1,1,44,0,0.0397767,"\int \frac{2}{3-\cos (4+6 x)} \, dx","Int[2/(3 - Cos[4 + 6*x]),x]","\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}}","\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,"x/Sqrt[2] + ArcTan[Sin[4 + 6*x]/(3 + 2*Sqrt[2] - Cos[4 + 6*x])]/(3*Sqrt[2])","A",2,2,14,0.1429,1,"{12, 2657}"
2,1,44,0,0.0375237,"\int \frac{2 \csc (4+6 x)}{-\cot (4+6 x)+3 \csc (4+6 x)} \, dx","Int[(2*Csc[4 + 6*x])/(-Cot[4 + 6*x] + 3*Csc[4 + 6*x]),x]","\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}}","\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,"x/Sqrt[2] + ArcTan[Sin[4 + 6*x]/(3 + 2*Sqrt[2] - Cos[4 + 6*x])]/(3*Sqrt[2])","A",3,3,27,0.1111,1,"{12, 3166, 2657}"
3,1,48,0,0.0215153,"\int \frac{1}{1+\sin ^2(2+3 x)} \, dx","Int[(1 + Sin[2 + 3*x]^2)^(-1),x]","\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}}","\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,"x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,2,12,0.1667,1,"{3181, 203}"
4,1,48,0,0.0198624,"\int \frac{1}{2-\cos ^2(2+3 x)} \, dx","Int[(2 - Cos[2 + 3*x]^2)^(-1),x]","\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}}","\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,"x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,2,14,0.1429,1,"{3181, 203}"
5,1,48,0,0.0255043,"\int \frac{1}{\cos ^2(2+3 x)+2 \sin ^2(2+3 x)} \, dx","Int[(Cos[2 + 3*x]^2 + 2*Sin[2 + 3*x]^2)^(-1),x]","\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}}","\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,"x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,1,21,0.04762,1,"{203}"
6,1,48,0,0.0436017,"\int \frac{\sec ^2(2+3 x)}{1+2 \tan ^2(2+3 x)} \, dx","Int[Sec[2 + 3*x]^2/(1 + 2*Tan[2 + 3*x]^2),x]","\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}}","\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,"x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,2,23,0.08696,1,"{3675, 203}"
7,1,48,0,0.041115,"\int \frac{\csc ^2(2+3 x)}{2+\cot ^2(2+3 x)} \, dx","Int[Csc[2 + 3*x]^2/(2 + Cot[2 + 3*x]^2),x]","\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}}","\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,"x/Sqrt[2] + ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Sin[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,2,21,0.09524,1,"{3675, 203}"
8,1,60,0,0.0260871,"\int \frac{2}{1-3 \cos (4+6 x)} \, dx","Int[2/(1 - 3*Cos[4 + 6*x]),x]","\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}}","\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,"Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])","A",3,3,14,0.2143,1,"{12, 2659, 207}"
9,1,60,0,0.046477,"\int \frac{2 \csc (4+6 x)}{-3 \cot (4+6 x)+\csc (4+6 x)} \, dx","Int[(2*Csc[4 + 6*x])/(-3*Cot[4 + 6*x] + Csc[4 + 6*x]),x]","\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}}","\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,"Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])","A",4,4,25,0.1600,1,"{12, 3166, 2659, 207}"
10,1,60,0,0.0195075,"\int \frac{1}{-1+3 \sin ^2(2+3 x)} \, dx","Int[(-1 + 3*Sin[2 + 3*x]^2)^(-1),x]","\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}}","\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,"Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,2,14,0.1429,1,"{3181, 207}"
11,1,60,0,0.0191408,"\int \frac{1}{2-3 \cos ^2(2+3 x)} \, dx","Int[(2 - 3*Cos[2 + 3*x]^2)^(-1),x]","\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}}","\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,"Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,2,14,0.1429,1,"{3181, 206}"
12,1,60,0,0.0304304,"\int \frac{1}{-\cos ^2(2+3 x)+2 \sin ^2(2+3 x)} \, dx","Int[(-Cos[2 + 3*x]^2 + 2*Sin[2 + 3*x]^2)^(-1),x]","\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}}","\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,"Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,1,23,0.04348,1,"{207}"
13,1,60,0,0.0453745,"\int \frac{\sec ^2(2+3 x)}{-1+2 \tan ^2(2+3 x)} \, dx","Int[Sec[2 + 3*x]^2/(-1 + 2*Tan[2 + 3*x]^2),x]","\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}}","\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,"Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,2,23,0.08696,1,"{3675, 207}"
14,1,60,0,0.0462344,"\int \frac{\csc ^2(2+3 x)}{2-\cot ^2(2+3 x)} \, dx","Int[Csc[2 + 3*x]^2/(2 - Cot[2 + 3*x]^2),x]","\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}}","\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,"Log[Cos[2 + 3*x] - Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Cos[2 + 3*x] + Sqrt[2]*Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,2,23,0.08696,1,"{3675, 206}"
15,1,42,0,0.036651,"\int \frac{2}{3+\cos (4+6 x)} \, dx","Int[2/(3 + Cos[4 + 6*x]),x]","\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}}","\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,"x/Sqrt[2] - ArcTan[Sin[4 + 6*x]/(3 + 2*Sqrt[2] + Cos[4 + 6*x])]/(3*Sqrt[2])","A",2,2,12,0.1667,1,"{12, 2657}"
16,1,42,0,0.0370735,"\int \frac{2 \csc (4+6 x)}{\cot (4+6 x)+3 \csc (4+6 x)} \, dx","Int[(2*Csc[4 + 6*x])/(Cot[4 + 6*x] + 3*Csc[4 + 6*x]),x]","\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}}","\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,"x/Sqrt[2] - ArcTan[Sin[4 + 6*x]/(3 + 2*Sqrt[2] + Cos[4 + 6*x])]/(3*Sqrt[2])","A",3,3,25,0.1200,1,"{12, 3166, 2657}"
17,1,48,0,0.0190109,"\int \frac{1}{2-\sin ^2(2+3 x)} \, dx","Int[(2 - Sin[2 + 3*x]^2)^(-1),x]","\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}}","\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,"x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,2,14,0.1429,1,"{3181, 203}"
18,1,48,0,0.0163445,"\int \frac{1}{1+\cos ^2(2+3 x)} \, dx","Int[(1 + Cos[2 + 3*x]^2)^(-1),x]","\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}}","\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,"x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,2,12,0.1667,1,"{3181, 203}"
19,1,48,0,0.026868,"\int \frac{1}{2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx","Int[(2*Cos[2 + 3*x]^2 + Sin[2 + 3*x]^2)^(-1),x]","\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}}","\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,"x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,1,21,0.04762,1,"{203}"
20,1,48,0,0.0422988,"\int \frac{\sec ^2(2+3 x)}{2+\tan ^2(2+3 x)} \, dx","Int[Sec[2 + 3*x]^2/(2 + Tan[2 + 3*x]^2),x]","\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}}","\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,"x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,2,21,0.09524,1,"{3675, 203}"
21,1,48,0,0.044736,"\int \frac{\csc ^2(2+3 x)}{1+2 \cot ^2(2+3 x)} \, dx","Int[Csc[2 + 3*x]^2/(1 + 2*Cot[2 + 3*x]^2),x]","\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}}","\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,"x/Sqrt[2] - ArcTan[(Cos[2 + 3*x]*Sin[2 + 3*x])/(1 + Sqrt[2] + Cos[2 + 3*x]^2)]/(3*Sqrt[2])","A",2,2,23,0.08696,1,"{3675, 203}"
22,1,61,0,0.0303837,"\int -\frac{2}{1+3 \cos (4+6 x)} \, dx","Int[-2/(1 + 3*Cos[4 + 6*x]),x]","\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}}","\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,"Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])","A",3,3,14,0.2143,1,"{12, 2659, 206}"
23,1,61,0,0.0457299,"\int -\frac{2 \csc (4+6 x)}{3 \cot (4+6 x)+\csc (4+6 x)} \, dx","Int[(-2*Csc[4 + 6*x])/(3*Cot[4 + 6*x] + Csc[4 + 6*x]),x]","\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}}","\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,"Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])","A",4,4,25,0.1600,1,"{12, 3166, 2659, 206}"
24,1,61,0,0.0201867,"\int \frac{1}{-2+3 \sin ^2(2+3 x)} \, dx","Int[(-2 + 3*Sin[2 + 3*x]^2)^(-1),x]","\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}}","\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,"Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,2,14,0.1429,1,"{3181, 207}"
25,1,61,0,0.0191013,"\int \frac{1}{1-3 \cos ^2(2+3 x)} \, dx","Int[(1 - 3*Cos[2 + 3*x]^2)^(-1),x]","\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}}","\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,"Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,2,14,0.1429,1,"{3181, 206}"
26,1,61,0,0.0307402,"\int \frac{1}{-2 \cos ^2(2+3 x)+\sin ^2(2+3 x)} \, dx","Int[(-2*Cos[2 + 3*x]^2 + Sin[2 + 3*x]^2)^(-1),x]","\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}}","\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,"Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,1,21,0.04762,1,"{207}"
27,1,61,0,0.0429416,"\int \frac{\sec ^2(2+3 x)}{-2+\tan ^2(2+3 x)} \, dx","Int[Sec[2 + 3*x]^2/(-2 + Tan[2 + 3*x]^2),x]","\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}}","\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,"Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,2,21,0.09524,1,"{3675, 207}"
28,1,61,0,0.046606,"\int \frac{\csc ^2(2+3 x)}{1-2 \cot ^2(2+3 x)} \, dx","Int[Csc[2 + 3*x]^2/(1 - 2*Cot[2 + 3*x]^2),x]","\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}}","\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,"Log[Sqrt[2]*Cos[2 + 3*x] - Sin[2 + 3*x]]/(6*Sqrt[2]) - Log[Sqrt[2]*Cos[2 + 3*x] + Sin[2 + 3*x]]/(6*Sqrt[2])","A",2,2,23,0.08696,1,"{3675, 206}"
29,1,30,0,0.0348256,"\int (x+\sin (x))^2 \, dx","Int[(x + Sin[x])^2,x]","\frac{x^3}{3}+\frac{x}{2}+2 \sin (x)-2 x \cos (x)-\frac{1}{2} \sin (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/2 + x^3/3 - 2*x*Cos[x] + 2*Sin[x] - (Cos[x]*Sin[x])/2","A",6,5,6,0.8333,1,"{6742, 3296, 2637, 2635, 8}"
30,1,56,0,0.0669768,"\int (x+\sin (x))^3 \, dx","Int[(x + Sin[x])^3,x]","\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)","\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,"(3*x^2)/4 + x^4/4 + 5*Cos[x] - 3*x^2*Cos[x] + Cos[x]^3/3 + 6*x*Sin[x] - (3*x*Cos[x]*Sin[x])/2 + (3*Sin[x]^2)/4","A",9,6,6,1.000,1,"{6742, 3296, 2638, 3310, 30, 2633}"
31,1,213,0,0.5363059,"\int \frac{\sin (a+b x)}{c+d x^2} \, dx","Int[Sin[a + b*x]/(c + d*x^2),x]","-\frac{\sin \left(a-\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{CosIntegral}\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{CosIntegral}\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}}","-\frac{\sin \left(a-\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{CosIntegral}\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{CosIntegral}\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,"-(CosIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x]*Sin[a - (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d]) + (CosIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x]*Sin[a + (b*Sqrt[-c])/Sqrt[d]])/(2*Sqrt[-c]*Sqrt[d]) - (Cos[a + (b*Sqrt[-c])/Sqrt[d]]*SinIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Cos[a - (b*Sqrt[-c])/Sqrt[d]]*SinIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])","A",8,4,16,0.2500,1,"{3333, 3303, 3299, 3302}"
32,1,271,0,0.8016345,"\int \frac{\sin (a+b x)}{c+d x+e x^2} \, dx","Int[Sin[a + b*x]/(c + d*x + e*x^2),x]","\frac{\sin \left(a-\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}\right) \text{CosIntegral}\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{CosIntegral}\left(\frac{b \left(\sqrt{d^2-4 c e}+d\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}}","\frac{\sin \left(a-\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}\right) \text{CosIntegral}\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{CosIntegral}\left(\frac{b \left(\sqrt{d^2-4 c e}+d\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) + b*x]*Sin[a - (b*(d - Sqrt[d^2 - 4*c*e]))/(2*e)])/Sqrt[d^2 - 4*c*e] - (CosIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x]*Sin[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)])/Sqrt[d^2 - 4*c*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])/Sqrt[d^2 - 4*c*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])/Sqrt[d^2 - 4*c*e]","A",8,4,19,0.2105,1,"{6728, 3303, 3299, 3302}"
33,1,10,0,0.0179978,"\int \frac{\sin \left(\sqrt{-7+x}\right)}{\sqrt{-7+x}} \, dx","Int[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",3,3,16,0.1875,1,"{3431, 15, 2638}"
34,1,28,0,0.4709823,"\int \frac{\sqrt{b-\frac{a}{x^2}} \sin (x)}{\sqrt{a-b x^2}} \, dx","Int[(Sqrt[b - a/x^2]*Sin[x])/Sqrt[a - b*x^2],x]","\frac{x \text{Si}(x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}}","\frac{x \text{Si}(x) \sqrt{b-\frac{a}{x^2}}}{\sqrt{a-b x^2}}",1,"(Sqrt[b - a/x^2]*x*SinIntegral[x])/Sqrt[a - b*x^2]","A",3,3,27,0.1111,1,"{6721, 23, 3299}"
35,1,12,0,0.0289448,"\int \frac{1}{x (1+\sin (\log (x)))} \, dx","Int[1/(x*(1 + Sin[Log[x]])),x]","-\frac{\cos (\log (x))}{\sin (\log (x))+1}","-\frac{\cos (\log (x))}{\sin (\log (x))+1}",1,"-(Cos[Log[x]]/(1 + Sin[Log[x]]))","A",2,1,11,0.09091,1,"{2648}"
36,1,100,0,0.1635255,"\int \sin \left(\frac{a+b x}{c+d x}\right) \, dx","Int[Sin[(a + b*x)/(c + d*x)],x]","\frac{\cos \left(\frac{b}{d}\right) (b c-a d) \text{CosIntegral}\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}","\frac{\cos \left(\frac{b}{d}\right) (b c-a d) \text{CosIntegral}\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,"((b*c - a*d)*Cos[b/d]*CosIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2 + ((c + d*x)*Sin[(a + b*x)/(c + d*x)])/d + ((b*c - a*d)*Sin[b/d]*SinIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2","A",5,5,14,0.3571,1,"{4563, 3297, 3303, 3299, 3302}"
37,1,107,0,0.1920036,"\int \sin ^2\left(\frac{a+b x}{c+d x}\right) \, dx","Int[Sin[(a + b*x)/(c + d*x)]^2,x]","\frac{\sin \left(\frac{2 b}{d}\right) (b c-a d) \text{CosIntegral}\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}","\frac{\sin \left(\frac{2 b}{d}\right) (b c-a d) \text{CosIntegral}\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,"((b*c - a*d)*CosIntegral[(2*(b*c - a*d))/(d*(c + d*x))]*Sin[(2*b)/d])/d^2 + ((c + d*x)*Sin[(a + b*x)/(c + d*x)]^2)/d - ((b*c - a*d)*Cos[(2*b)/d]*SinIntegral[(2*(b*c - a*d))/(d*(c + d*x))])/d^2","A",6,6,16,0.3750,1,"{4563, 3313, 12, 3303, 3299, 3302}"
38,1,194,0,0.3221274,"\int \sin ^3\left(\frac{a+b x}{c+d x}\right) \, dx","Int[Sin[(a + b*x)/(c + d*x)]^3,x]","\frac{3 \cos \left(\frac{b}{d}\right) (b c-a d) \text{CosIntegral}\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{CosIntegral}\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}","\frac{3 \cos \left(\frac{b}{d}\right) (b c-a d) \text{CosIntegral}\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{CosIntegral}\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 - a*d)*Cos[b/d]*CosIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Cos[(3*b)/d]*CosIntegral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2) + ((c + d*x)*Sin[(a + b*x)/(c + d*x)]^3)/d + (3*(b*c - a*d)*Sin[b/d]*SinIntegral[(b*c - a*d)/(d*(c + d*x))])/(4*d^2) - (3*(b*c - a*d)*Sin[(3*b)/d]*SinIntegral[(3*(b*c - a*d))/(d*(c + d*x))])/(4*d^2)","A",9,5,16,0.3125,1,"{4563, 3313, 3303, 3299, 3302}"
39,1,58,0,0.1107031,"\int \frac{\sin ^3\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[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)}{4 a}-\frac{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]])/(4*a) + SinIntegral[(3*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(4*a)","A",5,3,36,0.08333,1,"{6681, 3312, 3299}"
40,1,58,0,0.0817573,"\int \frac{\sin ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[Sin[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2),x]","\frac{\text{CosIntegral}\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}","\frac{\text{CosIntegral}\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[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(2*a)","A",4,3,36,0.08333,1,"{6681, 3312, 3302}"
41,1,26,0,0.0397187,"\int \frac{\sin \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[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",2,2,34,0.05882,1,"{6681, 3299}"
42,0,0,0,0.0369904,"\int \frac{\csc \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[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,"-(Defer[Subst][Defer[Int][Csc[x]/x, x], x, Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)","A",0,0,0,0,-1,"{}"
43,0,0,0,0.0806222,"\int \frac{\csc ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[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,"-(Defer[Subst][Defer[Int][Csc[x]^2/x, x], x, Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)","A",0,0,0,0,-1,"{}"
44,1,30,0,0.0344333,"\int (x+\cos (x))^2 \, dx","Int[(x + Cos[x])^2,x]","\frac{x^3}{3}+\frac{x}{2}+2 x \sin (x)+2 \cos (x)+\frac{1}{2} \sin (x) \cos (x)","\frac{x^3}{3}+\frac{x}{2}+2 x \sin (x)+2 \cos (x)+\frac{1}{2} \sin (x) \cos (x)",1,"x/2 + x^3/3 + 2*Cos[x] + 2*x*Sin[x] + (Cos[x]*Sin[x])/2","A",6,5,6,0.8333,1,"{6742, 3296, 2638, 2635, 8}"
45,1,56,0,0.0694327,"\int (x+\cos (x))^3 \, dx","Int[(x + Cos[x])^3,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)","\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,"(3*x^2)/4 + x^4/4 + 6*x*Cos[x] + (3*Cos[x]^2)/4 - 5*Sin[x] + 3*x^2*Sin[x] + (3*x*Cos[x]*Sin[x])/2 - Sin[x]^3/3","A",9,6,6,1.000,1,"{6742, 3296, 2637, 3310, 30, 2633}"
46,1,213,0,0.3078998,"\int \frac{\cos (a+b x)}{c+d x^2} \, dx","Int[Cos[a + b*x]/(c + d*x^2),x]","\frac{\cos \left(a+\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{CosIntegral}\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{CosIntegral}\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(\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}}","\frac{\cos \left(a+\frac{b \sqrt{-c}}{\sqrt{d}}\right) \text{CosIntegral}\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{CosIntegral}\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(\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,"(Cos[a + (b*Sqrt[-c])/Sqrt[d]]*CosIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) - (Cos[a - (b*Sqrt[-c])/Sqrt[d]]*CosIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d]) + (Sin[a + (b*Sqrt[-c])/Sqrt[d]]*SinIntegral[(b*Sqrt[-c])/Sqrt[d] - b*x])/(2*Sqrt[-c]*Sqrt[d]) + (Sin[a - (b*Sqrt[-c])/Sqrt[d]]*SinIntegral[(b*Sqrt[-c])/Sqrt[d] + b*x])/(2*Sqrt[-c]*Sqrt[d])","A",8,4,16,0.2500,1,"{3334, 3303, 3299, 3302}"
47,1,271,0,0.5623296,"\int \frac{\cos (a+b x)}{c+d x+e x^2} \, dx","Int[Cos[a + b*x]/(c + d*x + e*x^2),x]","\frac{\cos \left(a-\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}\right) \text{CosIntegral}\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{CosIntegral}\left(\frac{b \left(\sqrt{d^2-4 c e}+d\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}}","\frac{\cos \left(a-\frac{b \left(d-\sqrt{d^2-4 c e}\right)}{2 e}\right) \text{CosIntegral}\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{CosIntegral}\left(\frac{b \left(\sqrt{d^2-4 c e}+d\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) + b*x])/Sqrt[d^2 - 4*c*e] - (Cos[a - (b*(d + Sqrt[d^2 - 4*c*e]))/(2*e)]*CosIntegral[(b*(d + Sqrt[d^2 - 4*c*e]))/(2*e) + b*x])/Sqrt[d^2 - 4*c*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])/Sqrt[d^2 - 4*c*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])/Sqrt[d^2 - 4*c*e]","A",8,4,19,0.2105,1,"{6728, 3303, 3299, 3302}"
48,1,10,0,0.1358686,"\int \frac{x \cos \left(\sqrt{1+x^2}\right)}{\sqrt{1+x^2}} \, dx","Int[(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",4,4,21,0.1905,1,"{6715, 3432, 15, 2637}"
49,1,22,0,0.1892876,"\int \frac{x \cos \left(\sqrt{3} \sqrt{2+x^2}\right)}{\sqrt{2+x^2}} \, dx","Int[(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",4,4,27,0.1481,1,"{6715, 3432, 15, 2637}"
50,1,24,0,0.4930793,"\int \frac{(-1+2 x) \cos \left(\sqrt{6+3 (-1+2 x)^2}\right)}{\sqrt{6+3 (-1+2 x)^2}} \, dx","Int[((-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} \sqrt{(2 x-1)^2+2}\right)","\frac{1}{6} \sin \left(\sqrt{3} \sqrt{(2 x-1)^2+2}\right)",1,"Sin[Sqrt[3]*Sqrt[2 + (-1 + 2*x)^2]]/6","A",5,4,37,0.1081,1,"{6715, 3432, 15, 2637}"
51,1,101,0,0.132663,"\int \cos \left(\frac{a+b x}{c+d x}\right) \, dx","Int[Cos[(a + b*x)/(c + d*x)],x]","-\frac{\sin \left(\frac{b}{d}\right) (b c-a d) \text{CosIntegral}\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}","-\frac{\sin \left(\frac{b}{d}\right) (b c-a d) \text{CosIntegral}\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,"((c + d*x)*Cos[(a + b*x)/(c + d*x)])/d - ((b*c - a*d)*CosIntegral[(b*c - a*d)/(d*(c + d*x))]*Sin[b/d])/d^2 + ((b*c - a*d)*Cos[b/d]*SinIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2","A",5,5,14,0.3571,1,"{4564, 3297, 3303, 3299, 3302}"
52,1,107,0,0.1606481,"\int \cos ^2\left(\frac{a+b x}{c+d x}\right) \, dx","Int[Cos[(a + b*x)/(c + d*x)]^2,x]","-\frac{\sin \left(\frac{2 b}{d}\right) (b c-a d) \text{CosIntegral}\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}","-\frac{\sin \left(\frac{2 b}{d}\right) (b c-a d) \text{CosIntegral}\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,"((c + d*x)*Cos[(a + b*x)/(c + d*x)]^2)/d - ((b*c - a*d)*CosIntegral[(2*(b*c - a*d))/(d*(c + d*x))]*Sin[(2*b)/d])/d^2 + ((b*c - a*d)*Cos[(2*b)/d]*SinIntegral[(2*(b*c - a*d))/(d*(c + d*x))])/d^2","A",6,6,16,0.3750,1,"{4564, 3313, 12, 3303, 3299, 3302}"
53,1,58,0,0.110671,"\int \frac{\cos ^3\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^3/(1 - a^2*x^2),x]","-\frac{3 \text{CosIntegral}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}-\frac{\text{CosIntegral}\left(\frac{3 \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}","-\frac{3 \text{CosIntegral}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}-\frac{\text{CosIntegral}\left(\frac{3 \sqrt{1-a x}}{\sqrt{a x+1}}\right)}{4 a}",1,"(-3*CosIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]])/(4*a) - CosIntegral[(3*Sqrt[1 - a*x])/Sqrt[1 + a*x]]/(4*a)","A",5,3,36,0.08333,1,"{6681, 3312, 3302}"
54,1,58,0,0.0799495,"\int \frac{\cos ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]^2/(1 - a^2*x^2),x]","-\frac{\text{CosIntegral}\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}","-\frac{\text{CosIntegral}\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[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(2*a)","A",4,3,36,0.08333,1,"{6681, 3312, 3302}"
55,1,26,0,0.0377898,"\int \frac{\cos \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[Cos[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/(1 - a^2*x^2),x]","-\frac{\text{CosIntegral}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}","-\frac{\text{CosIntegral}\left(\frac{\sqrt{1-a x}}{\sqrt{a x+1}}\right)}{a}",1,"-(CosIntegral[Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)","A",2,2,34,0.05882,1,"{6681, 3302}"
56,0,0,0,0.0375202,"\int \frac{\sec \left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[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,"-(Defer[Subst][Defer[Int][Sec[x]/x, x], x, Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)","A",0,0,0,0,-1,"{}"
57,0,0,0,0.0842014,"\int \frac{\sec ^2\left(\frac{\sqrt{1-a x}}{\sqrt{1+a x}}\right)}{1-a^2 x^2} \, dx","Int[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,"-(Defer[Subst][Defer[Int][Sec[x]^2/x, x], x, Sqrt[1 - a*x]/Sqrt[1 + a*x]]/a)","A",0,0,0,0,-1,"{}"
58,1,9,0,0.0097141,"\int \frac{\tan \left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Int[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",2,2,12,0.1667,1,"{3747, 3475}"
59,1,16,0,0.0178689,"\int \frac{\tan ^2\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Int[Tan[Sqrt[x]]^2/Sqrt[x],x]","2 \tan \left(\sqrt{x}\right)-2 \sqrt{x}","2 \tan \left(\sqrt{x}\right)-2 \sqrt{x}",1,"-2*Sqrt[x] + 2*Tan[Sqrt[x]]","A",3,3,14,0.2143,1,"{3747, 3473, 8}"
60,1,70,0,0.0912281,"\int \sqrt{x} \tan \left(\sqrt{x}\right) \, dx","Int[Sqrt[x]*Tan[Sqrt[x]],x]","2 i \sqrt{x} \text{PolyLog}\left(2,-e^{2 i \sqrt{x}}\right)-\text{PolyLog}\left(3,-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{PolyLog}\left(2,-e^{2 i \sqrt{x}}\right)-\text{PolyLog}\left(3,-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",6,6,12,0.5000,1,"{3747, 3719, 2190, 2531, 2282, 6589}"
61,1,19,0,0.0176489,"\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","Int[(b*Tan[a + b*x + c*x^2])/(2*c) + x*Tan[a + b*x + c*x^2],x]","-\frac{\log \left(\cos \left(a+b x+c x^2\right)\right)}{2 c}","-\frac{\log \left(\cos \left(a+b x+c x^2\right)\right)}{2 c}",1,"-Log[Cos[a + b*x + c*x^2]]/(2*c)","A",2,1,33,0.03030,1,"{3763}"
62,1,16,0,0.0184914,"\int \frac{\cot ^2\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Int[Cot[Sqrt[x]]^2/Sqrt[x],x]","-2 \sqrt{x}-2 \cot \left(\sqrt{x}\right)","-2 \sqrt{x}-2 \cot \left(\sqrt{x}\right)",1,"-2*Sqrt[x] - 2*Cot[Sqrt[x]]","A",3,3,14,0.2143,1,"{3748, 3473, 8}"
63,1,92,0,0.1640958,"\int \frac{\sqrt{a+b \sec (c+d x)}}{1+\cos (c+d x)} \, dx","Int[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(\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)}}}","\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]/(1 + Sec[c + d*x])], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))])","A",2,2,25,0.08000,1,"{2829, 3968}"
64,1,35,0,0.0333557,"\int \sec (a+b x) \sec (2 a+2 b x) \, dx","Int[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",4,3,16,0.1875,1,"{4364, 1093, 207}"
65,1,35,0,0.0325854,"\int \sec (a+b x) \sec (2 (a+b x)) \, dx","Int[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",4,3,15,0.2000,1,"{4364, 1093, 207}"
66,1,15,0,0.0088231,"\int \sin (x) \sin (2 x) \, dx","Int[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",1,1,7,0.1429,1,"{4282}"
67,1,17,0,0.0082572,"\int \sin (x) \sin (3 x) \, dx","Int[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",1,1,7,0.1429,1,"{4282}"
68,1,17,0,0.0076908,"\int \sin (x) \sin (4 x) \, dx","Int[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",1,1,7,0.1429,1,"{4282}"
69,1,35,0,0.0308315,"\int \sin (x) \sin (m x) \, dx","Int[Sin[x]*Sin[m*x],x]","\frac{\sin ((1-m) x)}{2 (1-m)}-\frac{\sin ((m+1) x)}{2 (m+1)}","\frac{\sin ((1-m) x)}{2 (1-m)}-\frac{\sin ((m+1) x)}{2 (m+1)}",1,"Sin[(1 - m)*x]/(2*(1 - m)) - Sin[(1 + m)*x]/(2*(1 + m))","A",4,2,7,0.2857,1,"{4569, 2637}"
70,1,15,0,0.0081673,"\int \cos (2 x) \sin (x) \, dx","Int[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",1,1,7,0.1429,1,"{4284}"
71,1,17,0,0.0081178,"\int \cos (3 x) \sin (x) \, dx","Int[Cos[3*x]*Sin[x],x]","\frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x)","\frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x)",1,"Cos[2*x]/4 - Cos[4*x]/8","A",1,1,7,0.1429,1,"{4284}"
72,1,17,0,0.0080587,"\int \cos (4 x) \sin (x) \, dx","Int[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",1,1,7,0.1429,1,"{4284}"
73,1,35,0,0.0271964,"\int \cos (m x) \sin (x) \, dx","Int[Cos[m*x]*Sin[x],x]","-\frac{\cos ((1-m) x)}{2 (1-m)}-\frac{\cos ((m+1) x)}{2 (m+1)}","-\frac{\cos ((1-m) x)}{2 (1-m)}-\frac{\cos ((m+1) x)}{2 (m+1)}",1,"-Cos[(1 - m)*x]/(2*(1 - m)) - Cos[(1 + m)*x]/(2*(1 + m))","A",4,2,7,0.2857,1,"{4574, 2638}"
74,1,20,0,0.0230994,"\int \sin (x) \tan (2 x) \, dx","Int[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",4,3,7,0.4286,1,"{12, 321, 206}"
75,1,47,0,0.0519936,"\int \sin (x) \tan (3 x) \, dx","Int[Sin[x]*Tan[3*x],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)","-\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,"-Log[1 - 2*Sin[x]]/6 - Log[1 - Sin[x]]/6 + Log[1 + Sin[x]]/6 + Log[1 + 2*Sin[x]]/6 - Sin[x]","A",9,4,7,0.5714,1,"{1279, 1161, 616, 31}"
76,1,71,0,0.1093188,"\int \sin (x) \tan (4 x) \, dx","Int[Sin[x]*Tan[4*x],x]","-\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)","-\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]]])/4 + (Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[2]]])/4 - Sin[x]","A",5,3,7,0.4286,1,"{1279, 1166, 207}"
77,1,112,0,0.1695667,"\int \sin (x) \tan (5 x) \, dx","Int[Sin[x]*Tan[5*x],x]","-\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))","-\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,"ArcTanh[Sin[x]]/5 - ((1 - Sqrt[5])*Log[1 - Sqrt[5] - 4*Sin[x]])/20 - ((1 + Sqrt[5])*Log[1 + Sqrt[5] - 4*Sin[x]])/20 + ((1 - Sqrt[5])*Log[1 - Sqrt[5] + 4*Sin[x]])/20 + ((1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Sin[x]])/20 - Sin[x]","A",10,4,7,0.5714,1,"{2075, 207, 632, 31}"
78,1,89,0,0.2706879,"\int \sin (x) \tan (6 x) \, dx","Int[Sin[x]*Tan[6*x],x]","-\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)","-\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,"ArcTanh[Sqrt[2]*Sin[x]]/(3*Sqrt[2]) + (Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[3]]])/6 + (Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[3]]])/6 - Sin[x]","A",10,5,7,0.7143,1,"{12, 6742, 2073, 207, 1166}"
79,1,105,0,0.0773561,"\int \sin (x) \tan (n x) \, dx","Int[Sin[x]*Tan[n*x],x]","-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}","-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,"(I/2)/E^(I*x) + (I/2)*E^(I*x) - (I*Hypergeometric2F1[1, -1/(2*n), 1 - 1/(2*n), -E^((2*I)*n*x)])/E^(I*x) - I*E^(I*x)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -E^((2*I)*n*x)]","A",6,3,7,0.4286,1,"{4557, 2194, 2251}"
80,1,10,0,0.0218386,"\int \cot (2 x) \sin (x) \, dx","Int[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,"-ArcTanh[Sin[x]]/2 + Sin[x]","A",3,2,7,0.2857,1,"{388, 206}"
81,1,20,0,0.026517,"\int \cot (3 x) \sin (x) \, dx","Int[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",3,2,7,0.2857,1,"{388, 206}"
82,1,28,0,0.0511705,"\int \cot (4 x) \sin (x) \, dx","Int[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,"-ArcTanh[Sin[x]]/4 - ArcTanh[Sqrt[2]*Sin[x]]/(2*Sqrt[2]) + Sin[x]","A",6,3,7,0.4286,1,"{1676, 1166, 207}"
83,1,82,0,0.1960138,"\int \cot (5 x) \sin (x) \, dx","Int[Cot[5*x]*Sin[x],x]","\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)","\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[(5 + Sqrt[5])/2]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Sin[x]])/5 - (Sqrt[(5 - Sqrt[5])/2]*ArcTanh[Sqrt[(2*(5 + Sqrt[5]))/5]*Sin[x]])/5 + Sin[x]","A",6,3,7,0.4286,1,"{1676, 1166, 207}"
84,1,38,0,0.0823428,"\int \cot (6 x) \sin (x) \, dx","Int[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,"-ArcTanh[Sin[x]]/6 - ArcTanh[2*Sin[x]]/6 - ArcTanh[(2*Sin[x])/Sqrt[3]]/(2*Sqrt[3]) + Sin[x]","A",7,3,7,0.4286,1,"{12, 2073, 207}"
85,1,15,0,0.0153169,"\int \sec (2 x) \sin (x) \, dx","Int[Sec[2*x]*Sin[x],x]","\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{\sqrt{2}}","\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{\sqrt{2}}",1,"ArcTanh[Sqrt[2]*Cos[x]]/Sqrt[2]","A",2,2,7,0.2857,1,"{4357, 207}"
86,1,21,0,0.0273773,"\int \sec (3 x) \sin (x) \, dx","Int[Sec[3*x]*Sin[x],x]","\frac{1}{3} \log (\cos (x))-\frac{1}{6} \log \left(3-4 \cos ^2(x)\right)","\frac{1}{3} \log (\cos (x))-\frac{1}{6} \log \left(3-4 \cos ^2(x)\right)",1,"Log[Cos[x]]/3 - Log[3 - 4*Cos[x]^2]/6","A",5,5,7,0.7143,1,"{4357, 266, 36, 29, 31}"
87,1,71,0,0.0621501,"\int \sec (4 x) \sin (x) \, dx","Int[Sec[4*x]*Sin[x],x]","\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)}}","\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,"-ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) + ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])])","A",4,3,7,0.4286,1,"{4357, 1093, 207}"
88,1,62,0,0.0725292,"\int \sec (5 x) \sin (x) \, dx","Int[Sec[5*x]*Sin[x],x]","\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))","\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,"-Log[Cos[x]]/5 + ((1 + Sqrt[5])*Log[5 - Sqrt[5] - 8*Cos[x]^2])/20 + ((1 - Sqrt[5])*Log[5 + Sqrt[5] - 8*Cos[x]^2])/20","A",7,6,7,0.8571,1,"{4357, 1114, 705, 29, 632, 31}"
89,1,85,0,0.0631543,"\int \sec (6 x) \sin (x) \, dx","Int[Sec[6*x]*Sin[x],x]","-\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}}}","-\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,"-ArcTanh[Sqrt[2]*Cos[x]]/(3*Sqrt[2]) + ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) + ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])","A",7,4,7,0.5714,1,"{4357, 2057, 207, 1166}"
90,1,7,0,0.0119241,"\int \csc (2 x) \sin (x) \, dx","Int[Csc[2*x]*Sin[x],x]","\frac{1}{2} \tanh ^{-1}(\sin (x))","\frac{1}{2} \tanh ^{-1}(\sin (x))",1,"ArcTanh[Sin[x]]/2","A",2,2,7,0.2857,1,"{4288, 3770}"
91,1,45,0,0.0399368,"\int \csc (3 x) \sin (x) \, dx","Int[Csc[3*x]*Sin[x],x]","\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}}","\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,"-Log[Sqrt[3]*Cos[x] - Sin[x]]/(2*Sqrt[3]) + Log[Sqrt[3]*Cos[x] + Sin[x]]/(2*Sqrt[3])","A",2,1,7,0.1429,1,"{206}"
92,1,26,0,0.0250421,"\int \csc (4 x) \sin (x) \, dx","Int[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,"-ArcTanh[Sin[x]]/4 + ArcTanh[Sqrt[2]*Sin[x]]/(2*Sqrt[2])","A",4,2,7,0.2857,1,"{1093, 207}"
93,1,165,0,0.1409522,"\int \csc (5 x) \sin (x) \, dx","Int[Csc[5*x]*Sin[x],x]","-\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)","-\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])/2]*Log[Sqrt[5 - 2*Sqrt[5]]*Cos[x] - Sin[x]])/10 + (Sqrt[(5 + Sqrt[5])/2]*Log[Sqrt[5 + 2*Sqrt[5]]*Cos[x] - Sin[x]])/10 + (Sqrt[(5 - Sqrt[5])/2]*Log[Sqrt[5 - 2*Sqrt[5]]*Cos[x] + Sin[x]])/10 - (Sqrt[(5 + Sqrt[5])/2]*Log[Sqrt[5 + 2*Sqrt[5]]*Cos[x] + Sin[x]])/10","A",4,2,7,0.2857,1,"{1166, 207}"
94,1,36,0,0.0463048,"\int \csc (6 x) \sin (x) \, dx","Int[Csc[6*x]*Sin[x],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}}","\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]]/6 + ArcTanh[2*Sin[x]]/6 - ArcTanh[(2*Sin[x])/Sqrt[3]]/(2*Sqrt[3])","A",7,3,7,0.4286,1,"{12, 2057, 207}"
95,1,8,0,0.0303075,"\int \csc (x) \sin (3 x) \, dx","Int[Csc[x]*Sin[3*x],x]","x+2 \sin (x) \cos (x)","x+2 \sin (x) \cos (x)",1,"x + 2*Cos[x]*Sin[x]","A",3,2,7,0.2857,1,"{385, 203}"
96,1,8,0,0.0135801,"\int \csc (3 x) \sin (6 x) \, dx","Int[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",2,2,9,0.2222,1,"{4288, 2637}"
97,1,15,0,0.0086686,"\int \cos (x) \sin (2 x) \, dx","Int[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,"-Cos[x]/2 - Cos[3*x]/6","A",1,1,7,0.1429,1,"{4284}"
98,1,17,0,0.0084583,"\int \cos (x) \sin (3 x) \, dx","Int[Cos[x]*Sin[3*x],x]","-\frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x)","-\frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x)",1,"-Cos[2*x]/4 - Cos[4*x]/8","A",1,1,7,0.1429,1,"{4284}"
99,1,17,0,0.0082679,"\int \cos (x) \sin (4 x) \, dx","Int[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,"-Cos[3*x]/6 - Cos[5*x]/10","A",1,1,7,0.1429,1,"{4284}"
100,1,35,0,0.0302323,"\int \cos (x) \sin (m x) \, dx","Int[Cos[x]*Sin[m*x],x]","\frac{\cos ((1-m) x)}{2 (1-m)}-\frac{\cos ((m+1) x)}{2 (m+1)}","\frac{\cos ((1-m) x)}{2 (1-m)}-\frac{\cos ((m+1) x)}{2 (m+1)}",1,"Cos[(1 - m)*x]/(2*(1 - m)) - Cos[(1 + m)*x]/(2*(1 + m))","A",4,2,7,0.2857,1,"{4574, 2638}"
101,1,15,0,0.008827,"\int \cos (x) \cos (2 x) \, dx","Int[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",1,1,7,0.1429,1,"{4283}"
102,1,17,0,0.0088595,"\int \cos (x) \cos (3 x) \, dx","Int[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",1,1,7,0.1429,1,"{4283}"
103,1,17,0,0.0083703,"\int \cos (x) \cos (4 x) \, dx","Int[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",1,1,7,0.1429,1,"{4283}"
104,1,35,0,0.0288736,"\int \cos (x) \cos (m x) \, dx","Int[Cos[x]*Cos[m*x],x]","\frac{\sin ((1-m) x)}{2 (1-m)}+\frac{\sin ((m+1) x)}{2 (m+1)}","\frac{\sin ((1-m) x)}{2 (1-m)}+\frac{\sin ((m+1) x)}{2 (m+1)}",1,"Sin[(1 - m)*x]/(2*(1 - m)) + Sin[(1 + m)*x]/(2*(1 + m))","A",4,2,7,0.2857,1,"{4570, 2637}"
105,1,20,0,0.0254347,"\int \cos (x) \tan (2 x) \, dx","Int[Cos[x]*Tan[2*x],x]","\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{\sqrt{2}}-\cos (x)","\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{\sqrt{2}}-\cos (x)",1,"ArcTanh[Sqrt[2]*Cos[x]]/Sqrt[2] - Cos[x]","A",4,3,7,0.4286,1,"{12, 321, 207}"
106,1,21,0,0.0242572,"\int \cos (x) \tan (3 x) \, dx","Int[Cos[x]*Tan[3*x],x]","\frac{\tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{3}}\right)}{\sqrt{3}}-\cos (x)","\frac{\tanh ^{-1}\left(\frac{2 \cos (x)}{\sqrt{3}}\right)}{\sqrt{3}}-\cos (x)",1,"ArcTanh[(2*Cos[x])/Sqrt[3]]/Sqrt[3] - Cos[x]","A",3,2,7,0.2857,1,"{388, 206}"
107,1,71,0,0.0831244,"\int \cos (x) \tan (4 x) \, dx","Int[Cos[x]*Tan[4*x],x]","-\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)","-\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,"(Sqrt[2 - Sqrt[2]]*ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[2]]])/4 + (Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[2]]])/4 - Cos[x]","A",6,4,7,0.5714,1,"{12, 1279, 1166, 207}"
108,1,84,0,0.0979321,"\int \cos (x) \tan (5 x) \, dx","Int[Cos[x]*Tan[5*x],x]","-\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)","-\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,"(Sqrt[(5 + Sqrt[5])/2]*ArcTanh[2*Sqrt[2/(5 + Sqrt[5])]*Cos[x]])/5 + (Sqrt[(5 - Sqrt[5])/2]*ArcTanh[Sqrt[(2*(5 + Sqrt[5]))/5]*Cos[x]])/5 - Cos[x]","A",6,3,7,0.4286,1,"{1676, 1166, 207}"
109,1,89,0,0.2382264,"\int \cos (x) \tan (6 x) \, dx","Int[Cos[x]*Tan[6*x],x]","-\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)","-\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,"ArcTanh[Sqrt[2]*Cos[x]]/(3*Sqrt[2]) + (Sqrt[2 - Sqrt[3]]*ArcTanh[(2*Cos[x])/Sqrt[2 - Sqrt[3]]])/6 + (Sqrt[2 + Sqrt[3]]*ArcTanh[(2*Cos[x])/Sqrt[2 + Sqrt[3]]])/6 - Cos[x]","A",10,5,7,0.7143,1,"{12, 6742, 2073, 207, 1166}"
110,1,10,0,0.0203756,"\int \cos (x) \cot (2 x) \, dx","Int[Cos[x]*Cot[2*x],x]","\cos (x)-\frac{1}{2} \tanh ^{-1}(\cos (x))","\cos (x)-\frac{1}{2} \tanh ^{-1}(\cos (x))",1,"-ArcTanh[Cos[x]]/2 + Cos[x]","A",4,3,7,0.4286,1,"{12, 388, 206}"
