c + d*x]^2],x]","\frac{\cos (c+d x) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \sqrt{a \cos ^2(c+d x)}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"(Cos[c + d*x]*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]))/(d*Sqrt[a*Cos[c + d*x]^2])","A",1
852,1,8,8,0.0029123,"\int \frac{x \cos \left(x^2\right)}{\sqrt{\sin \left(x^2\right)}} \, dx","Integrate[(x*Cos[x^2])/Sqrt[Sin[x^2]],x]","\sqrt{\sin \left(x^2\right)}","\sqrt{\sin \left(x^2\right)}",1,"Sqrt[Sin[x^2]]","A",1
853,1,18,19,0.0142706,"\int \frac{\cos (x)}{\sqrt{1-\cos (2 x)}} \, dx","Integrate[Cos[x]/Sqrt[1 - Cos[2*x]],x]","\frac{\sin (x) \log (\sin (x))}{\sqrt{1-\cos (2 x)}}","\frac{\sin (x) \log (\sin (x))}{\sqrt{2} \sqrt{\sin ^2(x)}}",1,"(Log[Sin[x]]*Sin[x])/Sqrt[1 - Cos[2*x]]","A",1
854,1,16,29,0.0173257,"\int \frac{\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx","Integrate[(Cos[Log[x]]^2*Sin[Log[x]]^2)/x,x]","\frac{\log (x)}{8}-\frac{1}{32} \sin (4 \log (x))","\frac{\log (x)}{8}-\frac{1}{4} \sin (\log (x)) \cos ^3(\log (x))+\frac{1}{8} \sin (\log (x)) \cos (\log (x))",1,"Log[x]/8 - Sin[4*Log[x]]/32","A",1
855,1,29,29,0.0964333,"\int \frac{\sin ^3(x)}{\cos ^3(x)+\sin ^3(x)} \, dx","Integrate[Sin[x]^3/(Cos[x]^3 + Sin[x]^3),x]","\frac{x}{2}+\frac{1}{3} \log (2-\sin (2 x))-\frac{1}{6} \log (\sin (x)+\cos (x))","\frac{x}{2}+\frac{1}{3} \log (2-\sin (2 x))-\frac{1}{6} \log (\sin (x)+\cos (x))",1,"x/2 - Log[Cos[x] + Sin[x]]/6 + Log[2 - Sin[2*x]]/3","A",1
856,1,29,29,0.0730815,"\int \frac{\cos ^3(x)}{\cos ^3(x)+\sin ^3(x)} \, dx","Integrate[Cos[x]^3/(Cos[x]^3 + Sin[x]^3),x]","\frac{x}{2}-\frac{1}{3} \log (2-\sin (2 x))+\frac{1}{6} \log (\sin (x)+\cos (x))","\frac{x}{2}-\frac{1}{3} \log (2-\sin (2 x))+\frac{1}{6} \log (\sin (x)+\cos (x))",1,"x/2 + Log[Cos[x] + Sin[x]]/6 - Log[2 - Sin[2*x]]/3","A",1
857,1,38,44,0.0766468,"\int \frac{\sec (x)}{-5+\cos ^2(x)+4 \sin (x)} \, dx","Integrate[Sec[x]/(-5 + Cos[x]^2 + 4*Sin[x]),x]","\frac{1}{18} \left(-\frac{6}{\sin (x)-2}+9 \log (1-\sin (x))-8 \log (2-\sin (x))-\log (\sin (x)+1)\right)","\frac{1}{3 (2-\sin (x))}+\frac{1}{2} \log (1-\sin (x))-\frac{4}{9} \log (2-\sin (x))-\frac{1}{18} \log (\sin (x)+1)",1,"(9*Log[1 - Sin[x]] - 8*Log[2 - Sin[x]] - Log[1 + Sin[x]] - 6/(-2 + Sin[x]))/18","A",1
858,1,19,19,0.0631555,"\int \frac{1}{\cos ^{\frac{3}{2}}(x) \sqrt{3 \cos (x)+\sin (x)}} \, dx","Integrate[1/(Cos[x]^(3/2)*Sqrt[3*Cos[x] + Sin[x]]),x]","\frac{2 \sqrt{\sin (x)+3 \cos (x)}}{\sqrt{\cos (x)}}","\frac{2 \sqrt{\sin (x)+3 \cos (x)}}{\sqrt{\cos (x)}}",1,"(2*Sqrt[3*Cos[x] + Sin[x]])/Sqrt[Cos[x]]","A",1
859,1,68,44,0.3950576,"\int \frac{\csc (x) \sqrt{\cos (x)+\sin (x)}}{\cos ^{\frac{3}{2}}(x)} \, dx","Integrate[(Csc[x]*Sqrt[Cos[x] + Sin[x]])/Cos[x]^(3/2),x]","\frac{2 \left(\sin (x)+\cos (x)-\sqrt{\cos (x)} \sqrt{\sqrt{\sin ^2(x)}+\cos (x)} \coth ^{-1}\left(\frac{\sqrt{\sqrt{\sin ^2(x)}+\cos (x)}}{\sqrt{\cos (x)}}\right)\right)}{\sqrt{\cos (x)} \sqrt{\sin (x)+\cos (x)}}","-\log (\sin (x))+\frac{2 \sqrt{\sin (x)+\cos (x)}}{\sqrt{\cos (x)}}+2 \log \left(\sqrt{\sin (x)+\cos (x)}-\sqrt{\cos (x)}\right)",1,"(2*(Cos[x] + Sin[x] - ArcCoth[Sqrt[Cos[x] + Sqrt[Sin[x]^2]]/Sqrt[Cos[x]]]*Sqrt[Cos[x]]*Sqrt[Cos[x] + Sqrt[Sin[x]^2]]))/(Sqrt[Cos[x]]*Sqrt[Cos[x] + Sin[x]])","A",0
860,1,17,19,0.0132886,"\int \frac{\cos (x)+\sin (x)}{\sqrt{1+\sin (2 x)}} \, dx","Integrate[(Cos[x] + Sin[x])/Sqrt[1 + Sin[2*x]],x]","\frac{x (\sin (x)+\cos (x))}{\sqrt{\sin (2 x)+1}}","\frac{x \sqrt{\sin (2 x)+1}}{\sin (x)+\cos (x)}",1,"(x*(Cos[x] + Sin[x]))/Sqrt[1 + Sin[2*x]]","A",1
861,1,37,13,0.0420222,"\int \sec (x) \sqrt{\sec (x)+\tan (x)} \, dx","Integrate[Sec[x]*Sqrt[Sec[x] + Tan[x]],x]","2 \sqrt{\frac{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}}","2 \sqrt{(\sin (x)+1) \sec (x)}",1,"2*Sqrt[(Cos[x/2] + Sin[x/2])/(Cos[x/2] - Sin[x/2])]","B",1
862,1,14,14,0.0520018,"\int \sec (x) \sqrt{4+3 \sec (x)} \tan (x) \, dx","Integrate[Sec[x]*Sqrt[4 + 3*Sec[x]]*Tan[x],x]","\frac{2}{9} (3 \sec (x)+4)^{3/2}","\frac{2}{9} (3 \sec (x)+4)^{3/2}",1,"(2*(4 + 3*Sec[x])^(3/2))/9","A",1
863,1,30,25,0.1914918,"\int \sec (x) \sqrt{1+\sec (x)} \tan ^3(x) \, dx","Integrate[Sec[x]*Sqrt[1 + Sec[x]]*Tan[x]^3,x]","-\frac{8}{35} \cos ^4\left(\frac{x}{2}\right) (9 \cos (x)-5) \sec ^3(x) \sqrt{\sec (x)+1}","\frac{2}{7} (\sec (x)+1)^{7/2}-\frac{4}{5} (\sec (x)+1)^{5/2}",1,"(-8*Cos[x/2]^4*(-5 + 9*Cos[x])*Sec[x]^3*Sqrt[1 + Sec[x]])/35","A",1
864,1,18,25,0.0376463,"\int \cot ^3(x) \csc (x) \sqrt{1+\csc (x)} \, dx","Integrate[Cot[x]^3*Csc[x]*Sqrt[1 + Csc[x]],x]","-\frac{2}{35} (\csc (x)+1)^{5/2} (5 \csc (x)-9)","\frac{4}{5} (\csc (x)+1)^{5/2}-\frac{2}{7} (\csc (x)+1)^{7/2}",1,"(-2*(1 + Csc[x])^(5/2)*(-9 + 5*Csc[x]))/35","A",1
865,1,17,20,0.44297,"\int \sqrt{\csc (x)} (x \cos (x)-4 \sec (x) \tan (x)) \, dx","Integrate[Sqrt[Csc[x]]*(x*Cos[x] - 4*Sec[x]*Tan[x]),x]","\frac{2 (x \csc (x)-2 \sec (x))}{\csc ^{\frac{3}{2}}(x)}","\frac{2 x}{\sqrt{\csc (x)}}-\frac{4 \sec (x)}{\csc ^{\frac{3}{2}}(x)}",1,"(2*(x*Csc[x] - 2*Sec[x]))/Csc[x]^(3/2)","A",1
866,1,40,76,0.0887688,"\int \cot (x) \sqrt{-1+\csc ^2(x)} \left(1-\sin ^2(x)\right)^3 \, dx","Integrate[Cot[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3,x]","\frac{1}{384} \sqrt{\cot ^2(x)} \sec (x) (-840 x \sin (x)-525 \cos (x)+126 \cos (3 x)+14 \cos (5 x)+\cos (7 x))","-\frac{35}{16} \sqrt{\cot ^2(x)}+\frac{1}{6} \cos ^6(x) \sqrt{\cot ^2(x)}+\frac{7}{24} \cos ^4(x) \sqrt{\cot ^2(x)}+\frac{35}{48} \cos ^2(x) \sqrt{\cot ^2(x)}-\frac{35}{16} x \tan (x) \sqrt{\cot ^2(x)}",1,"(Sqrt[Cot[x]^2]*Sec[x]*(-525*Cos[x] + 126*Cos[3*x] + 14*Cos[5*x] + Cos[7*x] - 840*x*Sin[x]))/384","A",1
867,1,55,81,0.0612768,"\int \cos (x) \sqrt{-1+\csc ^2(x)} \left(1-\sin ^2(x)\right)^3 \, dx","Integrate[Cos[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3,x]","\frac{\tan (x) \sqrt{\cot ^2(x)} \left(9765 \cos (x)+1295 \cos (3 x)+189 \cos (5 x)+15 \cos (7 x)+6720 \log \left(\sin \left(\frac{x}{2}\right)\right)-6720 \log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{6720}","\sin (x) \sqrt{\cot ^2(x)}+\frac{1}{7} \sin (x) \cos ^6(x) \sqrt{\cot ^2(x)}+\frac{1}{5} \sin (x) \cos ^4(x) \sqrt{\cot ^2(x)}+\frac{1}{3} \sin (x) \cos ^2(x) \sqrt{\cot ^2(x)}-\tan (x) \sqrt{\cot ^2(x)} \tanh ^{-1}(\cos (x))",1,"(Sqrt[Cot[x]^2]*(9765*Cos[x] + 1295*Cos[3*x] + 189*Cos[5*x] + 15*Cos[7*x] - 6720*Log[Cos[x/2]] + 6720*Log[Sin[x/2]])*Tan[x])/6720","A",1
868,1,69,76,0.0576347,"\int \frac{x \csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}} \, dx","Integrate[(x*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2],x]","\frac{\sec (x) \left(i \text{Li}_2\left(-e^{i x}\right)-i \text{Li}_2\left(e^{i x}\right)+x \left(\log \left(1-e^{i x}\right)-\log \left(1+e^{i x}\right)\right)\right)}{\sqrt{a \sec ^2(x)}}","\frac{i \text{Li}_2\left(-e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{i \text{Li}_2\left(e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{2 x \sec (x) \tanh ^{-1}\left(e^{i x}\right)}{\sqrt{a \sec ^2(x)}}",1,"((x*(Log[1 - E^(I*x)] - Log[1 + E^(I*x)]) + I*PolyLog[2, -E^(I*x)] - I*PolyLog[2, E^(I*x)])*Sec[x])/Sqrt[a*Sec[x]^2]","A",1