111,1,45,0,0.0529215,"\int \cos (x) \cot (3 x) \, dx","Int[Cos[x]*Cot[3*x],x]","\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)","\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[1 - 2*Cos[x]]/6 + Log[1 - Cos[x]]/6 - Log[1 + Cos[x]]/6 - Log[1 + 2*Cos[x]]/6","A",9,4,7,0.5714,1,"{1279, 1161, 616, 31}"
112,1,28,0,0.0485803,"\int \cos (x) \cot (4 x) \, dx","Int[Cos[x]*Cot[4*x],x]","\cos (x)-\frac{1}{4} \tanh ^{-1}(\cos (x))-\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{2 \sqrt{2}}","\cos (x)-\frac{1}{4} \tanh ^{-1}(\cos (x))-\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{2 \sqrt{2}}",1,"-ArcTanh[Cos[x]]/4 - ArcTanh[Sqrt[2]*Cos[x]]/(2*Sqrt[2]) + Cos[x]","A",6,3,7,0.4286,1,"{1676, 1166, 207}"
113,1,110,0,0.1554897,"\int \cos (x) \cot (5 x) \, dx","Int[Cos[x]*Cot[5*x],x]","\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))","\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,"-ArcTanh[Cos[x]]/5 + Cos[x] + ((1 - Sqrt[5])*Log[1 - Sqrt[5] - 4*Cos[x]])/20 + ((1 + Sqrt[5])*Log[1 + Sqrt[5] - 4*Cos[x]])/20 - ((1 - Sqrt[5])*Log[1 - Sqrt[5] + 4*Cos[x]])/20 - ((1 + Sqrt[5])*Log[1 + Sqrt[5] + 4*Cos[x]])/20","A",10,4,7,0.5714,1,"{2075, 207, 632, 31}"
114,1,38,0,0.0712837,"\int \cos (x) \cot (6 x) \, dx","Int[Cos[x]*Cot[6*x],x]","\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}}","\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,"-ArcTanh[Cos[x]]/6 - ArcTanh[2*Cos[x]]/6 - ArcTanh[(2*Cos[x])/Sqrt[3]]/(2*Sqrt[3]) + Cos[x]","A",7,3,7,0.4286,1,"{12, 2073, 207}"
115,1,92,0,0.0908161,"\int \cos (x) \cot (n x) \, dx","Int[Cos[x]*Cot[n*x],x]","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}","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,"-1/(2*E^(I*x)) + E^(I*x)/2 + Hypergeometric2F1[1, -1/(2*n), 1 - 1/(2*n), E^((2*I)*n*x)]/E^(I*x) - E^(I*x)*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, E^((2*I)*n*x)]","A",6,3,7,0.4286,1,"{4558, 2194, 2251}"
116,1,15,0,0.0147158,"\int \cos (x) \sec (2 x) \, dx","Int[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",2,2,7,0.2857,1,"{4356, 206}"
117,1,44,0,0.0362144,"\int \cos (x) \sec (3 x) \, dx","Int[Cos[x]*Sec[3*x],x]","\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}}","\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,"-Log[Cos[x] - Sqrt[3]*Sin[x]]/(2*Sqrt[3]) + Log[Cos[x] + Sqrt[3]*Sin[x]]/(2*Sqrt[3])","A",2,1,7,0.1429,1,"{206}"
118,1,71,0,0.0457058,"\int \cos (x) \sec (4 x) \, dx","Int[Cos[x]*Sec[4*x],x]","\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)}}-\frac{\tanh ^{-1}\left(\frac{2 \sin (x)}{\sqrt{2+\sqrt{2}}}\right)}{2 \sqrt{2 \left(2+\sqrt{2}\right)}}",1,"ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[2]]]/(2*Sqrt[2*(2 - Sqrt[2])]) - ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[2]]]/(2*Sqrt[2*(2 + Sqrt[2])])","A",4,3,7,0.4286,1,"{4356, 1093, 207}"
119,1,163,0,0.1294488,"\int \cos (x) \sec (5 x) \, dx","Int[Cos[x]*Sec[5*x],x]","\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)","\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])/2]*Log[Cos[x] - Sqrt[5 - 2*Sqrt[5]]*Sin[x]])/10 - (Sqrt[(5 - Sqrt[5])/2]*Log[Cos[x] + Sqrt[5 - 2*Sqrt[5]]*Sin[x]])/10 - (Sqrt[(5 + Sqrt[5])/2]*Log[Cos[x] - Sqrt[5 + 2*Sqrt[5]]*Sin[x]])/10 + (Sqrt[(5 + Sqrt[5])/2]*Log[Cos[x] + Sqrt[5 + 2*Sqrt[5]]*Sin[x]])/10","A",4,2,7,0.2857,1,"{1166, 207}"
120,1,85,0,0.0606861,"\int \cos (x) \sec (6 x) \, dx","Int[Cos[x]*Sec[6*x],x]","-\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}}}","-\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,"-ArcTanh[Sqrt[2]*Sin[x]]/(3*Sqrt[2]) + ArcTanh[(2*Sin[x])/Sqrt[2 - Sqrt[3]]]/(6*Sqrt[2 - Sqrt[3]]) + ArcTanh[(2*Sin[x])/Sqrt[2 + Sqrt[3]]]/(6*Sqrt[2 + Sqrt[3]])","A",7,4,7,0.5714,1,"{4356, 2057, 207, 1166}"
121,1,10,0,0.0179098,"\int \cos (2 x) \sec (x) \, dx","Int[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",3,3,7,0.4286,1,"{4364, 388, 206}"
122,1,14,0,0.0199355,"\int \cos (4 x) \sec (2 x) \, dx","Int[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,"-ArcTanh[Sin[2*x]]/2 + Sin[2*x]","A",3,3,9,0.3333,1,"{4364, 388, 206}"
123,1,7,0,0.0112712,"\int \cos (x) \csc (2 x) \, dx","Int[Cos[x]*Csc[2*x],x]","-\frac{1}{2} \tanh ^{-1}(\cos (x))","-\frac{1}{2} \tanh ^{-1}(\cos (x))",1,"-ArcTanh[Cos[x]]/2","A",2,2,7,0.2857,1,"{4287, 3770}"
124,1,21,0,0.0257293,"\int \cos (x) \csc (3 x) \, dx","Int[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",5,5,7,0.7143,1,"{4356, 266, 36, 31, 29}"
125,1,26,0,0.0257339,"\int \cos (x) \csc (4 x) \, dx","Int[Cos[x]*Csc[4*x],x]","\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{2 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\cos (x))","\frac{\tanh ^{-1}\left(\sqrt{2} \cos (x)\right)}{2 \sqrt{2}}-\frac{1}{4} \tanh ^{-1}(\cos (x))",1,"-ArcTanh[Cos[x]]/4 + ArcTanh[Sqrt[2]*Cos[x]]/(2*Sqrt[2])","A",4,2,7,0.2857,1,"{1093, 206}"
126,1,62,0,0.0699113,"\int \cos (x) \csc (5 x) \, dx","Int[Cos[x]*Csc[5*x],x]","-\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))","-\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,"Log[Sin[x]]/5 - ((1 + Sqrt[5])*Log[5 - Sqrt[5] - 8*Sin[x]^2])/20 - ((1 - Sqrt[5])*Log[5 + Sqrt[5] - 8*Sin[x]^2])/20","A",7,6,7,0.8571,1,"{4356, 1114, 705, 29, 632, 31}"
127,1,36,0,0.0414389,"\int \cos (x) \csc (6 x) \, dx","Int[Cos[x]*Csc[6*x],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}}","-\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,"-ArcTanh[Cos[x]]/6 - ArcTanh[2*Cos[x]]/6 + ArcTanh[(2*Cos[x])/Sqrt[3]]/(2*Sqrt[3])","A",7,3,7,0.4286,1,"{12, 2057, 207}"
128,1,33,0,0.0314779,"\int \cos ^3(6 x) \sin (x) \, dx","Int[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",6,2,9,0.2222,1,"{4354, 2638}"
129,1,33,0,0.0328019,"\int \cos ^3(6 x) \sin (9 x) \, dx","Int[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,"-Cos[3*x]/8 + Cos[9*x]/72 - Cos[15*x]/40 - Cos[27*x]/216","A",6,2,11,0.1818,1,"{4354, 2638}"
130,1,25,0,0.027917,"\int \cos (2 x) \sin ^2(6 x) \, dx","Int[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",5,2,11,0.1818,1,"{4354, 2637}"
131,1,23,0,0.0263351,"\int \cos (x) \sin ^2(6 x) \, dx","Int[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",5,2,9,0.2222,1,"{4354, 2637}"
132,1,33,0,0.031273,"\int \cos (x) \sin ^3(6 x) \, dx","Int[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",6,2,9,0.2222,1,"{4354, 2638}"
133,1,31,0,0.0299518,"\int \cos (7 x) \sin ^3(6 x) \, dx","Int[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",6,2,11,0.1818,1,"{4354, 2638}"
134,1,41,0,0.0433548,"\int \cos ^2(3 x) \sin ^3(2 x) \, dx","Int[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",7,2,13,0.1538,1,"{4354, 2638}"
135,1,27,0,0.0243191,"\int \sin (a+b x) \sin (c+b x) \, dx","Int[Sin[a + b*x]*Sin[c + b*x],x]","\frac{1}{2} x \cos (a-c)-\frac{\sin (a+2 b x+c)}{4 b}","\frac{1}{2} x \cos (a-c)-\frac{\sin (a+2 b x+c)}{4 b}",1,"(x*Cos[a - c])/2 - Sin[a + c + 2*b*x]/(4*b)","A",3,2,13,0.1538,1,"{4569, 2637}"
136,1,27,0,0.025399,"\int \sin (c-b x) \sin (a+b x) \, dx","Int[Sin[c - b*x]*Sin[a + b*x],x]","\frac{\sin (a+2 b x-c)}{4 b}-\frac{1}{2} x \cos (a+c)","\frac{\sin (a+2 b x-c)}{4 b}-\frac{1}{2} x \cos (a+c)",1,"-(x*Cos[a + c])/2 + Sin[a - c + 2*b*x]/(4*b)","A",3,2,14,0.1429,1,"{4569, 2637}"
137,1,27,0,0.017493,"\int \cos (a+b x) \cos (c+b x) \, dx","Int[Cos[a + b*x]*Cos[c + b*x],x]","\frac{\sin (a+2 b x+c)}{4 b}+\frac{1}{2} x \cos (a-c)","\frac{\sin (a+2 b x+c)}{4 b}+\frac{1}{2} x \cos (a-c)",1,"(x*Cos[a - c])/2 + Sin[a + c + 2*b*x]/(4*b)","A",3,2,13,0.1538,1,"{4570, 2637}"
138,1,27,0,0.0186565,"\int \cos (c-b x) \cos (a+b x) \, dx","Int[Cos[c - b*x]*Cos[a + b*x],x]","\frac{\sin (a+2 b x-c)}{4 b}+\frac{1}{2} x \cos (a+c)","\frac{\sin (a+2 b x-c)}{4 b}+\frac{1}{2} x \cos (a+c)",1,"(x*Cos[a + c])/2 + Sin[a - c + 2*b*x]/(4*b)","A",3,2,14,0.1429,1,"{4570, 2637}"
139,1,39,0,0.065661,"\int \tan (a+b x) \tan (c+b x) \, dx","Int[Tan[a + b*x]*Tan[c + b*x],x]","-\frac{\cot (a-c) \log (\cos (a+b x))}{b}+\frac{\cot (a-c) \log (\cos (b x+c))}{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]])/b + (Cot[a - c]*Log[Cos[c + b*x]])/b","A",4,3,13,0.2308,1,"{4612, 4610, 3475}"
140,1,34,0,0.0678364,"\int \tan (c-b x) \tan (a+b x) \, dx","Int[Tan[c - b*x]*Tan[a + b*x],x]","-\frac{\cot (a+c) \log (\cos (c-b x))}{b}+\frac{\cot (a+c) \log (\cos (a+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]])/b + (Cot[a + c]*Log[Cos[a + b*x]])/b","A",4,3,14,0.2143,1,"{4612, 4610, 3475}"
141,1,39,0,0.0321744,"\int \cot (a+b x) \cot (c+b x) \, dx","Int[Cot[a + b*x]*Cot[c + b*x],x]","-\frac{\cot (a-c) \log (\sin (a+b x))}{b}+\frac{\cot (a-c) \log (\sin (b x+c))}{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]])/b + (Cot[a - c]*Log[Sin[c + b*x]])/b","A",4,3,13,0.2308,1,"{4613, 4611, 3475}"
142,1,34,0,0.034246,"\int \cot (c-b x) \cot (a+b x) \, dx","Int[Cot[c - b*x]*Cot[a + b*x],x]","-\frac{\cot (a+c) \log (\sin (c-b x))}{b}+\frac{\cot (a+c) \log (\sin (a+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]])/b + (Cot[a + c]*Log[Sin[a + b*x]])/b","A",4,3,14,0.2143,1,"{4613, 4611, 3475}"
143,1,36,0,0.019024,"\int \sec (a+b x) \sec (c+b x) \, dx","Int[Sec[a + b*x]*Sec[c + b*x],x]","\frac{\csc (a-c) \log (\cos (b x+c))}{b}-\frac{\csc (a-c) \log (\cos (a+b x))}{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]])/b) + (Csc[a - c]*Log[Cos[c + b*x]])/b","A",3,2,13,0.1538,1,"{4610, 3475}"
144,1,33,0,0.018378,"\int \sec (c-b x) \sec (a+b x) \, dx","Int[Sec[c - b*x]*Sec[a + b*x],x]","\frac{\csc (a+c) \log (\cos (c-b x))}{b}-\frac{\csc (a+c) \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]])/b - (Csc[a + c]*Log[Cos[a + b*x]])/b","A",3,2,14,0.1429,1,"{4610, 3475}"
145,1,36,0,0.0184961,"\int \csc (a+b x) \csc (c+b x) \, dx","Int[Csc[a + b*x]*Csc[c + b*x],x]","\frac{\csc (a-c) \log (\sin (b x+c))}{b}-\frac{\csc (a-c) \log (\sin (a+b x))}{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]])/b) + (Csc[a - c]*Log[Sin[c + b*x]])/b","A",3,2,13,0.1538,1,"{4611, 3475}"
146,1,33,0,0.0184097,"\int \csc (c-b x) \csc (a+b x) \, dx","Int[Csc[c - b*x]*Csc[a + b*x],x]","\frac{\csc (a+c) \log (\sin (a+b x))}{b}-\frac{\csc (a+c) \log (\sin (c-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]])/b) + (Csc[a + c]*Log[Sin[a + b*x]])/b","A",3,2,14,0.1429,1,"{4611, 3475}"
147,1,13,0,0.0321317,"\int \sqrt{\sin (x) \tan (x)} \, dx","Int[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",2,2,9,0.2222,1,"{4400, 2589}"
148,1,31,0,0.0534873,"\int (\sin (x) \tan (x))^{3/2} \, dx","Int[(Sin[x]*Tan[x])^(3/2),x]","\frac{8}{3} \csc (x) \sqrt{\sin (x) \tan (x)}-\frac{2}{3} \sin (x) \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,"(8*Csc[x]*Sqrt[Sin[x]*Tan[x]])/3 - (2*Sin[x]*Sqrt[Sin[x]*Tan[x]])/3","A",3,3,9,0.3333,1,"{4400, 2598, 2589}"
149,1,50,0,0.0753787,"\int (\sin (x) \tan (x))^{5/2} \, dx","Int[(Sin[x]*Tan[x])^(5/2),x]","-\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)}","-\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,"(64*Cot[x]*Sqrt[Sin[x]*Tan[x]])/15 + (16*Tan[x]*Sqrt[Sin[x]*Tan[x]])/15 - (2*Sin[x]^2*Tan[x]*Sqrt[Sin[x]*Tan[x]])/5","A",4,4,9,0.4444,1,"{4400, 2598, 2594, 2589}"
150,1,13,0,0.0381709,"\int \sqrt{\cos (x) \cot (x)} \, dx","Int[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",2,2,9,0.2222,1,"{4400, 2589}"
151,1,31,0,0.0683295,"\int (\cos (x) \cot (x))^{3/2} \, dx","Int[(Cos[x]*Cot[x])^(3/2),x]","\frac{2}{3} \cos (x) \sqrt{\cos (x) \cot (x)}-\frac{8}{3} \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*Cos[x]*Sqrt[Cos[x]*Cot[x]])/3 - (8*Sqrt[Cos[x]*Cot[x]]*Sec[x])/3","A",3,3,9,0.3333,1,"{4400, 2598, 2589}"
152,1,50,0,0.0944576,"\int (\cos (x) \cot (x))^{5/2} \, dx","Int[(Cos[x]*Cot[x])^(5/2),x]","\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)}","\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,"(-16*Cot[x]*Sqrt[Cos[x]*Cot[x]])/15 + (2*Cos[x]^2*Cot[x]*Sqrt[Cos[x]*Cot[x]])/5 - (64*Sqrt[Cos[x]*Cot[x]]*Tan[x])/15","A",4,4,9,0.4444,1,"{4400, 2598, 2594, 2589}"
153,1,58,0,0.0776166,"\int \frac{x \cos (x)}{(a+b \sin (x))^2} \, dx","Int[(x*Cos[x])/(a + b*Sin[x])^2,x]","\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))}","\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]])/(b*Sqrt[a^2 - b^2]) - x/(b*(a + b*Sin[x]))","A",4,4,12,0.3333,1,"{4422, 2660, 618, 204}"
154,1,85,0,0.105745,"\int \frac{x \cos (x)}{(a+b \sin (x))^3} \, dx","Int[(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{\cos (x)}{2 \left(a^2-b^2\right) (a+b \sin (x))}-\frac{x}{2 b (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/(2*b*(a + b*Sin[x])^2) + Cos[x]/(2*(a^2 - b^2)*(a + b*Sin[x]))","A",6,6,12,0.5000,1,"{4422, 2664, 12, 2660, 618, 204}"
155,1,59,0,0.0592501,"\int \frac{x \sin (x)}{(a+b \cos (x))^2} \, dx","Int[(x*Sin[x])/(a + b*Cos[x])^2,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}}","\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*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]) + x/(b*(a + b*Cos[x]))","A",3,3,12,0.2500,1,"{4423, 2659, 205}"
156,1,88,0,0.1032669,"\int \frac{x \sin (x)}{(a+b \cos (x))^3} \, dx","Int[(x*Sin[x])/(a + b*Cos[x])^3,x]","\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}}","\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*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*b*(a + b)^(3/2))) + x/(2*b*(a + b*Cos[x])^2) + Sin[x]/(2*(a^2 - b^2)*(a + b*Cos[x]))","A",5,5,12,0.4167,1,"{4423, 2664, 12, 2659, 205}"
157,1,50,0,0.0826108,"\int \frac{x \sec ^2(x)}{(a+b \tan (x))^2} \, dx","Int[(x*Sec[x]^2)/(a + b*Tan[x])^2,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))}","\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,"(a*x)/(b*(a^2 + b^2)) + Log[a*Cos[x] + b*Sin[x]]/(a^2 + b^2) - x/(b*(a + b*Tan[x]))","A",3,3,14,0.2143,1,"{4424, 3484, 3530}"
158,1,50,0,0.0817342,"\int \frac{x \csc ^2(x)}{(a+b \cot (x))^2} \, dx","Int[(x*Csc[x]^2)/(a + b*Cot[x])^2,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))}","-\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*(a^2 + b^2))) + x/(b*(a + b*Cot[x])) + Log[b*Cos[x] + a*Sin[x]]/(a^2 + b^2)","A",3,3,14,0.2143,1,"{4425, 3484, 3530}"
159,1,32,0,0.0547793,"\int \frac{\sec ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[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",2,2,23,0.08696,1,"{3675, 205}"
160,1,211,0,0.5283791,"\int \frac{x \sec ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[(x*Sec[c + d*x]^2)/(a + b*Tan[c + d*x]^2),x]","-\frac{\text{PolyLog}\left(2,-\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{PolyLog}\left(2,-\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}","-\frac{\text{PolyLog}\left(2,-\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{PolyLog}\left(2,-\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,"((-I/2)*x*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(Sqrt[a] - Sqrt[b])^2])/(Sqrt[a]*Sqrt[b]*d) + ((I/2)*x*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(Sqrt[a] + Sqrt[b])^2])/(Sqrt[a]*Sqrt[b]*d) - PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(Sqrt[a] - Sqrt[b])^2)]/(4*Sqrt[a]*Sqrt[b]*d^2) + PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(Sqrt[a] + Sqrt[b])^2)]/(4*Sqrt[a]*Sqrt[b]*d^2)","A",9,6,24,0.2500,1,"{4588, 3321, 2264, 2190, 2279, 2391}"
161,1,337,0,0.9004041,"\int \frac{x^2 \sec ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[(x^2*Sec[c + d*x]^2)/(a + b*Tan[c + d*x]^2),x]","-\frac{x \text{PolyLog}\left(2,-\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{PolyLog}\left(2,-\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 \text{PolyLog}\left(3,-\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{PolyLog}\left(3,-\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 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}","-\frac{x \text{PolyLog}\left(2,-\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{PolyLog}\left(2,-\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 \text{PolyLog}\left(3,-\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{PolyLog}\left(3,-\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 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/2)*x^2*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(Sqrt[a] - Sqrt[b])^2])/(Sqrt[a]*Sqrt[b]*d) + ((I/2)*x^2*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(Sqrt[a] + Sqrt[b])^2])/(Sqrt[a]*Sqrt[b]*d) - (x*PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(Sqrt[a] - Sqrt[b])^2)])/(2*Sqrt[a]*Sqrt[b]*d^2) + (x*PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(Sqrt[a] + Sqrt[b])^2)])/(2*Sqrt[a]*Sqrt[b]*d^2) + ((I/4)*PolyLog[3, -(((Sqrt[a] - Sqrt[b])*E^((2*I)*(c + d*x)))/(Sqrt[a] + Sqrt[b]))])/(Sqrt[a]*Sqrt[b]*d^3) - ((I/4)*PolyLog[3, -(((Sqrt[a] + Sqrt[b])*E^((2*I)*(c + d*x)))/(Sqrt[a] - Sqrt[b]))])/(Sqrt[a]*Sqrt[b]*d^3)","A",11,7,26,0.2692,1,"{4588, 3321, 2264, 2190, 2531, 2282, 6589}"
162,1,40,0,0.6054528,"\int \frac{\sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx","Int[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",2,1,33,0.03030,1,"{205}"
163,1,267,0,0.7186622,"\int \frac{x \sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx","Int[(x*Sec[c + d*x]^2)/(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2),x]","-\frac{\text{PolyLog}\left(2,-\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right)}{4 d^2 \sqrt{a+c} \sqrt{b+c}}+\frac{\text{PolyLog}\left(2,-\frac{(a-b) e^{2 i (c+d x)}}{2 \left(\sqrt{a+c} \sqrt{b+c}+c\right)+a+b}\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}}","-\frac{\text{PolyLog}\left(2,-\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right)}{4 d^2 \sqrt{a+c} \sqrt{b+c}}+\frac{\text{PolyLog}\left(2,-\frac{(a-b) e^{2 i (c+d x)}}{2 \left(\sqrt{a+c} \sqrt{b+c}+c\right)+a+b}\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,"((-I/2)*x*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c])])/(Sqrt[a + c]*Sqrt[b + c]*d) + ((I/2)*x*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c]))])/(Sqrt[a + c]*Sqrt[b + c]*d) - PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c]))]/(4*Sqrt[a + c]*Sqrt[b + c]*d^2) + PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c])))]/(4*Sqrt[a + c]*Sqrt[b + c]*d^2)","A",9,6,34,0.1765,1,"{4589, 3321, 2264, 2190, 2279, 2391}"
164,1,407,0,1.0750987,"\int \frac{x^2 \sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx","Int[(x^2*Sec[c + d*x]^2)/(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2),x]","-\frac{x \text{PolyLog}\left(2,-\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right)}{2 d^2 \sqrt{a+c} \sqrt{b+c}}+\frac{x \text{PolyLog}\left(2,-\frac{(a-b) e^{2 i (c+d x)}}{2 \left(\sqrt{a+c} \sqrt{b+c}+c\right)+a+b}\right)}{2 d^2 \sqrt{a+c} \sqrt{b+c}}-\frac{i \text{PolyLog}\left(3,-\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right)}{4 d^3 \sqrt{a+c} \sqrt{b+c}}+\frac{i \text{PolyLog}\left(3,-\frac{(a-b) e^{2 i (c+d x)}}{2 \left(\sqrt{a+c} \sqrt{b+c}+c\right)+a+b}\right)}{4 d^3 \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}}","-\frac{x \text{PolyLog}\left(2,-\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right)}{2 d^2 \sqrt{a+c} \sqrt{b+c}}+\frac{x \text{PolyLog}\left(2,-\frac{(a-b) e^{2 i (c+d x)}}{2 \left(\sqrt{a+c} \sqrt{b+c}+c\right)+a+b}\right)}{2 d^2 \sqrt{a+c} \sqrt{b+c}}-\frac{i \text{PolyLog}\left(3,-\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right)}{4 d^3 \sqrt{a+c} \sqrt{b+c}}+\frac{i \text{PolyLog}\left(3,-\frac{(a-b) e^{2 i (c+d x)}}{2 \left(\sqrt{a+c} \sqrt{b+c}+c\right)+a+b}\right)}{4 d^3 \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,"((-I/2)*x^2*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c])])/(Sqrt[a + c]*Sqrt[b + c]*d) + ((I/2)*x^2*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c]))])/(Sqrt[a + c]*Sqrt[b + c]*d) - (x*PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c]))])/(2*Sqrt[a + c]*Sqrt[b + c]*d^2) + (x*PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c])))])/(2*Sqrt[a + c]*Sqrt[b + c]*d^2) - ((I/4)*PolyLog[3, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c]))])/(Sqrt[a + c]*Sqrt[b + c]*d^3) + ((I/4)*PolyLog[3, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c])))])/(Sqrt[a + c]*Sqrt[b + c]*d^3)","A",11,7,36,0.1944,1,"{4589, 3321, 2264, 2190, 2531, 2282, 6589}"
165,1,155,0,0.1987273,"\int x^3 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \, dx","Int[x^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]],x]","\frac{3 x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}-\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{x^3 \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f}","\frac{3 x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}-\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{x^3 \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f}",1,"(-6*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^4 + (3*x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 - (6*x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 + (x^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f","A",5,3,33,0.09091,1,"{4604, 3296, 2638}"
166,1,118,0,0.1760693,"\int x^2 \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \, dx","Int[x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]],x]","\frac{2 x \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}-\frac{2 \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^3}+\frac{x^2 \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f}","\frac{2 x \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^2}-\frac{2 \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f^3}+\frac{x^2 \tan (e+f x) \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}{f}",1,"(2*x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 - (2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 + (x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f","A",4,3,33,0.09091,1,"{4604, 3296, 2637}"
167,1,74,0,0.1103537,"\int x \sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)} \, dx","Int[x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]],x]","\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}","\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[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 + (x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f","A",3,3,31,0.09677,1,"{4604, 3296, 2638}"
168,1,86,0,0.183241,"\int \frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x} \, dx","Int[(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x,x]","\cos (e) \text{CosIntegral}(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}","\cos (e) \text{CosIntegral}(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,"Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x]","A",4,4,33,0.1212,1,"{4604, 3303, 3299, 3302}"
169,1,123,0,0.1985076,"\int \frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x^2} \, dx","Int[(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x^2,x]","-f \sin (e) \text{CosIntegral}(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}","-f \sin (e) \text{CosIntegral}(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,"-((Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x) - f*CosIntegral[f*x]*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - f*Cos[e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x]","A",5,5,33,0.1515,1,"{4604, 3297, 3303, 3299, 3302}"
170,1,176,0,0.2256484,"\int \frac{\sqrt{a-a \sin (e+f x)} \sqrt{c+c \sin (e+f x)}}{x^3} \, dx","Int[(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x^3,x]","-\frac{1}{2} f^2 \cos (e) \text{CosIntegral}(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}","-\frac{1}{2} f^2 \cos (e) \text{CosIntegral}(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,"-(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x^2) - (f^2*Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/2 + (f^2*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x])/2 + (f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(2*x)","A",6,5,33,0.1515,1,"{4604, 3297, 3303, 3299, 3302}"
171,1,393,0,0.3752679,"\int x^3 \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2} \, dx","Int[x^3*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2),x]","\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{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{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}","\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{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{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,"(-6*c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^4 + (3*c*x^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 + (3*c*x*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(8*f^3) - (3*c*x^3*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f) - (3*c*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(8*f^4) + (3*c*x^2*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f^2) + (x^3*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(5/2))/(2*c*f) - (6*c*x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 - (3*c*x*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(4*f^3)","A",11,9,33,0.2727,1,"{4604, 4422, 3317, 3296, 2638, 3311, 30, 2635, 8}"
172,1,265,0,0.2736286,"\int x^2 \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2} \, dx","Int[x^2*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2),x]","\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{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{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}","\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{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{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,"(2*c*x*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 - (3*c*x^2*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f) + (c*x*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*f^2) + (x^2*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(5/2))/(2*c*f) - (2*c*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/f^3 - (c*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(4*f^3)","A",8,7,33,0.2121,1,"{4604, 4422, 3317, 3296, 2637, 3310, 30}"
173,1,168,0,0.1433358,"\int x \sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2} \, dx","Int[x*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2),x]","\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}","\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[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/f^2 - (3*c*x*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f) + (c*Sin[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(4*f^2) + (x*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(5/2))/(2*c*f)","A",3,3,31,0.09677,1,"{4604, 4422, 2644}"
174,1,186,0,0.6617547,"\int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x} \, dx","Int[(Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2))/x,x]","\frac{1}{2} c \sin (2 e) \text{CosIntegral}(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{CosIntegral}(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}","\frac{1}{2} c \sin (2 e) \text{CosIntegral}(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{CosIntegral}(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*Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] + (c*CosIntegral[2*f*x]*Sec[e + f*x]*Sin[2*e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/2 - c*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x] + (c*Cos[2*e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[2*f*x])/2","A",11,7,33,0.2121,1,"{4604, 6741, 12, 6742, 3303, 3299, 3302}"
175,1,273,0,0.6647949,"\int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^2} \, dx","Int[(Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2))/x^2,x]","-c f \sin (e) \text{CosIntegral}(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{CosIntegral}(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}","-c f \sin (e) \text{CosIntegral}(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{CosIntegral}(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[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/x) + c*f*Cos[2*e]*CosIntegral[2*f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - c*f*CosIntegral[f*x]*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - (c*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Sin[2*e + 2*f*x])/(2*x) - c*f*Cos[e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x] - c*f*Sec[e + f*x]*Sin[2*e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[2*f*x]","A",13,8,33,0.2424,1,"{4604, 6741, 12, 6742, 3297, 3303, 3299, 3302}"
176,1,385,0,0.7412894,"\int \frac{\sqrt{a-a \sin (e+f x)} (c+c \sin (e+f x))^{3/2}}{x^3} \, dx","Int[(Sqrt[a - a*Sin[e + f*x]]*(c + c*Sin[e + f*x])^(3/2))/x^3,x]","-c f^2 \sin (2 e) \text{CosIntegral}(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{CosIntegral}(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}","-c f^2 \sin (2 e) \text{CosIntegral}(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{CosIntegral}(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*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x^2) - (c*f*Cos[2*e + 2*f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/(2*x) - (c*f^2*Cos[e]*CosIntegral[f*x]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])/2 - c*f^2*CosIntegral[2*f*x]*Sec[e + f*x]*Sin[2*e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]] - (c*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Sin[2*e + 2*f*x])/(4*x^2) + (c*f^2*Sec[e + f*x]*Sin[e]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[f*x])/2 - c*f^2*Cos[2*e]*Sec[e + f*x]*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*SinIntegral[2*f*x] + (c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]*Tan[e + f*x])/(2*x)","A",15,8,33,0.2424,1,"{4604, 6741, 12, 6742, 3297, 3303, 3299, 3302}"
177,1,767,0,1.3513863,"\int \frac{(g+h x)^3 \sqrt{a-a \sin (e+f x)}}{\sqrt{c+c \sin (e+f x)}} \, dx","Int[((g + h*x)^3*Sqrt[a - a*Sin[e + f*x]])/Sqrt[c + c*Sin[e + f*x]],x]","-\frac{6 a h^2 (g+h x) \cos (e+f x) \text{PolyLog}\left(3,-i e^{i (e+f x)}\right)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{6 a h^2 (g+h x) \cos (e+f x) \text{PolyLog}\left(3,i e^{i (e+f x)}\right)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{3 a h^2 (g+h x) \cos (e+f x) \text{PolyLog}\left(3,-e^{2 i (e+f x)}\right)}{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 \cos (e+f x) \text{PolyLog}\left(2,-i e^{i (e+f x)}\right)}{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 \cos (e+f x) \text{PolyLog}\left(2,i e^{i (e+f x)}\right)}{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 \cos (e+f x) \text{PolyLog}\left(2,-e^{2 i (e+f x)}\right)}{2 f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{6 i a h^3 \cos (e+f x) \text{PolyLog}\left(4,-i e^{i (e+f x)}\right)}{f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{6 i a h^3 \cos (e+f x) \text{PolyLog}\left(4,i e^{i (e+f x)}\right)}{f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{3 i a h^3 \cos (e+f x) \text{PolyLog}\left(4,-e^{2 i (e+f x)}\right)}{4 f^4 \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}}","-\frac{6 a h^2 (g+h x) \cos (e+f x) \text{PolyLog}\left(3,-i e^{i (e+f x)}\right)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{6 a h^2 (g+h x) \cos (e+f x) \text{PolyLog}\left(3,i e^{i (e+f x)}\right)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{3 a h^2 (g+h x) \cos (e+f x) \text{PolyLog}\left(3,-e^{2 i (e+f x)}\right)}{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 \cos (e+f x) \text{PolyLog}\left(2,-i e^{i (e+f x)}\right)}{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 \cos (e+f x) \text{PolyLog}\left(2,i e^{i (e+f x)}\right)}{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 \cos (e+f x) \text{PolyLog}\left(2,-e^{2 i (e+f x)}\right)}{2 f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{6 i a h^3 \cos (e+f x) \text{PolyLog}\left(4,-i e^{i (e+f x)}\right)}{f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{6 i a h^3 \cos (e+f x) \text{PolyLog}\left(4,i e^{i (e+f x)}\right)}{f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{3 i a h^3 \cos (e+f x) \text{PolyLog}\left(4,-e^{2 i (e+f x)}\right)}{4 f^4 \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,"((-I/4)*a*(g + h*x)^4*Cos[e + f*x])/(h*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((2*I)*a*(g + h*x)^3*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*(g + h*x)^3*Cos[e + f*x]*Log[1 + E^((2*I)*(e + f*x))])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + ((3*I)*a*h*(g + h*x)^2*Cos[e + f*x]*PolyLog[2, (-I)*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((3*I)*a*h*(g + h*x)^2*Cos[e + f*x]*PolyLog[2, I*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (((3*I)/2)*a*h*(g + h*x)^2*Cos[e + f*x]*PolyLog[2, -E^((2*I)*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (6*a*h^2*(g + h*x)*Cos[e + f*x]*PolyLog[3, (-I)*E^(I*(e + f*x))])/(f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (6*a*h^2*(g + h*x)*Cos[e + f*x]*PolyLog[3, I*E^(I*(e + f*x))])/(f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (3*a*h^2*(g + h*x)*Cos[e + f*x]*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((6*I)*a*h^3*Cos[e + f*x]*PolyLog[4, (-I)*E^(I*(e + f*x))])/(f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + ((6*I)*a*h^3*Cos[e + f*x]*PolyLog[4, I*E^(I*(e + f*x))])/(f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (((3*I)/4)*a*h^3*Cos[e + f*x]*PolyLog[4, -E^((2*I)*(e + f*x))])/(f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])","A",20,11,37,0.2973,1,"{4604, 6741, 12, 6742, 4181, 2531, 6609, 2282, 6589, 3719, 2190}"
178,1,555,0,0.8769054,"\int \frac{(g+h x)^2 \sqrt{a-a \sin (e+f x)}}{\sqrt{c+c \sin (e+f x)}} \, dx","Int[((g + h*x)^2*Sqrt[a - a*Sin[e + f*x]])/Sqrt[c + c*Sin[e + f*x]],x]","\frac{2 i a h (g+h x) \cos (e+f x) \text{PolyLog}\left(2,-i e^{i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a h (g+h x) \cos (e+f x) \text{PolyLog}\left(2,i e^{i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h (g+h x) \cos (e+f x) \text{PolyLog}\left(2,-e^{2 i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 a h^2 \cos (e+f x) \text{PolyLog}\left(3,-i e^{i (e+f x)}\right)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a h^2 \cos (e+f x) \text{PolyLog}\left(3,i e^{i (e+f x)}\right)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a h^2 \cos (e+f x) \text{PolyLog}\left(3,-e^{2 i (e+f x)}\right)}{2 f^3 \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}}","\frac{2 i a h (g+h x) \cos (e+f x) \text{PolyLog}\left(2,-i e^{i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 i a h (g+h x) \cos (e+f x) \text{PolyLog}\left(2,i e^{i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h (g+h x) \cos (e+f x) \text{PolyLog}\left(2,-e^{2 i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{2 a h^2 \cos (e+f x) \text{PolyLog}\left(3,-i e^{i (e+f x)}\right)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{2 a h^2 \cos (e+f x) \text{PolyLog}\left(3,i e^{i (e+f x)}\right)}{f^3 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}+\frac{a h^2 \cos (e+f x) \text{PolyLog}\left(3,-e^{2 i (e+f x)}\right)}{2 f^3 \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,"((-I/3)*a*(g + h*x)^3*Cos[e + f*x])/(h*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((2*I)*a*(g + h*x)^2*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*(g + h*x)^2*Cos[e + f*x]*Log[1 + E^((2*I)*(e + f*x))])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + ((2*I)*a*h*(g + h*x)*Cos[e + f*x]*PolyLog[2, (-I)*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((2*I)*a*h*(g + h*x)*Cos[e + f*x]*PolyLog[2, I*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (I*a*h*(g + h*x)*Cos[e + f*x]*PolyLog[2, -E^((2*I)*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (2*a*h^2*Cos[e + f*x]*PolyLog[3, (-I)*E^(I*(e + f*x))])/(f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*h^2*Cos[e + f*x]*PolyLog[3, I*E^(I*(e + f*x))])/(f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*h^2*Cos[e + f*x]*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])","A",17,10,37,0.2703,1,"{4604, 6741, 12, 6742, 4181, 2531, 2282, 6589, 3719, 2190}"
179,1,355,0,0.5112593,"\int \frac{(g+h x) \sqrt{a-a \sin (e+f x)}}{\sqrt{c+c \sin (e+f x)}} \, dx","Int[((g + h*x)*Sqrt[a - a*Sin[e + f*x]])/Sqrt[c + c*Sin[e + f*x]],x]","\frac{i a h \cos (e+f x) \text{PolyLog}\left(2,-i e^{i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h \cos (e+f x) \text{PolyLog}\left(2,i e^{i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h \cos (e+f x) \text{PolyLog}\left(2,-e^{2 i (e+f x)}\right)}{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}}","\frac{i a h \cos (e+f x) \text{PolyLog}\left(2,-i e^{i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h \cos (e+f x) \text{PolyLog}\left(2,i e^{i (e+f x)}\right)}{f^2 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{i a h \cos (e+f x) \text{PolyLog}\left(2,-e^{2 i (e+f x)}\right)}{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/2)*a*(g + h*x)^2*Cos[e + f*x])/(h*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((2*I)*a*(g + h*x)*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*(g + h*x)*Cos[e + f*x]*Log[1 + E^((2*I)*(e + f*x))])/(f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (I*a*h*Cos[e + f*x]*PolyLog[2, (-I)*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (I*a*h*Cos[e + f*x]*PolyLog[2, I*E^(I*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((I/2)*a*h*Cos[e + f*x]*PolyLog[2, -E^((2*I)*(e + f*x))])/(f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])","A",14,9,35,0.2571,1,"{4604, 6741, 12, 6742, 4181, 2279, 2391, 3719, 2190}"