869,1,99,128,0.0661997,"\int \frac{x^2 \csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}} \, dx","Integrate[(x^2*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2],x]","\frac{\sec (x) \left(2 i x \text{Li}_2\left(-e^{i x}\right)-2 i x \text{Li}_2\left(e^{i x}\right)-2 \text{Li}_3\left(-e^{i x}\right)+2 \text{Li}_3\left(e^{i x}\right)+x^2 \log \left(1-e^{i x}\right)-x^2 \log \left(1+e^{i x}\right)\right)}{\sqrt{a \sec ^2(x)}}","\frac{2 i x \text{Li}_2\left(-e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{2 i x \text{Li}_2\left(e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{2 \text{Li}_3\left(-e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}+\frac{2 \text{Li}_3\left(e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{2 x^2 \sec (x) \tanh ^{-1}\left(e^{i x}\right)}{\sqrt{a \sec ^2(x)}}",1,"((x^2*Log[1 - E^(I*x)] - x^2*Log[1 + E^(I*x)] + (2*I)*x*PolyLog[2, -E^(I*x)] - (2*I)*x*PolyLog[2, E^(I*x)] - 2*PolyLog[3, -E^(I*x)] + 2*PolyLog[3, E^(I*x)])*Sec[x])/Sqrt[a*Sec[x]^2]","A",1
870,1,147,186,0.0955326,"\int \frac{x^3 \csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}} \, dx","Integrate[(x^3*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2],x]","-\frac{i \sec (x) \left(-24 x^2 \text{Li}_2\left(e^{-i x}\right)-24 x^2 \text{Li}_2\left(-e^{i x}\right)+48 i x \text{Li}_3\left(e^{-i x}\right)-48 i x \text{Li}_3\left(-e^{i x}\right)+48 \text{Li}_4\left(e^{-i x}\right)+48 \text{Li}_4\left(-e^{i x}\right)-2 x^4+8 i x^3 \log \left(1-e^{-i x}\right)-8 i x^3 \log \left(1+e^{i x}\right)+\pi ^4\right)}{8 \sqrt{a \sec ^2(x)}}","\frac{3 i x^2 \text{Li}_2\left(-e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{3 i x^2 \text{Li}_2\left(e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{6 x \text{Li}_3\left(-e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}+\frac{6 x \text{Li}_3\left(e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{6 i \text{Li}_4\left(-e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}+\frac{6 i \text{Li}_4\left(e^{i x}\right) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{2 x^3 \sec (x) \tanh ^{-1}\left(e^{i x}\right)}{\sqrt{a \sec ^2(x)}}",1,"((-1/8*I)*(Pi^4 - 2*x^4 + (8*I)*x^3*Log[1 - E^((-I)*x)] - (8*I)*x^3*Log[1 + E^(I*x)] - 24*x^2*PolyLog[2, E^((-I)*x)] - 24*x^2*PolyLog[2, -E^(I*x)] + (48*I)*x*PolyLog[3, E^((-I)*x)] - (48*I)*x*PolyLog[3, -E^(I*x)] + 48*PolyLog[4, E^((-I)*x)] + 48*PolyLog[4, -E^(I*x)])*Sec[x])/Sqrt[a*Sec[x]^2]","A",1
871,1,50,81,0.0397158,"\int \frac{x \csc (x) \sec (x)}{\sqrt{a \sec ^4(x)}} \, dx","Integrate[(x*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^4],x]","-\frac{i \sec ^2(x) \left(\text{Li}_2\left(e^{2 i x}\right)+x \left(x+2 i \log \left(1-e^{2 i x}\right)\right)\right)}{2 \sqrt{a \sec ^4(x)}}","-\frac{i \text{Li}_2\left(e^{2 i x}\right) \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}}-\frac{i x^2 \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}}+\frac{x \log \left(1-e^{2 i x}\right) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}",1,"((-1/2*I)*(x*(x + (2*I)*Log[1 - E^((2*I)*x)]) + PolyLog[2, E^((2*I)*x)])*Sec[x]^2)/Sqrt[a*Sec[x]^4]","A",1
872,1,75,109,0.0597158,"\int \frac{x^2 \csc (x) \sec (x)}{\sqrt{a \sec ^4(x)}} \, dx","Integrate[(x^2*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^4],x]","\frac{\sec ^2(x) \left(24 i x \text{Li}_2\left(e^{-2 i x}\right)+12 \text{Li}_3\left(e^{-2 i x}\right)+8 i x^3+24 x^2 \log \left(1-e^{-2 i x}\right)-i \pi ^3\right)}{24 \sqrt{a \sec ^4(x)}}","-\frac{i x \text{Li}_2\left(e^{2 i x}\right) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}+\frac{\text{Li}_3\left(e^{2 i x}\right) \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}}-\frac{i x^3 \sec ^2(x)}{3 \sqrt{a \sec ^4(x)}}+\frac{x^2 \log \left(1-e^{2 i x}\right) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}",1,"(((-I)*Pi^3 + (8*I)*x^3 + 24*x^2*Log[1 - E^((-2*I)*x)] + (24*I)*x*PolyLog[2, E^((-2*I)*x)] + 12*PolyLog[3, E^((-2*I)*x)])*Sec[x]^2)/(24*Sqrt[a*Sec[x]^4])","A",1
873,1,87,143,0.068489,"\int \frac{x^3 \csc (x) \sec (x)}{\sqrt{a \sec ^4(x)}} \, dx","Integrate[(x^3*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^4],x]","-\frac{i \sec ^2(x) \left(-96 x^2 \text{Li}_2\left(e^{-2 i x}\right)+96 i x \text{Li}_3\left(e^{-2 i x}\right)+48 \text{Li}_4\left(e^{-2 i x}\right)-16 x^4+64 i x^3 \log \left(1-e^{-2 i x}\right)+\pi ^4\right)}{64 \sqrt{a \sec ^4(x)}}","-\frac{3 i x^2 \text{Li}_2\left(e^{2 i x}\right) \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}}+\frac{3 x \text{Li}_3\left(e^{2 i x}\right) \sec ^2(x)}{2 \sqrt{a \sec ^4(x)}}+\frac{3 i \text{Li}_4\left(e^{2 i x}\right) \sec ^2(x)}{4 \sqrt{a \sec ^4(x)}}-\frac{i x^4 \sec ^2(x)}{4 \sqrt{a \sec ^4(x)}}+\frac{x^3 \log \left(1-e^{2 i x}\right) \sec ^2(x)}{\sqrt{a \sec ^4(x)}}",1,"((-1/64*I)*(Pi^4 - 16*x^4 + (64*I)*x^3*Log[1 - E^((-2*I)*x)] - 96*x^2*PolyLog[2, E^((-2*I)*x)] + (96*I)*x*PolyLog[3, E^((-2*I)*x)] + 48*PolyLog[4, E^((-2*I)*x)])*Sec[x]^2)/Sqrt[a*Sec[x]^4]","A",1
874,1,108,105,0.0773823,"\int x \csc (x) \sec (x) \sqrt{a \sec ^2(x)} \, dx","Integrate[x*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^2],x]","\sqrt{a \sec ^2(x)} \left(i \left(\text{Li}_2\left(-e^{i x}\right)-\text{Li}_2\left(e^{i x}\right)\right) \cos (x)+x+x \left(\log \left(1-e^{i x}\right)-\log \left(1+e^{i x}\right)\right) \cos (x)+\cos (x) \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\cos (x) \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)","i \text{Li}_2\left(-e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}-i \text{Li}_2\left(e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}+x \sqrt{a \sec ^2(x)}-2 x \cos (x) \tanh ^{-1}\left(e^{i x}\right) \sqrt{a \sec ^2(x)}-\cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))",1,"(x + x*Cos[x]*(Log[1 - E^(I*x)] - Log[1 + E^(I*x)]) + Cos[x]*Log[Cos[x/2] - Sin[x/2]] - Cos[x]*Log[Cos[x/2] + Sin[x/2]] + I*Cos[x]*(PolyLog[2, -E^(I*x)] - PolyLog[2, E^(I*x)]))*Sqrt[a*Sec[x]^2]","A",1
875,1,174,225,0.1221475,"\int x^2 \csc (x) \sec (x) \sqrt{a \sec ^2(x)} \, dx","Integrate[x^2*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^2],x]","\sqrt{a \sec ^2(x)} \left(2 i x \left(\text{Li}_2\left(-e^{i x}\right)-\text{Li}_2\left(e^{i x}\right)\right) \cos (x)+2 \left(\text{Li}_3\left(e^{i x}\right)-\text{Li}_3\left(-e^{i x}\right)\right) \cos (x)-2 \cos (x) \left(i \left(\text{Li}_2\left(-i e^{i x}\right)-\text{Li}_2\left(i e^{i x}\right)\right)+x \left(\log \left(1-i e^{i x}\right)-\log \left(1+i e^{i x}\right)\right)\right)+x^2+x^2 \left(\log \left(1-e^{i x}\right)-\log \left(1+e^{i x}\right)\right) \cos (x)\right)","2 i x \text{Li}_2\left(-e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}-2 i x \text{Li}_2\left(e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}-2 i \text{Li}_2\left(-i e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}+2 i \text{Li}_2\left(i e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}-2 \text{Li}_3\left(-e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}+2 \text{Li}_3\left(e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}+x^2 \sqrt{a \sec ^2(x)}-2 x^2 \cos (x) \tanh ^{-1}\left(e^{i x}\right) \sqrt{a \sec ^2(x)}+4 i x \cos (x) \tan ^{-1}\left(e^{i x}\right) \sqrt{a \sec ^2(x)}",1,"(x^2 + x^2*Cos[x]*(Log[1 - E^(I*x)] - Log[1 + E^(I*x)]) - 2*Cos[x]*(x*(Log[1 - I*E^(I*x)] - Log[1 + I*E^(I*x)]) + I*(PolyLog[2, (-I)*E^(I*x)] - PolyLog[2, I*E^(I*x)])) + (2*I)*x*Cos[x]*(PolyLog[2, -E^(I*x)] - PolyLog[2, E^(I*x)]) + 2*Cos[x]*(-PolyLog[3, -E^(I*x)] + PolyLog[3, E^(I*x)]))*Sqrt[a*Sec[x]^2]","A",1
876,1,290,341,0.4118638,"\int x^3 \csc (x) \sec (x) \sqrt{a \sec ^2(x)} \, dx","Integrate[x^3*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^2],x]","\frac{1}{8} \sqrt{a \sec ^2(x)} \left(24 i x^2 \text{Li}_2\left(e^{-i x}\right) \cos (x)+24 i x^2 \text{Li}_2\left(-e^{i x}\right) \cos (x)-48 i x \text{Li}_2\left(-i e^{i x}\right) \cos (x)+48 i x \text{Li}_2\left(i e^{i x}\right) \cos (x)+48 x \text{Li}_3\left(e^{-i x}\right) \cos (x)-48 x \text{Li}_3\left(-e^{i x}\right) \cos (x)+48 \text{Li}_3\left(-i e^{i x}\right) \cos (x)-48 \text{Li}_3\left(i e^{i x}\right) \cos (x)-48 i \text{Li}_4\left(e^{-i x}\right) \cos (x)-48 i \text{Li}_4\left(-e^{i x}\right) \cos (x)+2 i x^4 \cos (x)+8 x^3+8 x^3 \log \left(1-e^{-i x}\right) \cos (x)-8 x^3 \log \left(1+e^{i x}\right) \cos (x)-24 x^2 \log \left(1-i e^{i x}\right) \cos (x)+24 x^2 \log \left(1+i e^{i x}\right) \cos (x)-i \pi ^4 \cos (x)\right)","3 i x^2 \text{Li}_2\left(-e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}-3 i x^2 \text{Li}_2\left(e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}-6 i x \text{Li}_2\left(-i e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}+6 i x \text{Li}_2\left(i e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}-6 x \text{Li}_3\left(-e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}+6 x \text{Li}_3\left(e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}+6 \text{Li}_3\left(-i e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}-6 \text{Li}_3\left(i e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}-6 i \text{Li}_4\left(-e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}+6 i \text{Li}_4\left(e^{i x}\right) \cos (x) \sqrt{a \sec ^2(x)}+x^3 \sqrt{a \sec ^2(x)}-2 x^3 \cos (x) \tanh ^{-1}\left(e^{i x}\right) \sqrt{a \sec ^2(x)}+6 i x^2 \cos (x) \tan ^{-1}\left(e^{i x}\right) \sqrt{a \sec ^2(x)}",1,"((8*x^3 - I*Pi^4*Cos[x] + (2*I)*x^4*Cos[x] + 8*x^3*Cos[x]*Log[1 - E^((-I)*x)] - 24*x^2*Cos[x]*Log[1 - I*E^(I*x)] + 24*x^2*Cos[x]*Log[1 + I*E^(I*x)] - 8*x^3*Cos[x]*Log[1 + E^(I*x)] + (24*I)*x^2*Cos[x]*PolyLog[2, E^((-I)*x)] + (24*I)*x^2*Cos[x]*PolyLog[2, -E^(I*x)] - (48*I)*x*Cos[x]*PolyLog[2, (-I)*E^(I*x)] + (48*I)*x*Cos[x]*PolyLog[2, I*E^(I*x)] + 48*x*Cos[x]*PolyLog[3, E^((-I)*x)] - 48*x*Cos[x]*PolyLog[3, -E^(I*x)] + 48*Cos[x]*PolyLog[3, (-I)*E^(I*x)] - 48*Cos[x]*PolyLog[3, I*E^(I*x)] - (48*I)*Cos[x]*PolyLog[4, E^((-I)*x)] - (48*I)*Cos[x]*PolyLog[4, -E^(I*x)])*Sqrt[a*Sec[x]^2])/8","A",1
877,1,85,142,0.2332601,"\int x \csc (x) \sec (x) \sqrt{a \sec ^4(x)} \, dx","Integrate[x*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]","\frac{1}{2} \cos ^2(x) \sqrt{a \sec ^4(x)} \left(i \text{Li}_2\left(-e^{2 i x}\right)-i \text{Li}_2\left(e^{2 i x}\right)+2 x \log \left(1-e^{2 i x}\right)-2 x \log \left(1+e^{2 i x}\right)-\tan (x)+x \sec ^2(x)\right)","\frac{1}{2} i \text{Li}_2\left(-e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}-\frac{1}{2} i \text{Li}_2\left(e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x \sin ^2(x) \sqrt{a \sec ^4(x)}-2 x \cos ^2(x) \tanh ^{-1}\left(e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{1}{2} \sin (x) \cos (x) \sqrt{a \sec ^4(x)}",1,"(Cos[x]^2*Sqrt[a*Sec[x]^4]*(2*x*Log[1 - E^((2*I)*x)] - 2*x*Log[1 + E^((2*I)*x)] + I*PolyLog[2, -E^((2*I)*x)] - I*PolyLog[2, E^((2*I)*x)] + x*Sec[x]^2 - Tan[x]))/2","A",1
878,1,138,220,0.6341021,"\int x^2 \csc (x) \sec (x) \sqrt{a \sec ^4(x)} \, dx","Integrate[x^2*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]","\frac{1}{24} \cos ^2(x) \sqrt{a \sec ^4(x)} \left(24 i x \text{Li}_2\left(e^{-2 i x}\right)+24 i x \text{Li}_2\left(-e^{2 i x}\right)+12 \text{Li}_3\left(e^{-2 i x}\right)-12 \text{Li}_3\left(-e^{2 i x}\right)+16 i x^3+24 x^2 \log \left(1-e^{-2 i x}\right)-24 x^2 \log \left(1+e^{2 i x}\right)+12 x^2 \sec ^2(x)-24 x \tan (x)-24 \log (\cos (x))-i \pi ^3\right)","i x \text{Li}_2\left(-e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}-i x \text{Li}_2\left(e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}-\frac{1}{2} \text{Li}_3\left(-e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} \text{Li}_3\left(e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^2 \sin ^2(x) \sqrt{a \sec ^4(x)}-2 x^2 \cos ^2(x) \tanh ^{-1}\left(e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\cos ^2(x) \sqrt{a \sec ^4(x)} \log (\cos (x))-x \sin (x) \cos (x) \sqrt{a \sec ^4(x)}",1,"(Cos[x]^2*Sqrt[a*Sec[x]^4]*((-I)*Pi^3 + (16*I)*x^3 + 24*x^2*Log[1 - E^((-2*I)*x)] - 24*x^2*Log[1 + E^((2*I)*x)] - 24*Log[Cos[x]] + (24*I)*x*PolyLog[2, E^((-2*I)*x)] + (24*I)*x*PolyLog[2, -E^((2*I)*x)] + 12*PolyLog[3, E^((-2*I)*x)] - 12*PolyLog[3, -E^((2*I)*x)] + 12*x^2*Sec[x]^2 - 24*x*Tan[x]))/24","A",1
879,1,191,356,1.0688509,"\int x^3 \csc (x) \sec (x) \sqrt{a \sec ^4(x)} \, dx","Integrate[x^3*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]","\frac{1}{64} \cos ^2(x) \sqrt{a \sec ^4(x)} \left(96 i x^2 \text{Li}_2\left(e^{-2 i x}\right)+96 i \left(x^2+1\right) \text{Li}_2\left(-e^{2 i x}\right)+96 x \text{Li}_3\left(e^{-2 i x}\right)-96 x \text{Li}_3\left(-e^{2 i x}\right)-48 i \text{Li}_4\left(e^{-2 i x}\right)-48 i \text{Li}_4\left(-e^{2 i x}\right)+32 i x^4+64 x^3 \log \left(1-e^{-2 i x}\right)-64 x^3 \log \left(1+e^{2 i x}\right)+32 x^3 \sec ^2(x)+96 i x^2-96 x^2 \tan (x)-192 x \log \left(1+e^{2 i x}\right)-i \pi ^4\right)","\frac{3}{2} i x^2 \text{Li}_2\left(-e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}-\frac{3}{2} i x^2 \text{Li}_2\left(e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}-\frac{3}{2} x \text{Li}_3\left(-e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{3}{2} x \text{Li}_3\left(e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{3}{2} i \text{Li}_2\left(-e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}-\frac{3}{4} i \text{Li}_4\left(-e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{3}{4} i \text{Li}_4\left(e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \sin ^2(x) \sqrt{a \sec ^4(x)}-2 x^3 \cos ^2(x) \tanh ^{-1}\left(e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{3}{2} i x^2 \cos ^2(x) \sqrt{a \sec ^4(x)}-\frac{3}{2} x^2 \sin (x) \cos (x) \sqrt{a \sec ^4(x)}-3 x \log \left(1+e^{2 i x}\right) \cos ^2(x) \sqrt{a \sec ^4(x)}",1,"(Cos[x]^2*Sqrt[a*Sec[x]^4]*((-I)*Pi^4 + (96*I)*x^2 + (32*I)*x^4 + 64*x^3*Log[1 - E^((-2*I)*x)] - 192*x*Log[1 + E^((2*I)*x)] - 64*x^3*Log[1 + E^((2*I)*x)] + (96*I)*x^2*PolyLog[2, E^((-2*I)*x)] + (96*I)*(1 + x^2)*PolyLog[2, -E^((2*I)*x)] + 96*x*PolyLog[3, E^((-2*I)*x)] - 96*x*PolyLog[3, -E^((2*I)*x)] - (48*I)*PolyLog[4, E^((-2*I)*x)] - (48*I)*PolyLog[4, -E^((2*I)*x)] + 32*x^3*Sec[x]^2 - 96*x^2*Tan[x]))/64","A",1