180,0,0,0,0.6464462,"\int \frac{\sqrt{a-a \sin (e+f x)}}{(g+h x) \sqrt{c+c \sin (e+f x)}} \, dx","Int[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,"(a*Cos[e + f*x]*Defer[Int][Sec[e + f*x]/(g + h*x), x])/(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (a*Cos[e + f*x]*Defer[Int][Tan[e + f*x]/(g + h*x), x])/(Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])","A",0,0,0,0,-1,"{}"
181,1,536,0,3.5497439,"\int \frac{x^3 \sqrt{a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx","Int[(x^3*Sqrt[a - a*Sin[e + f*x]])/(c + c*Sin[e + f*x])^(3/2),x]","\frac{6 i a \cos (e+f x) \text{PolyLog}\left(2,-i e^{i (e+f x)}\right)}{c f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{6 i a \cos (e+f x) \text{PolyLog}\left(2,i e^{i (e+f x)}\right)}{c f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{3 i a \cos (e+f x) \text{PolyLog}\left(2,-e^{2 i (e+f x)}\right)}{c f^4 \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{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{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}}","\frac{6 i a \cos (e+f x) \text{PolyLog}\left(2,-i e^{i (e+f x)}\right)}{c f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{6 i a \cos (e+f x) \text{PolyLog}\left(2,i e^{i (e+f x)}\right)}{c f^4 \sqrt{a-a \sin (e+f x)} \sqrt{c \sin (e+f x)+c}}-\frac{3 i a \cos (e+f x) \text{PolyLog}\left(2,-e^{2 i (e+f x)}\right)}{c f^4 \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{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{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,"(-3*a*x^2)/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((3*I)*a*x^2*Cos[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((12*I)*a*x*ArcTan[E^(I*(e + f*x))]*Cos[e + f*x])/(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (6*a*x*Cos[e + f*x]*Log[1 + E^((2*I)*(e + f*x))])/(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + ((6*I)*a*Cos[e + f*x]*PolyLog[2, (-I)*E^(I*(e + f*x))])/(c*f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((6*I)*a*Cos[e + f*x]*PolyLog[2, I*E^(I*(e + f*x))])/(c*f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - ((3*I)*a*Cos[e + f*x]*PolyLog[2, -E^((2*I)*(e + f*x))])/(c*f^4*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (a*x^3*Sec[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (3*a*x^2*Sin[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*x^3*Tan[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])","A",51,17,33,0.5152,1,"{4604, 6741, 12, 6742, 4186, 4181, 2279, 2391, 2531, 6609, 2282, 6589, 3757, 4184, 3719, 2190, 4413}"
182,1,280,0,2.1551996,"\int \frac{x^2 \sqrt{a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx","Int[(x^2*Sqrt[a - a*Sin[e + f*x]])/(c + c*Sin[e + f*x])^(3/2),x]","\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{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{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}}","\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{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{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,"(-2*a*x)/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*ArcTanh[Sin[e + f*x]]*Cos[e + f*x])/(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*Cos[e + f*x]*Log[Cos[e + f*x]])/(c*f^3*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) - (a*x^2*Sec[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (2*a*x*Sin[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*x^2*Tan[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])","A",34,14,33,0.4242,1,"{4604, 6741, 12, 6742, 4186, 3770, 4181, 2531, 2282, 6589, 3757, 4184, 3475, 4413}"
183,1,171,0,0.9729359,"\int \frac{x \sqrt{a-a \sin (e+f x)}}{(c+c \sin (e+f x))^{3/2}} \, dx","Int[(x*Sqrt[a - a*Sin[e + f*x]])/(c + c*Sin[e + f*x])^(3/2),x]","\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}}","\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,"-(a/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])) - (a*x*Sec[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*Sin[e + f*x])/(c*f^2*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]]) + (a*x*Tan[e + f*x])/(c*f*Sqrt[a - a*Sin[e + f*x]]*Sqrt[c + c*Sin[e + f*x]])","A",26,12,31,0.3871,1,"{4604, 6741, 12, 6742, 4185, 4181, 2279, 2391, 3757, 3767, 8, 4413}"
184,1,300,0,0.4383416,"\int \frac{z^2 \sqrt{1+\cos (z)}}{\sqrt{1-\cos (z)}} \, dz","Int[(z^2*Sqrt[1 + Cos[z]])/Sqrt[1 - Cos[z]],z]","\frac{2 i z \sin (z) \text{PolyLog}\left(2,-e^{i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{2 i z \sin (z) \text{PolyLog}\left(2,e^{i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{i z \sin (z) \text{PolyLog}\left(2,e^{2 i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{2 \sin (z) \text{PolyLog}\left(3,-e^{i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}+\frac{2 \sin (z) \text{PolyLog}\left(3,e^{i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}+\frac{\sin (z) \text{PolyLog}\left(3,e^{2 i z}\right)}{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}}","\frac{2 i z \sin (z) \text{PolyLog}\left(2,-e^{i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{2 i z \sin (z) \text{PolyLog}\left(2,e^{i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{i z \sin (z) \text{PolyLog}\left(2,e^{2 i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}-\frac{2 \sin (z) \text{PolyLog}\left(3,-e^{i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}+\frac{2 \sin (z) \text{PolyLog}\left(3,e^{i z}\right)}{\sqrt{1-\cos (z)} \sqrt{\cos (z)+1}}+\frac{\sin (z) \text{PolyLog}\left(3,e^{2 i z}\right)}{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,"((-I/3)*z^3*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) - (2*z^2*ArcTanh[E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) + (z^2*Log[1 - E^((2*I)*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) + ((2*I)*z*PolyLog[2, -E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) - ((2*I)*z*PolyLog[2, E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) - (I*z*PolyLog[2, E^((2*I)*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) - (2*PolyLog[3, -E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) + (2*PolyLog[3, E^(I*z)]*Sin[z])/(Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]]) + (PolyLog[3, E^((2*I)*z)]*Sin[z])/(2*Sqrt[1 - Cos[z]]*Sqrt[1 + Cos[z]])","A",15,8,22,0.3636,1,"{4605, 6742, 3717, 2190, 2531, 2282, 6589, 4183}"
185,1,18,0,0.0983354,"\int (a+a \cos (x)) (A+B \sec (x)) \, dx","Int[(a + a*Cos[x])*(A + B*Sec[x]),x]","a x (A+B)+a A \sin (x)+a B \tanh ^{-1}(\sin (x))","a x (A+B)+a A \sin (x)+a B \tanh ^{-1}(\sin (x))",1,"a*(A + B)*x + a*B*ArcTanh[Sin[x]] + a*A*Sin[x]","A",5,5,13,0.3846,1,"{2828, 2968, 3023, 2735, 3770}"
186,1,57,0,0.2015731,"\int (a+a \cos (x))^2 (A+B \sec (x)) \, dx","Int[(a + a*Cos[x])^2*(A + B*Sec[x]),x]","\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))","\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*(3*A + 4*B)*x)/2 + a^2*B*ArcTanh[Sin[x]] + (a^2*(3*A + 2*B)*Sin[x])/2 + (A*(a^2 + a^2*Cos[x])*Sin[x])/2","A",6,6,15,0.4000,1,"{2828, 2976, 2968, 3023, 2735, 3770}"
187,1,75,0,0.2971537,"\int (a+a \cos (x))^3 (A+B \sec (x)) \, dx","Int[(a + a*Cos[x])^3*(A + B*Sec[x]),x]","\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","\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*(5*A + 7*B)*x)/2 + a^3*B*ArcTanh[Sin[x]] + (5*a^3*(A + B)*Sin[x])/2 + (a*A*(a + a*Cos[x])^2*Sin[x])/3 + ((5*A + 3*B)*(a^3 + a^3*Cos[x])*Sin[x])/6","A",7,6,15,0.4000,1,"{2828, 2976, 2968, 3023, 2735, 3770}"
188,1,104,0,0.4028114,"\int (a+a \cos (x))^4 (A+B \sec (x)) \, dx","Int[(a + a*Cos[x])^4*(A + B*Sec[x]),x]","\frac{1}{8} a^4 x (35 A+48 B)+\frac{5}{8} a^4 (7 A+8 B) \sin (x)+\frac{1}{12} (7 A+4 B) \sin (x) \left(a^2 \cos (x)+a^2\right)^2+\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}{4} a A \sin (x) (a \cos (x)+a)^3","\frac{1}{8} a^4 x (35 A+48 B)+\frac{5}{8} a^4 (7 A+8 B) \sin (x)+\frac{1}{12} (7 A+4 B) \sin (x) \left(a^2 \cos (x)+a^2\right)^2+\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}{4} a A \sin (x) (a \cos (x)+a)^3",1,"(a^4*(35*A + 48*B)*x)/8 + a^4*B*ArcTanh[Sin[x]] + (5*a^4*(7*A + 8*B)*Sin[x])/8 + (a*A*(a + a*Cos[x])^3*Sin[x])/4 + ((7*A + 4*B)*(a^2 + a^2*Cos[x])^2*Sin[x])/12 + ((35*A + 32*B)*(a^4 + a^4*Cos[x])*Sin[x])/24","A",8,6,15,0.4000,1,"{2828, 2976, 2968, 3023, 2735, 3770}"
189,1,25,0,0.0928542,"\int \frac{A+B \sec (x)}{a+a \cos (x)} \, dx","Int[(A + B*Sec[x])/(a + a*Cos[x]),x]","\frac{(A-B) \sin (x)}{a \cos (x)+a}+\frac{B \tanh ^{-1}(\sin (x))}{a}","\frac{(A-B) \sin (x)}{a \cos (x)+a}+\frac{B \tanh ^{-1}(\sin (x))}{a}",1,"(B*ArcTanh[Sin[x]])/a + ((A - B)*Sin[x])/(a + a*Cos[x])","A",4,4,15,0.2667,1,"{2828, 2978, 12, 3770}"
190,1,48,0,0.1846085,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^2} \, dx","Int[(A + B*Sec[x])/(a + a*Cos[x])^2,x]","\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}","\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,"(B*ArcTanh[Sin[x]])/a^2 + ((A - 4*B)*Sin[x])/(3*a^2*(1 + Cos[x])) + ((A - B)*Sin[x])/(3*(a + a*Cos[x])^2)","A",5,4,15,0.2667,1,"{2828, 2978, 12, 3770}"
191,1,75,0,0.3109554,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^3} \, dx","Int[(A + B*Sec[x])/(a + a*Cos[x])^3,x]","\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}","\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,"(B*ArcTanh[Sin[x]])/a^3 + ((A - B)*Sin[x])/(5*(a + a*Cos[x])^3) + ((2*A - 7*B)*Sin[x])/(15*a*(a + a*Cos[x])^2) + (2*(A - 11*B)*Sin[x])/(15*(a^3 + a^3*Cos[x]))","A",6,4,15,0.2667,1,"{2828, 2978, 12, 3770}"
192,1,96,0,0.414411,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^4} \, dx","Int[(A + B*Sec[x])/(a + a*Cos[x])^4,x]","\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}","\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,"(B*ArcTanh[Sin[x]])/a^4 + ((6*A - 55*B)*Sin[x])/(105*a^4*(1 + Cos[x])^2) + (2*(3*A - 80*B)*Sin[x])/(105*a^4*(1 + Cos[x])) + ((A - B)*Sin[x])/(7*(a + a*Cos[x])^4) + ((3*A - 10*B)*Sin[x])/(35*a*(a + a*Cos[x])^3)","A",7,4,15,0.2667,1,"{2828, 2978, 12, 3770}"
193,1,98,0,0.431637,"\int (a+a \cos (x))^{5/2} (A+B \sec (x)) \, dx","Int[(a + a*Cos[x])^(5/2)*(A + B*Sec[x]),x]","\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}+2 a^{5/2} B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)+\frac{2}{5} a A \sin (x) (a \cos (x)+a)^{3/2}","\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}+2 a^{5/2} B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)+\frac{2}{5} a A \sin (x) (a \cos (x)+a)^{3/2}",1,"2*a^(5/2)*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]] + (2*a^3*(32*A + 35*B)*Sin[x])/(15*Sqrt[a + a*Cos[x]]) + (2*a^2*(8*A + 5*B)*Sqrt[a + a*Cos[x]]*Sin[x])/15 + (2*a*A*(a + a*Cos[x])^(3/2)*Sin[x])/5","A",6,5,17,0.2941,1,"{2828, 2976, 2981, 2773, 206}"
194,1,72,0,0.2940851,"\int (a+a \cos (x))^{3/2} (A+B \sec (x)) \, dx","Int[(a + a*Cos[x])^(3/2)*(A + B*Sec[x]),x]","\frac{2 a^2 (4 A+3 B) \sin (x)}{3 \sqrt{a \cos (x)+a}}+2 a^{3/2} B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)+\frac{2}{3} a A \sin (x) \sqrt{a \cos (x)+a}","\frac{2 a^2 (4 A+3 B) \sin (x)}{3 \sqrt{a \cos (x)+a}}+2 a^{3/2} B \tanh ^{-1}\left(\frac{\sqrt{a} \sin (x)}{\sqrt{a \cos (x)+a}}\right)+\frac{2}{3} a A \sin (x) \sqrt{a \cos (x)+a}",1,"2*a^(3/2)*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]] + (2*a^2*(4*A + 3*B)*Sin[x])/(3*Sqrt[a + a*Cos[x]]) + (2*a*A*Sqrt[a + a*Cos[x]]*Sin[x])/3","A",5,5,17,0.2941,1,"{2828, 2976, 2981, 2773, 206}"
195,1,44,0,0.1603314,"\int \sqrt{a+a \cos (x)} (A+B \sec (x)) \, dx","Int[Sqrt[a + a*Cos[x]]*(A + B*Sec[x]),x]","\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)","\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,"2*Sqrt[a]*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]] + (2*a*A*Sin[x])/Sqrt[a + a*Cos[x]]","A",4,4,17,0.2353,1,"{2828, 2981, 2773, 206}"
196,1,68,0,0.1952222,"\int \frac{A+B \sec (x)}{\sqrt{a+a \cos (x)}} \, dx","Int[(A + B*Sec[x])/Sqrt[a + a*Cos[x]],x]","\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}}","\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*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]])/Sqrt[a] + (Sqrt[2]*(A - B)*ArcTanh[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[a + a*Cos[x]])])/Sqrt[a]","A",6,5,17,0.2941,1,"{2828, 2985, 2649, 206, 2773}"
197,1,92,0,0.3359555,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^{3/2}} \, dx","Int[(A + B*Sec[x])/(a + a*Cos[x])^(3/2),x]","\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}}","\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,"(2*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]])/a^(3/2) + ((A - 5*B)*ArcTanh[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[a + a*Cos[x]])])/(2*Sqrt[2]*a^(3/2)) + ((A - B)*Sin[x])/(2*(a + a*Cos[x])^(3/2))","A",7,6,17,0.3529,1,"{2828, 2978, 2985, 2649, 206, 2773}"
198,1,120,0,0.4818247,"\int \frac{A+B \sec (x)}{(a+a \cos (x))^{5/2}} \, dx","Int[(A + B*Sec[x])/(a + a*Cos[x])^(5/2),x]","\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}}","\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*B*ArcTanh[(Sqrt[a]*Sin[x])/Sqrt[a + a*Cos[x]]])/a^(5/2) + ((3*A - 43*B)*ArcTanh[(Sqrt[a]*Sin[x])/(Sqrt[2]*Sqrt[a + a*Cos[x]])])/(16*Sqrt[2]*a^(5/2)) + ((A - B)*Sin[x])/(4*(a + a*Cos[x])^(5/2)) + ((3*A - 11*B)*Sin[x])/(16*a*(a + a*Cos[x])^(3/2))","A",8,6,17,0.3529,1,"{2828, 2978, 2985, 2649, 206, 2773}"
199,1,25,0,0.0531446,"\int \frac{x (b+a \sin (x))}{(a+b \sin (x))^2} \, dx","Int[(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",3,3,16,0.1875,1,"{4592, 2668, 31}"
200,1,24,0,0.0555215,"\int \frac{x (b+a \cos (x))}{(a+b \cos (x))^2} \, dx","Int[(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",3,3,16,0.1875,1,"{4593, 2668, 31}"
201,1,8,0,0.0414272,"\int \frac{1+\sin ^2(x)}{1-\sin ^2(x)} \, dx","Int[(1 + Sin[x]^2)/(1 - Sin[x]^2),x]","2 \tan (x)-x","2 \tan (x)-x",1,"-x + 2*Tan[x]","A",4,4,17,0.2353,1,"{3171, 3175, 3767, 8}"
202,1,36,0,0.0410113,"\int \frac{1-\sin ^2(x)}{1+\sin ^2(x)} \, dx","Int[(1 - Sin[x]^2)/(1 + Sin[x]^2),x]","\sqrt{2} x-x+\sqrt{2} \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)","\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]*x + Sqrt[2]*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]","A",3,3,17,0.1765,1,"{3171, 3181, 203}"
203,1,8,0,0.0410208,"\int \frac{1+\cos ^2(x)}{1-\cos ^2(x)} \, dx","Int[(1 + Cos[x]^2)/(1 - Cos[x]^2),x]","-x-2 \cot (x)","-x-2 \cot (x)",1,"-x - 2*Cot[x]","A",4,4,17,0.2353,1,"{3171, 3175, 3767, 8}"
204,1,37,0,0.0383074,"\int \frac{1-\cos ^2(x)}{1+\cos ^2(x)} \, dx","Int[(1 - Cos[x]^2)/(1 + Cos[x]^2),x]","\sqrt{2} x-x-\sqrt{2} \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)","\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]*x - Sqrt[2]*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]","A",3,3,17,0.1765,1,"{3171, 3181, 203}"
205,1,14,0,0.1277962,"\int \frac{-1+\frac{c^2}{d^2}+\sin ^2(x)}{c+d \cos (x)} \, dx","Int[(-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",4,3,22,0.1364,1,"{4397, 3016, 2637}"
206,1,105,0,0.2599085,"\int \frac{a+b \sin ^2(x)}{c+d \cos (x)} \, dx","Int[(a + b*Sin[x]^2)/(c + d*Cos[x]),x]","\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}","\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)/d^2 + (2*a*ArcTan[(Sqrt[c - d]*Tan[x/2])/Sqrt[c + d]])/(Sqrt[c - d]*Sqrt[c + d]) - (2*b*Sqrt[c - d]*Sqrt[c + d]*ArcTan[(Sqrt[c - d]*Tan[x/2])/Sqrt[c + d]])/d^2 - (b*Sin[x])/d","A",8,5,17,0.2941,1,"{4401, 2659, 205, 2695, 2735}"
207,1,57,0,0.1341891,"\int \frac{a+b \sin ^2(x)}{c+c \cos ^2(x)} \, dx","Int[(a + b*Sin[x]^2)/(c + c*Cos[x]^2),x]","\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}","\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)*x)/(Sqrt[2]*c) - ((a + 2*b)*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)])/(Sqrt[2]*c)","A",5,3,19,0.1579,1,"{12, 1166, 203}"
208,1,15,0,0.0890223,"\int \frac{a+b \sin ^2(x)}{c-c \cos ^2(x)} \, dx","Int[(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",3,2,20,0.1000,1,"{453, 205}"
209,1,49,0,0.1498668,"\int \frac{a+b \sin ^2(x)}{c+d \cos ^2(x)} \, dx","Int[(a + b*Sin[x]^2)/(c + d*Cos[x]^2),x]","\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}","\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)/d) + ((a*d + b*(c + d))*ArcTan[(Sqrt[c]*Tan[x])/Sqrt[c + d]])/(Sqrt[c]*d*Sqrt[c + d])","A",4,3,19,0.1579,1,"{522, 203, 205}"
210,1,13,0,0.1193295,"\int \frac{-1+\frac{c^2}{d^2}+\cos ^2(x)}{c+d \sin (x)} \, dx","Int[(-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",4,3,22,0.1364,1,"{4397, 3016, 2638}"
211,1,100,0,0.2401047,"\int \frac{a+b \cos ^2(x)}{c+d \sin (x)} \, dx","Int[(a + b*Cos[x]^2)/(c + d*Sin[x]),x]","\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}","\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,"(b*c*x)/d^2 + (2*a*ArcTan[(d + c*Tan[x/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2] - (2*b*Sqrt[c^2 - d^2]*ArcTan[(d + c*Tan[x/2])/Sqrt[c^2 - d^2]])/d^2 + (b*Cos[x])/d","A",10,6,17,0.3529,1,"{4401, 2660, 618, 204, 2695, 2735}"
212,1,56,0,0.1966454,"\int \frac{a+b \cos ^2(x)}{c+c \sin ^2(x)} \, dx","Int[(a + b*Cos[x]^2)/(c + c*Sin[x]^2),x]","\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}","\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)*x)/(Sqrt[2]*c) + ((a + 2*b)*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)])/(Sqrt[2]*c)","A",4,2,19,0.1053,1,"{1166, 205}"
213,1,14,0,0.0576156,"\int \frac{a+b \cos ^2(x)}{c-c \sin ^2(x)} \, dx","Int[(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",3,3,20,0.1500,1,"{3175, 3012, 8}"
214,1,49,0,0.1609407,"\int \frac{a+b \cos ^2(x)}{c+d \sin ^2(x)} \, dx","Int[(a + b*Cos[x]^2)/(c + d*Sin[x]^2),x]","\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}","\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)/d) + ((a*d + b*(c + d))*ArcTan[(Sqrt[c + d]*Tan[x])/Sqrt[c]])/(Sqrt[c]*d*Sqrt[c + d])","A",4,3,19,0.1579,1,"{522, 203, 205}"
215,1,74,0,0.2456925,"\int \frac{a+b \sec ^2(x)}{c+d \cos (x)} \, dx","Int[(a + b*Sec[x]^2)/(c + d*Cos[x]),x]","\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}","\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)*ArcTan[(Sqrt[c - d]*Tan[x/2])/Sqrt[c + d]])/(c^2*Sqrt[c - d]*Sqrt[c + d]) - (b*d*ArcTanh[Sin[x]])/c^2 + (b*Tan[x])/c","A",6,6,17,0.3529,1,"{4234, 3056, 3001, 3770, 2659, 205}"
216,1,72,0,0.2378384,"\int \frac{a+b \csc ^2(x)}{c+d \sin (x)} \, dx","Int[(a + b*Csc[x]^2)/(c + d*Sin[x]),x]","\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}","\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,"(2*(a*c^2 + b*d^2)*ArcTan[(d + c*Tan[x/2])/Sqrt[c^2 - d^2]])/(c^2*Sqrt[c^2 - d^2]) + (b*d*ArcTanh[Cos[x]])/c^2 - (b*Cot[x])/c","A",7,7,17,0.4118,1,"{4233, 3056, 3001, 3770, 2660, 618, 204}"
217,1,136,0,0.0581775,"\int (a \cos (c+d x)+b \sin (c+d x))^n \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^n,x]","-\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)}}","-\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,"-((Cos[c + d*x - ArcTan[a, b]]^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/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]])/(d*(1 + n)*((a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2])^n*Sqrt[Sin[c + d*x - ArcTan[a, b]]^2]))","A",2,2,19,0.1053,1,"{3078, 2643}"
218,1,95,0,0.0489555,"\int (2 \cos (c+d x)+3 \sin (c+d x))^n \, dx","Int[(2*Cos[c + d*x] + 3*Sin[c + d*x])^n,x]","-\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)}}","-\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,"-((13^(n/2)*Cos[c + d*x - ArcTan[3/2]]^(1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x - ArcTan[3/2]]^2]*Sin[c + d*x - ArcTan[3/2]])/(d*(1 + n)*Sqrt[Sin[c + d*x - ArcTan[3/2]]^2]))","A",2,2,19,0.1053,1,"{3077, 2643}"
219,1,127,0,0.0778204,"\int (a \cos (c+d x)+b \sin (c+d x))^7 \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^7,x]","-\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}","-\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,"-(((a^2 + b^2)^3*(b*Cos[c + d*x] - a*Sin[c + d*x]))/d) + ((a^2 + b^2)^2*(b*Cos[c + d*x] - a*Sin[c + d*x])^3)/d - (3*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])^5)/(5*d) + (b*Cos[c + d*x] - a*Sin[c + d*x])^7/(7*d)","A",3,2,19,0.1053,1,"{3072, 194}"
220,1,161,0,0.0789943,"\int (a \cos (c+d x)+b \sin (c+d x))^6 \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^6,x]","-\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}","-\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,"(5*(a^2 + b^2)^3*x)/16 - (5*(a^2 + b^2)^2*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(16*d) - (5*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(24*d) - ((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^5)/(6*d)","A",4,2,19,0.1053,1,"{3073, 8}"
221,1,94,0,0.0452859,"\int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\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}","\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,"-(((a^2 + b^2)^2*(b*Cos[c + d*x] - a*Sin[c + d*x]))/d) + (2*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])^3)/(3*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])^5/(5*d)","A",3,2,19,0.1053,1,"{3072, 194}"
222,1,108,0,0.0444016,"\int (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\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}","-\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,"(3*(a^2 + b^2)^2*x)/8 - (3*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(8*d) - ((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(4*d)","A",3,2,19,0.1053,1,"{3073, 8}"
223,1,58,0,0.0236529,"\int (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\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}","\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,"-(((a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x]))/d) + (b*Cos[c + d*x] - a*Sin[c + d*x])^3/(3*d)","A",2,1,19,0.05263,1,"{3072}"
224,1,55,0,0.0192334,"\int (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\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}","\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,"((a^2 + b^2)*x)/2 - ((b*Cos[c + d*x] - a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(2*d)","A",2,2,19,0.1053,1,"{3073, 8}"
225,1,24,0,0.0142294,"\int (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[a*Cos[c + d*x] + b*Sin[c + d*x],x]","\frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d}","\frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d}",1,"-((b*Cos[c + d*x])/d) + (a*Sin[c + d*x])/d","A",3,2,17,0.1176,1,"{2637, 2638}"
226,1,47,0,0.0272597,"\int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-1),x]","-\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}}","-\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,"-(ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/(Sqrt[a^2 + b^2]*d))","A",2,2,19,0.1053,1,"{3074, 206}"
227,1,32,0,0.0164173,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Int[(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",1,1,19,0.05263,1,"{3075}"
228,1,103,0,0.0567913,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-3),x]","-\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}}","-\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,"-ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/(2*(a^2 + b^2)^(3/2)*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(2*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)","A",3,3,19,0.1579,1,"{3076, 3074, 206}"
229,1,98,0,0.041788,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-4),x]","\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}","\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,"-(b*Cos[c + d*x] - a*Sin[c + d*x])/(3*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (2*Sin[c + d*x])/(3*a*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","A",2,2,19,0.1053,1,"{3076, 3075}"
230,1,156,0,0.0860091,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^5} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-5),x]","-\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}}","-\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,"(-3*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(8*(a^2 + b^2)^(5/2)*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(4*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (3*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(8*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)","A",4,3,19,0.1579,1,"{3076, 3074, 206}"
231,1,151,0,0.0693007,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^6} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-6),x]","\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}","\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,"-(b*Cos[c + d*x] - a*Sin[c + d*x])/(5*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) - (4*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(15*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (8*Sin[c + d*x])/(15*a*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","A",3,2,19,0.1053,1,"{3076, 3075}"
232,1,186,0,0.0958722,"\int (a \cos (c+d x)+b \sin (c+d x))^{7/2} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(7/2),x]","\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}","\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,"(-10*(a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x])*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(21*d) - (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) + (10*(a^2 + b^2)^2*EllipticF[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(21*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])","A",4,3,21,0.1429,1,"{3073, 3078, 2641}"
233,1,131,0,0.057692,"\int (a \cos (c+d x)+b \sin (c+d x))^{5/2} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(5/2),x]","\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}","\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,"(-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) + (6*(a^2 + b^2)*EllipticE[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(5*d*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])","A",3,3,21,0.1429,1,"{3073, 3078, 2639}"
234,1,131,0,0.0580733,"\int (a \cos (c+d x)+b \sin (c+d x))^{3/2} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(3/2),x]","\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}","\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]])/(3*d) + (2*(a^2 + b^2)*EllipticF[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(3*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])","A",3,3,21,0.1429,1,"{3073, 3078, 2641}"
235,1,75,0,0.0296799,"\int \sqrt{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]],x]","\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}}}}","\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,"(2*EllipticE[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(d*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])","A",2,2,21,0.09524,1,"{3078, 2639}"
236,1,75,0,0.0296477,"\int \frac{1}{\sqrt{a \cos (c+d x)+b \sin (c+d x)}} \, dx","Int[1/Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]],x]","\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)}}","\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*EllipticF[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])","A",2,2,21,0.09524,1,"{3078, 2641}"
237,1,138,0,0.0576028,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^{3/2}} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-3/2),x]","-\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)}}","-\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] - a*Sin[c + d*x]))/((a^2 + b^2)*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]]) - (2*EllipticE[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])","A",3,3,21,0.1429,1,"{3076, 3078, 2639}"
238,1,142,0,0.0551344,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^{5/2}} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-5/2),x]","\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}}","\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]))/(3*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^(3/2)) + (2*EllipticF[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(3*(a^2 + b^2)*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])","A",3,3,21,0.1429,1,"{3076, 3078, 2641}"
239,1,197,0,0.0859234,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^{7/2}} \, dx","Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-7/2),x]","-\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}}","-\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*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(5*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^(5/2)) - (6*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(5*(a^2 + b^2)^2*d*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]]) - (6*EllipticE[(c + d*x - ArcTan[a, b])/2, 2]*Sqrt[a*Cos[c + d*x] + b*Sin[c + d*x]])/(5*(a^2 + b^2)^2*d*Sqrt[(a*Cos[c + d*x] + b*Sin[c + d*x])/Sqrt[a^2 + b^2]])","A",4,3,21,0.1429,1,"{3076, 3078, 2639}"
240,1,120,0,0.0699545,"\int (2 \cos (c+d x)+3 \sin (c+d x))^{7/2} \, dx","Int[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(7/2),x]","-\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}","-\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,"(130*13^(3/4)*EllipticF[(c + d*x - ArcTan[3/2])/2, 2])/(21*d) - (130*(3*Cos[c + d*x] - 2*Sin[c + d*x])*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]])/(21*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x])*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(5/2))/(7*d)","A",4,3,21,0.1429,1,"{3073, 3077, 2641}"
241,1,75,0,0.0445689,"\int (2 \cos (c+d x)+3 \sin (c+d x))^{5/2} \, dx","Int[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(5/2),x]","\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}","\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,"(78*13^(1/4)*EllipticE[(c + d*x - ArcTan[3/2])/2, 2])/(5*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x])*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2))/(5*d)","A",3,3,21,0.1429,1,"{3073, 3077, 2639}"
242,1,75,0,0.0426062,"\int (2 \cos (c+d x)+3 \sin (c+d x))^{3/2} \, dx","Int[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2),x]","\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}","\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*13^(3/4)*EllipticF[(c + d*x - ArcTan[3/2])/2, 2])/(3*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x])*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]])/(3*d)","A",3,3,21,0.1429,1,"{3073, 3077, 2641}"
243,1,27,0,0.0227896,"\int \sqrt{2 \cos (c+d x)+3 \sin (c+d x)} \, dx","Int[Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]],x]","\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}","\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,"(2*13^(1/4)*EllipticE[(c + d*x - ArcTan[3/2])/2, 2])/d","A",2,2,21,0.09524,1,"{3077, 2639}"
244,1,27,0,0.0240906,"\int \frac{1}{\sqrt{2 \cos (c+d x)+3 \sin (c+d x)}} \, dx","Int[1/Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]],x]","\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}","\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*EllipticF[(c + d*x - ArcTan[3/2])/2, 2])/(13^(1/4)*d)","A",2,2,21,0.09524,1,"{3077, 2641}"
245,1,73,0,0.0413938,"\int \frac{1}{(2 \cos (c+d x)+3 \sin (c+d x))^{3/2}} \, dx","Int[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(-3/2),x]","-\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}","-\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,"(-2*EllipticE[(c + d*x - ArcTan[3/2])/2, 2])/(13^(3/4)*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(13*d*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]])","A",3,3,21,0.1429,1,"{3076, 3077, 2639}"
246,1,75,0,0.0421979,"\int \frac{1}{(2 \cos (c+d x)+3 \sin (c+d x))^{5/2}} \, dx","Int[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(-5/2),x]","\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}}","\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,"(2*EllipticF[(c + d*x - ArcTan[3/2])/2, 2])/(39*13^(1/4)*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(39*d*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2))","A",3,3,21,0.1429,1,"{3076, 3077, 2641}"
247,1,120,0,0.0636797,"\int \frac{1}{(2 \cos (c+d x)+3 \sin (c+d x))^{7/2}} \, dx","Int[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(-7/2),x]","-\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}","-\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,"(-6*EllipticE[(c + d*x - ArcTan[3/2])/2, 2])/(65*13^(3/4)*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(65*d*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(5/2)) - (6*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(845*d*Sqrt[2*Cos[c + d*x] + 3*Sin[c + d*x]])","A",4,3,21,0.1429,1,"{3076, 3077, 2639}"
248,1,32,0,0.0154417,"\int (a \cos (c+d x)+i a \sin (c+d x))^n \, dx","Int[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n,x]","-\frac{i (a \cos (c+d x)+i a \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*a*Sin[c + d*x])^n)/(d*n)","A",1,1,22,0.04545,1,"{3071}"
249,1,31,0,0.0181858,"\int (a \cos (c+d x)+i a \sin (c+d x))^4 \, dx","Int[(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,"((-I/4)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^4)/d","A",1,1,22,0.04545,1,"{3071}"
250,1,31,0,0.0162244,"\int (a \cos (c+d x)+i a \sin (c+d x))^3 \, dx","Int[(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,"((-I/3)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3)/d","A",1,1,22,0.04545,1,"{3071}"
251,1,31,0,0.0143722,"\int (a \cos (c+d x)+i a \sin (c+d x))^2 \, dx","Int[(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,"((-I/2)*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2)/d","A",1,1,22,0.04545,1,"{3071}"
252,1,26,0,0.0140323,"\int (a \cos (c+d x)+i a \sin (c+d x)) \, dx","Int[a*Cos[c + d*x] + I*a*Sin[c + d*x],x]","\frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d}","\frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d}",1,"((-I)*a*Cos[c + d*x])/d + (a*Sin[c + d*x])/d","A",3,2,20,0.1000,1,"{2637, 2638}"
253,1,29,0,0.0147709,"\int \frac{1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[(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",1,1,22,0.04545,1,"{3071}"
254,1,31,0,0.0155649,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[(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",1,1,22,0.04545,1,"{3071}"
255,1,31,0,0.0153039,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[(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",1,1,22,0.04545,1,"{3071}"
256,1,31,0,0.0148451,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^4} \, dx","Int[(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",1,1,22,0.04545,1,"{3071}"
257,1,33,0,0.015664,"\int (a \cos (c+d x)+i a \sin (c+d x))^{5/2} \, dx","Int[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(5/2),x]","-\frac{2 i (a \cos (c+d x)+i a \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*a*Sin[c + d*x])^(5/2))/d","A",1,1,24,0.04167,1,"{3071}"
258,1,33,0,0.0160221,"\int (a \cos (c+d x)+i a \sin (c+d x))^{3/2} \, dx","Int[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(3/2),x]","-\frac{2 i (a \cos (c+d x)+i a \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*a*Sin[c + d*x])^(3/2))/d","A",1,1,24,0.04167,1,"{3071}"
259,1,31,0,0.0160164,"\int \sqrt{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Sqrt[a*Cos[c + d*x] + I*a*Sin[c + d*x]],x]","-\frac{2 i \sqrt{a \cos (c+d x)+i a \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*a*Sin[c + d*x]])/d","A",1,1,24,0.04167,1,"{3071}"
260,1,31,0,0.017475,"\int \frac{1}{\sqrt{a \cos (c+d x)+i a \sin (c+d x)}} \, dx","Int[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 a \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*a*Sin[c + d*x]])","A",1,1,24,0.04167,1,"{3071}"
261,1,33,0,0.0161539,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^{3/2}} \, dx","Int[(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 a \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*a*Sin[c + d*x])^(3/2))","A",1,1,24,0.04167,1,"{3071}"
262,1,33,0,0.0163566,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^{5/2}} \, dx","Int[(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 a \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*a*Sin[c + d*x])^(5/2))","A",1,1,24,0.04167,1,"{3071}"
263,1,149,0,0.2113748,"\int (a \sec (x)+b \tan (x))^5 \, dx","Int[(a*Sec[x] + b*Tan[x])^5,x]","-\frac{1}{8} a b^4 \left(7-\frac{3 a^2}{b^2}\right) \sin (x)-\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}{4} \sec ^4(x) (a \sin (x)+b) (a+b \sin (x))^4","-\frac{1}{8} a b^4 \left(7-\frac{3 a^2}{b^2}\right) \sin (x)-\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}{4} \sec ^4(x) (a \sin (x)+b) (a+b \sin (x))^4",1,"-((a + b)^3*(3*a^2 - 9*a*b + 8*b^2)*Log[1 - Sin[x]])/16 + ((a - b)^3*(3*a^2 + 9*a*b + 8*b^2)*Log[1 + Sin[x]])/16 - (a*(7 - (3*a^2)/b^2)*b^4*Sin[x])/8 + (Sec[x]^4*(b + a*Sin[x])*(a + b*Sin[x])^4)/4 + (Sec[x]^2*(a + b*Sin[x])^2*(2*b*(a^2 - 2*b^2) + a*(3*a^2 - 5*b^2)*Sin[x]))/8","A",8,7,11,0.6364,1,"{4391, 2668, 739, 819, 774, 633, 31}"
264,1,100,0,0.1962526,"\int (a \sec (x)+b \tan (x))^4 \, dx","Int[(a*Sec[x] + b*Tan[x])^4,x]","\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","\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,"b^4*x + (4*a*b*(a^2 - 2*b^2)*Cos[x])/3 + (b^2*(2*a^2 - 3*b^2)*Cos[x]*Sin[x])/3 + (Sec[x]^3*(b + a*Sin[x])*(a + b*Sin[x])^3)/3 - (Sec[x]*(a + b*Sin[x])^2*(a*b - (2*a^2 - 3*b^2)*Sin[x]))/3","A",4,4,11,0.3636,1,"{4391, 2691, 2861, 2734}"
265,1,75,0,0.1363357,"\int (a \sec (x)+b \tan (x))^3 \, dx","Int[(a*Sec[x] + b*Tan[x])^3,x]","\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","\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)*(a + b)^2*Log[1 - Sin[x]])/4 + ((a - b)^2*(a + 2*b)*Log[1 + Sin[x]])/4 + (a*b^2*Sin[x])/2 + (Sec[x]^2*(b + a*Sin[x])*(a + b*Sin[x])^2)/2","A",7,6,11,0.5455,1,"{4391, 2668, 739, 774, 633, 31}"
266,1,27,0,0.0538816,"\int (a \sec (x)+b \tan (x))^2 \, dx","Int[(a*Sec[x] + b*Tan[x])^2,x]","a b \cos (x)+\sec (x) (a \sin (x)+b) (a+b \sin (x))+b^2 (-x)","a b \cos (x)+\sec (x) (a \sin (x)+b) (a+b \sin (x))+b^2 (-x)",1,"-(b^2*x) + a*b*Cos[x] + Sec[x]*(b + a*Sin[x])*(a + b*Sin[x])","A",4,3,11,0.2727,1,"{4391, 2691, 2638}"
267,1,12,0,0.0074644,"\int (a \sec (x)+b \tan (x)) \, dx","Int[a*Sec[x] + b*Tan[x],x]","a \tanh ^{-1}(\sin (x))-b \log (\cos (x))","a \tanh ^{-1}(\sin (x))-b \log (\cos (x))",1,"a*ArcTanh[Sin[x]] - b*Log[Cos[x]]","A",3,2,9,0.2222,1,"{3770, 3475}"
268,1,11,0,0.0342235,"\int \frac{1}{a \sec (x)+b \tan (x)} \, dx","Int[(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",3,3,11,0.2727,1,"{3159, 2668, 31}"
269,1,66,0,0.1333387,"\int \frac{1}{(a \sec (x)+b \tan (x))^2} \, dx","Int[(a*Sec[x] + b*Tan[x])^(-2),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}","\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,"-(x/b^2) + (2*a*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]) - Cos[x]/(b*(a + b*Sin[x]))","A",6,6,11,0.5455,1,"{4391, 2693, 2735, 2660, 618, 204}"
270,1,51,0,0.0751398,"\int \frac{1}{(a \sec (x)+b \tan (x))^3} \, dx","Int[(a*Sec[x] + b*Tan[x])^(-3),x]","\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}","\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]]/b^3) + (a^2 - b^2)/(2*b^3*(a + b*Sin[x])^2) - (2*a)/(b^3*(a + b*Sin[x]))","A",4,3,11,0.2727,1,"{4391, 2668, 697}"
271,1,156,0,0.3369857,"\int \frac{1}{(a \sec (x)+b \tan (x))^4} \, dx","Int[(a*Sec[x] + b*Tan[x])^(-4),x]","-\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{a \cos ^3(x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^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}","-\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{a \cos ^3(x)}{2 b \left(a^2-b^2\right) (a+b \sin (x))^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,"x/b^4 - (a*(2*a^2 - 3*b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*(a^2 - b^2)^(3/2)) - Cos[x]^3/(3*b*(a + b*Sin[x])^3) + (a*Cos[x]^3)/(2*b*(a^2 - b^2)*(a + b*Sin[x])^2) + (Cos[x]*(2*(a^2 - b^2) + a*b*Sin[x]))/(2*b^3*(a^2 - b^2)*(a + b*Sin[x]))","A",8,8,11,0.7273,1,"{4391, 2693, 2864, 2863, 2735, 2660, 618, 204}"
272,1,101,0,0.1156661,"\int \frac{1}{(a \sec (x)+b \tan (x))^5} \, dx","Int[(a*Sec[x] + b*Tan[x])^(-5),x]","-\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}","-\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]]/b^5 - (a^2 - b^2)^2/(4*b^5*(a + b*Sin[x])^4) + (4*a*(a^2 - b^2))/(3*b^5*(a + b*Sin[x])^3) - (3*a^2 - b^2)/(b^5*(a + b*Sin[x])^2) + (4*a)/(b^5*(a + b*Sin[x]))","A",4,3,11,0.2727,1,"{4391, 2668, 697}"
273,1,30,0,0.0499651,"\int (\sec (x)+\tan (x))^5 \, dx","Int[(Sec[x] + Tan[x])^5,x]","-\frac{4}{1-\sin (x)}+\frac{2}{(1-\sin (x))^2}-\log (1-\sin (x))","-\frac{4}{1-\sin (x)}+\frac{2}{(1-\sin (x))^2}-\log (1-\sin (x))",1,"-Log[1 - Sin[x]] + 2/(1 - Sin[x])^2 - 4/(1 - Sin[x])","A",4,3,7,0.4286,1,"{4391, 2667, 43}"
274,1,30,0,0.1020199,"\int (\sec (x)+\tan (x))^4 \, dx","Int[(Sec[x] + Tan[x])^4,x]","x+\frac{2 \cos ^3(x)}{3 (1-\sin (x))^3}-\frac{2 \cos (x)}{1-\sin (x)}","x+\frac{2 \cos ^3(x)}{3 (1-\sin (x))^3}-\frac{2 \cos (x)}{1-\sin (x)}",1,"x + (2*Cos[x]^3)/(3*(1 - Sin[x])^3) - (2*Cos[x])/(1 - Sin[x])","A",5,4,7,0.5714,1,"{4391, 2670, 2680, 8}"
275,1,18,0,0.0438103,"\int (\sec (x)+\tan (x))^3 \, dx","Int[(Sec[x] + Tan[x])^3,x]","\frac{2}{1-\sin (x)}+\log (1-\sin (x))","\frac{2}{1-\sin (x)}+\log (1-\sin (x))",1,"Log[1 - Sin[x]] + 2/(1 - Sin[x])","A",4,3,7,0.4286,1,"{4391, 2667, 43}"
276,1,16,0,0.0698457,"\int (\sec (x)+\tan (x))^2 \, dx","Int[(Sec[x] + Tan[x])^2,x]","\frac{2 \cos (x)}{1-\sin (x)}-x","\frac{2 \cos (x)}{1-\sin (x)}-x",1,"-x + (2*Cos[x])/(1 - Sin[x])","A",4,4,7,0.5714,1,"{4391, 2670, 2680, 8}"
277,1,9,0,0.0056346,"\int (\sec (x)+\tan (x)) \, dx","Int[Sec[x] + Tan[x],x]","\tanh ^{-1}(\sin (x))-\log (\cos (x))","-2 \log \left(\cos \left(\frac{1}{4} (2 x+\pi )\right)\right)",1,"ArcTanh[Sin[x]] - Log[Cos[x]]","A",3,2,5,0.4000,1,"{3770, 3475}"
278,1,5,0,0.0244933,"\int \frac{1}{\sec (x)+\tan (x)} \, dx","Int[(Sec[x] + Tan[x])^(-1),x]","\log (\sin (x)+1)","\log (\sin (x)+1)",1,"Log[1 + Sin[x]]","A",3,3,7,0.4286,1,"{3159, 2667, 31}"
279,1,14,0,0.0414318,"\int \frac{1}{(\sec (x)+\tan (x))^2} \, dx","Int[(Sec[x] + Tan[x])^(-2),x]","-x-\frac{2 \cos (x)}{\sin (x)+1}","-x-\frac{2 \cos (x)}{\sin (x)+1}",1,"-x - (2*Cos[x])/(1 + Sin[x])","A",3,3,7,0.4286,1,"{4391, 2680, 8}"
280,1,16,0,0.0462508,"\int \frac{1}{(\sec (x)+\tan (x))^3} \, dx","Int[(Sec[x] + Tan[x])^(-3),x]","-\frac{2}{\sin (x)+1}-\log (\sin (x)+1)","-\frac{2}{\sin (x)+1}-\log (\sin (x)+1)",1,"-Log[1 + Sin[x]] - 2/(1 + Sin[x])","A",4,3,7,0.4286,1,"{4391, 2667, 43}"
281,1,26,0,0.0689964,"\int \frac{1}{(\sec (x)+\tan (x))^4} \, dx","Int[(Sec[x] + Tan[x])^(-4),x]","x-\frac{2 \cos ^3(x)}{3 (\sin (x)+1)^3}+\frac{2 \cos (x)}{\sin (x)+1}","x-\frac{2 \cos ^3(x)}{3 (\sin (x)+1)^3}+\frac{2 \cos (x)}{\sin (x)+1}",1,"x - (2*Cos[x]^3)/(3*(1 + Sin[x])^3) + (2*Cos[x])/(1 + Sin[x])","A",4,3,7,0.4286,1,"{4391, 2680, 8}"
282,1,22,0,0.0479451,"\int \frac{1}{(\sec (x)+\tan (x))^5} \, dx","Int[(Sec[x] + Tan[x])^(-5),x]","\frac{4}{\sin (x)+1}-\frac{2}{(\sin (x)+1)^2}+\log (\sin (x)+1)","\frac{4}{\sin (x)+1}-\frac{2}{(\sin (x)+1)^2}+\log (\sin (x)+1)",1,"Log[1 + Sin[x]] - 2/(1 + Sin[x])^2 + 4/(1 + Sin[x])","A",4,3,7,0.4286,1,"{4391, 2667, 43}"
283,1,152,0,0.2195008,"\int (a \cot (x)+b \csc (x))^5 \, dx","Int[(a*Cot[x] + b*Csc[x])^5,x]","\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))","\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,"(a^2*b*(7*a^2 - 3*b^2)*Cos[x])/8 + ((b + a*Cos[x])^2*(2*a*(2*a^2 - b^2) + b*(5*a^2 - 3*b^2)*Cos[x])*Csc[x]^2)/8 - ((b + a*Cos[x])^4*(a + b*Cos[x])*Csc[x]^4)/4 + ((a + b)^3*(8*a^2 - 9*a*b + 3*b^2)*Log[1 - Cos[x]])/16 + ((a - b)^3*(8*a^2 + 9*a*b + 3*b^2)*Log[1 + Cos[x]])/16","A",8,7,11,0.6364,1,"{4392, 2668, 739, 819, 774, 633, 31}"
284,1,101,0,0.2151159,"\int (a \cot (x)+b \csc (x))^4 \, dx","Int[(a*Cot[x] + b*Csc[x])^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)+a^4 x-\frac{1}{3} \csc ^3(x) (a \cos (x)+b)^3 (a+b \cos (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)+a^4 x-\frac{1}{3} \csc ^3(x) (a \cos (x)+b)^3 (a+b \cos (x))",1,"a^4*x + ((b + a*Cos[x])^2*(a*b + (3*a^2 - 2*b^2)*Cos[x])*Csc[x])/3 - ((b + a*Cos[x])^3*(a + b*Cos[x])*Csc[x]^3)/3 + (4*a*b*(2*a^2 - b^2)*Sin[x])/3 + (a^2*(3*a^2 - 2*b^2)*Cos[x]*Sin[x])/3","A",4,4,11,0.3636,1,"{4392, 2691, 2861, 2734}"
285,1,77,0,0.1357663,"\int (a \cot (x)+b \csc (x))^3 \, dx","Int[(a*Cot[x] + b*Csc[x])^3,x]","-\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))","-\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^2*b*Cos[x])/2 - ((b + a*Cos[x])^2*(a + b*Cos[x])*Csc[x]^2)/2 - ((2*a - b)*(a + b)^2*Log[1 - Cos[x]])/4 - ((a - b)^2*(2*a + b)*Log[1 + Cos[x]])/4","A",7,6,11,0.5455,1,"{4392, 2668, 739, 774, 633, 31}"
286,1,29,0,0.056306,"\int (a \cot (x)+b \csc (x))^2 \, dx","Int[(a*Cot[x] + b*Csc[x])^2,x]","a^2 (-x)-a b \sin (x)-\csc (x) (a \cos (x)+b) (a+b \cos (x))","a^2 (-x)-a b \sin (x)-\csc (x) (a \cos (x)+b) (a+b \cos (x))",1,"-(a^2*x) - (b + a*Cos[x])*(a + b*Cos[x])*Csc[x] - a*b*Sin[x]","A",4,3,11,0.2727,1,"{4392, 2691, 2637}"
287,1,12,0,0.0074097,"\int (a \cot (x)+b \csc (x)) \, dx","Int[a*Cot[x] + b*Csc[x],x]","a \log (\sin (x))-b \tanh ^{-1}(\cos (x))","a \log (\sin (x))-b \tanh ^{-1}(\cos (x))",1,"-(b*ArcTanh[Cos[x]]) + a*Log[Sin[x]]","A",3,2,9,0.2222,1,"{3475, 3770}"
288,1,12,0,0.0350371,"\int \frac{1}{a \cot (x)+b \csc (x)} \, dx","Int[(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",3,3,11,0.2727,1,"{3160, 2668, 31}"
289,1,67,0,0.1171845,"\int \frac{1}{(a \cot (x)+b \csc (x))^2} \, dx","Int[(a*Cot[x] + b*Csc[x])^(-2),x]","\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)}","\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,"-(x/a^2) + (2*b*ArcTanh[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]) + Sin[x]/(a*(b + a*Cos[x]))","A",5,5,11,0.4545,1,"{4392, 2693, 2735, 2659, 208}"
290,1,50,0,0.0787822,"\int \frac{1}{(a \cot (x)+b \csc (x))^3} \, dx","Int[(a*Cot[x] + b*Csc[x])^(-3),x]","\frac{a^2-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}+\frac{2 b}{a^3 (a \cos (x)+b)}+\frac{\log (a \cos (x)+b)}{a^3}",1,"(a^2 - b^2)/(2*a^3*(b + a*Cos[x])^2) + (2*b)/(a^3*(b + a*Cos[x])) + Log[b + a*Cos[x]]/a^3","A",4,3,11,0.2727,1,"{4392, 2668, 697}"
291,1,159,0,0.3384551,"\int \frac{1}{(a \cot (x)+b \csc (x))^4} \, dx","Int[(a*Cot[x] + b*Csc[x])^(-4),x]","\frac{b \sin ^3(x)}{2 a \left(a^2-b^2\right) (a \cos (x)+b)^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{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{x}{a^4}+\frac{\sin ^3(x)}{3 a (a \cos (x)+b)^3}","\frac{b \sin ^3(x)}{2 a \left(a^2-b^2\right) (a \cos (x)+b)^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{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{x}{a^4}+\frac{\sin ^3(x)}{3 a (a \cos (x)+b)^3}",1,"x/a^4 - (b*(3*a^2 - 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)) - ((2*(a^2 - b^2) - a*b*Cos[x])*Sin[x])/(2*a^3*(a^2 - b^2)*(b + a*Cos[x])) + Sin[x]^3/(3*a*(b + a*Cos[x])^3) + (b*Sin[x]^3)/(2*a*(a^2 - b^2)*(b + a*Cos[x])^2)","A",7,7,11,0.6364,1,"{4392, 2693, 2864, 2863, 2735, 2659, 208}"
292,1,100,0,0.1231814,"\int \frac{1}{(a \cot (x)+b \csc (x))^5} \, dx","Int[(a*Cot[x] + b*Csc[x])^(-5),x]","\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}-\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}-\frac{4 b}{a^5 (a \cos (x)+b)}-\frac{\log (a \cos (x)+b)}{a^5}",1,"(a^2 - b^2)^2/(4*a^5*(b + a*Cos[x])^4) + (4*b*(a^2 - b^2))/(3*a^5*(b + a*Cos[x])^3) - (a^2 - 3*b^2)/(a^5*(b + a*Cos[x])^2) - (4*b)/(a^5*(b + a*Cos[x])) - Log[b + a*Cos[x]]/a^5","A",4,3,11,0.2727,1,"{4392, 2668, 697}"
293,1,28,0,0.0510641,"\int (\cot (x)+\csc (x))^5 \, dx","Int[(Cot[x] + Csc[x])^5,x]","\frac{4}{1-\cos (x)}-\frac{2}{(1-\cos (x))^2}+\log (1-\cos (x))","\frac{4}{1-\cos (x)}-\frac{2}{(1-\cos (x))^2}+\log (1-\cos (x))",1,"-2/(1 - Cos[x])^2 + 4/(1 - Cos[x]) + Log[1 - Cos[x]]","A",4,3,7,0.4286,1,"{4392, 2667, 43}"
294,1,30,0,0.1010505,"\int (\cot (x)+\csc (x))^4 \, dx","Int[(Cot[x] + Csc[x])^4,x]","x-\frac{2 \sin ^3(x)}{3 (1-\cos (x))^3}+\frac{2 \sin (x)}{1-\cos (x)}","x-\frac{2 \sin ^3(x)}{3 (1-\cos (x))^3}+\frac{2 \sin (x)}{1-\cos (x)}",1,"x + (2*Sin[x])/(1 - Cos[x]) - (2*Sin[x]^3)/(3*(1 - Cos[x])^3)","A",5,4,7,0.5714,1,"{4392, 2670, 2680, 8}"
295,1,20,0,0.0480181,"\int (\cot (x)+\csc (x))^3 \, dx","Int[(Cot[x] + Csc[x])^3,x]","-\frac{2}{1-\cos (x)}-\log (1-\cos (x))","-\frac{2}{1-\cos (x)}-\log (1-\cos (x))",1,"-2/(1 - Cos[x]) - Log[1 - Cos[x]]","A",4,3,7,0.4286,1,"{4392, 2667, 43}"
296,1,16,0,0.0688719,"\int (\cot (x)+\csc (x))^2 \, dx","Int[(Cot[x] + Csc[x])^2,x]","-x-\frac{2 \sin (x)}{1-\cos (x)}","-x-\frac{2 \sin (x)}{1-\cos (x)}",1,"-x - (2*Sin[x])/(1 - Cos[x])","A",4,4,7,0.5714,1,"{4392, 2670, 2680, 8}"
297,1,9,0,0.0053846,"\int (\cot (x)+\csc (x)) \, dx","Int[Cot[x] + Csc[x],x]","\log (\sin (x))-\tanh ^{-1}(\cos (x))","\log (\sin (x))-\tanh ^{-1}(\cos (x))",1,"-ArcTanh[Cos[x]] + Log[Sin[x]]","A",3,2,5,0.4000,1,"{3475, 3770}"
298,1,7,0,0.0269082,"\int \frac{1}{\cot (x)+\csc (x)} \, dx","Int[(Cot[x] + Csc[x])^(-1),x]","-\log (\cos (x)+1)","-\log (\cos (x)+1)",1,"-Log[1 + Cos[x]]","A",3,3,7,0.4286,1,"{3160, 2667, 31}"
299,1,14,0,0.0413406,"\int \frac{1}{(\cot (x)+\csc (x))^2} \, dx","Int[(Cot[x] + Csc[x])^(-2),x]","\frac{2 \sin (x)}{\cos (x)+1}-x","\frac{2 \sin (x)}{\cos (x)+1}-x",1,"-x + (2*Sin[x])/(1 + Cos[x])","A",3,3,7,0.4286,1,"{4392, 2680, 8}"
300,1,14,0,0.0465222,"\int \frac{1}{(\cot (x)+\csc (x))^3} \, dx","Int[(Cot[x] + Csc[x])^(-3),x]","\frac{2}{\cos (x)+1}+\log (\cos (x)+1)","\frac{2}{\cos (x)+1}+\log (\cos (x)+1)",1,"2/(1 + Cos[x]) + Log[1 + Cos[x]]","A",4,3,7,0.4286,1,"{4392, 2667, 43}"
301,1,26,0,0.0711771,"\int \frac{1}{(\cot (x)+\csc (x))^4} \, dx","Int[(Cot[x] + Csc[x])^(-4),x]","x+\frac{2 \sin ^3(x)}{3 (\cos (x)+1)^3}-\frac{2 \sin (x)}{\cos (x)+1}","x+\frac{2 \sin ^3(x)}{3 (\cos (x)+1)^3}-\frac{2 \sin (x)}{\cos (x)+1}",1,"x - (2*Sin[x])/(1 + Cos[x]) + (2*Sin[x]^3)/(3*(1 + Cos[x])^3)","A",4,3,7,0.4286,1,"{4392, 2680, 8}"
302,1,24,0,0.0502716,"\int \frac{1}{(\cot (x)+\csc (x))^5} \, dx","Int[(Cot[x] + Csc[x])^(-5),x]","-\frac{4}{\cos (x)+1}+\frac{2}{(\cos (x)+1)^2}-\log (\cos (x)+1)","-\frac{4}{\cos (x)+1}+\frac{2}{(\cos (x)+1)^2}-\log (\cos (x)+1)",1,"2/(1 + Cos[x])^2 - 4/(1 + Cos[x]) - Log[1 + Cos[x]]","A",4,3,7,0.4286,1,"{4392, 2667, 43}"
303,1,44,0,0.0336686,"\int (\csc (x)-\sin (x))^4 \, dx","Int[(Csc[x] - Sin[x])^4,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)","\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 + (35*Cot[x])/8 - (35*Cot[x]^3)/24 + (7*Cos[x]^2*Cot[x]^3)/8 + (Cos[x]^4*Cot[x]^3)/4","A",6,3,9,0.3333,1,"{290, 325, 203}"
304,1,34,0,0.0458312,"\int (\csc (x)-\sin (x))^3 \, dx","Int[(Csc[x] - Sin[x])^3,x]","-\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))","-\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,"(5*ArcTanh[Cos[x]])/2 - (5*Cos[x])/2 - (5*Cos[x]^3)/6 - (Cos[x]^3*Cot[x]^2)/2","A",6,5,9,0.5556,1,"{4397, 2592, 288, 302, 206}"
305,1,22,0,0.0238772,"\int (\csc (x)-\sin (x))^2 \, dx","Int[(Csc[x] - Sin[x])^2,x]","-\frac{3 x}{2}-\frac{3 \cot (x)}{2}+\frac{1}{2} \cos ^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 - (3*Cot[x])/2 + (Cos[x]^2*Cot[x])/2","A",4,3,9,0.3333,1,"{290, 325, 203}"
306,1,8,0,0.0054189,"\int (\csc (x)-\sin (x)) \, dx","Int[Csc[x] - Sin[x],x]","\cos (x)-\tanh ^{-1}(\cos (x))","\cos (x)-\tanh ^{-1}(\cos (x))",1,"-ArcTanh[Cos[x]] + Cos[x]","A",3,2,7,0.2857,1,"{3770, 2638}"
307,1,2,0,0.0187821,"\int \frac{1}{\csc (x)-\sin (x)} \, dx","Int[(Csc[x] - Sin[x])^(-1),x]","\sec (x)","\sec (x)",1,"Sec[x]","A",3,3,9,0.3333,1,"{4397, 2606, 8}"
308,1,8,0,0.014444,"\int \frac{1}{(\csc (x)-\sin (x))^2} \, dx","Int[(Csc[x] - Sin[x])^(-2),x]","\frac{\tan ^3(x)}{3}","\frac{\tan ^3(x)}{3}",1,"Tan[x]^3/3","A",2,1,9,0.1111,1,"{30}"
309,1,17,0,0.0380005,"\int \frac{1}{(\csc (x)-\sin (x))^3} \, dx","Int[(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,"-Sec[x]^3/3 + Sec[x]^5/5","A",4,3,9,0.3333,1,"{4397, 2606, 14}"
310,1,17,0,0.0177155,"\int \frac{1}{(\csc (x)-\sin (x))^4} \, dx","Int[(Csc[x] - Sin[x])^(-4),x]","\frac{\tan ^7(x)}{7}+\frac{\tan ^5(x)}{5}","\frac{\tan ^7(x)}{7}+\frac{\tan ^5(x)}{5}",1,"Tan[x]^5/5 + Tan[x]^7/7","A",2,0,9,0,1,"{}"
311,1,25,0,0.0398882,"\int \frac{1}{(\csc (x)-\sin (x))^5} \, dx","Int[(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",4,3,9,0.3333,1,"{4397, 2606, 270}"
312,1,25,0,0.0219173,"\int \frac{1}{(\csc (x)-\sin (x))^6} \, dx","Int[(Csc[x] - Sin[x])^(-6),x]","\frac{\tan ^{11}(x)}{11}+\frac{2 \tan ^9(x)}{9}+\frac{\tan ^7(x)}{7}","\frac{\tan ^{11}(x)}{11}+\frac{2 \tan ^9(x)}{9}+\frac{\tan ^7(x)}{7}",1,"Tan[x]^7/7 + (2*Tan[x]^9)/9 + Tan[x]^11/11","A",3,1,9,0.1111,1,"{270}"
313,1,33,0,0.0426652,"\int \frac{1}{(\csc (x)-\sin (x))^7} \, dx","Int[(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,"-Sec[x]^7/7 + Sec[x]^9/3 - (3*Sec[x]^11)/11 + Sec[x]^13/13","A",4,3,9,0.3333,1,"{4397, 2606, 270}"
314,1,73,0,0.1483913,"\int (\csc (x)-\sin (x))^{7/2} \, dx","Int[(Csc[x] - Sin[x])^(7/2),x]","\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)}","\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,"(8*Cos[x]*Cot[x]^2*Sqrt[Cos[x]*Cot[x]])/7 + (2*Cos[x]^3*Cot[x]^2*Sqrt[Cos[x]*Cot[x]])/7 - (64*Cot[x]*Sqrt[Cos[x]*Cot[x]]*Csc[x])/35 + (256*Sqrt[Cos[x]*Cot[x]]*Sec[x])/35","A",6,5,11,0.4545,1,"{4397, 4400, 2598, 2594, 2589}"
315,1,50,0,0.1117005,"\int (\csc (x)-\sin (x))^{5/2} \, dx","Int[(Csc[x] - Sin[x])^(5/2),x]","\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)}","\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,"(-16*Cot[x]*Sqrt[Cos[x]*Cot[x]])/15 + (2*Cos[x]^2*Cot[x]*Sqrt[Cos[x]*Cot[x]])/5 - (64*Sqrt[Cos[x]*Cot[x]]*Tan[x])/15","A",5,5,11,0.4545,1,"{4397, 4400, 2598, 2594, 2589}"
316,1,31,0,0.0812269,"\int (\csc (x)-\sin (x))^{3/2} \, dx","Int[(Csc[x] - Sin[x])^(3/2),x]","\frac{2}{3} \cos (x) \sqrt{\cos (x) \cot (x)}-\frac{8}{3} \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*Cos[x]*Sqrt[Cos[x]*Cot[x]])/3 - (8*Sqrt[Cos[x]*Cot[x]]*Sec[x])/3","A",4,4,11,0.3636,1,"{4397, 4400, 2598, 2589}"
317,1,13,0,0.0486451,"\int \sqrt{\csc (x)-\sin (x)} \, dx","Int[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",3,3,11,0.2727,1,"{4397, 4400, 2589}"
318,1,60,0,0.0906745,"\int \frac{1}{\sqrt{\csc (x)-\sin (x)}} \, dx","Int[1/Sqrt[Csc[x] - Sin[x]],x]","\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)}}","\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[Sqrt[-Sin[x]]]*Cos[x])/(Sqrt[Cos[x]*Cot[x]]*Sqrt[-Sin[x]]) - (ArcTanh[Sqrt[-Sin[x]]]*Cos[x])/(Sqrt[Cos[x]*Cot[x]]*Sqrt[-Sin[x]])","A",8,8,11,0.7273,1,"{4397, 4400, 2601, 2564, 329, 298, 203, 206}"
319,1,80,0,0.1155203,"\int \frac{1}{(\csc (x)-\sin (x))^{3/2}} \, dx","Int[(Csc[x] - Sin[x])^(-3/2),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)}}","\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,"Sec[x]/(2*Sqrt[Cos[x]*Cot[x]]) + (ArcTan[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(4*Sqrt[Cos[x]*Cot[x]]) + (ArcTanh[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(4*Sqrt[Cos[x]*Cot[x]])","A",9,9,11,0.8182,1,"{4397, 4400, 2597, 2601, 2564, 329, 212, 206, 203}"
320,1,99,0,0.1512835,"\int \frac{1}{(\csc (x)-\sin (x))^{5/2}} \, dx","Int[(Csc[x] - Sin[x])^(-5/2),x]","-\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)}}","-\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,"(-3*ArcTan[Sqrt[-Sin[x]]]*Cos[x])/(32*Sqrt[Cos[x]*Cot[x]]*Sqrt[-Sin[x]]) + (3*ArcTanh[Sqrt[-Sin[x]]]*Cos[x])/(32*Sqrt[Cos[x]*Cot[x]]*Sqrt[-Sin[x]]) - (3*Tan[x])/(16*Sqrt[Cos[x]*Cot[x]]) + (Sec[x]^2*Tan[x])/(4*Sqrt[Cos[x]*Cot[x]])","A",10,10,11,0.9091,1,"{4397, 4400, 2597, 2599, 2601, 2564, 329, 298, 203, 206}"
321,1,118,0,0.1795685,"\int \frac{1}{(\csc (x)-\sin (x))^{7/2}} \, dx","Int[(Csc[x] - Sin[x])^(-7/2),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)}}","-\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,"(5*Sec[x])/(192*Sqrt[Cos[x]*Cot[x]]) - (5*Sec[x]^3)/(48*Sqrt[Cos[x]*Cot[x]]) - (5*ArcTan[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(128*Sqrt[Cos[x]*Cot[x]]) - (5*ArcTanh[Sqrt[-Sin[x]]]*Cot[x]*Sqrt[-Sin[x]])/(128*Sqrt[Cos[x]*Cot[x]]) + (Sec[x]^3*Tan[x]^2)/(6*Sqrt[Cos[x]*Cot[x]])","A",11,10,11,0.9091,1,"{4397, 4400, 2597, 2599, 2601, 2564, 329, 212, 206, 203}"
322,1,44,0,0.0307354,"\int (-\cos (x)+\sec (x))^4 \, dx","Int[(-Cos[x] + Sec[x])^4,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)","\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 - (35*Tan[x])/8 + (35*Tan[x]^3)/24 - (7*Sin[x]^2*Tan[x]^3)/8 - (Sin[x]^4*Tan[x]^3)/4","A",6,3,9,0.3333,1,"{288, 302, 203}"
323,1,34,0,0.0415915,"\int (-\cos (x)+\sec (x))^3 \, dx","Int[(-Cos[x] + Sec[x])^3,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))","\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*Sin[x])/2 + (5*Sin[x]^3)/6 + (Sin[x]^3*Tan[x]^2)/2","A",6,5,9,0.5556,1,"{4397, 2592, 288, 302, 206}"
324,1,22,0,0.0205319,"\int (-\cos (x)+\sec (x))^2 \, dx","Int[(-Cos[x] + Sec[x])^2,x]","-\frac{3 x}{2}+\frac{3 \tan (x)}{2}-\frac{1}{2} \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 + (3*Tan[x])/2 - (Sin[x]^2*Tan[x])/2","A",4,3,9,0.3333,1,"{288, 321, 203}"
325,1,8,0,0.004773,"\int (-\cos (x)+\sec (x)) \, dx","Int[-Cos[x] + Sec[x],x]","\tanh ^{-1}(\sin (x))-\sin (x)","\tanh ^{-1}(\sin (x))-\sin (x)",1,"ArcTanh[Sin[x]] - Sin[x]","A",3,2,7,0.2857,1,"{2637, 3770}"
326,1,4,0,0.0175251,"\int \frac{1}{-\cos (x)+\sec (x)} \, dx","Int[(-Cos[x] + Sec[x])^(-1),x]","-\csc (x)","-\csc (x)",1,"-Csc[x]","A",3,3,9,0.3333,1,"{4397, 2606, 8}"
327,1,8,0,0.0140351,"\int \frac{1}{(-\cos (x)+\sec (x))^2} \, dx","Int[(-Cos[x] + Sec[x])^(-2),x]","-\frac{1}{3} \cot ^3(x)","-\frac{1}{3} \cot ^3(x)",1,"-Cot[x]^3/3","A",2,1,9,0.1111,1,"{30}"
328,1,17,0,0.0379673,"\int \frac{1}{(-\cos (x)+\sec (x))^3} \, dx","Int[(-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",4,3,9,0.3333,1,"{4397, 2606, 14}"
329,1,17,0,0.0171948,"\int \frac{1}{(-\cos (x)+\sec (x))^4} \, dx","Int[(-Cos[x] + Sec[x])^(-4),x]","-\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5}","-\frac{1}{7} \cot ^7(x)-\frac{\cot ^5(x)}{5}",1,"-Cot[x]^5/5 - Cot[x]^7/7","A",2,0,9,0,1,"{}"
330,1,25,0,0.0410418,"\int \frac{1}{(-\cos (x)+\sec (x))^5} \, dx","Int[(-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,"-Csc[x]^5/5 + (2*Csc[x]^7)/7 - Csc[x]^9/9","A",4,3,9,0.3333,1,"{4397, 2606, 270}"
331,1,25,0,0.0205889,"\int \frac{1}{(-\cos (x)+\sec (x))^6} \, dx","Int[(-Cos[x] + Sec[x])^(-6),x]","-\frac{1}{11} \cot ^{11}(x)-\frac{2 \cot ^9(x)}{9}-\frac{\cot ^7(x)}{7}","-\frac{1}{11} \cot ^{11}(x)-\frac{2 \cot ^9(x)}{9}-\frac{\cot ^7(x)}{7}",1,"-Cot[x]^7/7 - (2*Cot[x]^9)/9 - Cot[x]^11/11","A",3,1,9,0.1111,1,"{270}"
332,1,33,0,0.0422955,"\int \frac{1}{(-\cos (x)+\sec (x))^7} \, dx","Int[(-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",4,3,9,0.3333,1,"{4397, 2606, 270}"
333,1,73,0,0.1125315,"\int (-\cos (x)+\sec (x))^{7/2} \, dx","Int[(-Cos[x] + Sec[x])^(7/2),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)}","-\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,"(-256*Csc[x]*Sqrt[Sin[x]*Tan[x]])/35 + (64*Sec[x]*Tan[x]*Sqrt[Sin[x]*Tan[x]])/35 - (8*Sin[x]*Tan[x]^2*Sqrt[Sin[x]*Tan[x]])/7 - (2*Sin[x]^3*Tan[x]^2*Sqrt[Sin[x]*Tan[x]])/7","A",6,5,11,0.4545,1,"{4397, 4400, 2598, 2594, 2589}"
334,1,50,0,0.085911,"\int (-\cos (x)+\sec (x))^{5/2} \, dx","Int[(-Cos[x] + Sec[x])^(5/2),x]","-\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)}","-\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,"(64*Cot[x]*Sqrt[Sin[x]*Tan[x]])/15 + (16*Tan[x]*Sqrt[Sin[x]*Tan[x]])/15 - (2*Sin[x]^2*Tan[x]*Sqrt[Sin[x]*Tan[x]])/5","A",5,5,11,0.4545,1,"{4397, 4400, 2598, 2594, 2589}"
335,1,31,0,0.0639886,"\int (-\cos (x)+\sec (x))^{3/2} \, dx","Int[(-Cos[x] + Sec[x])^(3/2),x]","\frac{8}{3} \csc (x) \sqrt{\sin (x) \tan (x)}-\frac{2}{3} \sin (x) \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,"(8*Csc[x]*Sqrt[Sin[x]*Tan[x]])/3 - (2*Sin[x]*Sqrt[Sin[x]*Tan[x]])/3","A",4,4,11,0.3636,1,"{4397, 4400, 2598, 2589}"
336,1,13,0,0.0410275,"\int \sqrt{-\cos (x)+\sec (x)} \, dx","Int[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",3,3,11,0.2727,1,"{4397, 4400, 2589}"
337,1,52,0,0.0780825,"\int \frac{1}{\sqrt{-\cos (x)+\sec (x)}} \, dx","Int[1/Sqrt[-Cos[x] + Sec[x]],x]","\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)}}","\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[Sqrt[Cos[x]]]*Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) - (ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]])","A",8,8,11,0.7273,1,"{4397, 4400, 2601, 2565, 329, 298, 203, 206}"
338,1,72,0,0.093649,"\int \frac{1}{(-\cos (x)+\sec (x))^{3/2}} \, dx","Int[(-Cos[x] + Sec[x])^(-3/2),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)}}","\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,"-Csc[x]/(2*Sqrt[Sin[x]*Tan[x]]) + (ArcTan[Sqrt[Cos[x]]]*Sin[x])/(4*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) + (ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(4*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]])","A",9,9,11,0.8182,1,"{4397, 4400, 2597, 2601, 2565, 329, 212, 206, 203}"
339,1,91,0,0.1208491,"\int \frac{1}{(-\cos (x)+\sec (x))^{5/2}} \, dx","Int[(-Cos[x] + Sec[x])^(-5/2),x]","-\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)}}","-\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,"(3*Cot[x])/(16*Sqrt[Sin[x]*Tan[x]]) - (Cot[x]*Csc[x]^2)/(4*Sqrt[Sin[x]*Tan[x]]) - (3*ArcTan[Sqrt[Cos[x]]]*Sin[x])/(32*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) + (3*ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(32*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]])","A",10,10,11,0.9091,1,"{4397, 4400, 2597, 2599, 2601, 2565, 329, 298, 203, 206}"
340,1,110,0,0.1403668,"\int \frac{1}{(-\cos (x)+\sec (x))^{7/2}} \, dx","Int[(-Cos[x] + Sec[x])^(-7/2),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)}}","-\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,"(-5*Csc[x])/(192*Sqrt[Sin[x]*Tan[x]]) + (5*Csc[x]^3)/(48*Sqrt[Sin[x]*Tan[x]]) - (Cot[x]^2*Csc[x]^3)/(6*Sqrt[Sin[x]*Tan[x]]) - (5*ArcTan[Sqrt[Cos[x]]]*Sin[x])/(128*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]]) - (5*ArcTanh[Sqrt[Cos[x]]]*Sin[x])/(128*Sqrt[Cos[x]]*Sqrt[Sin[x]*Tan[x]])","A",11,10,11,0.9091,1,"{4397, 4400, 2597, 2599, 2601, 2565, 329, 212, 206, 203}"
341,1,55,0,0.1096142,"\int (\sin (x)+\tan (x))^4 \, dx","Int[(Sin[x] + Tan[x])^4,x]","-\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)","-\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,"(-61*x)/8 - 2*ArcTanh[Sin[x]] + (19*Cos[x]*Sin[x])/8 + (Cos[x]^3*Sin[x])/4 - (4*Sin[x]^3)/3 + 5*Tan[x] + 2*Sec[x]*Tan[x] + Tan[x]^3/3","A",18,9,7,1.286,1,"{4397, 2709, 2637, 2635, 8, 2633, 3770, 3767, 3768}"
342,1,38,0,0.049694,"\int (\sin (x)+\tan (x))^3 \, dx","Int[(Sin[x] + Tan[x])^3,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))","\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,"2*Cos[x] + (3*Cos[x]^2)/2 + Cos[x]^3/3 - 2*Log[Cos[x]] + 3*Sec[x] + Sec[x]^2/2","A",4,3,7,0.4286,1,"{4397, 2707, 75}"
343,1,25,0,0.0625847,"\int (\sin (x)+\tan (x))^2 \, dx","Int[(Sin[x] + Tan[x])^2,x]","-\frac{x}{2}-2 \sin (x)+\tan (x)+2 \tanh ^{-1}(\sin (x))-\frac{1}{2} \sin (x) \cos (x)","-\frac{x}{2}-2 \sin (x)+\tan (x)+2 \tanh ^{-1}(\sin (x))-\frac{1}{2} \sin (x) \cos (x)",1,"-x/2 + 2*ArcTanh[Sin[x]] - 2*Sin[x] - (Cos[x]*Sin[x])/2 + Tan[x]","A",9,7,7,1.000,1,"{4397, 2709, 2637, 2635, 8, 3770, 3767}"
344,1,10,0,0.0049923,"\int (\sin (x)+\tan (x)) \, dx","Int[Sin[x] + Tan[x],x]","-\cos (x)-\log (\cos (x))","-\cos (x)-\log (\cos (x))",1,"-Cos[x] - Log[Cos[x]]","A",3,2,5,0.4000,1,"{2638, 3475}"
345,1,24,0,0.0583856,"\int \frac{1}{\sin (x)+\tan (x)} \, dx","Int[(Sin[x] + Tan[x])^(-1),x]","-\frac{1}{2} \csc ^2(x)-\frac{1}{2} \tanh ^{-1}(\cos (x))+\frac{1}{2} \cot (x) \csc (x)","-\frac{1}{2} \csc ^2(x)-\frac{1}{2} \tanh ^{-1}(\cos (x))+\frac{1}{2} \cot (x) \csc (x)",1,"-ArcTanh[Cos[x]]/2 + (Cot[x]*Csc[x])/2 - Csc[x]^2/2","A",6,6,7,0.8571,1,"{4397, 2706, 2606, 30, 2611, 3770}"
346,1,33,0,0.1266044,"\int \frac{1}{(\sin (x)+\tan (x))^2} \, dx","Int[(Sin[x] + Tan[x])^(-2),x]","-\frac{2}{5} \cot ^5(x)-\frac{\cot ^3(x)}{3}+\frac{2 \csc ^5(x)}{5}-\frac{2 \csc ^3(x)}{3}","-\frac{2}{5} \cot ^5(x)-\frac{\cot ^3(x)}{3}+\frac{2 \csc ^5(x)}{5}-\frac{2 \csc ^3(x)}{3}",1,"-Cot[x]^3/3 - (2*Cot[x]^5)/5 - (2*Csc[x]^3)/3 + (2*Csc[x]^5)/5","A",11,6,7,0.8571,1,"{4397, 2711, 2607, 30, 2606, 14}"
347,1,60,0,0.0716931,"\int \frac{1}{(\sin (x)+\tan (x))^3} \, dx","Int[(Sin[x] + Tan[x])^(-3),x]","-\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))","-\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,"ArcTanh[Cos[x]]/32 - 1/(32*(1 - Cos[x])) - 1/(16*(1 + Cos[x])^4) + 1/(6*(1 + Cos[x])^3) - 3/(32*(1 + Cos[x])^2) - 1/(16*(1 + Cos[x]))","A",5,4,7,0.5714,1,"{4397, 2707, 88, 207}"
348,1,65,0,0.2106768,"\int \frac{1}{(\sin (x)+\tan (x))^4} \, dx","Int[(Sin[x] + Tan[x])^(-4),x]","-\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}","-\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]^5/5 - (9*Cot[x]^7)/7 - (16*Cot[x]^9)/9 - (8*Cot[x]^11)/11 - (4*Csc[x]^5)/5 + (16*Csc[x]^7)/7 - (20*Csc[x]^9)/9 + (8*Csc[x]^11)/11","A",18,6,7,0.8571,1,"{4397, 2711, 2607, 14, 2606, 270}"
349,1,74,0,0.0627072,"\int \frac{A+C \sin (x)}{b \cos (x)+c \sin (x)} \, dx","Int[(A + C*Sin[x])/(b*Cos[x] + c*Sin[x]),x]","-\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}","-\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,"(c*C*x)/(b^2 + c^2) - (A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/Sqrt[b^2 + c^2] - (b*C*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",3,3,18,0.1667,1,"{3137, 3074, 206}"
350,1,75,0,0.0601355,"\int \frac{A+C \sin (x)}{(b \cos (x)+c \sin (x))^2} \, dx","Int[(A + C*Sin[x])/(b*Cos[x] + c*Sin[x])^2,x]","\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}}","\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,"-((c*C*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2)) + (b*C - A*c*Cos[x] + A*b*Sin[x])/((b^2 + c^2)*(b*Cos[x] + c*Sin[x]))","A",3,3,18,0.1667,1,"{3154, 3074, 206}"
351,1,116,0,0.1103965,"\int \frac{A+C \sin (x)}{(b \cos (x)+c \sin (x))^3} \, dx","Int[(A + C*Sin[x])/(b*Cos[x] + c*Sin[x])^3,x]","\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))}","\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,"-(A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(2*(b^2 + c^2)^(3/2)) + (b*C - A*c*Cos[x] + A*b*Sin[x])/(2*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2) - (c^2*C*Cos[x] - b*c*C*Sin[x])/((b^2 + c^2)^2*(b*Cos[x] + c*Sin[x]))","A",4,4,18,0.2222,1,"{3157, 3153, 3074, 206}"
352,1,73,0,0.0528934,"\int \frac{A+B \cos (x)}{b \cos (x)+c \sin (x)} \, dx","Int[(A + B*Cos[x])/(b*Cos[x] + c*Sin[x]),x]","-\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}","-\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,"(b*B*x)/(b^2 + c^2) - (A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/Sqrt[b^2 + c^2] + (B*c*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",3,3,18,0.1667,1,"{3138, 3074, 206}"
353,1,76,0,0.0533076,"\int \frac{A+B \cos (x)}{(b \cos (x)+c \sin (x))^2} \, dx","Int[(A + B*Cos[x])/(b*Cos[x] + c*Sin[x])^2,x]","-\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}}","-\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,"-((b*B*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2)) - (B*c + A*c*Cos[x] - A*b*Sin[x])/((b^2 + c^2)*(b*Cos[x] + c*Sin[x]))","A",3,3,18,0.1667,1,"{3155, 3074, 206}"
354,1,116,0,0.1080698,"\int \frac{A+B \cos (x)}{(b \cos (x)+c \sin (x))^3} \, dx","Int[(A + B*Cos[x])/(b*Cos[x] + c*Sin[x])^3,x]","-\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))}","-\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,"-(A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(2*(b^2 + c^2)^(3/2)) - (B*c + A*c*Cos[x] - A*b*Sin[x])/(2*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2) - (b*B*c*Cos[x] - b^2*B*Sin[x])/((b^2 + c^2)^2*(b*Cos[x] + c*Sin[x]))","A",4,4,18,0.2222,1,"{3158, 3153, 3074, 206}"
355,1,246,0,0.1693143,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^4 \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4,x]","\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","\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,"(35*(b^2 + c^2)^2*x)/8 - (35*c*(b^2 + c^2)^(3/2)*Cos[d + e*x])/(8*e) + (35*b*(b^2 + c^2)^(3/2)*Sin[d + e*x])/(8*e) - (35*(b^2 + c^2)*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]))/(24*e) - (7*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2)/(12*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3)/(4*e)","A",6,3,30,0.1000,1,"{3113, 2637, 2638}"
356,1,178,0,0.1018319,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^3 \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3,x]","\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}","\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,"(5*(b^2 + c^2)^(3/2)*x)/2 - (5*c*(b^2 + c^2)*Cos[d + e*x])/(2*e) + (5*b*(b^2 + c^2)*Sin[d + e*x])/(2*e) - (5*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]))/(6*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*e)","A",5,3,30,0.1000,1,"{3113, 2637, 2638}"
357,1,116,0,0.0584102,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^2 \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2,x]","\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)","\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,"(3*(b^2 + c^2)*x)/2 - (3*c*Sqrt[b^2 + c^2]*Cos[d + e*x])/(2*e) + (3*b*Sqrt[b^2 + c^2]*Sin[d + e*x])/(2*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]))/(2*e)","A",4,3,30,0.1000,1,"{3113, 2637, 2638}"
358,1,37,0,0.0153018,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right) \, dx","Int[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x],x]","x \sqrt{b^2+c^2}+\frac{b \sin (d+e x)}{e}-\frac{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]*x - (c*Cos[d + e*x])/e + (b*Sin[d + e*x])/e","A",3,2,28,0.07143,1,"{2637, 2638}"
359,1,49,0,0.036152,"\int \frac{1}{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-1),x]","-\frac{c-\sqrt{b^2+c^2} \sin (d+e x)}{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",1,1,30,0.03333,1,"{3114}"
360,1,129,0,0.0852958,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^2} \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-2),x]","-\frac{c \cos (d+e x)-b \sin (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))}","-\frac{c \cos (d+e x)-b \sin (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,"-(c*Cos[d + e*x] - b*Sin[d + e*x])/(3*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (c - Sqrt[b^2 + c^2]*Sin[d + e*x])/(3*c*Sqrt[b^2 + c^2]*e*(c*Cos[d + e*x] - b*Sin[d + e*x]))","A",2,2,30,0.06667,1,"{3116, 3114}"
361,1,191,0,0.1326849,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^3} \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3),x]","-\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{c \cos (d+e x)-b \sin (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))}","-\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{c \cos (d+e x)-b \sin (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,"-(c*Cos[d + e*x] - b*Sin[d + e*x])/(5*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(15*(b^2 + c^2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (2*(c - Sqrt[b^2 + c^2]*Sin[d + e*x]))/(15*c*(b^2 + c^2)*e*(c*Cos[d + e*x] - b*Sin[d + e*x]))","A",3,2,30,0.06667,1,"{3116, 3114}"
362,1,259,0,0.1892689,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^4} \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]","-\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{c \cos (d+e x)-b \sin (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))}","-\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{c \cos (d+e x)-b \sin (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,"-(c*Cos[d + e*x] - b*Sin[d + e*x])/(7*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^4) - (3*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^3) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(35*(b^2 + c^2)^(3/2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^2) - (2*(c - Sqrt[b^2 + c^2]*Sin[d + e*x]))/(35*c*(b^2 + c^2)^(3/2)*e*(c*Cos[d + e*x] - b*Sin[d + e*x]))","A",4,2,30,0.06667,1,"{3116, 3114}"
363,1,157,0,0.1428044,"\int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^3,x]","\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}","\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,"4*a*(5*a^2 + 3*c^2)*x - (4*c*(15*a^2 + 4*c^2)*Cos[d + e*x])/(3*e) + (4*a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(3*e) - (20*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x])*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))/(3*e) - (8*(c*Cos[d + e*x] - a*Sin[d + e*x])*(a + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*e)","A",5,4,24,0.1667,1,"{3120, 3146, 2637, 2638}"
364,1,81,0,0.0495218,"\int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2,x]","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}","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,"2*(3*a^2 + c^2)*x - (6*a*c*Cos[d + e*x])/e + (6*a^2*Sin[d + e*x])/e - (2*(c*Cos[d + e*x] - a*Sin[d + e*x])*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))/e","A",4,3,24,0.1250,1,"{3120, 2637, 2638}"
365,1,29,0,0.015509,"\int (2 a+2 a \cos (d+e x)+2 c \sin (d+e x)) \, dx","Int[2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x],x]","\frac{2 a \sin (d+e x)}{e}+2 a x-\frac{2 c \cos (d+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 + e*x])/e + (2*a*Sin[d + e*x])/e","A",3,2,22,0.09091,1,"{2637, 2638}"