880,1,25,25,0.0103295,"\int \sin (x) \sin (2 x) \sin (3 x) \, dx","Integrate[Sin[x]*Sin[2*x]*Sin[3*x],x]","-\frac{1}{8} \cos (2 x)-\frac{1}{16} \cos (4 x)+\frac{1}{24} \cos (6 x)","-\frac{1}{8} \cos (2 x)-\frac{1}{16} \cos (4 x)+\frac{1}{24} \cos (6 x)",1,"-1/8*Cos[2*x] - Cos[4*x]/16 + Cos[6*x]/24","A",1
881,1,30,30,0.0101864,"\int \cos (x) \cos (2 x) \cos (3 x) \, dx","Integrate[Cos[x]*Cos[2*x]*Cos[3*x],x]","\frac{x}{4}+\frac{1}{8} \sin (2 x)+\frac{1}{16} \sin (4 x)+\frac{1}{24} \sin (6 x)","\frac{x}{4}+\frac{1}{8} \sin (2 x)+\frac{1}{16} \sin (4 x)+\frac{1}{24} \sin (6 x)",1,"x/4 + Sin[2*x]/8 + Sin[4*x]/16 + Sin[6*x]/24","A",1
882,1,30,30,0.0083094,"\int \cos (x) \sin (2 x) \sin (3 x) \, dx","Integrate[Cos[x]*Sin[2*x]*Sin[3*x],x]","\frac{x}{4}+\frac{1}{8} \sin (2 x)-\frac{1}{16} \sin (4 x)-\frac{1}{24} \sin (6 x)","\frac{x}{4}+\frac{1}{8} \sin (2 x)-\frac{1}{16} \sin (4 x)-\frac{1}{24} \sin (6 x)",1,"x/4 + Sin[2*x]/8 - Sin[4*x]/16 - Sin[6*x]/24","A",1
883,1,25,25,0.0082045,"\int \cos (2 x) \cos (3 x) \sin (x) \, dx","Integrate[Cos[2*x]*Cos[3*x]*Sin[x],x]","-\frac{1}{8} \cos (2 x)+\frac{1}{16} \cos (4 x)-\frac{1}{24} \cos (6 x)","-\frac{1}{8} \cos (2 x)+\frac{1}{16} \cos (4 x)-\frac{1}{24} \cos (6 x)",1,"-1/8*Cos[2*x] + Cos[4*x]/16 - Cos[6*x]/24","A",1
884,1,8,8,0.0056082,"\int x \sin \left(x^2\right) \, dx","Integrate[x*Sin[x^2],x]","-\frac{1}{2} \cos \left(x^2\right)","-\frac{1}{2} \cos \left(x^2\right)",1,"-1/2*Cos[x^2]","A",1
885,1,25,11,0.0770671,"\int (-\cos (x)+\sin (x)) (\cos (x)+\sin (x))^5 \, dx","Integrate[(-Cos[x] + Sin[x])*(Cos[x] + Sin[x])^5,x]","-\frac{5}{8} \sin (2 x)+\frac{1}{24} \sin (6 x)+\frac{1}{4} \cos (4 x)","-\frac{1}{6} (\sin (x)+\cos (x))^6",1,"Cos[4*x]/4 - (5*Sin[2*x])/8 + Sin[6*x]/24","B",1
886,1,18,11,0.0106241,"\int 2 x \sec ^2(x) \tan (x) \, dx","Integrate[2*x*Sec[x]^2*Tan[x],x]","2 \left(\frac{1}{2} x \sec ^2(x)-\frac{\tan (x)}{2}\right)","x \sec ^2(x)-\tan (x)",1,"2*((x*Sec[x]^2)/2 - Tan[x]/2)","A",1
887,1,12,12,0.0159914,"\int \frac{1+\cos ^2(x)}{1+\cos (2 x)} \, dx","Integrate[(1 + Cos[x]^2)/(1 + Cos[2*x]),x]","\frac{x}{2}+\frac{\tan (x)}{2}","\frac{x}{2}+\frac{\tan (x)}{2}",1,"x/2 + Tan[x]/2","A",1
888,1,17,12,0.0144437,"\int \frac{\sin (x)}{\cos ^3(x)-\cos ^5(x)} \, dx","Integrate[Sin[x]/(Cos[x]^3 - Cos[x]^5),x]","\frac{\sec ^2(x)}{2}+\log (\sin (x))-\log (\cos (x))","\frac{\tan ^2(x)}{2}+\log (\tan (x))",1,"-Log[Cos[x]] + Log[Sin[x]] + Sec[x]^2/2","A",1
889,1,19,19,0.012858,"\int \sec (x) \left(5-11 \sec ^5(x)\right)^2 \tan (x) \, dx","Integrate[Sec[x]*(5 - 11*Sec[x]^5)^2*Tan[x],x]","11 \sec ^{11}(x)-\frac{55 \sec ^6(x)}{3}+25 \sec (x)","11 \sec ^{11}(x)-\frac{55 \sec ^6(x)}{3}+25 \sec (x)",1,"25*Sec[x] - (55*Sec[x]^6)/3 + 11*Sec[x]^11","A",1
890,1,52,44,0.0442785,"\int \sin ^3(5 x) \tan ^3(5 x) \, dx","Integrate[Sin[5*x]^3*Tan[5*x]^3,x]","-\frac{1}{15} \sin ^3(5 x) \tan ^2(5 x)-\frac{1}{3} \sin (5 x) \tan ^2(5 x)-\frac{1}{2} \tanh ^{-1}(\sin (5 x))+\frac{1}{2} \tan (5 x) \sec (5 x)","\frac{1}{6} \sin ^3(5 x)+\frac{1}{2} \sin (5 x)+\frac{1}{10} \sin ^3(5 x) \tan ^2(5 x)-\frac{1}{2} \tanh ^{-1}(\sin (5 x))",1,"-1/2*ArcTanh[Sin[5*x]] + (Sec[5*x]*Tan[5*x])/2 - (Sin[5*x]*Tan[5*x]^2)/3 - (Sin[5*x]^3*Tan[5*x]^2)/15","A",1
891,1,35,37,0.0285157,"\int \sin ^3(5 x) \tan ^4(5 x) \, dx","Integrate[Sin[5*x]^3*Tan[5*x]^4,x]","-\frac{11}{20} \cos (5 x)+\frac{1}{60} \cos (15 x)+\frac{1}{15} \sec ^3(5 x)-\frac{3}{5} \sec (5 x)","\frac{1}{15} \cos ^3(5 x)-\frac{3}{5} \cos (5 x)+\frac{1}{15} \sec ^3(5 x)-\frac{3}{5} \sec (5 x)",1,"(-11*Cos[5*x])/20 + Cos[15*x]/60 - (3*Sec[5*x])/5 + Sec[5*x]^3/15","A",1
892,1,68,54,0.0959972,"\int \sin ^5(6 x) \tan ^3(6 x) \, dx","Integrate[Sin[6*x]^5*Tan[6*x]^3,x]","-\frac{1}{30} \sin ^5(6 x) \tan ^2(6 x)-\frac{7}{90} \sin ^3(6 x) \tan ^2(6 x)-\frac{7}{18} \sin (6 x) \tan ^2(6 x)-\frac{7}{12} \tanh ^{-1}(\sin (6 x))+\frac{7}{12} \tan (6 x) \sec (6 x)","\frac{7}{60} \sin ^5(6 x)+\frac{7}{36} \sin ^3(6 x)+\frac{7}{12} \sin (6 x)+\frac{1}{12} \sin ^5(6 x) \tan ^2(6 x)-\frac{7}{12} \tanh ^{-1}(\sin (6 x))",1,"(-7*ArcTanh[Sin[6*x]])/12 + (7*Sec[6*x]*Tan[6*x])/12 - (7*Sin[6*x]*Tan[6*x]^2)/18 - (7*Sin[6*x]^3*Tan[6*x]^2)/90 - (Sin[6*x]^5*Tan[6*x]^2)/30","A",1
893,1,37,37,0.027065,"\int \left(-1+\sec ^2(2 x)\right)^3 \sin (2 x) \, dx","Integrate[(-1 + Sec[2*x]^2)^3*Sin[2*x],x]","\frac{1}{2} \cos (2 x)+\frac{1}{10} \sec ^5(2 x)-\frac{1}{2} \sec ^3(2 x)+\frac{3}{2} \sec (2 x)","\frac{1}{2} \cos (2 x)+\frac{1}{10} \sec ^5(2 x)-\frac{1}{2} \sec ^3(2 x)+\frac{3}{2} \sec (2 x)",1,"Cos[2*x]/2 + (3*Sec[2*x])/2 - Sec[2*x]^3/2 + Sec[2*x]^5/10","A",1
894,1,42,34,0.0089027,"\int \sin (x) \tan ^5(x) \, dx","Integrate[Sin[x]*Tan[x]^5,x]","-\sin (x) \tan ^4(x)+\frac{15}{8} \tanh ^{-1}(\sin (x))-\frac{15}{4} \tan (x) \sec ^3(x)+5 \tan ^3(x) \sec (x)+\frac{15}{8} \tan (x) \sec (x)","-\frac{15 \sin (x)}{8}+\frac{1}{4} \sin (x) \tan ^4(x)-\frac{5}{8} \sin (x) \tan ^2(x)+\frac{15}{8} \tanh ^{-1}(\sin (x))",1,"(15*ArcTanh[Sin[x]])/8 + (15*Sec[x]*Tan[x])/8 - (15*Sec[x]^3*Tan[x])/4 + 5*Sec[x]*Tan[x]^3 - Sin[x]*Tan[x]^4","A",1
895,1,43,43,0.0301957,"\int \cos ^5(2 x) \cot ^4(2 x) \, dx","Integrate[Cos[2*x]^5*Cot[2*x]^4,x]","\frac{1}{10} \sin ^5(2 x)-\frac{2}{3} \sin ^3(2 x)+3 \sin (2 x)-\frac{1}{6} \csc ^3(2 x)+2 \csc (2 x)","\frac{1}{10} \sin ^5(2 x)-\frac{2}{3} \sin ^3(2 x)+3 \sin (2 x)-\frac{1}{6} \csc ^3(2 x)+2 \csc (2 x)",1,"2*Csc[2*x] - Csc[2*x]^3/6 + 3*Sin[2*x] - (2*Sin[2*x]^3)/3 + Sin[2*x]^5/10","A",1
896,1,87,87,0.0622001,"\int \cos (3 x) \left(-1+\csc ^2(3 x)\right)^3 \left(1-\sin ^2(3 x)\right)^5 \, dx","Integrate[Cos[3*x]*(-1 + Csc[3*x]^2)^3*(1 - Sin[3*x]^2)^5,x]","\frac{1}{33} \sin ^{11}(3 x)-\frac{8}{27} \sin ^9(3 x)+\frac{4}{3} \sin ^7(3 x)-\frac{56}{15} \sin ^5(3 x)+\frac{70}{9} \sin ^3(3 x)-\frac{56}{3} \sin (3 x)-\frac{1}{15} \csc ^5(3 x)+\frac{8}{9} \csc ^3(3 x)-\frac{28}{3} \csc (3 x)","\frac{1}{33} \sin ^{11}(3 x)-\frac{8}{27} \sin ^9(3 x)+\frac{4}{3} \sin ^7(3 x)-\frac{56}{15} \sin ^5(3 x)+\frac{70}{9} \sin ^3(3 x)-\frac{56}{3} \sin (3 x)-\frac{1}{15} \csc ^5(3 x)+\frac{8}{9} \csc ^3(3 x)-\frac{28}{3} \csc (3 x)",1,"(-28*Csc[3*x])/3 + (8*Csc[3*x]^3)/9 - Csc[3*x]^5/15 - (56*Sin[3*x])/3 + (70*Sin[3*x]^3)/9 - (56*Sin[3*x]^5)/15 + (4*Sin[3*x]^7)/3 - (8*Sin[3*x]^9)/27 + Sin[3*x]^11/33","A",1