366,1,25,0,0.0223817,"\int \frac{1}{2 a+2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-1),x]","\frac{\log \left(a+c \tan \left(\frac{1}{2} (d+e x)\right)\right)}{2 c e}","\frac{\log \left(a+c \tan \left(\frac{1}{2} (d+e x)\right)\right)}{2 c e}",1,"Log[a + c*Tan[(d + e*x)/2]]/(2*c*e)","A",2,2,24,0.08333,1,"{3124, 31}"
367,1,75,0,0.0487294,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-2),x]","-\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))}","-\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,"-(a*Log[a + c*Tan[(d + e*x)/2]])/(4*c^3*e) - (c*Cos[d + e*x] - a*Sin[d + e*x])/(4*c^2*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 12, 3124, 31}"
368,1,134,0,0.1116933,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-3),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{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{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}","\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{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{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,"((3*a^2 + c^2)*Log[a + c*Tan[(d + e*x)/2]])/(16*c^5*e) - (c*Cos[d + e*x] - a*Sin[d + e*x])/(16*c^2*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x])^2) + (3*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x]))/(16*c^4*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 3153, 3124, 31}"
369,1,207,0,0.248415,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]","-\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{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{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}","-\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{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{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,"-(a*(5*a^2 + 3*c^2)*Log[a + c*Tan[(d + e*x)/2]])/(32*c^7*e) - (c*Cos[d + e*x] - a*Sin[d + e*x])/(48*c^2*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] - a^2*Sin[d + e*x]))/(96*c^4*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x])^2) - (c*(15*a^2 + 4*c^2)*Cos[d + e*x] - a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(96*c^6*e*(a + a*Cos[d + e*x] + c*Sin[d + e*x]))","A",5,5,24,0.2083,1,"{3129, 3156, 3153, 3124, 31}"
370,1,23,0,0.0213659,"\int \frac{1}{2 a+2 a \cos (d+e x)+2 a \sin (d+e x)} \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-1),x]","\frac{\log \left(\tan \left(\frac{1}{2} (d+e x)\right)+1\right)}{2 a e}","\frac{\log \left(\tan \left(\frac{1}{2} (d+e x)\right)+1\right)}{2 a e}",1,"Log[1 + Tan[(d + e*x)/2]]/(2*a*e)","A",2,2,24,0.08333,1,"{3124, 31}"
371,1,75,0,0.0479758,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^2} \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-2),x]","-\frac{\log \left(\tan \left(\frac{1}{2} (d+e x)\right)+1\right)}{4 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}-\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)}",1,"-Log[1 + Tan[(d + e*x)/2]]/(4*a^2*e) - (a*Cos[d + e*x] - a*Sin[d + e*x])/(4*e*(a^3 + a^3*Cos[d + e*x] + a^3*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 12, 3124, 31}"
372,1,123,0,0.1073176,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^3} \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-3),x]","\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}","\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,"Log[1 + Tan[(d + e*x)/2]]/(4*a^3*e) - (a*Cos[d + e*x] - a*Sin[d + e*x])/(16*e*(a^2 + a^2*Cos[d + e*x] + a^2*Sin[d + e*x])^2) + (3*(Cos[d + e*x] - Sin[d + e*x]))/(16*e*(a^3 + a^3*Cos[d + e*x] + a^3*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 3153, 3124, 31}"
373,1,168,0,0.1864934,"\int \frac{1}{(2 a+2 a \cos (d+e x)+2 a \sin (d+e x))^4} \, dx","Int[(2*a + 2*a*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-4),x]","-\frac{\log \left(\tan \left(\frac{1}{2} (d+e x)\right)+1\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{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}","-\frac{\log \left(\tan \left(\frac{1}{2} (d+e x)\right)+1\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{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[1 + Tan[(d + e*x)/2]]/(4*a^4*e) - (Cos[d + e*x] - Sin[d + e*x])/(48*a*e*(a + a*Cos[d + e*x] + a*Sin[d + e*x])^3) + (5*(Cos[d + e*x] - Sin[d + e*x]))/(96*e*(a^2 + a^2*Cos[d + e*x] + a^2*Sin[d + e*x])^2) - (19*(a*Cos[d + e*x] - a*Sin[d + e*x]))/(96*e*(a^5 + a^5*Cos[d + e*x] + a^5*Sin[d + e*x]))","A",5,5,24,0.2083,1,"{3129, 3156, 3153, 3124, 31}"
374,1,157,0,0.1338248,"\int (2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^3 \, dx","Int[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^3,x]","-\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}","-\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,"4*a*(5*a^2 + 3*c^2)*x - (4*c*(15*a^2 + 4*c^2)*Cos[d + e*x])/(3*e) - (4*a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(3*e) - (20*(a*c*Cos[d + e*x] + a^2*Sin[d + e*x])*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))/(3*e) - (8*(c*Cos[d + e*x] + a*Sin[d + e*x])*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*e)","A",5,4,24,0.1667,1,"{3120, 3146, 2637, 2638}"
375,1,81,0,0.0469545,"\int (2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2 \, dx","Int[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^2,x]","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}","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,"2*(3*a^2 + c^2)*x - (6*a*c*Cos[d + e*x])/e - (6*a^2*Sin[d + e*x])/e - (2*(c*Cos[d + e*x] + a*Sin[d + e*x])*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))/e","A",4,3,24,0.1250,1,"{3120, 2637, 2638}"
376,1,29,0,0.014309,"\int (2 a-2 a \cos (d+e x)+2 c \sin (d+e x)) \, dx","Int[2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x],x]","-\frac{2 a \sin (d+e x)}{e}+2 a x-\frac{2 c \cos (d+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 + e*x])/e - (2*a*Sin[d + e*x])/e","A",3,2,22,0.09091,1,"{2637, 2638}"
377,1,25,0,0.0210431,"\int \frac{1}{2 a-2 a \cos (d+e x)+2 c \sin (d+e x)} \, dx","Int[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-1),x]","-\frac{\log \left(a+c \cot \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[a + c*Cot[(d + e*x)/2]]/(2*c*e)","A",2,2,24,0.08333,1,"{3121, 31}"
378,1,75,0,0.0531722,"\int \frac{1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^2} \, dx","Int[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-2),x]","\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))}","\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,"(a*Log[a + c*Cot[(d + e*x)/2]])/(4*c^3*e) - (c*Cos[d + e*x] + a*Sin[d + e*x])/(4*c^2*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 12, 3121, 31}"
379,1,134,0,0.1126311,"\int \frac{1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^3} \, dx","Int[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-3),x]","\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}","\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,"-((3*a^2 + c^2)*Log[a + c*Cot[(d + e*x)/2]])/(16*c^5*e) - (c*Cos[d + e*x] + a*Sin[d + e*x])/(16*c^2*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^2) + (3*(a*c*Cos[d + e*x] + a^2*Sin[d + e*x]))/(16*c^4*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 3153, 3121, 31}"
380,1,207,0,0.24045,"\int \frac{1}{(2 a-2 a \cos (d+e x)+2 c \sin (d+e x))^4} \, dx","Int[(2*a - 2*a*Cos[d + e*x] + 2*c*Sin[d + e*x])^(-4),x]","-\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{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 \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}","-\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{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 \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,"(a*(5*a^2 + 3*c^2)*Log[a + c*Cot[(d + e*x)/2]])/(32*c^7*e) - (c*Cos[d + e*x] + a*Sin[d + e*x])/(48*c^2*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] + a^2*Sin[d + e*x]))/(96*c^4*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x])^2) - (c*(15*a^2 + 4*c^2)*Cos[d + e*x] + a*(15*a^2 + 4*c^2)*Sin[d + e*x])/(96*c^6*e*(a - a*Cos[d + e*x] + c*Sin[d + e*x]))","A",5,5,24,0.2083,1,"{3129, 3156, 3153, 3121, 31}"
381,1,157,0,0.1400003,"\int (2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3 \, dx","Int[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^3,x]","\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}","\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,"4*a*(5*a^2 + 3*b^2)*x - (4*a*(15*a^2 + 4*b^2)*Cos[d + e*x])/(3*e) + (4*b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(3*e) - (8*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^2*(a*Cos[d + e*x] - b*Sin[d + e*x]))/(3*e) - (20*(a + b*Cos[d + e*x] + a*Sin[d + e*x])*(a^2*Cos[d + e*x] - a*b*Sin[d + e*x]))/(3*e)","A",5,4,24,0.1667,1,"{3120, 3146, 2637, 2638}"
382,1,81,0,0.0466635,"\int (2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^2 \, dx","Int[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^2,x]","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}","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,"2*(3*a^2 + b^2)*x - (6*a^2*Cos[d + e*x])/e + (6*a*b*Sin[d + e*x])/e - (2*(a + b*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - b*Sin[d + e*x]))/e","A",4,3,24,0.1250,1,"{3120, 2637, 2638}"
383,1,29,0,0.0162816,"\int (2 a+2 b \cos (d+e x)+2 a \sin (d+e x)) \, dx","Int[2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x],x]","-\frac{2 a \cos (d+e x)}{e}+2 a x+\frac{2 b \sin (d+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 + e*x])/e + (2*b*Sin[d + e*x])/e","A",3,2,22,0.09091,1,"{2637, 2638}"
384,1,33,0,0.0217124,"\int \frac{1}{2 a+2 b \cos (d+e x)+2 a \sin (d+e x)} \, dx","Int[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-1),x]","-\frac{\log \left(a+b \cot \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{2 b e}","-\frac{\log \left(a+b \cot \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\right)\right)}{2 b e}",1,"-Log[a + b*Cot[d/2 + Pi/4 + (e*x)/2]]/(2*b*e)","A",2,2,24,0.08333,1,"{3123, 31}"
385,1,83,0,0.0495658,"\int \frac{1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^2} \, dx","Int[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-2),x]","\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))}","\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[a + b*Cot[d/2 + Pi/4 + (e*x)/2]])/(4*b^3*e) - (a*Cos[d + e*x] - b*Sin[d + e*x])/(4*b^2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 12, 3123, 31}"
386,1,142,0,0.112195,"\int \frac{1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^3} \, dx","Int[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-3),x]","\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}","\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,"-((3*a^2 + b^2)*Log[a + b*Cot[d/2 + Pi/4 + (e*x)/2]])/(16*b^5*e) - (a*Cos[d + e*x] - b*Sin[d + e*x])/(16*b^2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^2) + (3*(a^2*Cos[d + e*x] - a*b*Sin[d + e*x]))/(16*b^4*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 3153, 3123, 31}"
387,1,215,0,0.2440989,"\int \frac{1}{(2 a+2 b \cos (d+e x)+2 a \sin (d+e x))^4} \, dx","Int[(2*a + 2*b*Cos[d + e*x] + 2*a*Sin[d + e*x])^(-4),x]","\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(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 \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 \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}","\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(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 \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 \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,"(a*(5*a^2 + 3*b^2)*Log[a + b*Cot[d/2 + Pi/4 + (e*x)/2]])/(32*b^7*e) - (a*Cos[d + e*x] - b*Sin[d + e*x])/(48*b^2*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^3) + (5*(a^2*Cos[d + e*x] - a*b*Sin[d + e*x]))/(96*b^4*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x])^2) - (a*(15*a^2 + 4*b^2)*Cos[d + e*x] - b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(96*b^6*e*(a + b*Cos[d + e*x] + a*Sin[d + e*x]))","A",5,5,24,0.2083,1,"{3129, 3156, 3153, 3123, 31}"
388,1,157,0,0.1357512,"\int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3 \, dx","Int[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^3,x]","\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}","\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,"4*a*(5*a^2 + 3*b^2)*x + (4*a*(15*a^2 + 4*b^2)*Cos[d + e*x])/(3*e) + (4*b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(3*e) + (8*(a + b*Cos[d + e*x] - a*Sin[d + e*x])^2*(a*Cos[d + e*x] + b*Sin[d + e*x]))/(3*e) + (20*(a + b*Cos[d + e*x] - a*Sin[d + e*x])*(a^2*Cos[d + e*x] + a*b*Sin[d + e*x]))/(3*e)","A",5,4,24,0.1667,1,"{3120, 3146, 2637, 2638}"
389,1,81,0,0.0469208,"\int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^2 \, dx","Int[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^2,x]","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}","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,"2*(3*a^2 + b^2)*x + (6*a^2*Cos[d + e*x])/e + (6*a*b*Sin[d + e*x])/e + (2*(a + b*Cos[d + e*x] - a*Sin[d + e*x])*(a*Cos[d + e*x] + b*Sin[d + e*x]))/e","A",4,3,24,0.1250,1,"{3120, 2637, 2638}"
390,1,29,0,0.0144019,"\int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx","Int[2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x],x]","\frac{2 a \cos (d+e x)}{e}+2 a x+\frac{2 b \sin (d+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 + e*x])/e + (2*b*Sin[d + e*x])/e","A",3,2,22,0.09091,1,"{2637, 2638}"
391,1,33,0,0.0217288,"\int \frac{1}{2 a+2 b \cos (d+e x)-2 a \sin (d+e x)} \, dx","Int[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^(-1),x]","\frac{\log \left(a+b \tan \left(\frac{d}{2}+\frac{e x}{2}+\frac{\pi }{4}\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,"Log[a + b*Tan[d/2 + Pi/4 + (e*x)/2]]/(2*b*e)","A",2,2,24,0.08333,1,"{3122, 31}"
392,1,83,0,0.0529481,"\int \frac{1}{(2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^2} \, dx","Int[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^(-2),x]","\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}","\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[a + b*Tan[d/2 + Pi/4 + (e*x)/2]])/(4*b^3*e) + (a*Cos[d + e*x] + b*Sin[d + e*x])/(4*b^2*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 12, 3122, 31}"
393,1,142,0,0.1076581,"\int \frac{1}{(2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^3} \, dx","Int[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^(-3),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{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{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}","\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{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{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,"((3*a^2 + b^2)*Log[a + b*Tan[d/2 + Pi/4 + (e*x)/2]])/(16*b^5*e) + (a*Cos[d + e*x] + b*Sin[d + e*x])/(16*b^2*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x])^2) - (3*(a^2*Cos[d + e*x] + a*b*Sin[d + e*x]))/(16*b^4*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x]))","A",4,4,24,0.1667,1,"{3129, 3153, 3122, 31}"
394,1,215,0,0.2351646,"\int \frac{1}{(2 a+2 b \cos (d+e x)-2 a \sin (d+e x))^4} \, dx","Int[(2*a + 2*b*Cos[d + e*x] - 2*a*Sin[d + e*x])^(-4),x]","-\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{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{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}","-\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{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{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,"-(a*(5*a^2 + 3*b^2)*Log[a + b*Tan[d/2 + Pi/4 + (e*x)/2]])/(32*b^7*e) + (a*Cos[d + e*x] + b*Sin[d + e*x])/(48*b^2*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x])^3) - (5*(a^2*Cos[d + e*x] + a*b*Sin[d + e*x]))/(96*b^4*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x])^2) + (a*(15*a^2 + 4*b^2)*Cos[d + e*x] + b*(15*a^2 + 4*b^2)*Sin[d + e*x])/(96*b^6*e*(a + b*Cos[d + e*x] - a*Sin[d + e*x]))","A",5,5,24,0.2083,1,"{3129, 3156, 3153, 3122, 31}"
395,1,260,0,0.3996865,"\int (a+b \cos (d+e x)+c \sin (d+e x))^4 \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^4,x]","\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(24 a^2 \left(b^2+c^2\right)+8 a^4+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}","\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(24 a^2 \left(b^2+c^2\right)+8 a^4+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,"((8*a^4 + 24*a^2*(b^2 + c^2) + 3*(b^2 + c^2)^2)*x)/8 - (5*a*c*(10*a^2 + 11*(b^2 + c^2))*Cos[d + e*x])/(24*e) + (5*a*b*(10*a^2 + 11*(b^2 + c^2))*Sin[d + e*x])/(24*e) - (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) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3)/(4*e) - ((a + b*Cos[d + e*x] + c*Sin[d + e*x])*(c*(26*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - b*(26*a^2 + 9*(b^2 + c^2))*Sin[d + e*x]))/(24*e)","A",6,4,20,0.2000,1,"{3120, 3146, 2637, 2638}"
396,1,170,0,0.1862885,"\int (a+b \cos (d+e x)+c \sin (d+e x))^3 \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3,x]","\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}","\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,"(a*(2*a^2 + 3*(b^2 + c^2))*x)/2 - (c*(11*a^2 + 4*(b^2 + c^2))*Cos[d + e*x])/(6*e) + (b*(11*a^2 + 4*(b^2 + c^2))*Sin[d + e*x])/(6*e) - (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) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*e)","A",5,4,20,0.2000,1,"{3120, 3146, 2637, 2638}"
397,1,91,0,0.0463345,"\int (a+b \cos (d+e x)+c \sin (d+e x))^2 \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2,x]","\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}","\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*a^2 + b^2 + c^2)*x)/2 - (3*a*c*Cos[d + e*x])/(2*e) + (3*a*b*Sin[d + e*x])/(2*e) - ((c*Cos[d + e*x] - b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))/(2*e)","A",4,3,20,0.1500,1,"{3120, 2637, 2638}"
398,1,27,0,0.0157105,"\int (a+b \cos (d+e x)+c \sin (d+e x)) \, dx","Int[a + b*Cos[d + e*x] + c*Sin[d + e*x],x]","a x+\frac{b \sin (d+e x)}{e}-\frac{c \cos (d+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 + e*x])/e + (b*Sin[d + e*x])/e","A",3,2,18,0.1111,1,"{2637, 2638}"
399,1,61,0,0.0837496,"\int \frac{1}{a+b \cos (d+e x)+c \sin (d+e x)} \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-1),x]","\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}}","\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*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*e)","A",3,3,20,0.1500,1,"{3124, 618, 204}"
400,1,121,0,0.1079529,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^2} \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-2),x]","\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))}","\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*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(3/2)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/((a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))","A",5,5,20,0.2500,1,"{3129, 12, 3124, 618, 204}"
401,1,197,0,0.1979373,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^3} \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3),x]","\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}","\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*a^2 + b^2 + c^2)*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(5/2)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(2*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (3*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(2*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))","A",5,5,20,0.2500,1,"{3129, 3153, 3124, 618, 204}"
402,1,292,0,0.3761033,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^4} \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-4),x]","\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}","\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,"(a*(2*a^2 + 3*(b^2 + c^2))*ArcTan[(c + (a - b)*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2 - c^2)^(7/2)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(3*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^3) + (5*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(6*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^2) + (c*(11*a^2 + 4*(b^2 + c^2))*Cos[d + e*x] - b*(11*a^2 + 4*(b^2 + c^2))*Sin[d + e*x])/(6*(a^2 - b^2 - c^2)^3*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x]))","A",6,6,20,0.3000,1,"{3129, 3156, 3153, 3124, 618, 204}"
403,1,185,0,0.2679044,"\int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx","Int[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2),x]","-\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}","-\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,"(796*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(15*e) + (64*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e) - (32*(5*Cos[d + e*x] - 3*Sin[d + e*x])*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])/(15*e) - (2*(5*Cos[d + e*x] - 3*Sin[d + e*x])*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2))/(5*e)","A",7,7,22,0.3182,1,"{3120, 3146, 3149, 3118, 2653, 3126, 2661}"
404,1,139,0,0.1386601,"\int (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2} \, dx","Int[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2),x]","-\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}","-\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,"(16*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(3*e) + (20*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e) - (2*(5*Cos[d + e*x] - 3*Sin[d + e*x])*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])/(3*e)","A",6,6,22,0.2727,1,"{3120, 3149, 3118, 2653, 3126, 2661}"
405,1,45,0,0.0305556,"\int \sqrt{2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx","Int[Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]],x]","\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}","\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,"(2*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/e","A",2,2,22,0.09091,1,"{3118, 2653}"
406,1,45,0,0.0372017,"\int \frac{1}{\sqrt{2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx","Int[1/Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]],x]","\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}","\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,"(2*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(Sqrt[2 + Sqrt[34]]*e)","A",2,2,22,0.09091,1,"{3126, 2661}"
407,1,94,0,0.0535315,"\int \frac{1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx","Int[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-3/2),x]","-\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}","-\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,"-(Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(15*e) - (5*Cos[d + e*x] - 3*Sin[d + e*x])/(15*e*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])","A",3,3,22,0.1364,1,"{3128, 3118, 2653}"
408,1,187,0,0.1996578,"\int \frac{1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx","Int[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-5/2),x]","\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}","\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,"(4*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(675*e) + EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15]/(45*Sqrt[2 + Sqrt[34]]*e) - (5*Cos[d + e*x] - 3*Sin[d + e*x])/(45*e*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2)) + (4*(5*Cos[d + e*x] - 3*Sin[d + e*x]))/(675*e*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])","A",7,7,22,0.3182,1,"{3129, 3156, 3149, 3118, 2653, 3126, 2661}"
409,1,233,0,0.2595479,"\int \frac{1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{7/2}} \, dx","Int[(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(-7/2),x]","-\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}","-\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,"(-199*Sqrt[2 + Sqrt[34]]*EllipticE[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(101250*e) - (8*EllipticF[(d + e*x - ArcTan[5/3])/2, (2*(17 - Sqrt[34]))/15])/(3375*Sqrt[2 + Sqrt[34]]*e) - (5*Cos[d + e*x] - 3*Sin[d + e*x])/(75*e*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(5/2)) + (8*(5*Cos[d + e*x] - 3*Sin[d + e*x]))/(3375*e*(2 + 3*Cos[d + e*x] + 5*Sin[d + e*x])^(3/2)) - (199*(5*Cos[d + e*x] - 3*Sin[d + e*x]))/(101250*e*Sqrt[2 + 3*Cos[d + e*x] + 5*Sin[d + e*x]])","A",8,7,22,0.3182,1,"{3129, 3156, 3149, 3118, 2653, 3126, 2661}"
410,1,347,0,0.5318041,"\int (a+b \cos (d+e x)+c \sin (d+e x))^{5/2} \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2),x]","-\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}","-\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,"(-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) - (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) + (2*(23*a^2 + 9*(b^2 + c^2))*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(15*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (16*a*(a^2 - b^2 - c^2)*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(15*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])","A",7,7,22,0.3182,1,"{3120, 3146, 3149, 3119, 2653, 3127, 2661}"
411,1,283,0,0.2818046,"\int (a+b \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2),x]","-\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}","-\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,"(-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) + (8*a*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(3*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (2*(a^2 - b^2 - c^2)*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(3*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])","A",6,6,22,0.2727,1,"{3120, 3149, 3119, 2653, 3127, 2661}"
412,1,108,0,0.0708033,"\int \sqrt{a+b \cos (d+e x)+c \sin (d+e x)} \, dx","Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","\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}}}}","\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*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])","A",2,2,22,0.09091,1,"{3119, 2653}"
413,1,108,0,0.0702036,"\int \frac{1}{\sqrt{a+b \cos (d+e x)+c \sin (d+e x)}} \, dx","Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","\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)}}","\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*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])","A",2,2,22,0.09091,1,"{3127, 2661}"
414,1,186,0,0.1039608,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^{3/2}} \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3/2),x]","\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)}}","\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,"(2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/((a^2 - b^2 - c^2)*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + (2*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/((a^2 - b^2 - c^2)*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])","A",3,3,22,0.1364,1,"{3128, 3119, 2653}"
415,1,382,0,0.3628053,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^{5/2}} \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-5/2),x]","-\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}}","-\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,"(2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(3*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + (8*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(3*(a^2 - b^2 - c^2)^2*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + (8*a*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(3*(a^2 - b^2 - c^2)^2*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (2*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(3*(a^2 - b^2 - c^2)*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])","A",7,7,22,0.3182,1,"{3129, 3156, 3149, 3119, 2653, 3127, 2661}"
416,1,490,0,0.6192594,"\int \frac{1}{(a+b \cos (d+e x)+c \sin (d+e x))^{7/2}} \, dx","Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(-7/2),x]","-\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}}","-\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,"(2*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(5*(a^2 - b^2 - c^2)*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) + (16*(a*c*Cos[d + e*x] - a*b*Sin[d + e*x]))/(15*(a^2 - b^2 - c^2)^2*e*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2)) + (2*(23*a^2 + 9*(b^2 + c^2))*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(15*(a^2 - b^2 - c^2)^3*e*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (16*a*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])])/(15*(a^2 - b^2 - c^2)^2*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]) + (2*(c*(23*a^2 + 9*(b^2 + c^2))*Cos[d + e*x] - b*(23*a^2 + 9*(b^2 + c^2))*Sin[d + e*x]))/(15*(a^2 - b^2 - c^2)^3*e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])","A",8,7,22,0.3182,1,"{3129, 3156, 3149, 3119, 2653, 3127, 2661}"
417,1,139,0,0.0648625,"\int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx","Int[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2),x]","-\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}}","-\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,"(-320*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (16*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))/(5*e)","A",3,2,22,0.09091,1,"{3113, 3112}"
418,1,93,0,0.0399338,"\int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx","Int[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2),x]","-\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}}","-\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,"(-40*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e)","A",2,2,22,0.09091,1,"{3113, 3112}"
419,1,44,0,0.0182359,"\int \sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx","Int[Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]","-\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}}","-\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] - 4*Sin[d + e*x]))/(e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])","A",1,1,22,0.04545,1,"{3112}"
420,1,48,0,0.0648061,"\int \frac{1}{\sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx","Int[1/Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]","\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}","\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,"(Sqrt[2/5]*ArcTanh[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[1 + Cos[d + e*x - ArcTan[3/4]]])])/e","A",3,3,22,0.1364,1,"{3115, 2649, 206}"
421,1,96,0,0.0532055,"\int \frac{1}{(5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx","Int[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-3/2),x]","\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}}","\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,"ArcTanh[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[1 + Cos[d + e*x - ArcTan[3/4]]])]/(10*Sqrt[10]*e) - (3*Cos[d + e*x] - 4*Sin[d + e*x])/(10*e*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))","A",4,4,22,0.1818,1,"{3116, 3115, 2649, 206}"
422,1,142,0,0.0767014,"\int \frac{1}{(5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx","Int[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-5/2),x]","-\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}","-\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,"(3*ArcTanh[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[1 + Cos[d + e*x - ArcTan[3/4]]])])/(400*Sqrt[10]*e) - (3*Cos[d + e*x] - 4*Sin[d + e*x])/(20*e*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2)) - (3*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(400*e*(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))","A",5,4,22,0.1818,1,"{3116, 3115, 2649, 206}"
423,1,185,0,0.0935227,"\int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{7/2} \, dx","Int[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(7/2),x]","-\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}}","-\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,"(6400*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(7*e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (320*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(7*e) + (24*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))/(7*e) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2))/(7*e)","A",4,2,22,0.09091,1,"{3113, 3112}"
424,1,139,0,0.0739644,"\int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx","Int[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2),x]","-\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}}","-\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,"(-320*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) + (16*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))/(5*e)","A",3,2,22,0.09091,1,"{3113, 3112}"
425,1,93,0,0.0383577,"\int (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx","Int[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2),x]","\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}","\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,"(40*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e)","A",2,2,22,0.09091,1,"{3113, 3112}"
426,1,44,0,0.0173385,"\int \sqrt{-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx","Int[Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]","-\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}}","-\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] - 4*Sin[d + e*x]))/(e*Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])","A",1,1,22,0.04545,1,"{3112}"
427,1,49,0,0.0608601,"\int \frac{1}{\sqrt{-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx","Int[1/Sqrt[-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]],x]","-\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}","-\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,"-((Sqrt[2/5]*ArcTan[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[-1 + Cos[d + e*x - ArcTan[3/4]]])])/e)","A",3,3,22,0.1364,1,"{3115, 2649, 204}"
428,1,96,0,0.0523508,"\int \frac{1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx","Int[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-3/2),x]","\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}","\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,"ArcTan[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[-1 + Cos[d + e*x - ArcTan[3/4]]])]/(10*Sqrt[10]*e) + (3*Cos[d + e*x] - 4*Sin[d + e*x])/(10*e*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))","A",4,4,22,0.1818,1,"{3116, 3115, 2649, 204}"
429,1,142,0,0.0751394,"\int \frac{1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2}} \, dx","Int[(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(-5/2),x]","-\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}","-\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,"(-3*ArcTan[Sin[d + e*x - ArcTan[3/4]]/(Sqrt[2]*Sqrt[-1 + Cos[d + e*x - ArcTan[3/4]]])])/(400*Sqrt[10]*e) + (3*Cos[d + e*x] - 4*Sin[d + e*x])/(20*e*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2)) - (3*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(400*e*(-5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))","A",5,4,22,0.1818,1,"{3116, 3115, 2649, 204}"
430,1,258,0,0.1786053,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{7/2} \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(7/2),x]","-\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)}}","-\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,"(-256*(b^2 + c^2)^(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]]) - (64*(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]])/(35*e) - (24*Sqrt[b^2 + c^2]*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))/(35*e) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2))/(7*e)","A",4,2,32,0.06250,1,"{3113, 3112}"
431,1,190,0,0.1221108,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2} \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2),x]","-\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)}}","-\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,"(-64*(b^2 + c^2)*(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]]) - (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) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))/(5*e)","A",3,2,32,0.06250,1,"{3113, 3112}"
432,1,126,0,0.0747549,"\int \left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2} \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2),x]","-\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)}}","-\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,"(-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]]) - (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)","A",2,2,32,0.06250,1,"{3113, 3112}"
433,1,55,0,0.0332414,"\int \sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx","Int[Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","-\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)}}","-\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,"(-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]])","A",1,1,32,0.03125,1,"{3112}"