897,1,42,42,0.04135,"\int \cot (2 x) \left(-1+\csc ^2(2 x)\right)^2 \left(1-\sin ^2(2 x)\right)^2 \, dx","Integrate[Cot[2*x]*(-1 + Csc[2*x]^2)^2*(1 - Sin[2*x]^2)^2,x]","\frac{1}{8} \sin ^4(2 x)-\sin ^2(2 x)-\frac{1}{8} \csc ^4(2 x)+\csc ^2(2 x)+3 \log (\sin (2 x))","\frac{1}{8} \sin ^4(2 x)-\sin ^2(2 x)-\frac{1}{8} \csc ^4(2 x)+\csc ^2(2 x)+3 \log (\sin (2 x))",1,"Csc[2*x]^2 - Csc[2*x]^4/8 + 3*Log[Sin[2*x]] - Sin[2*x]^2 + Sin[2*x]^4/8","A",1
898,1,63,63,0.0339448,"\int \cos (2 x) \left(-1+\csc ^2(2 x)\right)^4 \left(1-\sin ^2(2 x)\right)^2 \, dx","Integrate[Cos[2*x]*(-1 + Csc[2*x]^2)^4*(1 - Sin[2*x]^2)^2,x]","\frac{1}{10} \sin ^5(2 x)-\sin ^3(2 x)+\frac{15}{2} \sin (2 x)-\frac{1}{14} \csc ^7(2 x)+\frac{3}{5} \csc ^5(2 x)-\frac{5}{2} \csc ^3(2 x)+10 \csc (2 x)","\frac{1}{10} \sin ^5(2 x)-\sin ^3(2 x)+\frac{15}{2} \sin (2 x)-\frac{1}{14} \csc ^7(2 x)+\frac{3}{5} \csc ^5(2 x)-\frac{5}{2} \csc ^3(2 x)+10 \csc (2 x)",1,"10*Csc[2*x] - (5*Csc[2*x]^3)/2 + (3*Csc[2*x]^5)/5 - Csc[2*x]^7/14 + (15*Sin[2*x])/2 - Sin[2*x]^3 + Sin[2*x]^5/10","A",1
899,1,52,60,0.1270114,"\int \cot (3 x) \left(-1+\csc ^2(3 x)\right)^3 \left(1-\sin ^2(3 x)\right)^2 \, dx","Integrate[Cot[3*x]*(-1 + Csc[3*x]^2)^3*(1 - Sin[3*x]^2)^2,x]","\frac{1}{36} \left(-3 \sin ^4(3 x)+30 \sin ^2(3 x)-2 \csc ^6(3 x)+15 \csc ^4(3 x)-60 \csc ^2(3 x)-120 \log (\sin (3 x))\right)","-\frac{1}{12} \sin ^4(3 x)+\frac{5}{6} \sin ^2(3 x)-\frac{1}{18} \csc ^6(3 x)+\frac{5}{12} \csc ^4(3 x)-\frac{5}{3} \csc ^2(3 x)-\frac{10}{3} \log (\sin (3 x))",1,"(-60*Csc[3*x]^2 + 15*Csc[3*x]^4 - 2*Csc[3*x]^6 - 120*Log[Sin[3*x]] + 30*Sin[3*x]^2 - 3*Sin[3*x]^4)/36","A",1
900,1,59,47,0.0489143,"\int \left(1+\cot ^2(9 x)\right)^2 \left(1+\tan ^2(9 x)\right)^3 \, dx","Integrate[(1 + Cot[9*x]^2)^2*(1 + Tan[9*x]^2)^3,x]","\frac{73}{135} \tan (9 x)-\frac{11}{27} \cot (9 x)-\frac{1}{27} \cot (9 x) \csc ^2(9 x)+\frac{1}{45} \tan (9 x) \sec ^4(9 x)+\frac{14}{135} \tan (9 x) \sec ^2(9 x)","\frac{1}{45} \tan ^5(9 x)+\frac{4}{27} \tan ^3(9 x)+\frac{2}{3} \tan (9 x)-\frac{1}{27} \cot ^3(9 x)-\frac{4}{9} \cot (9 x)",1,"(-11*Cot[9*x])/27 - (Cot[9*x]*Csc[9*x]^2)/27 + (73*Tan[9*x])/135 + (14*Sec[9*x]^2*Tan[9*x])/135 + (Sec[9*x]^4*Tan[9*x])/45","A",1
901,1,71,43,0.0224842,"\int \frac{\cos (x) \left(9-7 \sin ^3(x)\right)^2}{1-\sin ^2(x)} \, dx","Integrate[(Cos[x]*(9 - 7*Sin[x]^3)^2)/(1 - Sin[x]^2),x]","-\frac{49}{5} \sin ^5(x)-\frac{49 \sin ^3(x)}{3}-49 \sin (x)-63 \cos ^2(x)+49 \tanh ^{-1}(\sin (x))+126 \log (\cos (x))-81 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+81 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","-\frac{49}{5} \sin ^5(x)-\frac{49 \sin ^3(x)}{3}+63 \sin ^2(x)-49 \sin (x)-2 \log (1-\sin (x))+128 \log (\sin (x)+1)",1,"49*ArcTanh[Sin[x]] - 63*Cos[x]^2 + 126*Log[Cos[x]] - 81*Log[Cos[x/2] - Sin[x/2]] + 81*Log[Cos[x/2] + Sin[x/2]] - 49*Sin[x] - (49*Sin[x]^3)/3 - (49*Sin[x]^5)/5","A",1
902,1,42,42,0.0286162,"\int \cos ^4(2 x) \cot ^5(2 x) \, dx","Integrate[Cos[2*x]^4*Cot[2*x]^5,x]","\frac{1}{8} \sin ^4(2 x)-\sin ^2(2 x)-\frac{1}{8} \csc ^4(2 x)+\csc ^2(2 x)+3 \log (\sin (2 x))","\frac{1}{8} \sin ^4(2 x)-\sin ^2(2 x)-\frac{1}{8} \csc ^4(2 x)+\csc ^2(2 x)+3 \log (\sin (2 x))",1,"Csc[2*x]^2 - Csc[2*x]^4/8 + 3*Log[Sin[2*x]] - Sin[2*x]^2 + Sin[2*x]^4/8","A",1
903,1,63,74,0.0832571,"\int \frac{\sec (x) \tan ^2(x)}{4+3 \sec (x)} \, dx","Integrate[(Sec[x]*Tan[x]^2)/(4 + 3*Sec[x]),x]","\frac{1}{9} \left(3 \tan (x)+2 \sqrt{7} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)}{\sqrt{7}}\right)+4 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-4 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)","\frac{\tan (x)}{3}-\frac{4}{9} \tanh ^{-1}(\sin (x))-\frac{1}{9} \sqrt{7} \log \left(\sqrt{7} \cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\frac{1}{9} \sqrt{7} \log \left(\sin \left(\frac{x}{2}\right)+\sqrt{7} \cos \left(\frac{x}{2}\right)\right)",1,"(2*Sqrt[7]*ArcTanh[Tan[x/2]/Sqrt[7]] + 4*Log[Cos[x/2] - Sin[x/2]] - 4*Log[Cos[x/2] + Sin[x/2]] + 3*Tan[x])/9","A",1
904,1,47,14,0.0401844,"\int x \sec (1+x) \tan (1+x) \, dx","Integrate[x*Sec[1 + x]*Tan[1 + x],x]","x \sec (x+1)+\log \left(\cos \left(\frac{x+1}{2}\right)-\sin \left(\frac{x+1}{2}\right)\right)-\log \left(\sin \left(\frac{x+1}{2}\right)+\cos \left(\frac{x+1}{2}\right)\right)","x \sec (x+1)-\tanh ^{-1}(\sin (x+1))",1,"Log[Cos[(1 + x)/2] - Sin[(1 + x)/2]] - Log[Cos[(1 + x)/2] + Sin[(1 + x)/2]] + x*Sec[1 + x]","B",1
905,1,14,14,0.0151246,"\int \frac{\sin (2 x)}{\sqrt{9-\sin ^2(x)}} \, dx","Integrate[Sin[2*x]/Sqrt[9 - Sin[x]^2],x]","-2 \sqrt{9-\sin ^2(x)}","-2 \sqrt{9-\sin ^2(x)}",1,"-2*Sqrt[9 - Sin[x]^2]","A",1
906,1,11,11,0.0163766,"\int \frac{\sin (2 x)}{\sqrt{9-\cos ^4(x)}} \, dx","Integrate[Sin[2*x]/Sqrt[9 - Cos[x]^4],x]","-\sin ^{-1}\left(\frac{\cos ^2(x)}{3}\right)","-\sin ^{-1}\left(\frac{\cos ^2(x)}{3}\right)",1,"-ArcSin[Cos[x]^2/3]","A",1
907,1,32,34,0.0280575,"\int \frac{\cos \left(\frac{1}{x}\right)}{x^5} \, dx","Integrate[Cos[x^(-1)]/x^5,x]","\frac{3 \left(2 x^2-1\right) \cos \left(\frac{1}{x}\right)}{x^2}+\frac{\left(6 x^2-1\right) \sin \left(\frac{1}{x}\right)}{x^3}","-\frac{\sin \left(\frac{1}{x}\right)}{x^3}-\frac{3 \cos \left(\frac{1}{x}\right)}{x^2}+\frac{6 \sin \left(\frac{1}{x}\right)}{x}+6 \cos \left(\frac{1}{x}\right)",1,"(3*(-1 + 2*x^2)*Cos[x^(-1)])/x^2 + ((-1 + 6*x^2)*Sin[x^(-1)])/x^3","A",1
908,1,25,21,0.0130055,"\int \cos ^3(1+x) \sin ^3(1+x) \, dx","Integrate[Cos[1 + x]^3*Sin[1 + x]^3,x]","\frac{1}{8} \left(\frac{1}{24} \cos (6 (x+1))-\frac{3}{8} \cos (2 (x+1))\right)","\frac{1}{4} \sin ^4(x+1)-\frac{1}{6} \sin ^6(x+1)",1,"((-3*Cos[2*(1 + x)])/8 + Cos[6*(1 + x)]/24)/8","A",1
909,1,55,99,0.2293071,"\int (1+2 x)^3 \sin ^2(1+2 x) \, dx","Integrate[(1 + 2*x)^3*Sin[1 + 2*x]^2,x]","\frac{1}{32} \left(2 (2 x+1) \left(\left(-8 x^2-8 x+1\right) \sin (4 x+2)+(2 x+1)^3\right)-3 \left(8 x^2+8 x+1\right) \cos (4 x+2)\right)","-\frac{3 x^2}{4}+\frac{1}{16} (2 x+1)^4-\frac{3 x}{4}+\frac{3}{8} (2 x+1)^2 \sin ^2(2 x+1)-\frac{3}{16} \sin ^2(2 x+1)-\frac{1}{4} (2 x+1)^3 \sin (2 x+1) \cos (2 x+1)+\frac{3}{8} (2 x+1) \sin (2 x+1) \cos (2 x+1)",1,"(-3*(1 + 8*x + 8*x^2)*Cos[2 + 4*x] + 2*(1 + 2*x)*((1 + 2*x)^3 + (1 - 8*x - 8*x^2)*Sin[2 + 4*x]))/32","A",1
910,1,40,37,0.0590842,"\int \frac{-1+\sec (x)}{1-\tan (x)} \, dx","Integrate[(-1 + Sec[x])/(1 - Tan[x]),x]","\frac{1}{2} \left(-x+(2-2 i) \sqrt[4]{-1} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)+1}{\sqrt{2}}\right)+\log (\cos (x)-\sin (x))\right)","-\frac{x}{2}+\frac{1}{2} \log (\cos (x)-\sin (x))+\frac{\tanh ^{-1}\left(\frac{\cos (x) (\tan (x)+1)}{\sqrt{2}}\right)}{\sqrt{2}}",1,"(-x + (2 - 2*I)*(-1)^(1/4)*ArcTanh[(1 + Tan[x/2])/Sqrt[2]] + Log[Cos[x] - Sin[x]])/2","C",1
911,1,49,57,0.0952477,"\int x^2 \cos (3 x) \cos (5 x) \, dx","Integrate[x^2*Cos[3*x]*Cos[5*x],x]","\frac{1}{512} \left(128 x^2 \sin (2 x)+32 x^2 \sin (8 x)-64 \sin (2 x)-\sin (8 x)+128 x \cos (2 x)+8 x \cos (8 x)\right)","\frac{1}{4} x^2 \sin (2 x)+\frac{1}{16} x^2 \sin (8 x)-\frac{1}{8} \sin (2 x)-\frac{1}{512} \sin (8 x)+\frac{1}{4} x \cos (2 x)+\frac{1}{64} x \cos (8 x)",1,"(128*x*Cos[2*x] + 8*x*Cos[8*x] - 64*Sin[2*x] + 128*x^2*Sin[2*x] - Sin[8*x] + 32*x^2*Sin[8*x])/512","A",1