434,1,88,0,0.1195671,"\int \frac{1}{\sqrt{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx","Int[1/Sqrt[Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","\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}}","\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,"(Sqrt[2]*ArcTanh[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])])/((b^2 + c^2)^(1/4)*e)","A",3,3,32,0.09375,1,"{3115, 2649, 206}"
435,1,160,0,0.1329448,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}} \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3/2),x]","\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}}","\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,"ArcTanh[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])]/(2*Sqrt[2]*(b^2 + c^2)^(3/4)*e) - (c*Cos[d + e*x] - b*Sin[d + e*x])/(2*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))","A",4,4,32,0.1250,1,"{3116, 3115, 2649, 206}"
436,1,226,0,0.1858233,"\int \frac{1}{\left(\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2}} \, dx","Int[(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-5/2),x]","\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}}","\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,"(3*ArcTanh[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])])/(16*Sqrt[2]*(b^2 + c^2)^(5/4)*e) - (c*Cos[d + e*x] - b*Sin[d + e*x])/(4*Sqrt[b^2 + c^2]*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) - (3*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(16*(b^2 + c^2)*e*(Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))","A",5,4,32,0.1250,1,"{3116, 3115, 2649, 206}"
437,1,196,0,0.1337819,"\int \left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2} \, dx","Int[(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2),x]","-\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)}}","-\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,"(-64*(b^2 + c^2)*(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]]) + (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) - (2*(c*Cos[d + e*x] - b*Sin[d + e*x])*(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))/(5*e)","A",3,2,34,0.05882,1,"{3113, 3112}"
438,1,130,0,0.0807832,"\int \left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2} \, dx","Int[(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2),x]","\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}","\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,"(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]]) - (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)","A",2,2,34,0.05882,1,"{3113, 3112}"
439,1,57,0,0.0381648,"\int \sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx","Int[Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","-\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)}}","-\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,"(-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]])","A",1,1,34,0.02941,1,"{3112}"
440,1,91,0,0.0978455,"\int \frac{1}{\sqrt{-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)}} \, dx","Int[1/Sqrt[-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x]],x]","-\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}}","-\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,"-((Sqrt[2]*ArcTan[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])])/((b^2 + c^2)^(1/4)*e))","A",3,3,34,0.08824,1,"{3115, 2649, 204}"
441,1,164,0,0.1256276,"\int \frac{1}{\left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{3/2}} \, dx","Int[(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-3/2),x]","\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}}","\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,"ArcTan[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])]/(2*Sqrt[2]*(b^2 + c^2)^(3/4)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(2*Sqrt[b^2 + c^2]*e*(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))","A",4,4,34,0.1176,1,"{3116, 3115, 2649, 204}"
442,1,232,0,0.1704737,"\int \frac{1}{\left(-\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)\right)^{5/2}} \, dx","Int[(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(-5/2),x]","-\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}}","-\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,"(-3*ArcTan[((b^2 + c^2)^(1/4)*Sin[d + e*x - ArcTan[b, c]])/(Sqrt[2]*Sqrt[-Sqrt[b^2 + c^2] + Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]]])])/(16*Sqrt[2]*(b^2 + c^2)^(5/4)*e) + (c*Cos[d + e*x] - b*Sin[d + e*x])/(4*Sqrt[b^2 + c^2]*e*(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(5/2)) - (3*(c*Cos[d + e*x] - b*Sin[d + e*x]))/(16*(b^2 + c^2)*e*(-Sqrt[b^2 + c^2] + b*Cos[d + e*x] + c*Sin[d + e*x])^(3/2))","A",5,4,34,0.1176,1,"{3116, 3115, 2649, 204}"
443,1,101,0,0.0964729,"\int \frac{\sin (x)}{a+b \cos (x)+c \sin (x)} \, dx","Int[Sin[x]/(a + b*Cos[x] + c*Sin[x]),x]","-\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}","-\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)/(b^2 + c^2) - (2*a*c*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) - (b*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",4,4,15,0.2667,1,"{3137, 3124, 618, 204}"
444,1,30,0,0.0306626,"\int \frac{\sin (x)}{1+\cos (x)+\sin (x)} \, dx","Int[Sin[x]/(1 + Cos[x] + Sin[x]),x]","\frac{x}{2}-\frac{1}{2} \log \left(\tan \left(\frac{x}{2}\right)+1\right)-\frac{1}{2} \log (\sin (x)+\cos (x)+1)","\frac{x}{2}-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)",1,"x/2 - Log[1 + Cos[x] + Sin[x]]/2 - Log[1 + Tan[x/2]]/2","A",3,3,11,0.2727,1,"{3137, 3124, 31}"
445,1,97,0,0.1270243,"\int \frac{1}{a+c \sec (x)+b \tan (x)} \, dx","Int[(a + c*Sec[x] + b*Tan[x])^(-1),x]","\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}","\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)/(a^2 + b^2) + (2*a*c*ArcTanh[(b - (a - c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/((a^2 + b^2)*Sqrt[a^2 + b^2 - c^2]) + (b*Log[c + a*Cos[x] + b*Sin[x]])/(a^2 + b^2)","A",5,5,12,0.4167,1,"{3159, 3138, 3124, 618, 206}"
446,1,51,0,0.0760878,"\int \frac{\sec (x)}{a+c \sec (x)+b \tan (x)} \, dx","Int[Sec[x]/(a + c*Sec[x] + b*Tan[x]),x]","-\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}}","-\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",4,4,15,0.2667,1,"{3165, 3124, 618, 206}"
447,1,142,0,0.5146893,"\int \frac{\sec ^2(x)}{a+c \sec (x)+b \tan (x)} \, dx","Int[Sec[x]^2/(a + c*Sec[x] + b*Tan[x]),x]","-\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(-(a-c) \tan ^2\left(\frac{x}{2}\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}","-\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(-(a-c) \tan ^2\left(\frac{x}{2}\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]])/((b^2 - c^2)*Sqrt[a^2 + b^2 - c^2]) - Log[1 - Tan[x/2]]/(b + c) - Log[1 + Tan[x/2]]/(b - c) + (b*Log[a + c + 2*b*Tan[x/2] - (a - c)*Tan[x/2]^2])/(b^2 - c^2)","A",10,8,17,0.4706,1,"{4397, 1075, 634, 618, 206, 628, 633, 31}"
448,1,371,0,0.4465653,"\int \frac{(a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{\sec ^{\frac{3}{2}}(d+e x)} \, dx","Int[(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)/Sec[d + e*x]^(3/2),x]","\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))}","\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,"(-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[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])) + (8*b*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(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])*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*EllipticF[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(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])^2)","A",7,7,33,0.2121,1,"{3167, 3120, 3149, 3119, 2653, 3127, 2661}"
449,1,118,0,0.1440395,"\int \frac{\sqrt{a+b \sec (d+e x)+c \tan (d+e x)}}{\sqrt{\sec (d+e x)}} \, dx","Int[Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]]/Sqrt[Sec[d + e*x]],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}}}","\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*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*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])/(b + Sqrt[a^2 + c^2])])","A",3,3,33,0.09091,1,"{3167, 3119, 2653}"
450,1,118,0,0.1662847,"\int \frac{\sqrt{\sec (d+e x)}}{\sqrt{a+b \sec (d+e x)+c \tan (d+e x)}} \, dx","Int[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)} \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)}}","\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*EllipticF[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[Sec[d + e*x]]*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])","A",3,3,33,0.09091,1,"{3167, 3127, 2661}"
451,1,240,0,0.2163177,"\int \frac{\sec ^{\frac{3}{2}}(d+e x)}{(a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx","Int[Sec[d + e*x]^(3/2)/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2),x]","-\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}}","-\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,"(-2*Sec[d + e*x]^(3/2)*(c*Cos[d + e*x] - a*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)) - (2*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sec[d + e*x]^(3/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))","A",4,4,33,0.1212,1,"{3167, 3128, 3119, 2653}"
452,1,492,0,0.5190643,"\int \frac{\sec ^{\frac{5}{2}}(d+e x)}{(a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx","Int[Sec[d + e*x]^(5/2)/(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2),x]","\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}}","\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,"(-2*Sec[d + e*x]^(5/2)*(c*Cos[d + e*x] - a*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (8*Sec[d + e*x]^(5/2)*(b*c*Cos[d + e*x] - a*b*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*(a^2 - b^2 + c^2)^2*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (8*b*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sec[d + e*x]^(5/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^3)/(3*(a^2 - b^2 + c^2)^2*e*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (2*EllipticF[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sec[d + e*x]^(5/2)*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*(a^2 - b^2 + c^2)*e*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2))","A",8,8,33,0.2424,1,"{3167, 3129, 3156, 3149, 3119, 2653, 3127, 2661}"
453,1,371,0,0.3898211,"\int \cos ^{\frac{3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} \, dx","Int[Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2),x]","\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))}","\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,"(-2*Cos[d + e*x]^(3/2)*(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*(b + a*Cos[d + e*x] + c*Sin[d + e*x])) + (8*b*Cos[d + e*x]^(3/2)*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*e*(b + a*Cos[d + e*x] + c*Sin[d + e*x])*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*Cos[d + e*x]^(3/2)*EllipticF[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))/(3*e*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)","A",7,7,33,0.2121,1,"{3163, 3120, 3149, 3119, 2653, 3127, 2661}"
454,1,118,0,0.1466484,"\int \sqrt{\cos (d+e x)} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)} \, dx","Int[Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]],x]","\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}}}","\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,"(2*Sqrt[Cos[d + e*x]]*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])/(e*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])","A",3,3,33,0.09091,1,"{3163, 3119, 2653}"
455,1,118,0,0.1519469,"\int \frac{1}{\sqrt{\cos (d+e x)} \sqrt{a+b \sec (d+e x)+c \tan (d+e x)}} \, dx","Int[1/(Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]]),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)}}","\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,"(2*EllipticF[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[Cos[d + e*x]]*Sqrt[a + b*Sec[d + e*x] + c*Tan[d + e*x]])","A",3,3,33,0.09091,1,"{3163, 3127, 2661}"
456,1,240,0,0.2112255,"\int \frac{1}{\cos ^{\frac{3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}} \, dx","Int[1/(Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)),x]","-\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}}","-\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,"(-2*(c*Cos[d + e*x] - a*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*Cos[d + e*x]^(3/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2)) - (2*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*Cos[d + e*x]^(3/2)*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(3/2))","A",4,4,33,0.1212,1,"{3163, 3128, 3119, 2653}"
457,1,492,0,0.4947091,"\int \frac{1}{\cos ^{\frac{5}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{5/2}} \, dx","Int[1/(Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)),x]","\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}}","\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,"(-2*(c*Cos[d + e*x] - a*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)*e*Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (8*(b*c*Cos[d + e*x] - a*b*Sin[d + e*x])*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2)/(3*(a^2 - b^2 + c^2)^2*e*Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (8*b*EllipticE[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^3)/(3*(a^2 - b^2 + c^2)^2*e*Cos[d + e*x]^(5/2)*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2)) + (2*EllipticF[(d + e*x - ArcTan[a, c])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + a*Cos[d + e*x] + c*Sin[d + e*x])^2*Sqrt[(b + a*Cos[d + e*x] + c*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*(a^2 - b^2 + c^2)*e*Cos[d + e*x]^(5/2)*(a + b*Sec[d + e*x] + c*Tan[d + e*x])^(5/2))","A",8,8,33,0.2424,1,"{3163, 3129, 3156, 3149, 3119, 2653, 3127, 2661}"
458,1,98,0,0.1026959,"\int \frac{1}{a+b \cot (x)+c \csc (x)} \, dx","Int[(a + b*Cot[x] + c*Csc[x])^(-1),x]","\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}","\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)/(a^2 + b^2) + (2*a*c*ArcTanh[(a - (b - c)*Tan[x/2])/Sqrt[a^2 + b^2 - c^2]])/((a^2 + b^2)*Sqrt[a^2 + b^2 - c^2]) - (b*Log[c + b*Cos[x] + a*Sin[x]])/(a^2 + b^2)","A",5,5,12,0.4167,1,"{3160, 3137, 3124, 618, 206}"
459,1,51,0,0.0717676,"\int \frac{\csc (x)}{a+b \cot (x)+c \csc (x)} \, dx","Int[Csc[x]/(a + b*Cot[x] + c*Csc[x]),x]","-\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}}","-\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",4,4,15,0.2667,1,"{3166, 3124, 618, 206}"
460,1,120,0,0.5331737,"\int \frac{\csc ^2(x)}{a+b \cot (x)+c \csc (x)} \, dx","Int[Csc[x]^2/(a + b*Cot[x] + c*Csc[x]),x]","-\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)-(b-c) \tan ^2\left(\frac{x}{2}\right)+b+c\right)}{b^2-c^2}+\frac{\log \left(\tan \left(\frac{x}{2}\right)\right)}{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)-(b-c) \tan ^2\left(\frac{x}{2}\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]])/((b^2 - c^2)*Sqrt[a^2 + b^2 - c^2]) + Log[Tan[x/2]]/(b + c) - (b*Log[b + c + 2*a*Tan[x/2] - (b - c)*Tan[x/2]^2])/(b^2 - c^2)","A",9,7,17,0.4118,1,"{4397, 12, 1628, 634, 618, 206, 628}"
461,1,21,0,0.0478138,"\int \frac{\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx","Int[Csc[x]/(2 + 2*Cot[x] + 3*Csc[x]),x]","x+2 \tan ^{-1}\left(\frac{\cos (x)-\sin (x)}{\sin (x)+\cos (x)+2}\right)","x+2 \tan ^{-1}\left(\frac{\cos (x)-\sin (x)}{\sin (x)+\cos (x)+2}\right)",1,"x + 2*ArcTan[(Cos[x] - Sin[x])/(2 + Cos[x] + Sin[x])]","A",4,4,15,0.2667,1,"{3166, 3124, 618, 204}"
462,1,371,0,0.4321179,"\int \frac{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2}}{\csc ^{\frac{3}{2}}(d+e x)} \, dx","Int[(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)/Csc[d + e*x]^(3/2),x]","\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))}","\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,"(8*b*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])])/(3*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2) - (2*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3*e*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x]))","A",7,7,33,0.2121,1,"{3168, 3120, 3149, 3119, 2653, 3127, 2661}"
463,1,118,0,0.1436902,"\int \frac{\sqrt{a+c \cot (d+e x)+b \csc (d+e x)}}{\sqrt{\csc (d+e x)}} \, dx","Int[Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]/Sqrt[Csc[d + e*x]],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}}}","\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*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[Csc[d + e*x]]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])","A",3,3,33,0.09091,1,"{3168, 3119, 2653}"
464,1,118,0,0.1657041,"\int \frac{\sqrt{\csc (d+e x)}}{\sqrt{a+c \cot (d+e x)+b \csc (d+e x)}} \, dx","Int[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)} \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)}}","\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*Sqrt[Csc[d + e*x]]*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]])","A",3,3,33,0.09091,1,"{3168, 3127, 2661}"
465,1,240,0,0.2122656,"\int \frac{\csc ^{\frac{3}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2}} \, dx","Int[Csc[d + e*x]^(3/2)/(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2),x]","-\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}}","-\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,"(-2*Csc[d + e*x]^(3/2)*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) - (2*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2))","A",4,4,33,0.1212,1,"{3168, 3128, 3119, 2653}"
466,1,492,0,0.4969983,"\int \frac{\csc ^{\frac{5}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2}} \, dx","Int[Csc[d + e*x]^(5/2)/(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2),x]","\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}}","\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,"(8*b*Csc[d + e*x]^(5/2)*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^3)/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*Csc[d + e*x]^(5/2)*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) - (2*Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)) + (8*Csc[d + e*x]^(5/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*(a*b*Cos[d + e*x] - b*c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2))","A",8,8,33,0.2424,1,"{3168, 3129, 3156, 3149, 3119, 2653, 3127, 2661}"
467,1,371,0,0.3834814,"\int (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x) \, dx","Int[(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2),x]","\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))}","\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,"(8*b*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sin[d + e*x]^(3/2))/(3*e*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*(a^2 - b^2 + c^2)*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sin[d + e*x]^(3/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*e*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2) - (2*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3*e*(b + c*Cos[d + e*x] + a*Sin[d + e*x]))","A",7,7,33,0.2121,1,"{3164, 3120, 3149, 3119, 2653, 3127, 2661}"
468,1,118,0,0.1407328,"\int \sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sqrt{\sin (d+e x)} \, dx","Int[Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]],x]","\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}}}","\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,"(2*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[Sin[d + e*x]])/(e*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])","A",3,3,33,0.09091,1,"{3164, 3119, 2653}"
469,1,118,0,0.1488605,"\int \frac{1}{\sqrt{a+c \cot (d+e x)+b \csc (d+e x)} \sqrt{\sin (d+e x)}} \, dx","Int[1/(Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]]),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)}}","\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,"(2*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(e*Sqrt[a + c*Cot[d + e*x] + b*Csc[d + e*x]]*Sqrt[Sin[d + e*x]])","A",3,3,33,0.09091,1,"{3164, 3127, 2661}"
470,1,240,0,0.2053106,"\int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)} \, dx","Int[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)),x]","-\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}}","-\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,"(-2*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) - (2*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2))","A",4,4,33,0.1212,1,"{3164, 3128, 3119, 2653}"
471,1,492,0,0.4914159,"\int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)} \, dx","Int[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)),x]","\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}}","\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,"(8*b*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^3)/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) + (2*EllipticF[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*Sqrt[(b + c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])])/(3*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)) - (2*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d + e*x] - c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2)) + (8*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*(a*b*Cos[d + e*x] - b*c*Sin[d + e*x]))/(3*(a^2 - b^2 + c^2)^2*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(5/2)*Sin[d + e*x]^(5/2))","A",8,8,33,0.2424,1,"{3164, 3129, 3156, 3149, 3119, 2653, 3127, 2661}"
472,1,1,0,0.0092337,"\int \frac{1}{\cos ^2(x)+\sin ^2(x)} \, dx","Int[(Cos[x]^2 + Sin[x]^2)^(-1),x]","x","x",1,"x","A",2,2,11,0.1818,1,"{4380, 8}"
473,1,1,0,0.0093517,"\int \frac{1}{\left(\cos ^2(x)+\sin ^2(x)\right)^2} \, dx","Int[(Cos[x]^2 + Sin[x]^2)^(-2),x]","x","x",1,"x","A",2,2,11,0.1818,1,"{4380, 8}"
474,1,1,0,0.0087911,"\int \frac{1}{\left(\cos ^2(x)+\sin ^2(x)\right)^3} \, dx","Int[(Cos[x]^2 + Sin[x]^2)^(-3),x]","x","x",1,"x","A",2,2,11,0.1818,1,"{4380, 8}"
475,1,11,0,0.0154245,"\int \frac{1}{\cos ^2(x)-\sin ^2(x)} \, dx","Int[(Cos[x]^2 - Sin[x]^2)^(-1),x]","\frac{1}{2} \tanh ^{-1}(2 \sin (x) \cos (x))","\frac{1}{2} \tanh ^{-1}(2 \sin (x) \cos (x))",1,"ArcTanh[2*Cos[x]*Sin[x]]/2","A",2,1,13,0.07692,1,"{206}"
476,1,13,0,0.0229803,"\int \frac{1}{\left(\cos ^2(x)-\sin ^2(x)\right)^2} \, dx","Int[(Cos[x]^2 - Sin[x]^2)^(-2),x]","\frac{\tan (x)}{1-\tan ^2(x)}","\frac{\tan (x)}{1-\tan ^2(x)}",1,"Tan[x]/(1 - Tan[x]^2)","A",2,1,13,0.07692,1,"{383}"
477,1,32,0,0.0273345,"\int \frac{1}{\left(\cos ^2(x)-\sin ^2(x)\right)^3} \, dx","Int[(Cos[x]^2 - Sin[x]^2)^(-3),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))","\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[2*Cos[x]*Sin[x]]/4 + (Sec[x]^2*Tan[x])/(2*(1 - Tan[x]^2)^2)","A",4,3,13,0.2308,1,"{413, 21, 206}"
478,1,9,0,0.0182644,"\int \frac{1}{\cos ^2(x)+a^2 \sin ^2(x)} \, dx","Int[(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",2,1,15,0.06667,1,"{203}"
479,1,11,0,0.0197121,"\int \frac{1}{b^2 \cos ^2(x)+\sin ^2(x)} \, dx","Int[(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",2,1,15,0.06667,1,"{203}"
480,1,15,0,0.0257435,"\int \frac{1}{b^2 \cos ^2(x)+a^2 \sin ^2(x)} \, dx","Int[(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",2,1,19,0.05263,1,"{205}"
481,1,53,0,0.0373063,"\int \frac{1}{4 \cos ^2(1+2 x)+3 \sin ^2(1+2 x)} \, dx","Int[(4*Cos[1 + 2*x]^2 + 3*Sin[1 + 2*x]^2)^(-1),x]","\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}}","\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,"x/(2*Sqrt[3]) - ArcTan[(Cos[1 + 2*x]*Sin[1 + 2*x])/(3 + 2*Sqrt[3] + Cos[1 + 2*x]^2)]/(4*Sqrt[3])","A",2,1,23,0.04348,1,"{203}"
482,1,43,0,0.149493,"\int \frac{\sin ^2(x)}{a \cos ^2(x)+b \sin ^2(x)} \, dx","Int[Sin[x]^2/(a*Cos[x]^2 + b*Sin[x]^2),x]","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{b} (a-b)}-\frac{x}{a-b}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tan (x)}{\sqrt{a}}\right)}{\sqrt{b} (a-b)}-\frac{x}{a-b}",1,"-(x/(a - b)) + (Sqrt[a]*ArcTan[(Sqrt[b]*Tan[x])/Sqrt[a]])/((a - b)*Sqrt[b])","A",4,3,20,0.1500,1,"{481, 203, 205}"
483,1,43,0,0.109112,"\int \frac{\cos ^2(x)}{a \cos ^2(x)+b \sin ^2(x)} \, dx","Int[Cos[x]^2/(a*Cos[x]^2 + b*Sin[x]^2),x]","\frac{x}{a-b}-\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/(a - b) - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[x])/Sqrt[a]])/(Sqrt[a]*(a - b))","A",4,3,20,0.1500,1,"{391, 203, 205}"
484,1,36,0,0.0277595,"\int \frac{1}{\sec ^2(x)+\tan ^2(x)} \, dx","Int[(Sec[x]^2 + Tan[x]^2)^(-1),x]","\sqrt{2} x-x+\sqrt{2} \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\sin ^2(x)+\sqrt{2}+1}\right)","\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]*x + Sqrt[2]*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]","A",4,2,11,0.1818,1,"{1093, 203}"
485,1,49,0,0.0452656,"\int \frac{1}{\left(\sec ^2(x)+\tan ^2(x)\right)^2} \, dx","Int[(Sec[x]^2 + Tan[x]^2)^(-2),x]","-\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}}","-\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,"x - x/Sqrt[2] - ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)]/Sqrt[2] + Tan[x]/(1 + 2*Tan[x]^2)","A",6,4,11,0.3636,1,"{414, 12, 481, 203}"
486,1,74,0,0.0546415,"\int \frac{1}{\left(\sec ^2(x)+\tan ^2(x)\right)^3} \, dx","Int[(Sec[x]^2 + Tan[x]^2)^(-3),x]","\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}}","\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,"-x + (7*x)/(4*Sqrt[2]) + (7*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Sin[x]^2)])/(4*Sqrt[2]) + Tan[x]/(2*(1 + 2*Tan[x]^2)^2) - Tan[x]/(4*(1 + 2*Tan[x]^2))","A",6,4,11,0.3636,1,"{414, 527, 522, 203}"
487,1,1,0,0.012288,"\int \frac{1}{\sec ^2(x)-\tan ^2(x)} \, dx","Int[(Sec[x]^2 - Tan[x]^2)^(-1),x]","x","x",1,"x","A",2,2,13,0.1538,1,"{4381, 8}"
488,1,1,0,0.0125699,"\int \frac{1}{\left(\sec ^2(x)-\tan ^2(x)\right)^2} \, dx","Int[(Sec[x]^2 - Tan[x]^2)^(-2),x]","x","x",1,"x","A",2,2,13,0.1538,1,"{4381, 8}"
489,1,1,0,0.0123656,"\int \frac{1}{\left(\sec ^2(x)-\tan ^2(x)\right)^3} \, dx","Int[(Sec[x]^2 - Tan[x]^2)^(-3),x]","x","x",1,"x","A",2,2,13,0.1538,1,"{4381, 8}"
490,1,37,0,0.0310605,"\int \frac{1}{\cot ^2(x)+\csc ^2(x)} \, dx","Int[(Cot[x]^2 + Csc[x]^2)^(-1),x]","\sqrt{2} x-x-\sqrt{2} \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)","\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]*x - Sqrt[2]*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]","A",4,2,11,0.1818,1,"{1130, 203}"
491,1,47,0,0.0400156,"\int \frac{1}{\left(\cot ^2(x)+\csc ^2(x)\right)^2} \, dx","Int[(Cot[x]^2 + Csc[x]^2)^(-2),x]","-\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}}","-\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,"x - x/Sqrt[2] + ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)]/Sqrt[2] - Tan[x]/(2 + Tan[x]^2)","A",6,4,11,0.3636,1,"{470, 12, 391, 203}"
492,1,72,0,0.0755278,"\int \frac{1}{\left(\cot ^2(x)+\csc ^2(x)\right)^3} \, dx","Int[(Cot[x]^2 + Csc[x]^2)^(-3),x]","\frac{7 x}{4 \sqrt{2}}-x-\frac{\tan ^3(x)}{2 \left(\tan ^2(x)+2\right)^2}+\frac{\tan (x)}{4 \left(\tan ^2(x)+2\right)}-\frac{7 \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{4 \sqrt{2}}","\frac{7 x}{4 \sqrt{2}}-x-\frac{\tan ^3(x)}{2 \left(\tan ^2(x)+2\right)^2}+\frac{\tan (x)}{4 \left(\tan ^2(x)+2\right)}-\frac{7 \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{4 \sqrt{2}}",1,"-x + (7*x)/(4*Sqrt[2]) - (7*ArcTan[(Cos[x]*Sin[x])/(1 + Sqrt[2] + Cos[x]^2)])/(4*Sqrt[2]) - Tan[x]^3/(2*(2 + Tan[x]^2)^2) + Tan[x]/(4*(2 + Tan[x]^2))","A",6,4,11,0.3636,1,"{470, 578, 522, 203}"
493,1,3,0,0.0131306,"\int \frac{1}{\cot ^2(x)-\csc ^2(x)} \, dx","Int[(Cot[x]^2 - Csc[x]^2)^(-1),x]","-x","-x",1,"-x","A",2,2,13,0.1538,1,"{4382, 8}"
494,1,1,0,0.0129495,"\int \frac{1}{\left(\cot ^2(x)-\csc ^2(x)\right)^2} \, dx","Int[(Cot[x]^2 - Csc[x]^2)^(-2),x]","x","x",1,"x","A",2,2,13,0.1538,1,"{4382, 8}"
495,1,3,0,0.0131694,"\int \frac{1}{\left(\cot ^2(x)-\csc ^2(x)\right)^3} \, dx","Int[(Cot[x]^2 - Csc[x]^2)^(-3),x]","-x","-x",1,"-x","A",2,2,13,0.1538,1,"{4382, 8}"
496,1,33,0,0.0503087,"\int \frac{1}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx","Int[(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",2,1,16,0.06250,1,"{205}"
497,1,239,0,0.4916285,"\int \frac{x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx","Int[x/(a + b*Cos[x]^2 + c*Sin[x]^2),x]","-\frac{\text{PolyLog}\left(2,-\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right)}{4 \sqrt{a+b} \sqrt{a+c}}+\frac{\text{PolyLog}\left(2,-\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+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}}","-\frac{\text{PolyLog}\left(2,-\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right)}{4 \sqrt{a+b} \sqrt{a+c}}+\frac{\text{PolyLog}\left(2,-\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+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,"((-I/2)*x*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c])])/(Sqrt[a + b]*Sqrt[a + c]) + ((I/2)*x*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c])])/(Sqrt[a + b]*Sqrt[a + c]) - PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*Sqrt[a + b]*Sqrt[a + c]) + PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*Sqrt[a + b]*Sqrt[a + c])","A",9,6,18,0.3333,1,"{4587, 3321, 2264, 2190, 2279, 2391}"
498,1,365,0,0.738861,"\int \frac{x^2}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx","Int[x^2/(a + b*Cos[x]^2 + c*Sin[x]^2),x]","-\frac{x \text{PolyLog}\left(2,-\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{x \text{PolyLog}\left(2,-\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 \text{PolyLog}\left(3,-\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right)}{4 \sqrt{a+b} \sqrt{a+c}}+\frac{i \text{PolyLog}\left(3,-\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+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}}","-\frac{x \text{PolyLog}\left(2,-\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{x \text{PolyLog}\left(2,-\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 \text{PolyLog}\left(3,-\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right)}{4 \sqrt{a+b} \sqrt{a+c}}+\frac{i \text{PolyLog}\left(3,-\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+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,"((-I/2)*x^2*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c])])/(Sqrt[a + b]*Sqrt[a + c]) + ((I/2)*x^2*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c])])/(Sqrt[a + b]*Sqrt[a + c]) - (x*PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))])/(2*Sqrt[a + b]*Sqrt[a + c]) + (x*PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))])/(2*Sqrt[a + b]*Sqrt[a + c]) - ((I/4)*PolyLog[3, -(((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))])/(Sqrt[a + b]*Sqrt[a + c]) + ((I/4)*PolyLog[3, -(((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))])/(Sqrt[a + b]*Sqrt[a + c])","A",11,7,20,0.3500,1,"{4587, 3321, 2264, 2190, 2531, 2282, 6589}"
499,1,195,0,0.3926808,"\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","Int[(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{b \left(69 a^2 b^2+32 a^4+4 b^4\right) \cos (d+e x)}{10 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{a \left(82 a^2 b^2+15 a^4+8 b^4\right) \sin (d+e x) \cos (d+e x)}{40 e}+\frac{3}{8} a x \left(12 a^2 b^2+a^4+8 b^4\right)-\frac{b \cos (d+e x) (a \sin (d+e x)+b)^4}{5 e}","-\frac{b \left(69 a^2 b^2+32 a^4+4 b^4\right) \cos (d+e x)}{10 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{a \left(82 a^2 b^2+15 a^4+8 b^4\right) \sin (d+e x) \cos (d+e x)}{40 e}+\frac{3}{8} a x \left(12 a^2 b^2+a^4+8 b^4\right)-\frac{b \cos (d+e x) (a \sin (d+e x)+b)^4}{5 e}",1,"(3*a*(a^4 + 12*a^2*b^2 + 8*b^4)*x)/8 - (b*(32*a^4 + 69*a^2*b^2 + 4*b^4)*Cos[d + e*x])/(10*e) - (a*(15*a^4 + 82*a^2*b^2 + 8*b^4)*Cos[d + e*x]*Sin[d + e*x])/(40*e) - (b*(17*a^2 + 4*b^2)*Cos[d + e*x]*(b + a*Sin[d + e*x])^2)/(20*e) - ((5*a^2 + 4*b^2)*Cos[d + e*x]*(b + a*Sin[d + e*x])^3)/(20*e) - (b*Cos[d + e*x]*(b + a*Sin[d + e*x])^4)/(5*e)","A",5,3,39,0.07692,1,"{3288, 2753, 2734}"
500,1,109,0,0.0987093,"\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","Int[(a + b*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2),x]","\frac{\left(-8 a^2 b^2+a^4-3 b^4\right) \cos (d+e x)}{3 b 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(-8 a^2 b^2+a^4-3 b^4\right) \cos (d+e x)}{3 b 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}",1,"(a*(a^2 + 4*b^2)*x)/2 + ((a^4 - 8*a^2*b^2 - 3*b^4)*Cos[d + e*x])/(3*b*e) + (a*(a^2 - 6*b^2)*Cos[d + e*x]*Sin[d + e*x])/(6*e) - (a^2*Cos[d + e*x]*(a + b*Sin[d + e*x])^2)/(3*b*e)","A",2,2,37,0.05405,1,"{3023, 2734}"
501,1,23,0,0.0893867,"\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","Int[(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",3,3,39,0.07692,1,"{3288, 2754, 8}"
502,1,157,0,0.4153491,"\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","Int[(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{\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}","-\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,"(2*a*b*ArcTanh[(a + b*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*e) - Cos[d + e*x]/(3*e*(b + a*Sin[d + e*x])^3) + (b*Cos[d + e*x])/(3*(a^2 - b^2)*e*(b + a*Sin[d + e*x])^2) - ((2*a^2 + b^2)*Cos[d + e*x])/(3*(a^2 - b^2)^2*e*(b + a*Sin[d + e*x]))","A",9,6,39,0.1538,1,"{3288, 2754, 12, 2660, 618, 206}"
503,1,242,0,0.940384,"\int \frac{d+e \sin (x)}{a+b \sin (x)+c \sin ^2(x)} \, dx","Int[(d + e*Sin[x])/(a + b*Sin[x] + c*Sin[x]^2),x]","\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}}","\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,"(Sqrt[2]*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b - Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) - b*Sqrt[b^2 - 4*a*c]] + (Sqrt[2]*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2*c + (b + Sqrt[b^2 - 4*a*c])*Tan[x/2])/(Sqrt[2]*Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]])])/Sqrt[b^2 - 2*c*(a + c) + b*Sqrt[b^2 - 4*a*c]]","A",7,4,21,0.1905,1,"{3292, 2660, 618, 204}"
504,1,331,0,0.322718,"\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","Int[(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{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{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{\left(28 a^2 b^2+4 a^4+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}","\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{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{\left(28 a^2 b^2+4 a^4+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,"-(b*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(4*e) - ((4*a^4 + 28*a^2*b^2 + 3*b^4)*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(6*e*(b + a*Sin[d + e*x])^3) - ((4*a^2 + 3*b^2)*Cos[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(12*e*(b + a*Sin[d + e*x])) + (5*a^4*b*(3*a^2 + 4*b^2)*x*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(8*(a*b + a^2*Sin[d + e*x])^3) - (a^4*b*(29*a^2 + 6*b^2)*Cos[d + e*x]*Sin[d + e*x]*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))/(24*e*(a*b + a^2*Sin[d + e*x])^3)","A",4,3,41,0.07317,1,"{3290, 2753, 2734}"
505,1,185,0,0.1094083,"\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","Int[(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{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)}","\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,"-(((a^2 + b^2)*Cos[d + e*x]*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2])/(e*(b + a*Sin[d + e*x]))) + (3*a^2*b*x*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2])/(2*(a*b + a^2*Sin[d + e*x])) - (a^2*b*Cos[d + e*x]*Sin[d + e*x]*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2])/(2*e*(a*b + a^2*Sin[d + e*x]))","A",2,2,41,0.04878,1,"{3290, 2734}"
506,1,137,0,0.198201,"\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","Int[(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{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}}","\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*x*(b + a*Sin[d + e*x]))/(a*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2]) - (2*Sqrt[a^2 - b^2]*ArcTanh[(a + b*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Sin[d + e*x]))/(a*e*Sqrt[b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2])","A",5,5,41,0.1220,1,"{3290, 2735, 2660, 618, 206}"
507,1,239,0,0.2718686,"\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","Int[(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{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^3 b+a^4 \sin (d+e x)\right) \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}-\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^3 b+a^4 \sin (d+e x)\right) \left(a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right)^{3/2}}-\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}}",1,"-(Cos[d + e*x]*(b + a*Sin[d + e*x]))/(2*e*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2)) - (ArcTanh[(a + b*Tan[(d + e*x)/2])/Sqrt[a^2 - b^2]]*(a*b + a^2*Sin[d + e*x])^3)/(a^2*(a^2 - b^2)^(3/2)*e*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2)) + (b*Cos[d + e*x]*(a*b + a^2*Sin[d + e*x])^3)/(2*(a^2 - b^2)*e*(a^3*b + a^4*Sin[d + e*x])*(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2)^(3/2))","A",8,6,41,0.1463,1,"{3290, 2754, 12, 2660, 618, 206}"