912,1,68,57,0.0583389,"\int \frac{\cos (x)+\sin (x)}{\sqrt{\cos (x)} \sqrt{\sin (x)}} \, dx","Integrate[(Cos[x] + Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]]),x]","\frac{2 \sqrt{\sin (x)} \sqrt[4]{\cos ^2(x)} \left(\sin (x) \sqrt{\cos ^2(x)} \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\sin ^2(x)\right)+3 \cos (x) \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\sin ^2(x)\right)\right)}{3 \cos ^{\frac{3}{2}}(x)}","\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)-\sqrt{2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right)",1,"(2*(Cos[x]^2)^(1/4)*Sqrt[Sin[x]]*(3*Cos[x]*Hypergeometric2F1[1/4, 1/4, 5/4, Sin[x]^2] + Sqrt[Cos[x]^2]*Hypergeometric2F1[3/4, 3/4, 7/4, Sin[x]^2]*Sin[x]))/(3*Cos[x]^(3/2))","C",1
913,1,5,5,0.0038532,"\int \sec ^2(x) (1+\sin (x)) \, dx","Integrate[Sec[x]^2*(1 + Sin[x]),x]","\tan (x)+\sec (x)","\tan (x)+\sec (x)",1,"Sec[x] + Tan[x]","A",1
914,1,11,11,0.3081209,"\int \left(10 x^9 \cos \left(x^5 \log (x)\right)-x^{10} \left(x^4+5 x^4 \log (x)\right) \sin \left(x^5 \log (x)\right)\right) \, dx","Integrate[10*x^9*Cos[x^5*Log[x]] - x^10*(x^4 + 5*x^4*Log[x])*Sin[x^5*Log[x]],x]","x^{10} \cos \left(x^5 \log (x)\right)","x^{10} \cos \left(x^5 \log (x)\right)",1,"x^10*Cos[x^5*Log[x]]","A",1
915,1,24,27,0.1665749,"\int \cos ^2\left(\frac{x}{2}\right) \tan \left(\frac{\pi }{4}+\frac{x}{2}\right) \, dx","Integrate[Cos[x/2]^2*Tan[Pi/4 + x/2],x]","\frac{1}{2} \left(x-\cos (x)-\log (\cos (x))+2 \tanh ^{-1}\left(\cot \left(\frac{x}{2}\right)\right)\right)","\frac{x}{2}-\frac{\cos (x)}{2}-\log \left(\cos \left(\frac{x}{2}+\frac{\pi }{4}\right)\right)",1,"(x + 2*ArcTanh[Cot[x/2]] - Cos[x] - Log[Cos[x]])/2","A",1
916,1,50,65,0.0899357,"\int (2+3 x)^2 \sin ^3(x) \, dx","Integrate[(2 + 3*x)^2*Sin[x]^3,x]","\frac{1}{12} \left(-9 \left(9 x^2+12 x-14\right) \cos (x)+\left(9 x^2+12 x+2\right) \cos (3 x)-2 (3 x+2) (\sin (3 x)-27 \sin (x))\right)","\frac{2}{3} (3 x+2) \sin ^3(x)+4 (3 x+2) \sin (x)-\frac{2}{3} \cos ^3(x)-\frac{2}{3} (3 x+2)^2 \cos (x)+14 \cos (x)-\frac{1}{3} (3 x+2)^2 \sin ^2(x) \cos (x)",1,"(-9*(-14 + 12*x + 9*x^2)*Cos[x] + (2 + 12*x + 9*x^2)*Cos[3*x] - 2*(2 + 3*x)*(-27*Sin[x] + Sin[3*x]))/12","A",1
917,1,8,8,0.0148661,"\int \sec ^{1+m}(x) \sin (x) \, dx","Integrate[Sec[x]^(1 + m)*Sin[x],x]","\frac{\sec ^m(x)}{m}","\frac{\sec ^m(x)}{m}",1,"Sec[x]^m/m","A",1
918,1,32,32,0.0786297,"\int \cos ^n(a+b x) \sin ^{-2-n}(a+b x) \, dx","Integrate[Cos[a + b*x]^n*Sin[a + b*x]^(-2 - n),x]","-\frac{\sin ^{-n-1}(a+b x) \cos ^{n+1}(a+b x)}{b (n+1)}","-\frac{\sin ^{-n-1}(a+b x) \cos ^{n+1}(a+b x)}{b (n+1)}",1,"-((Cos[a + b*x]^(1 + n)*Sin[a + b*x]^(-1 - n))/(b*(1 + n)))","A",1
919,1,3,3,0.0175037,"\int \frac{1}{\sec (x)+\sin (x) \tan (x)} \, dx","Integrate[(Sec[x] + Sin[x]*Tan[x])^(-1),x]","\tan ^{-1}(\sin (x))","\tan ^{-1}(\sin (x))",1,"ArcTan[Sin[x]]","A",1
920,1,32,35,0.0418114,"\int \left(a+b x+c x^2\right) \sin (x) \, dx","Integrate[(a + b*x + c*x^2)*Sin[x],x]","-a \cos (x)+b \sin (x)-b x \cos (x)-c \left(x^2-2\right) \cos (x)+2 c x \sin (x)","-a \cos (x)+b \sin (x)-b x \cos (x)-c x^2 \cos (x)+2 c x \sin (x)+2 c \cos (x)",1,"-(a*Cos[x]) - b*x*Cos[x] - c*(-2 + x^2)*Cos[x] + b*Sin[x] + 2*c*x*Sin[x]","A",1
921,1,8,8,0.0020078,"\int \frac{\sin \left(x^5\right)}{x} \, dx","Integrate[Sin[x^5]/x,x]","\frac{\text{Si}\left(x^5\right)}{5}","\frac{\text{Si}\left(x^5\right)}{5}",1,"SinIntegral[x^5]/5","A",1
922,1,29,37,0.0711485,"\int \frac{\sin \left(2^x\right)}{1+2^x} \, dx","Integrate[Sin[2^x]/(1 + 2^x),x]","\frac{\sin (1) \text{Ci}\left(1+2^x\right)+\text{Si}\left(2^x\right)-\cos (1) \text{Si}\left(1+2^x\right)}{\log (2)}","\frac{\sin (1) \text{Ci}\left(1+2^x\right)}{\log (2)}+\frac{\text{Si}\left(2^x\right)}{\log (2)}-\frac{\cos (1) \text{Si}\left(1+2^x\right)}{\log (2)}",1,"(CosIntegral[1 + 2^x]*Sin[1] + SinIntegral[2^x] - Cos[1]*SinIntegral[1 + 2^x])/Log[2]","A",1
923,1,14,14,0.0063952,"\int x \cos \left(2 x^2\right) \sin ^{\frac{3}{4}}\left(2 x^2\right) \, dx","Integrate[x*Cos[2*x^2]*Sin[2*x^2]^(3/4),x]","\frac{1}{7} \sin ^{\frac{7}{4}}\left(2 x^2\right)","\frac{1}{7} \sin ^{\frac{7}{4}}\left(2 x^2\right)",1,"Sin[2*x^2]^(7/4)/7","A",1
924,1,10,10,0.0033277,"\int x \sec ^2\left(x^2\right) \tan ^2\left(x^2\right) \, dx","Integrate[x*Sec[x^2]^2*Tan[x^2]^2,x]","\frac{1}{6} \tan ^3\left(x^2\right)","\frac{1}{6} \tan ^3\left(x^2\right)",1,"Tan[x^2]^3/6","A",1
925,1,17,17,0.0219179,"\int x^2 \cos ^7\left(a+b x^3\right) \sin \left(a+b x^3\right) \, dx","Integrate[x^2*Cos[a + b*x^3]^7*Sin[a + b*x^3],x]","-\frac{\cos ^8\left(a+b x^3\right)}{24 b}","-\frac{\cos ^8\left(a+b x^3\right)}{24 b}",1,"-1/24*Cos[a + b*x^3]^8/b","A",1
926,1,120,129,0.5650823,"\int x^5 \cos ^7\left(a+b x^3\right) \sin \left(a+b x^3\right) \, dx","Integrate[x^5*Cos[a + b*x^3]^7*Sin[a + b*x^3],x]","\frac{672 \sin \left(2 \left(a+b x^3\right)\right)+168 \sin \left(4 \left(a+b x^3\right)\right)+32 \sin \left(6 \left(a+b x^3\right)\right)+3 \sin \left(8 \left(a+b x^3\right)\right)-1344 b x^3 \cos \left(2 \left(a+b x^3\right)\right)-672 b x^3 \cos \left(4 \left(a+b x^3\right)\right)-192 b x^3 \cos \left(6 \left(a+b x^3\right)\right)-24 b x^3 \cos \left(8 \left(a+b x^3\right)\right)}{73728 b^2}","\frac{\sin \left(a+b x^3\right) \cos ^7\left(a+b x^3\right)}{192 b^2}+\frac{7 \sin \left(a+b x^3\right) \cos ^5\left(a+b x^3\right)}{1152 b^2}+\frac{35 \sin \left(a+b x^3\right) \cos ^3\left(a+b x^3\right)}{4608 b^2}+\frac{35 \sin \left(a+b x^3\right) \cos \left(a+b x^3\right)}{3072 b^2}-\frac{x^3 \cos ^8\left(a+b x^3\right)}{24 b}+\frac{35 x^3}{3072 b}",1,"(-1344*b*x^3*Cos[2*(a + b*x^3)] - 672*b*x^3*Cos[4*(a + b*x^3)] - 192*b*x^3*Cos[6*(a + b*x^3)] - 24*b*x^3*Cos[8*(a + b*x^3)] + 672*Sin[2*(a + b*x^3)] + 168*Sin[4*(a + b*x^3)] + 32*Sin[6*(a + b*x^3)] + 3*Sin[8*(a + b*x^3)])/(73728*b^2)","A",1