508,1,11,0,0.0808742,"\int \frac{a+b \cos (x)}{b^2+2 a b \cos (x)+a^2 \cos ^2(x)} \, dx","Int[(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",3,3,27,0.1111,1,"{3289, 2754, 8}"
509,1,246,0,0.7890152,"\int \frac{d+e \cos (x)}{a+b \cos (x)+c \cos ^2(x)} \, dx","Int[(d + e*Cos[x])/(a + b*Cos[x] + c*Cos[x]^2),x]","\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}}","\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,"(2*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) + (2*(e - (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Tan[x/2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b - 2*c + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])","A",5,3,21,0.1429,1,"{3293, 2659, 205}"
510,1,144,0,0.2685323,"\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","Int[(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{\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{a \left(a^4-b^4\right) \tan (d+e x)}{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}","\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{a \left(a^4-b^4\right) \tan (d+e x)}{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,"a*(a^2 - 3*b^2)*(a^2 + b^2)*x + (b*(3*a^2 - b^2)*(a^2 + b^2)*Log[Cos[d + e*x]])/e - (a*(a^4 - b^4)*Tan[d + e*x])/e + (b*(a^2 + b^2)*(b + a*Tan[d + e*x])^2)/(2*e) + ((a^2 + b^2)*(b + a*Tan[d + e*x])^3)/(3*e) + (b*(b + a*Tan[d + e*x])^4)/(4*e)","A",7,5,39,0.1282,1,"{3708, 3528, 12, 3525, 3475}"
511,1,72,0,0.0755105,"\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","Int[(a + b*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2),x]","-\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}","-\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*(a^2 + b^2)*x) - (b*(a^2 + b^2)*Log[Cos[d + e*x]])/e + (2*a*b^2*Tan[d + e*x])/e + (a^2*(a + b*Tan[d + e*x])^2)/(2*b*e)","A",3,3,37,0.08108,1,"{3630, 3525, 3475}"
512,1,101,0,0.2575064,"\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","Int[(a + b*Tan[d + e*x])/(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2),x]","-\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}","-\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,"-((a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^2) + (b*(3*a^2 - b^2)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]])/((a^2 + b^2)^2*e) - (a^2 - b^2)/((a^2 + b^2)*e*(b + a*Tan[d + e*x]))","A",4,4,39,0.1026,1,"{3708, 3529, 3531, 3530}"
513,1,197,0,0.5353381,"\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","Int[(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{a^2-b^2}{3 e \left(a^2+b^2\right) (a \tan (d+e x)+b)^3}+\frac{-6 a^2 b^2+a^4+b^4}{e \left(a^2+b^2\right)^3 (a \tan (d+e x)+b)}-\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{b \left(-10 a^2 b^2+5 a^4+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(-10 a^2 b^2+a^4+5 b^4\right)}{\left(a^2+b^2\right)^4}","-\frac{a^2-b^2}{3 e \left(a^2+b^2\right) (a \tan (d+e x)+b)^3}+\frac{-6 a^2 b^2+a^4+b^4}{e \left(a^2+b^2\right)^3 (a \tan (d+e x)+b)}-\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{b \left(-10 a^2 b^2+5 a^4+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(-10 a^2 b^2+a^4+5 b^4\right)}{\left(a^2+b^2\right)^4}",1,"(a*(a^4 - 10*a^2*b^2 + 5*b^4)*x)/(a^2 + b^2)^4 - (b*(5*a^4 - 10*a^2*b^2 + b^4)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]])/((a^2 + b^2)^4*e) - (a^2 - b^2)/(3*(a^2 + b^2)*e*(b + a*Tan[d + e*x])^3) - (b*(3*a^2 - b^2))/(2*(a^2 + b^2)^2*e*(b + a*Tan[d + e*x])^2) + (a^4 - 6*a^2*b^2 + b^4)/((a^2 + b^2)^3*e*(b + a*Tan[d + e*x]))","A",6,4,39,0.1026,1,"{3708, 3529, 3531, 3530}"
514,1,284,0,0.2261393,"\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","Int[(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{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{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{\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}","-\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{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{\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,"(b*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(3*e) + ((a^4 - b^4)*Log[Cos[d + e*x]]*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(e*(b + a*Tan[d + e*x])^3) + ((a^2 + b^2)*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(2*e*(b + a*Tan[d + e*x])) - (2*a^4*b*(a^2 + b^2)*x*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(a*b + a^2*Tan[d + e*x])^3 + (a^4*b*(a^2 + b^2)*Tan[d + e*x]*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))/(e*(a*b + a^2*Tan[d + e*x])^3)","A",6,5,41,0.1220,1,"{3710, 3528, 12, 3525, 3475}"
515,1,122,0,0.1007761,"\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","Int[(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^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)}","\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,"-(((a^2 + b^2)*Log[Cos[d + e*x]]*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2])/(e*(b + a*Tan[d + e*x]))) + (a^2*b*Tan[d + e*x]*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2])/(e*(a*b + a^2*Tan[d + e*x]))","A",3,3,41,0.07317,1,"{3710, 3525, 3475}"
516,1,138,0,0.1881731,"\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","Int[(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{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}}","\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,"((a^2 - b^2)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]]*(b + a*Tan[d + e*x]))/((a^2 + b^2)*e*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2]) + (2*b*x*(a*b + a^2*Tan[d + e*x]))/((a^2 + b^2)*Sqrt[b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2])","A",3,3,41,0.07317,1,"{3710, 3531, 3530}"
517,1,316,0,0.4015707,"\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","Int[(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{\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{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^3 b+a^4 \tan (d+e x)\right) \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}-\frac{\left(-6 a^2 b^2+a^4+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{\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{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^3 b+a^4 \tan (d+e x)\right) \left(a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right)^{3/2}}-\frac{\left(-6 a^2 b^2+a^4+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}}",1,"-((a^2 - b^2)*(b + a*Tan[d + e*x]))/(2*(a^2 + b^2)*e*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2)) - ((a^4 - 6*a^2*b^2 + b^4)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]]*(b + a*Tan[d + e*x])^3)/((a^2 + b^2)^3*e*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2)) - (4*b*(a^2 - b^2)*x*(a*b + a^2*Tan[d + e*x])^3)/(a^2*(a^2 + b^2)^3*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2)) - (b*(3*a^2 - b^2)*(a*b + a^2*Tan[d + e*x])^3)/((a^2 + b^2)^2*e*(a^3*b + a^4*Tan[d + e*x])*(b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2))","A",5,4,41,0.09756,1,"{3710, 3529, 3531, 3530}"
518,1,184,0,0.4298606,"\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","Int[(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{a \left(50 a^2 b^2+4 a^4+19 b^4\right) \tan (d+e x)}{6 e}+\frac{b \left(56 a^2 b^2+19 a^4+8 b^4\right) \tanh ^{-1}(\sin (d+e x))}{8 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}+a b^4 x","\frac{a \left(50 a^2 b^2+4 a^4+19 b^4\right) \tan (d+e x)}{6 e}+\frac{b \left(56 a^2 b^2+19 a^4+8 b^4\right) \tanh ^{-1}(\sin (d+e x))}{8 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}+a b^4 x",1,"a*b^4*x + (b*(19*a^4 + 56*a^2*b^2 + 8*b^4)*ArcTanh[Sin[d + e*x]])/(8*e) + (a*(4*a^4 + 50*a^2*b^2 + 19*b^4)*Tan[d + e*x])/(6*e) + (a^2*b*(41*a^2 + 26*b^2)*Sec[d + e*x]*Tan[d + e*x])/(24*e) + ((4*a^2 + 7*b^2)*(a*b + a^2*Sec[d + e*x])^2*Tan[d + e*x])/(12*a*e) + (b*(a*b + a^2*Sec[d + e*x])^3*Tan[d + e*x])/(4*a^2*e)","A",8,7,39,0.1795,1,"{4172, 3918, 4056, 4048, 3770, 3767, 8}"
519,1,76,0,0.0783303,"\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","Int[(a + b*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2),x]","\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","\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,"a*b^2*x + (b*(5*a^2 + 2*b^2)*ArcTanh[Sin[d + e*x]])/(2*e) + (a*(a^2 + 2*b^2)*Tan[d + e*x])/e + (a^2*b*Sec[d + e*x]*Tan[d + e*x])/(2*e)","A",5,4,37,0.1081,1,"{4048, 3770, 3767, 8}"
520,1,92,0,0.3038615,"\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","Int[(a + b*Sec[d + e*x])/(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2),x]","-\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}","-\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,"(a*x)/b^2 - (2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]])/(b^2*e) - (a^2*Tan[d + e*x])/(b*e*(a*b + a^2*Sec[d + e*x]))","A",6,6,39,0.1538,1,"{4172, 3923, 3919, 3831, 2659, 205}"
521,1,230,0,0.8327683,"\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","Int[(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{\left(a^2-2 b^2\right) \left(-a^2 b^2+2 a^4+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(-11 a^2 b^2+6 a^4+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 \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{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}","-\frac{\left(a^2-2 b^2\right) \left(-a^2 b^2+2 a^4+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(-11 a^2 b^2+6 a^4+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 \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{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*x)/b^4 - ((a^2 - 2*b^2)*(2*a^4 - a^2*b^2 + b^4)*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*e) - (a*(3*a^2 - 5*b^2)*Tan[d + e*x])/(6*b^2*(a^2 - b^2)*e*(b + a*Sec[d + e*x])^2) - (a*(6*a^4 - 11*a^2*b^2 + 11*b^4)*Tan[d + e*x])/(6*b^3*(a^2 - b^2)^2*e*(b + a*Sec[d + e*x])) - (a^4*Tan[d + e*x])/(3*b*e*(a*b + a^2*Sec[d + e*x])^3)","A",8,7,39,0.1795,1,"{4172, 3923, 4060, 3919, 3831, 2659, 205}"
522,1,359,0,0.2870141,"\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","Int[(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{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{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{b \tan (d+e x) \left(a^2 b+a^3 \sec (d+e x)\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}+\frac{\left(9 a^2 b^2+a^4+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{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{b \tan (d+e x) \left(a^2 b+a^3 \sec (d+e x)\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}+\frac{\left(9 a^2 b^2+a^4+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}",1,"((a^4 + 9*a^2*b^2 + 2*b^4)*ArcTanh[Sin[d + e*x]]*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2))/(2*e*(b + a*Sec[d + e*x])^3) + (a^4*b^3*x*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2))/(a*b + a^2*Sec[d + e*x])^3 + (a^4*b*(11*a^2 + 8*b^2)*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)*Tan[d + e*x])/(3*e*(a*b + a^2*Sec[d + e*x])^3) + (a^5*(3*a^2 + 5*b^2)*Sec[d + e*x]*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)*Tan[d + e*x])/(6*e*(a*b + a^2*Sec[d + e*x])^3) + (b*(a^2*b + a^3*Sec[d + e*x])^2*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)*Tan[d + e*x])/(3*e*(a*b + a^2*Sec[d + e*x])^3)","A",7,6,41,0.1463,1,"{4174, 3918, 4048, 3770, 3767, 8}"
523,1,173,0,0.1189008,"\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","Int[(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{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)}","\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,"((a^2 + b^2)*ArcTanh[Sin[d + e*x]]*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2])/(e*(b + a*Sec[d + e*x])) + (a^2*b*x*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2])/(a*b + a^2*Sec[d + e*x]) + (a^2*b*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2]*Tan[d + e*x])/(e*(a*b + a^2*Sec[d + e*x]))","A",5,5,41,0.1220,1,"{4174, 3914, 3767, 8, 3770}"
524,1,142,0,0.2130628,"\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","Int[(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{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}}","\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,"(-2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]]*(b + a*Sec[d + e*x]))/(b*e*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2]) + (x*(a*b + a^2*Sec[d + e*x]))/(b*Sqrt[b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2])","A",5,5,41,0.1220,1,"{4174, 3919, 3831, 2659, 205}"
525,1,330,0,0.5663333,"\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","Int[(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{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(-3 a^2 b^2+2 a^4+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{\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{\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^2 b+a^3 \sec (d+e x)\right) \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(-3 a^2 b^2+2 a^4+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{\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{\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^2 b+a^3 \sec (d+e x)\right) \left(a^2 \sec ^2(d+e x)+2 a b \sec (d+e x)+b^2\right)^{3/2}}",1,"-(((2*a^4 - 3*a^2*b^2 + 2*b^4)*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]]*(b + a*Sec[d + e*x])^3)/((a - b)^(3/2)*b^3*(a + b)^(3/2)*e*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2))) + (x*(a*b + a^2*Sec[d + e*x])^3)/(a^2*b^3*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)) - ((a*b + a^2*Sec[d + e*x])*Tan[d + e*x])/(2*b*e*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2)) - ((2*a^2 - 3*b^2)*(a*b + a^2*Sec[d + e*x])^3*Tan[d + e*x])/(2*b^2*(a^2 - b^2)*e*(a^2*b + a^3*Sec[d + e*x])*(b^2 + 2*a*b*Sec[d + e*x] + a^2*Sec[d + e*x]^2)^(3/2))","A",7,7,41,0.1707,1,"{4174, 3923, 4060, 3919, 3831, 2659, 205}"
526,1,17,0,0.0400325,"\int \frac{\cos (x)-i \sin (x)}{\cos (x)+i \sin (x)} \, dx","Int[(Cos[x] - I*Sin[x])/(Cos[x] + I*Sin[x]),x]","\frac{1}{2} i (\cos (x)-i \sin (x))^2","\frac{1}{2} i (\cos (x)-i \sin (x))^2",1,"(I/2)*(Cos[x] - I*Sin[x])^2","A",1,1,21,0.04762,1,"{4385}"
527,1,17,0,0.0365433,"\int \frac{\cos (x)+i \sin (x)}{\cos (x)-i \sin (x)} \, dx","Int[(Cos[x] + I*Sin[x])/(Cos[x] - I*Sin[x]),x]","-\frac{i}{2 (\cos (x)-i \sin (x))^2}","-\frac{i}{2 (\cos (x)-i \sin (x))^2}",1,"(-I/2)/(Cos[x] - I*Sin[x])^2","A",1,1,21,0.04762,1,"{4385}"
528,1,6,0,0.0225762,"\int \frac{\cos (x)-\sin (x)}{\cos (x)+\sin (x)} \, dx","Int[(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",1,1,15,0.06667,1,"{3133}"
529,1,47,0,0.0405832,"\int \frac{B \cos (x)+C \sin (x)}{b \cos (x)+c \sin (x)} \, dx","Int[(B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x]),x]","\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}","\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^2 + c^2) + ((B*c - b*C)*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",1,1,21,0.04762,1,"{3133}"
530,1,74,0,0.0678324,"\int \frac{B \cos (x)+C \sin (x)}{(b \cos (x)+c \sin (x))^2} \, dx","Int[(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{(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}}","-\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,"-(((b*B + c*C)*ArcTanh[(c*Cos[x] - b*Sin[x])/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",3,3,21,0.1429,1,"{3153, 3074, 206}"
531,1,66,0,0.056569,"\int \frac{B \cos (x)+C \sin (x)}{(b \cos (x)+c \sin (x))^3} \, dx","Int[(B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^3,x]","\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}","\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*c - b*C)/(2*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2) + ((b*B + c*C)*Sin[x])/(b*(b^2 + c^2)*(b*Cos[x] + c*Sin[x]))","A",3,3,21,0.1429,1,"{3156, 12, 3075}"
532,1,84,0,0.0584858,"\int \frac{A+B \cos (x)+C \sin (x)}{b \cos (x)+c \sin (x)} \, dx","Int[(A + B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x]),x]","-\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}","-\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)/(b^2 + c^2) - (A*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/Sqrt[b^2 + c^2] + ((B*c - b*C)*Log[b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",3,3,22,0.1364,1,"{3136, 3074, 206}"
533,1,85,0,0.0569903,"\int \frac{A+B \cos (x)+C \sin (x)}{(b \cos (x)+c \sin (x))^2} \, dx","Int[(A + B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^2,x]","-\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}}","-\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,"-(((b*B + c*C)*ArcTanh[(c*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(b^2 + c^2)^(3/2)) - (B*c - b*C + A*c*Cos[x] - A*b*Sin[x])/((b^2 + c^2)*(b*Cos[x] + c*Sin[x]))","A",3,3,22,0.1364,1,"{3153, 3074, 206}"
534,1,129,0,0.1244751,"\int \frac{A+B \cos (x)+C \sin (x)}{(b \cos (x)+c \sin (x))^3} \, dx","Int[(A + B*Cos[x] + C*Sin[x])/(b*Cos[x] + c*Sin[x])^3,x]","-\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))}","-\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*Cos[x] - b*Sin[x])/Sqrt[b^2 + c^2]])/(2*(b^2 + c^2)^(3/2)) - (B*c - b*C + A*c*Cos[x] - A*b*Sin[x])/(2*(b^2 + c^2)*(b*Cos[x] + c*Sin[x])^2) - (c*(b*B + c*C)*Cos[x] - b*(b*B + c*C)*Sin[x])/((b^2 + c^2)^2*(b*Cos[x] + c*Sin[x]))","A",4,4,22,0.1818,1,"{3156, 3153, 3074, 206}"
535,1,115,0,0.1297075,"\int \frac{A+B \cos (x)}{a+b \cos (x)+c \sin (x)} \, dx","Int[(A + B*Cos[x])/(a + b*Cos[x] + c*Sin[x]),x]","-\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}","-\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,"(b*B*x)/(b^2 + c^2) - (2*(a*b*B - A*(b^2 + c^2))*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) + (B*c*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",4,4,19,0.2105,1,"{3138, 3124, 618, 204}"
536,1,113,0,0.1063406,"\int \frac{A+B \cos (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Int[(A + B*Cos[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","\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))}","\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)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(3/2) + (B*c + A*c*Cos[x] - (A*b - a*B)*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x]))","A",4,4,19,0.2105,1,"{3155, 3124, 618, 204}"
537,1,200,0,0.2548989,"\int \frac{A+B \cos (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx","Int[(A + B*Cos[x])/(a + b*Cos[x] + c*Sin[x])^3,x]","\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}","\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))*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(5/2) + (B*c + A*c*Cos[x] - (A*b - a*B)*Sin[x])/(2*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^2) + (a*B*c + (3*a*A - 2*b*B)*c*Cos[x] - (3*a*A*b - a^2*B - 2*b^2*B)*Sin[x])/(2*(a^2 - b^2 - c^2)^2*(a + b*Cos[x] + c*Sin[x]))","A",5,5,19,0.2632,1,"{3158, 3153, 3124, 618, 204}"
538,1,84,0,0.0449218,"\int \frac{A+B \cos (x)}{a+b \cos (x)+i b \sin (x)} \, dx","Int[(A + B*Cos[x])/(a + b*Cos[x] + I*b*Sin[x]),x]","\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}","\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*B)*x)/(2*a^2) + ((I/2)*B*Cos[x])/a + ((I/2)*(2*a*A*b - a^2*B - b^2*B)*Log[a + b*Cos[x] + I*b*Sin[x]])/(a^2*b) + (B*Sin[x])/(2*a)","A",1,1,22,0.04545,1,"{3132}"
539,1,84,0,0.0424567,"\int \frac{A+B \cos (x)}{a+b \cos (x)-i b \sin (x)} \, dx","Int[(A + B*Cos[x])/(a + b*Cos[x] - I*b*Sin[x]),x]","-\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}","-\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*B)*x)/(2*a^2) - ((I/2)*B*Cos[x])/a - ((I/2)*(2*a*A*b - a^2*B - b^2*B)*Log[a + b*Cos[x] - I*b*Sin[x]])/(a^2*b) + (B*Sin[x])/(2*a)","A",1,1,22,0.04545,1,"{3132}"
540,1,116,0,0.1078794,"\int \frac{A+C \sin (x)}{a+b \cos (x)+c \sin (x)} \, dx","Int[(A + C*Sin[x])/(a + b*Cos[x] + c*Sin[x]),x]","\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}","\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,"(c*C*x)/(b^2 + c^2) + (2*(A*(b^2 + c^2) - a*c*C)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) - (b*C*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",4,4,19,0.2105,1,"{3137, 3124, 618, 204}"
541,1,114,0,0.1005464,"\int \frac{A+C \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Int[(A + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","\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))}","\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)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(3/2) - (b*C - (A*c - a*C)*Cos[x] + A*b*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x]))","A",4,4,19,0.2105,1,"{3154, 3124, 618, 204}"
542,1,200,0,0.2494374,"\int \frac{A+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx","Int[(A + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^3,x]","\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}","\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)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(5/2) - (b*C - (A*c - a*C)*Cos[x] + A*b*Sin[x])/(2*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^2) - (a*b*C - (3*a*A*c - a^2*C - 2*c^2*C)*Cos[x] + b*(3*a*A - 2*c*C)*Sin[x])/(2*(a^2 - b^2 - c^2)^2*(a + b*Cos[x] + c*Sin[x]))","A",5,5,19,0.2632,1,"{3157, 3153, 3124, 618, 204}"
543,1,85,0,0.0458097,"\int \frac{A+C \sin (x)}{a+b \cos (x)+i b \sin (x)} \, dx","Int[(A + C*Sin[x])/(a + b*Cos[x] + I*b*Sin[x]),x]","\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}","\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 - I*b*C)*x)/(2*a^2) - (C*Cos[x])/(2*a) + (((2*I)*a*A*b - a^2*C + b^2*C)*Log[a + b*Cos[x] + I*b*Sin[x]])/(2*a^2*b) + ((I/2)*C*Sin[x])/a","A",1,1,22,0.04545,1,"{3131}"
544,1,85,0,0.0462409,"\int \frac{A+C \sin (x)}{a+b \cos (x)-i b \sin (x)} \, dx","Int[(A + C*Sin[x])/(a + b*Cos[x] - I*b*Sin[x]),x]","-\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}","-\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 + I*b*C)*x)/(2*a^2) - (C*Cos[x])/(2*a) - (((2*I)*a*A*b + a^2*C - b^2*C)*Log[a + b*Cos[x] - I*b*Sin[x]])/(2*a^2*b) - ((I/2)*C*Sin[x])/a","A",1,1,22,0.04545,1,"{3131}"
545,1,119,0,0.1134328,"\int \frac{B \cos (x)+C \sin (x)}{a+b \cos (x)+c \sin (x)} \, dx","Int[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x]),x]","-\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}","-\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)/(b^2 + c^2) - (2*a*(b*B + c*C)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) + ((B*c - b*C)*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",4,4,22,0.1818,1,"{3136, 3124, 618, 204}"
546,1,110,0,0.0963721,"\int \frac{B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Int[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","\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}}","\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)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(3/2) + (B*c - b*C - a*C*Cos[x] + a*B*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x]))","A",4,4,22,0.1818,1,"{3153, 3124, 618, 204}"
547,1,197,0,0.2324214,"\int \frac{B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx","Int[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^3,x]","-\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}","-\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)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(5/2) + (B*c - b*C - a*C*Cos[x] + a*B*Sin[x])/(2*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^2) + (a*(B*c - b*C) - (2*b*B*c + (a^2 + 2*c^2)*C)*Cos[x] + (a^2*B + 2*b*(b*B + c*C))*Sin[x])/(2*(a^2 - b^2 - c^2)^2*(a + b*Cos[x] + c*Sin[x]))","A",5,5,22,0.2273,1,"{3156, 3153, 3124, 618, 204}"
548,1,87,0,0.0775119,"\int \frac{B \cos (x)+C \sin (x)}{a+b \cos (x)+i b \sin (x)} \, dx","Int[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + I*b*Sin[x]),x]","-\frac{\left(\frac{i b^2 (B+i C)}{a^2}+i B+C\right) \log (a+i b \sin (x)+b \cos (x))}{2 b}-\frac{b x (B+i C)}{2 a^2}+\frac{(-C+i B) (\cos (x)-i \sin (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,"-(b*(B + I*C)*x)/(2*a^2) - ((I*B + (I*b^2*(B + I*C))/a^2 + C)*Log[a + b*Cos[x] + I*b*Sin[x]])/(2*b) + ((I*B - C)*(Cos[x] - I*Sin[x]))/(2*a)","A",1,1,25,0.04000,1,"{3130}"
549,1,85,0,0.07837,"\int \frac{B \cos (x)+C \sin (x)}{a+b \cos (x)-i b \sin (x)} \, dx","Int[(B*Cos[x] + C*Sin[x])/(a + b*Cos[x] - I*b*Sin[x]),x]","-\frac{b x (B-i C)}{2 a^2}+\frac{1}{2} \left(\frac{b (C+i B)}{a^2}+\frac{i (B+i C)}{b}\right) \log (a-i b \sin (x)+b \cos (x))-\frac{(C+i B) (\cos (x)+i \sin (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,"-(b*(B - I*C)*x)/(2*a^2) + (((I*(B + I*C))/b + (b*(I*B + C))/a^2)*Log[a + b*Cos[x] - I*b*Sin[x]])/2 - ((I*B + C)*(Cos[x] + I*Sin[x]))/(2*a)","A",1,1,25,0.04000,1,"{3130}"
550,1,131,0,0.1263169,"\int \frac{A+B \cos (x)+C \sin (x)}{a+b \cos (x)+c \sin (x)} \, dx","Int[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x]),x]","\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}","\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)/(b^2 + c^2) + (2*(A*(b^2 + c^2) - a*(b*B + c*C))*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(Sqrt[a^2 - b^2 - c^2]*(b^2 + c^2)) + ((B*c - b*C)*Log[a + b*Cos[x] + c*Sin[x]])/(b^2 + c^2)","A",4,4,23,0.1739,1,"{3136, 3124, 618, 204}"
551,1,127,0,0.1234139,"\int \frac{A+B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Int[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","\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))}","\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)*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(3/2) + (B*c - b*C + (A*c - a*C)*Cos[x] - (A*b - a*B)*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x]))","A",4,4,23,0.1739,1,"{3153, 3124, 618, 204}"
552,1,237,0,0.2765037,"\int \frac{A+B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx","Int[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + c*Sin[x])^3,x]","\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}","\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))*ArcTan[(c + (a - b)*Tan[x/2])/Sqrt[a^2 - b^2 - c^2]])/(a^2 - b^2 - c^2)^(5/2) + (B*c - b*C + (A*c - a*C)*Cos[x] - (A*b - a*B)*Sin[x])/(2*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^2) + (a*(B*c - b*C) + (3*a*A*c - a^2*C - 2*c*(b*B + c*C))*Cos[x] - (3*a*A*b - a^2*B - 2*b*(b*B + c*C))*Sin[x])/(2*(a^2 - b^2 - c^2)^2*(a + b*Cos[x] + c*Sin[x]))","A",5,5,23,0.2174,1,"{3156, 3153, 3124, 618, 204}"
553,1,105,0,0.0736377,"\int \frac{A+B \cos (x)+C \sin (x)}{a+b \cos (x)+i b \sin (x)} \, dx","Int[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] + I*b*Sin[x]),x]","\frac{i \left(a^2 (-(B-i C))+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}","\frac{i \left(a^2 (-(B-i C))+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*(B + I*C))*x)/(2*a^2) + ((I/2)*(2*a*A*b - a^2*(B - I*C) - b^2*(B + I*C))*Log[a + b*Cos[x] + I*b*Sin[x]])/(a^2*b) + ((I*B - C)*(Cos[x] - I*Sin[x]))/(2*a)","A",1,1,26,0.03846,1,"{3130}"
554,1,103,0,0.0732889,"\int \frac{A+B \cos (x)+C \sin (x)}{a+b \cos (x)-i b \sin (x)} \, dx","Int[(A + B*Cos[x] + C*Sin[x])/(a + b*Cos[x] - I*b*Sin[x]),x]","-\frac{i \left(a^2 (-(B+i C))+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}","-\frac{i \left(a^2 (-(B+i C))+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*b*C)*x)/(2*a^2) - ((I/2)*(2*a*A*b - b^2*(B - I*C) - a^2*(B + I*C))*Log[a + b*Cos[x] - I*b*Sin[x]])/(a^2*b) - ((I*B + C)*(Cos[x] + I*Sin[x]))/(2*a)","A",1,1,26,0.03846,1,"{3130}"
555,1,68,0,0.0675144,"\int \frac{b^2+c^2+a b \cos (x)+a c \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx","Int[(b^2 + c^2 + a*b*Cos[x] + a*c*Sin[x])/(a + b*Cos[x] + c*Sin[x])^2,x]","-\frac{c \cos (x) \left(a^2-b^2-c^2\right)-b \sin (x) \left(a^2-b^2-c^2\right)}{\left(a^2-b^2-c^2\right) (a+b \cos (x)+c \sin (x))}","-\frac{c \cos (x)-b \sin (x)}{a+b \cos (x)+c \sin (x)}",1,"-((c*(a^2 - b^2 - c^2)*Cos[x] - b*(a^2 - b^2 - c^2)*Sin[x])/((a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])))","B",1,1,30,0.03333,1,"{3150}"
556,1,390,0,0.8876844,"\int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx","Int[(a + b*Cos[x] + c*Sin[x])^(5/2)*(d + b*e*Cos[x] + c*e*Sin[x]),x]","-\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(161 a^2 d+15 a^3 e+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))","-\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(161 a^2 d+15 a^3 e+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,"(2*(161*a^2*d + 63*(b^2 + c^2)*d + 15*a^3*e + 145*a*(b^2 + c^2)*e)*EllipticE[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(105*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) - (2*(a^2 - b^2 - c^2)*(56*a*d + 15*a^2*e + 25*(b^2 + c^2)*e)*EllipticF[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(105*Sqrt[a + b*Cos[x] + c*Sin[x]]) - (2*(a + b*Cos[x] + c*Sin[x])^(5/2)*(c*e*Cos[x] - b*e*Sin[x]))/7 - (2*(a + b*Cos[x] + c*Sin[x])^(3/2)*(c*(7*d + 5*a*e)*Cos[x] - b*(7*d + 5*a*e)*Sin[x]))/35 - (2*Sqrt[a + b*Cos[x] + c*Sin[x]]*(c*(56*a*d + 15*a^2*e + 25*(b^2 + c^2)*e)*Cos[x] - b*(56*a*d + 15*a^2*e + 25*(b^2 + c^2)*e)*Sin[x]))/105","A",8,6,27,0.2222,1,"{3146, 3149, 3119, 2653, 3127, 2661}"
557,1,294,0,0.5552425,"\int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx","Int[(a + b*Cos[x] + c*Sin[x])^(3/2)*(d + b*e*Cos[x] + c*e*Sin[x]),x]","-\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))","-\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,"(2*(20*a*d + 3*a^2*e + 9*(b^2 + c^2)*e)*EllipticE[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(15*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) - (2*(a^2 - b^2 - c^2)*(5*d + 3*a*e)*EllipticF[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(15*Sqrt[a + b*Cos[x] + c*Sin[x]]) - (2*(a + b*Cos[x] + c*Sin[x])^(3/2)*(c*e*Cos[x] - b*e*Sin[x]))/5 - (2*Sqrt[a + b*Cos[x] + c*Sin[x]]*(c*(5*d + 3*a*e)*Cos[x] - b*(5*d + 3*a*e)*Sin[x]))/15","A",7,6,27,0.2222,1,"{3146, 3149, 3119, 2653, 3127, 2661}"
558,1,229,0,0.3321877,"\int \sqrt{a+b \cos (x)+c \sin (x)} (d+b e \cos (x)+c e \sin (x)) \, dx","Int[Sqrt[a + b*Cos[x] + c*Sin[x]]*(d + b*e*Cos[x] + c*e*Sin[x]),x]","-\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))","-\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,"(2*(3*d + a*e)*EllipticE[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(3*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) - (2*(a^2 - b^2 - c^2)*e*EllipticF[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(3*Sqrt[a + b*Cos[x] + c*Sin[x]]) - (2*Sqrt[a + b*Cos[x] + c*Sin[x]]*(c*e*Cos[x] - b*e*Sin[x]))/3","A",6,6,27,0.2222,1,"{3146, 3149, 3119, 2653, 3127, 2661}"
559,1,180,0,0.1867915,"\int \frac{d+b e \cos (x)+c e \sin (x)}{\sqrt{a+b \cos (x)+c \sin (x)}} \, dx","Int[(d + b*e*Cos[x] + c*e*Sin[x])/Sqrt[a + b*Cos[x] + c*Sin[x]],x]","\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}}}}","\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*e*EllipticE[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])] + (2*(d - a*e)*EllipticF[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/Sqrt[a + b*Cos[x] + c*Sin[x]]","A",5,5,27,0.1852,1,"{3149, 3119, 2653, 3127, 2661}"
560,1,250,0,0.3230513,"\int \frac{d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{3/2}} \, dx","Int[(d + b*e*Cos[x] + c*e*Sin[x])/(a + b*Cos[x] + c*Sin[x])^(3/2),x]","\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)}}","\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,"(2*(d - a*e)*EllipticE[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/((a^2 - b^2 - c^2)*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) + (2*e*EllipticF[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/Sqrt[a + b*Cos[x] + c*Sin[x]] + (2*(c*(d - a*e)*Cos[x] - b*(d - a*e)*Sin[x]))/((a^2 - b^2 - c^2)*Sqrt[a + b*Cos[x] + c*Sin[x]])","A",6,6,27,0.2222,1,"{3156, 3149, 3119, 2653, 3127, 2661}"
561,1,378,0,0.5611255,"\int \frac{d+b e \cos (x)+c e \sin (x)}{(a+b \cos (x)+c \sin (x))^{5/2}} \, dx","Int[(d + b*e*Cos[x] + c*e*Sin[x])/(a + b*Cos[x] + c*Sin[x])^(5/2),x]","-\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)}}","-\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,"(2*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*EllipticE[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(3*(a^2 - b^2 - c^2)^2*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])]) - (2*(d - a*e)*EllipticF[(x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(3*(a^2 - b^2 - c^2)*Sqrt[a + b*Cos[x] + c*Sin[x]]) + (2*(c*(d - a*e)*Cos[x] - b*(d - a*e)*Sin[x]))/(3*(a^2 - b^2 - c^2)*(a + b*Cos[x] + c*Sin[x])^(3/2)) + (2*(c*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*Cos[x] - b*(4*a*d - a^2*e - 3*(b^2 + c^2)*e)*Sin[x]))/(3*(a^2 - b^2 - c^2)^2*Sqrt[a + b*Cos[x] + c*Sin[x]])","A",7,6,27,0.2222,1,"{3156, 3149, 3119, 2653, 3127, 2661}"
562,1,84,0,0.1518609,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{a+c \sin (d+e x)} \, dx","Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x]),x]","\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}","\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*x)/c + (2*(A*c - a*C)*ArcTan[(c + a*Tan[(d + e*x)/2])/Sqrt[a^2 - c^2]])/(c*Sqrt[a^2 - c^2]*e) + (B*Log[a + c*Sin[d + e*x]])/(c*e)","A",7,7,31,0.2258,1,"{4376, 2735, 2660, 618, 204, 2668, 31}"
563,1,118,0,0.1566256,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^2} \, dx","Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^2,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)}{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))}","\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)*e) - B/(c*e*(a + c*Sin[d + e*x])) + ((A*c - a*C)*Cos[d + e*x])/((a^2 - c^2)*e*(a + c*Sin[d + e*x]))","A",8,8,31,0.2581,1,"{4376, 2754, 12, 2660, 618, 204, 2668, 32}"
564,1,185,0,0.2461841,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^3} \, dx","Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^3,x]","\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}","\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*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)*e) - B/(2*c*e*(a + c*Sin[d + e*x])^2) + ((A*c - a*C)*Cos[d + e*x])/(2*(a^2 - c^2)*e*(a + c*Sin[d + e*x])^2) + ((3*a*A*c - a^2*C - 2*c^2*C)*Cos[d + e*x])/(2*(a^2 - c^2)^2*e*(a + c*Sin[d + e*x]))","A",9,8,31,0.2581,1,"{4376, 2754, 12, 2660, 618, 204, 2668, 32}"
565,1,258,0,0.4046319,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+c \sin (d+e x))^4} \, dx","Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + c*Sin[d + e*x])^4,x]","\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(11 a^2 A c-2 a^3 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{\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{B}{3 c e (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(11 a^2 A c-2 a^3 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{\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{B}{3 c e (a+c \sin (d+e x))^3}",1,"((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)*e) - B/(3*c*e*(a + c*Sin[d + e*x])^3) + ((A*c - a*C)*Cos[d + e*x])/(3*(a^2 - c^2)*e*(a + c*Sin[d + e*x])^3) + ((5*a*A*c - 2*a^2*C - 3*c^2*C)*Cos[d + e*x])/(6*(a^2 - c^2)^2*e*(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])/(6*(a^2 - c^2)^3*e*(a + c*Sin[d + e*x]))","A",10,8,31,0.2581,1,"{4376, 2754, 12, 2660, 618, 204, 2668, 32}"
566,1,131,0,0.1122085,"\int (a+b \cos (c+d x) \sin (c+d x))^m \, dx","Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^m,x]","-\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}}","-\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[1/2, 1/2, -m, 3/2, (1 - Sin[2*c + 2*d*x])/2, (b*(1 - Sin[2*c + 2*d*x]))/(2*a + b)]*Cos[2*c + 2*d*x]*(a + (b*Sin[2*c + 2*d*x])/2)^m)/(Sqrt[2]*d*Sqrt[1 + Sin[2*c + 2*d*x]]*((2*a + b*Sin[2*c + 2*d*x])/(2*a + b))^m))","A",4,4,18,0.2222,1,"{2666, 2665, 139, 138}"
567,1,107,0,0.0823962,"\int (a+b \cos (c+d x) \sin (c+d x))^3 \, dx","Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^3,x]","-\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}","-\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,"(a*(8*a^2 + 3*b^2)*x)/8 - (b*(16*a^2 + b^2)*Cos[2*c + 2*d*x])/(24*d) - (5*a*b^2*Cos[2*c + 2*d*x]*Sin[2*c + 2*d*x])/(48*d) - (b*Cos[2*c + 2*d*x]*(2*a + b*Sin[2*c + 2*d*x])^2)/(48*d)","A",3,3,18,0.1667,1,"{2666, 2656, 2734}"
568,1,61,0,0.0342727,"\int (a+b \cos (c+d x) \sin (c+d x))^2 \, dx","Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^2,x]","\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}","\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,"((8*a^2 + b^2)*x)/8 - (a*b*Cos[2*c + 2*d*x])/(2*d) - (b^2*Cos[2*c + 2*d*x]*Sin[2*c + 2*d*x])/(16*d)","A",2,2,18,0.1111,1,"{2666, 2644}"
569,1,20,0,0.0155313,"\int (a+b \cos (c+d x) \sin (c+d x)) \, dx","Int[a + b*Cos[c + d*x]*Sin[c + d*x],x]","a x+\frac{b \sin ^2(c+d x)}{2 d}","a x+\frac{b \sin ^2(c+d x)}{2 d}",1,"a*x + (b*Sin[c + d*x]^2)/(2*d)","A",3,2,16,0.1250,1,"{2564, 30}"
570,1,48,0,0.0661284,"\int \frac{1}{a+b \cos (c+d x) \sin (c+d x)} \, dx","Int[(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",4,4,18,0.2222,1,"{2666, 2660, 618, 204}"