927,1,352,110,0.8709945,"\int x^5 \sec ^7\left(a+b x^3\right) \tan \left(a+b x^3\right) \, dx","Integrate[x^5*Sec[a + b*x^3]^7*Tan[a + b*x^3],x]","\frac{\sec ^7\left(a+b x^3\right) \left(-566 \sin \left(2 \left(a+b x^3\right)\right)-200 \sin \left(4 \left(a+b x^3\right)\right)-30 \sin \left(6 \left(a+b x^3\right)\right)+105 \cos \left(5 \left(a+b x^3\right)\right) \log \left(\cos \left(\frac{1}{2} \left(a+b x^3\right)\right)-\sin \left(\frac{1}{2} \left(a+b x^3\right)\right)\right)+15 \cos \left(7 \left(a+b x^3\right)\right) \log \left(\cos \left(\frac{1}{2} \left(a+b x^3\right)\right)-\sin \left(\frac{1}{2} \left(a+b x^3\right)\right)\right)+525 \cos \left(a+b x^3\right) \left(\log \left(\cos \left(\frac{1}{2} \left(a+b x^3\right)\right)-\sin \left(\frac{1}{2} \left(a+b x^3\right)\right)\right)-\log \left(\sin \left(\frac{1}{2} \left(a+b x^3\right)\right)+\cos \left(\frac{1}{2} \left(a+b x^3\right)\right)\right)\right)+315 \cos \left(3 \left(a+b x^3\right)\right) \left(\log \left(\cos \left(\frac{1}{2} \left(a+b x^3\right)\right)-\sin \left(\frac{1}{2} \left(a+b x^3\right)\right)\right)-\log \left(\sin \left(\frac{1}{2} \left(a+b x^3\right)\right)+\cos \left(\frac{1}{2} \left(a+b x^3\right)\right)\right)\right)-105 \cos \left(5 \left(a+b x^3\right)\right) \log \left(\sin \left(\frac{1}{2} \left(a+b x^3\right)\right)+\cos \left(\frac{1}{2} \left(a+b x^3\right)\right)\right)-15 \cos \left(7 \left(a+b x^3\right)\right) \log \left(\sin \left(\frac{1}{2} \left(a+b x^3\right)\right)+\cos \left(\frac{1}{2} \left(a+b x^3\right)\right)\right)+3072 b x^3\right)}{64512 b^2}","-\frac{5 \tanh ^{-1}\left(\sin \left(a+b x^3\right)\right)}{336 b^2}-\frac{\tan \left(a+b x^3\right) \sec ^5\left(a+b x^3\right)}{126 b^2}-\frac{5 \tan \left(a+b x^3\right) \sec ^3\left(a+b x^3\right)}{504 b^2}-\frac{5 \tan \left(a+b x^3\right) \sec \left(a+b x^3\right)}{336 b^2}+\frac{x^3 \sec ^7\left(a+b x^3\right)}{21 b}",1,"(Sec[a + b*x^3]^7*(3072*b*x^3 + 105*Cos[5*(a + b*x^3)]*Log[Cos[(a + b*x^3)/2] - Sin[(a + b*x^3)/2]] + 15*Cos[7*(a + b*x^3)]*Log[Cos[(a + b*x^3)/2] - Sin[(a + b*x^3)/2]] + 525*Cos[a + b*x^3]*(Log[Cos[(a + b*x^3)/2] - Sin[(a + b*x^3)/2]] - Log[Cos[(a + b*x^3)/2] + Sin[(a + b*x^3)/2]]) + 315*Cos[3*(a + b*x^3)]*(Log[Cos[(a + b*x^3)/2] - Sin[(a + b*x^3)/2]] - Log[Cos[(a + b*x^3)/2] + Sin[(a + b*x^3)/2]]) - 105*Cos[5*(a + b*x^3)]*Log[Cos[(a + b*x^3)/2] + Sin[(a + b*x^3)/2]] - 15*Cos[7*(a + b*x^3)]*Log[Cos[(a + b*x^3)/2] + Sin[(a + b*x^3)/2]] - 566*Sin[2*(a + b*x^3)] - 200*Sin[4*(a + b*x^3)] - 30*Sin[6*(a + b*x^3)]))/(64512*b^2)","B",1
928,1,6,6,0.0178953,"\int \frac{\sec ^2\left(\frac{1}{x}\right)}{x^2} \, dx","Integrate[Sec[x^(-1)]^2/x^2,x]","-\tan \left(\frac{1}{x}\right)","-\tan \left(\frac{1}{x}\right)",1,"-Tan[x^(-1)]","A",1
929,1,4,4,0.0016635,"\int 3 x^2 \cos \left(x^3\right) \, dx","Integrate[3*x^2*Cos[x^3],x]","\sin \left(x^3\right)","\sin \left(x^3\right)",1,"Sin[x^3]","A",1
930,1,30,27,0.0145377,"\int (1+2 x) \sec ^2(1+2 x) \, dx","Integrate[(1 + 2*x)*Sec[1 + 2*x]^2,x]","x \tan (2 x+1)+\frac{1}{2} \tan (2 x+1)+\frac{1}{2} \log (\cos (2 x+1))","\frac{1}{2} (2 x+1) \tan (2 x+1)+\frac{1}{2} \log (\cos (2 x+1))",1,"Log[Cos[1 + 2*x]]/2 + Tan[1 + 2*x]/2 + x*Tan[1 + 2*x]","A",1
931,1,26,26,0.4475514,"\int \left(\frac{x^4}{b \sqrt{x^3+3 \sin (a+b x)}}+\frac{x^2 \cos (a+b x)}{\sqrt{x^3+3 \sin (a+b x)}}+\frac{4 x \sqrt{x^3+3 \sin (a+b x)}}{3 b}\right) \, dx","Integrate[x^4/(b*Sqrt[x^3 + 3*Sin[a + b*x]]) + (x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]] + (4*x*Sqrt[x^3 + 3*Sin[a + b*x]])/(3*b),x]","\frac{2 x^2 \sqrt{3 \sin (a+b x)+x^3}}{3 b}","\frac{2 x^2 \sqrt{3 \sin (a+b x)+x^3}}{3 b}",1,"(2*x^2*Sqrt[x^3 + 3*Sin[a + b*x]])/(3*b)","A",1
932,0,0,29,7.5680688,"\int \frac{x^2 \cos (a+b x)}{\sqrt{x^3+3 \sin (a+b x)}} \, dx","Integrate[(x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]],x]","\int \frac{x^2 \cos (a+b x)}{\sqrt{x^3+3 \sin (a+b x)}} \, dx","\text{Int}\left(\frac{x^2 \cos (a+b x)}{\sqrt{3 \sin (a+b x)+x^3}},x\right)",0,"Integrate[(x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]], x]","A",-1
933,1,9,9,0.1171521,"\int \frac{\cos (x)+\sin (x)}{e^{-x}+\sin (x)} \, dx","Integrate[(Cos[x] + Sin[x])/(E^(-x) + Sin[x]),x]","\log \left(e^x \sin (x)+1\right)","\log \left(e^x \sin (x)+1\right)",1,"Log[1 + E^x*Sin[x]]","A",1
934,1,76,77,0.1516386,"\int \sin (c+d x) \left(a \sin ^2(c+d x)+b \sin ^3(c+d x)\right) \, dx","Integrate[Sin[c + d*x]*(a*Sin[c + d*x]^2 + b*Sin[c + d*x]^3),x]","-\frac{3 a \cos (c+d x)}{4 d}+\frac{a \cos (3 (c+d x))}{12 d}+\frac{3 b (c+d x)}{8 d}-\frac{b \sin (2 (c+d x))}{4 d}+\frac{b \sin (4 (c+d x))}{32 d}","\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8}",1,"(3*b*(c + d*x))/(8*d) - (3*a*Cos[c + d*x])/(4*d) + (a*Cos[3*(c + d*x)])/(12*d) - (b*Sin[2*(c + d*x)])/(4*d) + (b*Sin[4*(c + d*x)])/(32*d)","A",1
935,1,134,161,0.2101361,"\int \sin (c+d x) \left(a \sin ^2(c+d x)+b \sin ^3(c+d x)\right)^2 \, dx","Integrate[Sin[c + d*x]*(a*Sin[c + d*x]^2 + b*Sin[c + d*x]^3)^2,x]","\frac{-525 \left(8 a^2+7 b^2\right) \cos (c+d x)+35 \left(20 a^2+21 b^2\right) \cos (3 (c+d x))-84 a^2 \cos (5 (c+d x))-3150 a b \sin (2 (c+d x))+630 a b \sin (4 (c+d x))-70 a b \sin (6 (c+d x))+4200 a b c+4200 a b d x-147 b^2 \cos (5 (c+d x))+15 b^2 \cos (7 (c+d x))}{6720 d}","-\frac{\left(a^2+3 b^2\right) \cos ^5(c+d x)}{5 d}+\frac{\left(2 a^2+3 b^2\right) \cos ^3(c+d x)}{3 d}-\frac{\left(a^2+b^2\right) \cos (c+d x)}{d}-\frac{a b \sin ^5(c+d x) \cos (c+d x)}{3 d}-\frac{5 a b \sin ^3(c+d x) \cos (c+d x)}{12 d}-\frac{5 a b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a b x}{8}+\frac{b^2 \cos ^7(c+d x)}{7 d}",1,"(4200*a*b*c + 4200*a*b*d*x - 525*(8*a^2 + 7*b^2)*Cos[c + d*x] + 35*(20*a^2 + 21*b^2)*Cos[3*(c + d*x)] - 84*a^2*Cos[5*(c + d*x)] - 147*b^2*Cos[5*(c + d*x)] + 15*b^2*Cos[7*(c + d*x)] - 3150*a*b*Sin[2*(c + d*x)] + 630*a*b*Sin[4*(c + d*x)] - 70*a*b*Sin[6*(c + d*x)])/(6720*d)","A",1
936,1,105,89,0.1505398,"\int \sin (c+d x) \left(a \sin (c+d x)+b \sin ^2(c+d x)+c \sin ^3(c+d x)\right) \, dx","Integrate[Sin[c + d*x]*(a*Sin[c + d*x] + b*Sin[c + d*x]^2 + c*Sin[c + d*x]^3),x]","\frac{a (c+d x)}{2 d}-\frac{a \sin (2 (c+d x))}{4 d}-\frac{3 b \cos (c+d x)}{4 d}+\frac{b \cos (3 (c+d x))}{12 d}+\frac{3 c (c+d x)}{8 d}-\frac{c \sin (2 (c+d x))}{4 d}+\frac{c \sin (4 (c+d x))}{32 d}","-\frac{(4 a+3 c) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 a+3 c)+\frac{b \cos ^3(c+d x)}{3 d}-\frac{b \cos (c+d x)}{d}-\frac{c \sin ^3(c+d x) \cos (c+d x)}{4 d}",1,"(a*(c + d*x))/(2*d) + (3*c*(c + d*x))/(8*d) - (3*b*Cos[c + d*x])/(4*d) + (b*Cos[3*(c + d*x)])/(12*d) - (a*Sin[2*(c + d*x)])/(4*d) - (c*Sin[2*(c + d*x)])/(4*d) + (c*Sin[4*(c + d*x)])/(32*d)","A",1
937,1,167,288,0.4485384,"\int \sin (c+d x) \left(a \sin (c+d x)+b \sin ^2(c+d x)+c \sin ^3(c+d x)\right)^2 \, dx","Integrate[Sin[c + d*x]*(a*Sin[c + d*x] + b*Sin[c + d*x]^2 + c*Sin[c + d*x]^3)^2,x]","\frac{-105 \left(48 a^2+80 a c+40 b^2+35 c^2\right) \cos (c+d x)+35 \left(16 a^2+40 a c+20 b^2+21 c^2\right) \cos (3 (c+d x))-21 \left(c (8 a+7 c)+4 b^2\right) \cos (5 (c+d x))+840 b (6 a+5 c) (c+d x)-210 b (16 a+15 c) \sin (2 (c+d x))+210 b (2 a+3 c) \sin (4 (c+d x))-70 b c \sin (6 (c+d x))+15 c^2 \cos (7 (c+d x))}{6720 d}","\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{\left(2 a c+b^2\right) \cos ^5(c+d x)}{5 d}+\frac{2 \left(2 a c+b^2\right) \cos ^3(c+d x)}{3 d}-\frac{\left(2 a c+b^2\right) \cos (c+d x)}{d}-\frac{a b \sin ^3(c+d x) \cos (c+d x)}{2 d}-\frac{3 a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a b x}{4}-\frac{b c \sin ^5(c+d x) \cos (c+d x)}{3 d}-\frac{5 b c \sin ^3(c+d x) \cos (c+d x)}{12 d}-\frac{5 b c \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 b c x}{8}+\frac{c^2 \cos ^7(c+d x)}{7 d}-\frac{3 c^2 \cos ^5(c+d x)}{5 d}+\frac{c^2 \cos ^3(c+d x)}{d}-\frac{c^2 \cos (c+d x)}{d}",1,"(840*b*(6*a + 5*c)*(c + d*x) - 105*(48*a^2 + 40*b^2 + 80*a*c + 35*c^2)*Cos[c + d*x] + 35*(16*a^2 + 20*b^2 + 40*a*c + 21*c^2)*Cos[3*(c + d*x)] - 21*(4*b^2 + c*(8*a + 7*c))*Cos[5*(c + d*x)] + 15*c^2*Cos[7*(c + d*x)] - 210*b*(16*a + 15*c)*Sin[2*(c + d*x)] + 210*b*(2*a + 3*c)*Sin[4*(c + d*x)] - 70*b*c*Sin[6*(c + d*x)])/(6720*d)","A",1
938,1,55,61,0.1793574,"\int \sin (c+d x) \left(a+\frac{b}{\sqrt{\sin (c+d x)}}+c \sin (c+d x)\right) \, dx","Integrate[Sin[c + d*x]*(a + b/Sqrt[Sin[c + d*x]] + c*Sin[c + d*x]),x]","\frac{-4 a \cos (c+d x)-8 b E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+c (-\sin (2 (c+d x))+2 c+2 d x)}{4 d}","-\frac{a \cos (c+d x)}{d}+\frac{2 b E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d}-\frac{c \sin (c+d x) \cos (c+d x)}{2 d}+\frac{c x}{2}",1,"(-4*a*Cos[c + d*x] - 8*b*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + c*(2*c + 2*d*x - Sin[2*(c + d*x)]))/(4*d)","A",1
939,1,137,148,0.2810297,"\int \sin (c+d x) \left(a+\frac{b}{\sqrt{\sin (c+d x)}}+c \sin (c+d x)\right)^2 \, dx","Integrate[Sin[c + d*x]*(a + b/Sqrt[Sin[c + d*x]] + c*Sin[c + d*x])^2,x]","\frac{-12 a^2 \cos (c+d x)-48 a b E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+12 a c^2+12 a c d x-6 a c \sin (2 (c+d x))+12 b^2 c+12 b^2 d x-16 b c F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-16 b c \sqrt{\sin (c+d x)} \cos (c+d x)-9 c^2 \cos (c+d x)+c^2 \cos (3 (c+d x))}{12 d}","-\frac{a^2 \cos (c+d x)}{d}+\frac{4 a b E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d}-\frac{a c \sin (c+d x) \cos (c+d x)}{d}+a c x+b^2 x+\frac{4 b c F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d}-\frac{4 b c \sqrt{\sin (c+d x)} \cos (c+d x)}{3 d}+\frac{c^2 \cos ^3(c+d x)}{3 d}-\frac{c^2 \cos (c+d x)}{d}",1,"(12*b^2*c + 12*a*c^2 + 12*b^2*d*x + 12*a*c*d*x - 12*a^2*Cos[c + d*x] - 9*c^2*Cos[c + d*x] + c^2*Cos[3*(c + d*x)] - 48*a*b*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] - 16*b*c*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] - 16*b*c*Cos[c + d*x]*Sqrt[Sin[c + d*x]] - 6*a*c*Sin[2*(c + d*x)])/(12*d)","A",1
940,1,43,34,0.1002538,"\int f^{a+b x} (\cos (c+d x)+i \sin (c+d x))^n \, dx","Integrate[f^(a + b*x)*(Cos[c + d*x] + I*Sin[c + d*x])^n,x]","-\frac{i f^{a+b x} (\cos (c+d x)+i \sin (c+d x))^n}{d n-i b \log (f)}","\frac{f^{a+b x} \left(e^{i (c+d x)}\right)^n}{b \log (f)+i d n}",1,"((-I)*f^(a + b*x)*(Cos[c + d*x] + I*Sin[c + d*x])^n)/(d*n - I*b*Log[f])","A",1
941,1,43,36,0.0892054,"\int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx","Integrate[f^(a + b*x)*(Cos[c + d*x] - I*Sin[c + d*x])^n,x]","\frac{i f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n}{d n+i b \log (f)}","-\frac{f^{a+b x} \left(e^{-i (c+d x)}\right)^n}{-b \log (f)+i d n}",1,"(I*f^(a + b*x)*(Cos[c + d*x] - I*Sin[c + d*x])^n)/(d*n + I*b*Log[f])","A",1
942,1,73,120,0.5701371,"\int \frac{\cos ^5(a+b x)-\sin ^5(a+b x)}{\cos ^5(a+b x)+\sin ^5(a+b x)} \, dx","Integrate[(Cos[a + b*x]^5 - Sin[a + b*x]^5)/(Cos[a + b*x]^5 + Sin[a + b*x]^5),x]","\frac{-\left(\sqrt{5}-1\right) \log \left(\sin (2 (a+b x))-\sqrt{5}+1\right)+\left(1+\sqrt{5}\right) \log \left(\sin (2 (a+b x))+\sqrt{5}+1\right)+\log (\sin (a+b x)+\cos (a+b x))}{5 b}","-\frac{4 \log \left(2 \tan ^2(a+b x)-\left(1-\sqrt{5}\right) \tan (a+b x)+2\right)}{5 \left(1-\sqrt{5}\right) b}-\frac{4 \log \left(2 \tan ^2(a+b x)-\left(1+\sqrt{5}\right) \tan (a+b x)+2\right)}{5 \left(1+\sqrt{5}\right) b}+\frac{\log (\tan (a+b x)+1)}{5 b}+\frac{\log (\cos (a+b x))}{b}",1,"(Log[Cos[a + b*x] + Sin[a + b*x]] - (-1 + Sqrt[5])*Log[1 - Sqrt[5] + Sin[2*(a + b*x)]] + (1 + Sqrt[5])*Log[1 + Sqrt[5] + Sin[2*(a + b*x)]])/(5*b)","A",1
943,1,25,72,0.0319492,"\int \frac{\cos ^4(a+b x)-\sin ^4(a+b x)}{\cos ^4(a+b x)+\sin ^4(a+b x)} \, dx","Integrate[(Cos[a + b*x]^4 - Sin[a + b*x]^4)/(Cos[a + b*x]^4 + Sin[a + b*x]^4),x]","\frac{\tanh ^{-1}\left(\frac{\sin (2 a+2 b x)}{\sqrt{2}}\right)}{\sqrt{2} b}","\frac{\log \left(\tan ^2(a+b x)+\sqrt{2} \tan (a+b x)+1\right)}{2 \sqrt{2} b}-\frac{\log \left(\tan ^2(a+b x)-\sqrt{2} \tan (a+b x)+1\right)}{2 \sqrt{2} b}",1,"ArcTanh[Sin[2*a + 2*b*x]/Sqrt[2]]/(Sqrt[2]*b)","A",1
944,1,42,55,0.2002553,"\int \frac{\cos ^3(a+b x)-\sin ^3(a+b x)}{\cos ^3(a+b x)+\sin ^3(a+b x)} \, dx","Integrate[(Cos[a + b*x]^3 - Sin[a + b*x]^3)/(Cos[a + b*x]^3 + Sin[a + b*x]^3),x]","\frac{\log (\sin (a+b x)+\cos (a+b x))}{3 b}-\frac{2 \log (2-\sin (2 (a+b x)))}{3 b}","-\frac{2 \log \left(\tan ^2(a+b x)-\tan (a+b x)+1\right)}{3 b}+\frac{\log (\tan (a+b x)+1)}{3 b}-\frac{\log (\cos (a+b x))}{b}",1,"Log[Cos[a + b*x] + Sin[a + b*x]]/(3*b) - (2*Log[2 - Sin[2*(a + b*x)]])/(3*b)","A",1
945,1,33,16,0.011954,"\int \frac{\cos ^2(a+b x)-\sin ^2(a+b x)}{\cos ^2(a+b x)+\sin ^2(a+b x)} \, dx","Integrate[(Cos[a + b*x]^2 - Sin[a + b*x]^2)/(Cos[a + b*x]^2 + Sin[a + b*x]^2),x]","\frac{\sin (2 a) \cos (2 b x)}{2 b}+\frac{\cos (2 a) \sin (2 b x)}{2 b}","\frac{\sin (a+b x) \cos (a+b x)}{b}",1,"(Cos[2*b*x]*Sin[2*a])/(2*b) + (Cos[2*a]*Sin[2*b*x])/(2*b)","B",1
946,1,18,18,0.0435061,"\int \frac{\cos (a+b x)-\sin (a+b x)}{\cos (a+b x)+\sin (a+b x)} \, dx","Integrate[(Cos[a + b*x] - Sin[a + b*x])/(Cos[a + b*x] + Sin[a + b*x]),x]","\frac{\log (\sin (a+b x)+\cos (a+b x))}{b}","\frac{\log (\sin (a+b x)+\cos (a+b x))}{b}",1,"Log[Cos[a + b*x] + Sin[a + b*x]]/b","A",1
947,1,19,19,0.0594136,"\int \frac{-\csc (a+b x)+\sec (a+b x)}{\csc (a+b x)+\sec (a+b x)} \, dx","Integrate[(-Csc[a + b*x] + Sec[a + b*x])/(Csc[a + b*x] + Sec[a + b*x]),x]","-\frac{\log (\sin (a+b x)+\cos (a+b x))}{b}","-\frac{\log (\sin (a+b x)+\cos (a+b x))}{b}",1,"-(Log[Cos[a + b*x] + Sin[a + b*x]]/b)","A",1
948,1,33,17,0.0148424,"\int \frac{-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx","Integrate[(-Csc[a + b*x]^2 + Sec[a + b*x]^2)/(Csc[a + b*x]^2 + Sec[a + b*x]^2),x]","-\frac{\sin (2 a) \cos (2 b x)}{2 b}-\frac{\cos (2 a) \sin (2 b x)}{2 b}","-\frac{\sin (a+b x) \cos (a+b x)}{b}",1,"-1/2*(Cos[2*b*x]*Sin[2*a])/b - (Cos[2*a]*Sin[2*b*x])/(2*b)","A",1
949,1,42,54,0.2349966,"\int \frac{-\csc ^3(a+b x)+\sec ^3(a+b x)}{\csc ^3(a+b x)+\sec ^3(a+b x)} \, dx","Integrate[(-Csc[a + b*x]^3 + Sec[a + b*x]^3)/(Csc[a + b*x]^3 + Sec[a + b*x]^3),x]","\frac{2 \log (2-\sin (2 (a+b x)))}{3 b}-\frac{\log (\sin (a+b x)+\cos (a+b x))}{3 b}","\frac{2 \log \left(\tan ^2(a+b x)-\tan (a+b x)+1\right)}{3 b}-\frac{\log (\tan (a+b x)+1)}{3 b}+\frac{\log (\cos (a+b x))}{b}",1,"-1/3*Log[Cos[a + b*x] + Sin[a + b*x]]/b + (2*Log[2 - Sin[2*(a + b*x)]])/(3*b)","A",1
950,1,26,72,0.0238183,"\int \frac{-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx","Integrate[(-Csc[a + b*x]^4 + Sec[a + b*x]^4)/(Csc[a + b*x]^4 + Sec[a + b*x]^4),x]","-\frac{\tanh ^{-1}\left(\frac{\sin (2 a+2 b x)}{\sqrt{2}}\right)}{\sqrt{2} b}","\frac{\log \left(\tan ^2(a+b x)-\sqrt{2} \tan (a+b x)+1\right)}{2 \sqrt{2} b}-\frac{\log \left(\tan ^2(a+b x)+\sqrt{2} \tan (a+b x)+1\right)}{2 \sqrt{2} b}",1,"-(ArcTanh[Sin[2*a + 2*b*x]/Sqrt[2]]/(Sqrt[2]*b))","A",1