571,1,95,0,0.1086014,"\int \frac{1}{(a+b \cos (c+d x) \sin (c+d x))^2} \, dx","Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^(-2),x]","\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))}","\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,"(8*a*ArcTan[(b + 2*a*Tan[c + d*x])/Sqrt[4*a^2 - b^2]])/((4*a^2 - b^2)^(3/2)*d) + (2*b*Cos[2*c + 2*d*x])/((4*a^2 - b^2)*d*(2*a + b*Sin[2*c + 2*d*x]))","A",6,6,18,0.3333,1,"{2666, 2664, 12, 2660, 618, 204}"
572,1,149,0,0.177361,"\int \frac{1}{(a+b \cos (c+d x) \sin (c+d x))^3} \, dx","Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^(-3),x]","\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}","\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,"(4*(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)*d) + (2*b*Cos[2*c + 2*d*x])/((4*a^2 - b^2)*d*(2*a + b*Sin[2*c + 2*d*x])^2) + (12*a*b*Cos[2*c + 2*d*x])/((4*a^2 - b^2)^2*d*(2*a + b*Sin[2*c + 2*d*x]))","A",7,7,18,0.3889,1,"{2666, 2664, 2754, 12, 2660, 618, 204}"
573,1,265,0,0.3658479,"\int (a+b \cos (c+d x) \sin (c+d x))^{5/2} \, dx","Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^(5/2),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}","-\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*Sqrt[2]*a*b*Cos[2*c + 2*d*x]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(15*d) - (b*Cos[2*c + 2*d*x]*(2*a + b*Sin[2*c + 2*d*x])^(3/2))/(20*Sqrt[2]*d) + ((92*a^2 + 9*b^2)*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(60*Sqrt[2]*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)]) - (2*Sqrt[2]*a*(4*a^2 - b^2)*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])/(15*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]])","A",8,8,20,0.4000,1,"{2666, 2656, 2753, 2752, 2663, 2661, 2655, 2653}"
574,1,212,0,0.2198182,"\int (a+b \cos (c+d x) \sin (c+d x))^{3/2} \, dx","Int[(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+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}}}","-\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 + 2*d*x]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(6*Sqrt[2]*d) + (2*Sqrt[2]*a*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(3*d*Sqrt[(2*a + b*Sin[2*c + 2*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 + 2*d*x])/(2*a + b)])/(6*Sqrt[2]*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]])","A",7,7,20,0.3500,1,"{2666, 2656, 2752, 2663, 2661, 2655, 2653}"
575,1,76,0,0.0608275,"\int \sqrt{a+b \cos (c+d x) \sin (c+d x)} \, dx","Int[Sqrt[a + b*Cos[c + d*x]*Sin[c + d*x]],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}}}","\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,"(EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(Sqrt[2]*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])","A",3,3,20,0.1500,1,"{2666, 2655, 2653}"
576,1,76,0,0.0665995,"\int \frac{1}{\sqrt{a+b \cos (c+d x) \sin (c+d x)}} \, dx","Int[1/Sqrt[a + b*Cos[c + d*x]*Sin[c + d*x]],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)}}","\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,"(Sqrt[2]*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])/(d*Sqrt[2*a + b*Sin[2*c + 2*d*x]])","A",3,3,20,0.1500,1,"{2666, 2663, 2661}"
577,1,143,0,0.0942034,"\int \frac{1}{(a+b \cos (c+d x) \sin (c+d x))^{3/2}} \, dx","Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^(-3/2),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}}}","\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*Sqrt[2]*b*Cos[2*c + 2*d*x])/((4*a^2 - b^2)*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]]) + (2*Sqrt[2]*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/((4*a^2 - b^2)*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])","A",5,5,20,0.2500,1,"{2666, 2664, 21, 2655, 2653}"
578,1,295,0,0.3006712,"\int \frac{1}{(a+b \cos (c+d x) \sin (c+d x))^{5/2}} \, dx","Int[(a + b*Cos[c + d*x]*Sin[c + d*x])^(-5/2),x]","\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}}}","\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]*b*Cos[2*c + 2*d*x])/(3*(4*a^2 - b^2)*d*(2*a + b*Sin[2*c + 2*d*x])^(3/2)) + (32*Sqrt[2]*a*b*Cos[2*c + 2*d*x])/(3*(4*a^2 - b^2)^2*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]]) + (32*Sqrt[2]*a*EllipticE[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[2*a + b*Sin[2*c + 2*d*x]])/(3*(4*a^2 - b^2)^2*d*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)]) - (4*Sqrt[2]*EllipticF[c - Pi/4 + d*x, (2*b)/(2*a + b)]*Sqrt[(2*a + b*Sin[2*c + 2*d*x])/(2*a + b)])/(3*(4*a^2 - b^2)*d*Sqrt[2*a + b*Sin[2*c + 2*d*x]])","A",8,8,20,0.4000,1,"{2666, 2664, 2754, 2752, 2663, 2661, 2655, 2653}"
579,1,461,0,0.6296727,"\int \frac{x^3}{a+b \cos (x) \sin (x)} \, dx","Int[x^3/(a + b*Cos[x]*Sin[x]),x]","-\frac{3 x^2 \text{PolyLog}\left(2,\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{PolyLog}\left(2,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{2 \sqrt{4 a^2-b^2}}-\frac{3 i x \text{PolyLog}\left(3,\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{PolyLog}\left(3,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{2 \sqrt{4 a^2-b^2}}+\frac{3 \text{PolyLog}\left(4,\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{PolyLog}\left(4,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\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}}","-\frac{3 x^2 \text{PolyLog}\left(2,\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{PolyLog}\left(2,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{2 \sqrt{4 a^2-b^2}}-\frac{3 i x \text{PolyLog}\left(3,\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{PolyLog}\left(3,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{2 \sqrt{4 a^2-b^2}}+\frac{3 \text{PolyLog}\left(4,\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{PolyLog}\left(4,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\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,"((-I)*x^3*Log[1 - (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] + (I*x^3*Log[1 - (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] - (3*x^2*PolyLog[2, (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])])/(2*Sqrt[4*a^2 - b^2]) + (3*x^2*PolyLog[2, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])])/(2*Sqrt[4*a^2 - b^2]) - (((3*I)/2)*x*PolyLog[3, (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] + (((3*I)/2)*x*PolyLog[3, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])])/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]) - (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",13,8,14,0.5714,1,"{4584, 3323, 2264, 2190, 2531, 6609, 2282, 6589}"
580,1,340,0,0.5365794,"\int \frac{x^2}{a+b \cos (x) \sin (x)} \, dx","Int[x^2/(a + b*Cos[x]*Sin[x]),x]","-\frac{x \text{PolyLog}\left(2,\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{PolyLog}\left(2,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{\sqrt{4 a^2-b^2}}-\frac{i \text{PolyLog}\left(3,\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{PolyLog}\left(3,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\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}}","-\frac{x \text{PolyLog}\left(2,\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{PolyLog}\left(2,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\right)}{\sqrt{4 a^2-b^2}}-\frac{i \text{PolyLog}\left(3,\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{PolyLog}\left(3,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\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,"((-I)*x^2*Log[1 - (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] + (I*x^2*Log[1 - (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] - (x*PolyLog[2, (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] + (x*PolyLog[2, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] - ((I/2)*PolyLog[3, (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] + ((I/2)*PolyLog[3, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2]","A",11,7,14,0.5000,1,"{4584, 3323, 2264, 2190, 2531, 2282, 6589}"
581,1,225,0,0.3189764,"\int \frac{x}{a+b \cos (x) \sin (x)} \, dx","Int[x/(a + b*Cos[x]*Sin[x]),x]","-\frac{\text{PolyLog}\left(2,\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{PolyLog}\left(2,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\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}}","-\frac{\text{PolyLog}\left(2,\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{PolyLog}\left(2,\frac{i b e^{2 i x}}{\sqrt{4 a^2-b^2}+2 a}\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,"((-I)*x*Log[1 - (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] + (I*x*Log[1 - (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])])/Sqrt[4*a^2 - b^2] - PolyLog[2, (I*b*E^((2*I)*x))/(2*a - Sqrt[4*a^2 - b^2])]/(2*Sqrt[4*a^2 - b^2]) + PolyLog[2, (I*b*E^((2*I)*x))/(2*a + Sqrt[4*a^2 - b^2])]/(2*Sqrt[4*a^2 - b^2])","A",9,6,12,0.5000,1,"{4584, 3323, 2264, 2190, 2279, 2391}"
582,0,0,0,0.0853994,"\int \frac{1}{x (a+b \cos (x) \sin (x))} \, dx","Int[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,"Defer[Int][1/(x*(a + (b*Sin[2*x])/2)), x]","A",0,0,0,0,-1,"{}"
583,0,0,0,0.1551937,"\int \frac{(b x)^{2-n} \sin ^n(a x)}{(a c x \cos (a x)-c \sin (a x))^2} \, dx","Int[((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,"(b*(b*x)^(1 - n)*Sin[a*x]^(-1 + n))/(a^2*(a*c^2*x*Cos[a*x] - c^2*Sin[a*x])) + (b^2*(1 - n)*Defer[Int][Sin[a*x]^(-2 + n)/(b*x)^n, x])/(a^2*c^2)","A",0,0,0,0,-1,"{}"
584,0,0,0,0.1460154,"\int \frac{(b x)^{2-n} \cos ^n(a x)}{(c \cos (a x)+a c x \sin (a x))^2} \, dx","Int[((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,"-((b*(b*x)^(1 - n)*Cos[a*x]^(-1 + n))/(a^2*(c^2*Cos[a*x] + a*c^2*x*Sin[a*x]))) + (b^2*(1 - n)*Defer[Int][Cos[a*x]^(-2 + n)/(b*x)^n, x])/(a^2*c^2)","A",0,0,0,0,-1,"{}"
585,1,175,0,0.2965648,"\int \frac{\sin ^6(a x)}{x^4 (a x \cos (a x)-\sin (a x))^2} \, dx","Int[Sin[a*x]^6/(x^4*(a*x*Cos[a*x] - Sin[a*x])^2),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{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{\sin ^3(a x) \cos (a x)}{a x^4}+\frac{a \sin (a x) \cos (a x)}{x^2}","-\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{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{\sin ^3(a x) \cos (a x)}{a x^4}+\frac{a \sin (a x) \cos (a x)}{x^2}",1,"a^2/x + (a*Cos[a*x]*Sin[a*x])/x^2 + Sin[a*x]^2/x^3 - (10*a^2*Sin[a*x]^2)/x + (Cos[a*x]*Sin[a*x]^3)/(a*x^4) - (8*a*Cos[a*x]*Sin[a*x]^3)/(3*x^2) + Sin[a*x]^4/(a^2*x^5) - (4*Sin[a*x]^4)/(3*x^3) + (32*a^2*Sin[a*x]^4)/(3*x) + Sin[a*x]^5/(a^2*x^5*(a*x*Cos[a*x] - Sin[a*x])) - (2*a^3*SinIntegral[2*a*x])/3 + (16*a^3*SinIntegral[4*a*x])/3","A",15,6,26,0.2308,1,"{4598, 3314, 30, 3313, 12, 3299}"
586,1,131,0,0.2273891,"\int \frac{\sin ^5(a x)}{x^3 (a x \cos (a x)-\sin (a x))^2} \, dx","Int[Sin[a*x]^5/(x^3*(a*x*Cos[a*x] - Sin[a*x])^2),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{3 \sin ^3(a x)}{2 x^2}+\frac{\sin (a x)}{x^2}+\frac{\sin ^2(a x) \cos (a x)}{a x^3}+\frac{a \cos (a x)}{x}-\frac{9 a \sin ^2(a x) \cos (a x)}{2 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{3 \sin ^3(a x)}{2 x^2}+\frac{\sin (a x)}{x^2}+\frac{\sin ^2(a x) \cos (a x)}{a x^3}+\frac{a \cos (a x)}{x}-\frac{9 a \sin ^2(a x) \cos (a x)}{2 x}",1,"(a*Cos[a*x])/x + Sin[a*x]/x^2 + (Cos[a*x]*Sin[a*x]^2)/(a*x^3) - (9*a*Cos[a*x]*Sin[a*x]^2)/(2*x) + Sin[a*x]^3/(a^2*x^4) - (3*Sin[a*x]^3)/(2*x^2) + Sin[a*x]^4/(a^2*x^4*(a*x*Cos[a*x] - Sin[a*x])) - (a^2*SinIntegral[a*x])/8 + (27*a^2*SinIntegral[3*a*x])/8","A",11,5,26,0.1923,1,"{4598, 3314, 3297, 3299, 3312}"
587,1,80,0,0.130805,"\int \frac{\sin ^4(a x)}{x^2 (a x \cos (a x)-\sin (a x))^2} \, dx","Int[Sin[a*x]^4/(x^2*(a*x*Cos[a*x] - Sin[a*x])^2),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}","\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,"x^(-1) + (Cos[a*x]*Sin[a*x])/(a*x^2) + Sin[a*x]^2/(a^2*x^3) - (2*Sin[a*x]^2)/x + Sin[a*x]^3/(a^2*x^3*(a*x*Cos[a*x] - Sin[a*x])) + 2*a*SinIntegral[2*a*x]","A",6,6,26,0.2308,1,"{4598, 3314, 30, 3313, 12, 3299}"
588,1,56,0,0.1016011,"\int \frac{\sin ^3(a x)}{x (a x \cos (a x)-\sin (a x))^2} \, dx","Int[Sin[a*x]^3/(x*(a*x*Cos[a*x] - Sin[a*x])^2),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}","\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,"Cos[a*x]/(a*x) + Sin[a*x]/(a^2*x^2) + Sin[a*x]^2/(a^2*x^2*(a*x*Cos[a*x] - Sin[a*x])) + SinIntegral[a*x]","A",4,3,26,0.1154,1,"{4598, 3297, 3299}"
589,1,35,0,0.0241799,"\int \frac{\sin ^2(a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx","Int[Sin[a*x]^2/(a*x*Cos[a*x] - Sin[a*x])^2,x]","\frac{1}{a^2 x}+\frac{\sin (a x)}{a^2 x (a x \cos (a x)-\sin (a x))}","\frac{1}{a^2 x}+\frac{\sin (a x)}{a^2 x (a x \cos (a x)-\sin (a x))}",1,"1/(a^2*x) + Sin[a*x]/(a^2*x*(a*x*Cos[a*x] - Sin[a*x]))","A",1,1,23,0.04348,1,"{4596}"
590,1,20,0,0.037818,"\int \frac{x \sin (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx","Int[(x*Sin[a*x])/(a*x*Cos[a*x] - Sin[a*x])^2,x]","\frac{1}{a^2 (a x \cos (a x)-\sin (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",1,1,22,0.04545,1,"{6686}"
591,1,35,0,0.0388193,"\int \frac{x^2}{(a x \cos (a x)-\sin (a x))^2} \, dx","Int[x^2/(a*x*Cos[a*x] - Sin[a*x])^2,x]","\frac{x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\cot (a x)}{a^3}","\frac{x \csc (a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\frac{\cot (a x)}{a^3}",1,"-(Cot[a*x]/a^3) + (x*Csc[a*x])/(a^2*(a*x*Cos[a*x] - Sin[a*x]))","A",3,3,20,0.1500,1,"{4594, 3767, 8}"
592,1,104,0,0.0912974,"\int \frac{x^3 \csc (a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx","Int[(x^3*Csc[a*x])/(a*x*Cos[a*x] - Sin[a*x])^2,x]","\frac{i \text{PolyLog}\left(2,-e^{i a x}\right)}{a^4}-\frac{i \text{PolyLog}\left(2,e^{i a x}\right)}{a^4}+\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\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{i \text{PolyLog}\left(2,-e^{i a x}\right)}{a^4}-\frac{i \text{PolyLog}\left(2,e^{i a x}\right)}{a^4}+\frac{x^2 \csc ^2(a x)}{a^2 (a x \cos (a x)-\sin (a x))}-\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}",1,"(-2*x*ArcTanh[E^(I*a*x)])/a^3 - Csc[a*x]/a^4 - (x*Cot[a*x]*Csc[a*x])/a^3 + (I*PolyLog[2, -E^(I*a*x)])/a^4 - (I*PolyLog[2, E^(I*a*x)])/a^4 + (x^2*Csc[a*x]^2)/(a^2*(a*x*Cos[a*x] - Sin[a*x]))","A",7,5,24,0.2083,1,"{4600, 4185, 4183, 2279, 2391}"
593,1,127,0,0.1814628,"\int \frac{x^4 \csc ^2(a x)}{(a x \cos (a x)-\sin (a x))^2} \, dx","Int[(x^4*Csc[a*x]^2)/(a*x*Cos[a*x] - Sin[a*x])^2,x]","-\frac{2 i \text{PolyLog}\left(2,e^{2 i a x}\right)}{a^5}-\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))}+\frac{4 x \log \left(1-e^{2 i a x}\right)}{a^4}-\frac{\cot (a x)}{a^5}-\frac{x \csc ^2(a x)}{a^4}","-\frac{2 i \text{PolyLog}\left(2,e^{2 i a x}\right)}{a^5}-\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))}+\frac{4 x \log \left(1-e^{2 i a x}\right)}{a^4}-\frac{\cot (a x)}{a^5}-\frac{x \csc ^2(a x)}{a^4}",1,"((-2*I)*x^2)/a^3 - Cot[a*x]/a^5 - (2*x^2*Cot[a*x])/a^3 - (x*Csc[a*x]^2)/a^4 - (x^2*Cot[a*x]*Csc[a*x]^2)/a^3 + (4*x*Log[1 - E^((2*I)*a*x)])/a^4 - ((2*I)*PolyLog[2, E^((2*I)*a*x)])/a^5 + (x^3*Csc[a*x]^3)/(a^2*(a*x*Cos[a*x] - Sin[a*x]))","A",9,9,26,0.3462,1,"{4600, 4186, 3767, 8, 4184, 3717, 2190, 2279, 2391}"
594,1,176,0,0.2992068,"\int \frac{\cos ^6(a x)}{x^4 (\cos (a x)+a x \sin (a x))^2} \, dx","Int[Cos[a*x]^6/(x^4*(Cos[a*x] + a*x*Sin[a*x])^2),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{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{\sin (a x) \cos ^3(a x)}{a x^4}-\frac{a \sin (a x) \cos (a x)}{x^2}","\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{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{\sin (a x) \cos ^3(a x)}{a x^4}-\frac{a \sin (a x) \cos (a x)}{x^2}",1,"a^2/x + Cos[a*x]^2/x^3 - (10*a^2*Cos[a*x]^2)/x + Cos[a*x]^4/(a^2*x^5) - (4*Cos[a*x]^4)/(3*x^3) + (32*a^2*Cos[a*x]^4)/(3*x) - (a*Cos[a*x]*Sin[a*x])/x^2 - (Cos[a*x]^3*Sin[a*x])/(a*x^4) + (8*a*Cos[a*x]^3*Sin[a*x])/(3*x^2) - Cos[a*x]^5/(a^2*x^5*(Cos[a*x] + a*x*Sin[a*x])) + (2*a^3*SinIntegral[2*a*x])/3 + (16*a^3*SinIntegral[4*a*x])/3","A",15,6,24,0.2500,1,"{4599, 3314, 30, 3313, 12, 3299}"
595,1,132,0,0.2260772,"\int \frac{\cos ^5(a x)}{x^3 (\cos (a x)+a x \sin (a x))^2} \, dx","Int[Cos[a*x]^5/(x^3*(Cos[a*x] + a*x*Sin[a*x])^2),x]","-\frac{1}{8} a^2 \text{CosIntegral}(a x)-\frac{27}{8} a^2 \text{CosIntegral}(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{3 \cos ^3(a x)}{2 x^2}+\frac{\cos (a x)}{x^2}-\frac{\sin (a x) \cos ^2(a x)}{a x^3}-\frac{a \sin (a x)}{x}+\frac{9 a \sin (a x) \cos ^2(a x)}{2 x}","-\frac{1}{8} a^2 \text{CosIntegral}(a x)-\frac{27}{8} a^2 \text{CosIntegral}(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{3 \cos ^3(a x)}{2 x^2}+\frac{\cos (a x)}{x^2}-\frac{\sin (a x) \cos ^2(a x)}{a x^3}-\frac{a \sin (a x)}{x}+\frac{9 a \sin (a x) \cos ^2(a x)}{2 x}",1,"Cos[a*x]/x^2 + Cos[a*x]^3/(a^2*x^4) - (3*Cos[a*x]^3)/(2*x^2) - (a^2*CosIntegral[a*x])/8 - (27*a^2*CosIntegral[3*a*x])/8 - (a*Sin[a*x])/x - (Cos[a*x]^2*Sin[a*x])/(a*x^3) + (9*a*Cos[a*x]^2*Sin[a*x])/(2*x) - Cos[a*x]^4/(a^2*x^4*(Cos[a*x] + a*x*Sin[a*x]))","A",11,5,24,0.2083,1,"{4599, 3314, 3297, 3302, 3312}"
596,1,80,0,0.1284256,"\int \frac{\cos ^4(a x)}{x^2 (\cos (a x)+a x \sin (a x))^2} \, dx","Int[Cos[a*x]^4/(x^2*(Cos[a*x] + a*x*Sin[a*x])^2),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}","\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,"x^(-1) + Cos[a*x]^2/(a^2*x^3) - (2*Cos[a*x]^2)/x - (Cos[a*x]*Sin[a*x])/(a*x^2) - Cos[a*x]^3/(a^2*x^3*(Cos[a*x] + a*x*Sin[a*x])) - 2*a*SinIntegral[2*a*x]","A",6,6,24,0.2500,1,"{4599, 3314, 30, 3313, 12, 3299}"
597,1,56,0,0.09317,"\int \frac{\cos ^3(a x)}{x (\cos (a x)+a x \sin (a x))^2} \, dx","Int[Cos[a*x]^3/(x*(Cos[a*x] + a*x*Sin[a*x])^2),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{CosIntegral}(a x)-\frac{\sin (a x)}{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{CosIntegral}(a x)-\frac{\sin (a x)}{a x}",1,"Cos[a*x]/(a^2*x^2) + CosIntegral[a*x] - Sin[a*x]/(a*x) - Cos[a*x]^2/(a^2*x^2*(Cos[a*x] + a*x*Sin[a*x]))","A",4,3,24,0.1250,1,"{4599, 3297, 3302}"
598,1,34,0,0.0224119,"\int \frac{\cos ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx","Int[Cos[a*x]^2/(Cos[a*x] + a*x*Sin[a*x])^2,x]","\frac{1}{a^2 x}-\frac{\cos (a x)}{a^2 x (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,"1/(a^2*x) - Cos[a*x]/(a^2*x*(Cos[a*x] + a*x*Sin[a*x]))","A",1,1,21,0.04762,1,"{4597}"
599,1,19,0,0.0559611,"\int \frac{x \cos (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx","Int[(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",1,1,20,0.05000,1,"{6686}"
600,1,33,0,0.0379801,"\int \frac{x^2}{(\cos (a x)+a x \sin (a x))^2} \, dx","Int[x^2/(Cos[a*x] + a*x*Sin[a*x])^2,x]","\frac{\tan (a x)}{a^3}-\frac{x \sec (a x)}{a^2 (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,"-((x*Sec[a*x])/(a^2*(Cos[a*x] + a*x*Sin[a*x]))) + Tan[a*x]/a^3","A",3,3,18,0.1667,1,"{4595, 3767, 8}"
601,1,110,0,0.0934006,"\int \frac{x^3 \sec (a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx","Int[(x^3*Sec[a*x])/(Cos[a*x] + a*x*Sin[a*x])^2,x]","\frac{i \text{PolyLog}\left(2,-i e^{i a x}\right)}{a^4}-\frac{i \text{PolyLog}\left(2,i e^{i a x}\right)}{a^4}-\frac{x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}-\frac{2 i x \tan ^{-1}\left(e^{i a x}\right)}{a^3}-\frac{\sec (a x)}{a^4}+\frac{x \tan (a x) \sec (a x)}{a^3}","\frac{i \text{PolyLog}\left(2,-i e^{i a x}\right)}{a^4}-\frac{i \text{PolyLog}\left(2,i e^{i a x}\right)}{a^4}-\frac{x^2 \sec ^2(a x)}{a^2 (a x \sin (a x)+\cos (a x))}-\frac{2 i x \tan ^{-1}\left(e^{i a x}\right)}{a^3}-\frac{\sec (a x)}{a^4}+\frac{x \tan (a x) \sec (a x)}{a^3}",1,"((-2*I)*x*ArcTan[E^(I*a*x)])/a^3 + (I*PolyLog[2, (-I)*E^(I*a*x)])/a^4 - (I*PolyLog[2, I*E^(I*a*x)])/a^4 - Sec[a*x]/a^4 - (x^2*Sec[a*x]^2)/(a^2*(Cos[a*x] + a*x*Sin[a*x])) + (x*Sec[a*x]*Tan[a*x])/a^3","A",7,5,22,0.2273,1,"{4601, 4185, 4181, 2279, 2391}"
602,1,124,0,0.1832911,"\int \frac{x^4 \sec ^2(a x)}{(\cos (a x)+a x \sin (a x))^2} \, dx","Int[(x^4*Sec[a*x]^2)/(Cos[a*x] + a*x*Sin[a*x])^2,x]","-\frac{2 i \text{PolyLog}\left(2,-e^{2 i a x}\right)}{a^5}-\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))}+\frac{4 x \log \left(1+e^{2 i a x}\right)}{a^4}+\frac{\tan (a x)}{a^5}-\frac{x \sec ^2(a x)}{a^4}","-\frac{2 i \text{PolyLog}\left(2,-e^{2 i a x}\right)}{a^5}-\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))}+\frac{4 x \log \left(1+e^{2 i a x}\right)}{a^4}+\frac{\tan (a x)}{a^5}-\frac{x \sec ^2(a x)}{a^4}",1,"((-2*I)*x^2)/a^3 + (4*x*Log[1 + E^((2*I)*a*x)])/a^4 - ((2*I)*PolyLog[2, -E^((2*I)*a*x)])/a^5 - (x*Sec[a*x]^2)/a^4 - (x^3*Sec[a*x]^3)/(a^2*(Cos[a*x] + a*x*Sin[a*x])) + Tan[a*x]/a^5 + (2*x^2*Tan[a*x])/a^3 + (x^2*Sec[a*x]^2*Tan[a*x])/a^3","A",9,9,24,0.3750,1,"{4601, 4186, 3767, 8, 4184, 3719, 2190, 2279, 2391}"
603,1,157,0,0.4451674,"\int \sec ^4(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Int[Sec[2*(a + b*x)]^4*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\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}}","\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,"(-2*c*Tan[2*a + 2*b*x])/(5*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c*Sec[2*a + 2*b*x]^3*Tan[2*a + 2*b*x])/(7*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (4*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(35*b) - (6*(-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(35*b*c)","A",5,5,31,0.1613,1,"{4397, 3803, 3800, 4001, 3792}"
604,1,110,0,0.2762,"\int \sec ^3(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Int[Sec[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\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}}","\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,"(7*c*Tan[2*a + 2*b*x])/(15*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (2*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(15*b) + ((-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(5*b*c)","A",4,4,31,0.1290,1,"{4397, 3800, 4001, 3792}"
605,1,72,0,0.1989441,"\int \sec ^2(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Int[Sec[2*(a + b*x)]^2*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\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}}","\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,"-(c*Tan[2*a + 2*b*x])/(3*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(3*b)","A",3,3,31,0.09677,1,"{4397, 3798, 3792}"
606,1,33,0,0.0649942,"\int \sec (2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Int[Sec[2*(a + b*x)]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\frac{c \tan (2 a+2 b x)}{b \sqrt{c \sec (2 a+2 b x)-c}}","\frac{c \tan (2 a+2 b x)}{b \sqrt{c \sec (2 a+2 b x)-c}}",1,"(c*Tan[2*a + 2*b*x])/(b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",2,2,29,0.06897,1,"{4397, 3792}"
607,1,45,0,0.0404964,"\int \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Int[Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","-\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}","-\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,"-((Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/b)","A",3,3,20,0.1500,1,"{4397, 3774, 207}"
608,1,84,0,0.13946,"\int \cos (2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Int[Cos[2*(a + b*x)]*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\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}}","\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,"(Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(2*b) - (c*Sin[2*a + 2*b*x])/(2*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",4,4,29,0.1379,1,"{4397, 3805, 3774, 207}"
609,1,129,0,0.2143314,"\int \cos ^2(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Int[Cos[2*(a + b*x)]^2*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\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}","\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[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(8*b) + (3*c*Sin[2*a + 2*b*x])/(8*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (c*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(4*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",5,4,31,0.1290,1,"{4397, 3805, 3774, 207}"
610,1,176,0,0.2881505,"\int \cos ^3(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx","Int[Cos[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","-\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}","-\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,"(5*Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(16*b) - (5*c*Sin[2*a + 2*b*x])/(16*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (5*c*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(24*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (c*Cos[2*a + 2*b*x]^2*Sin[2*a + 2*b*x])/(6*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",6,4,31,0.1290,1,"{4397, 3805, 3774, 207}"
611,1,208,0,0.5276788,"\int \sec ^4(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Int[Sec[2*(a + b*x)]^4*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\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}","\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,"(34*c^2*Tan[2*a + 2*b*x])/(45*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (17*c^2*Sec[2*a + 2*b*x]^3*Tan[2*a + 2*b*x])/(63*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c^2*Sec[2*a + 2*b*x]^4*Tan[2*a + 2*b*x])/(9*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (68*c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(315*b) + (34*(-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(105*b)","A",7,7,31,0.2258,1,"{4397, 3814, 21, 3803, 3800, 4001, 3792}"
612,1,148,0,0.347543,"\int \sec ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Int[Sec[2*(a + b*x)]^3*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\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}","-\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,"(-76*c^2*Tan[2*a + 2*b*x])/(105*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (19*c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(105*b) + (2*(-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(35*b) + ((-c + c*Sec[2*a + 2*b*x])^(5/2)*Tan[2*a + 2*b*x])/(7*b*c)","A",5,5,31,0.1613,1,"{4397, 3800, 4001, 3793, 3792}"
613,1,110,0,0.268276,"\int \sec ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Int[Sec[2*(a + b*x)]^2*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\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}","\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,"(4*c^2*Tan[2*a + 2*b*x])/(5*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(5*b) + ((-c + c*Sec[2*a + 2*b*x])^(3/2)*Tan[2*a + 2*b*x])/(5*b)","A",4,4,31,0.1290,1,"{4397, 3798, 3793, 3792}"
614,1,75,0,0.1095762,"\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Int[Sec[2*(a + b*x)]*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\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}}","\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,"(-4*c^2*Tan[2*a + 2*b*x])/(3*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(3*b)","A",3,3,29,0.1034,1,"{4397, 3793, 3792}"
615,1,80,0,0.0597835,"\int (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Int[(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{c^2 \tan (2 a+2 b x)}{b \sqrt{c \sec (2 a+2 b x)-c}}+\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}}+\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}",1,"(c^(3/2)*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/b + (c^2*Tan[2*a + 2*b*x])/(b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",5,5,20,0.2500,1,"{4397, 3775, 21, 3774, 207}"
616,1,86,0,0.2222636,"\int \cos (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Int[Cos[2*(a + b*x)]*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\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}","\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,"(-3*c^(3/2)*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(2*b) + (c^2*Sin[2*a + 2*b*x])/(2*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",6,6,29,0.2069,1,"{4397, 3814, 21, 3805, 3774, 207}"
617,1,133,0,0.2564413,"\int \cos ^2(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Int[Cos[2*(a + b*x)]^2*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\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}}+\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}}+\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}",1,"(7*c^(3/2)*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(8*b) - (7*c^2*Sin[2*a + 2*b*x])/(8*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c^2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(4*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",6,6,31,0.1935,1,"{4397, 3813, 21, 3805, 3774, 207}"
618,1,182,0,0.3120897,"\int \cos ^3(2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx","Int[Cos[2*(a + b*x)]^3*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\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}}-\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}}-\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}",1,"(-11*c^(3/2)*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(16*b) + (11*c^2*Sin[2*a + 2*b*x])/(16*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (11*c^2*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(24*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c^2*Cos[2*a + 2*b*x]^2*Sin[2*a + 2*b*x])/(6*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",7,6,31,0.1935,1,"{4397, 3813, 21, 3805, 3774, 207}"
619,1,175,0,0.5997766,"\int \frac{\sec ^4(2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Int[Sec[2*(a + b*x)]^4/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],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}}","\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,"-(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c])) + (14*Tan[2*a + 2*b*x])/(15*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Sec[2*a + 2*b*x]^2*Tan[2*a + 2*b*x])/(5*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(15*b*c)","A",6,6,31,0.1935,1,"{4397, 3822, 4010, 4001, 3795, 207}"
620,1,129,0,0.3594696,"\int \frac{\sec ^3(2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Int[Sec[2*(a + b*x)]^3/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]","\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}}","\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,"-(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c])) + (2*Tan[2*a + 2*b*x])/(3*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(3*b*c)","A",5,5,31,0.1613,1,"{4397, 3800, 4001, 3795, 207}"
621,1,88,0,0.2377191,"\int \frac{\sec ^2(2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Int[Sec[2*(a + b*x)]^2/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],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}}","\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,"-(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c])) + Tan[2*a + 2*b*x]/(b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",4,4,31,0.1290,1,"{4397, 3798, 3795, 207}"
622,1,55,0,0.0761244,"\int \frac{\sec (2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Int[Sec[2*(a + b*x)]/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],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}}","-\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,"-(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c]))","A",3,3,29,0.1034,1,"{4397, 3795, 207}"
623,1,100,0,0.0899419,"\int \frac{1}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Int[1/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],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}}","\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,"ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]]/(b*Sqrt[c]) - ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c])","A",6,5,20,0.2500,1,"{4397, 3776, 3774, 207, 3795}"
624,1,138,0,0.2788312,"\int \frac{\cos (2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Int[Cos[2*(a + b*x)]/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],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}}","\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,"ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]]/(2*b*Sqrt[c]) - ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c]) + Sin[2*a + 2*b*x]/(2*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",7,6,29,0.2069,1,"{4397, 3823, 3904, 3887, 481, 206}"
625,1,182,0,0.3697532,"\int \frac{\cos ^2(2 (a+b x))}{\sqrt{c \tan (a+b x) \tan (2 (a+b x))}} \, dx","Int[Cos[2*(a + b*x)]^2/Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],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}}","\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,"(7*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(8*b*Sqrt[c]) - ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(Sqrt[2]*b*Sqrt[c]) + Sin[2*a + 2*b*x]/(8*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(4*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",8,7,31,0.2258,1,"{4397, 3823, 4022, 3920, 3774, 207, 3795}"
626,1,180,0,0.5130033,"\int \frac{\sec ^4(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Int[Sec[2*(a + b*x)]^4/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","\frac{7 \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{12 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{\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}}","\frac{7 \tan (2 a+2 b x) \sqrt{c \sec (2 a+2 b x)-c}}{12 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{\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,"(-11*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - (Sec[2*a + 2*b*x]^2*Tan[2*a + 2*b*x])/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2)) + (13*Tan[2*a + 2*b*x])/(6*b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (7*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b*x])/(12*b*c^2)","A",6,6,31,0.1935,1,"{4397, 3816, 4010, 4001, 3795, 207}"
627,1,128,0,0.3063614,"\int \frac{\sec ^3(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Int[Sec[2*(a + b*x)]^3/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\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}}","-\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,"(-7*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - Tan[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2)) + Tan[2*a + 2*b*x]/(b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",5,5,31,0.1613,1,"{4397, 3799, 4001, 3795, 207}"
628,1,93,0,0.2361043,"\int \frac{\sec ^2(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Int[Sec[2*(a + b*x)]^2/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\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}}","-\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,"(-3*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - Tan[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2))","A",4,4,31,0.1290,1,"{4397, 3797, 3795, 207}"
629,1,93,0,0.1212398,"\int \frac{\sec (2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Int[Sec[2*(a + b*x)]/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),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)}{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}}","\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,"ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])]/(4*Sqrt[2]*b*c^(3/2)) - Tan[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2))","A",4,4,29,0.1379,1,"{4397, 3796, 3795, 207}"
630,1,138,0,0.142754,"\int \frac{1}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Int[(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(-3/2),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}}","-\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,"-(ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]]/(b*c^(3/2))) + (5*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - Tan[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2))","A",7,6,20,0.3000,1,"{4397, 3777, 3920, 3774, 207, 3795}"
631,1,178,0,0.3196221,"\int \frac{\cos (2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Int[Cos[2*(a + b*x)]/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\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}}","-\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*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(2*b*c^(3/2)) + (9*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - Sin[2*a + 2*b*x]/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2)) - (3*Sin[2*a + 2*b*x])/(4*b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",8,7,29,0.2414,1,"{4397, 3817, 4022, 3920, 3774, 207, 3795}"
632,1,234,0,0.4967409,"\int \frac{\cos ^2(2 (a+b x))}{(c \tan (a+b x) \tan (2 (a+b x)))^{3/2}} \, dx","Int[Cos[2*(a + b*x)]^2/(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]","-\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}}","-\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,"(-19*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(8*b*c^(3/2)) + (13*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/(Sqrt[2]*Sqrt[-c + c*Sec[2*a + 2*b*x]])])/(4*Sqrt[2]*b*c^(3/2)) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(4*b*(-c + c*Sec[2*a + 2*b*x])^(3/2)) - (7*Sin[2*a + 2*b*x])/(8*b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]]) - (Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(2*b*c*Sqrt[-c + c*Sec[2*a + 2*b*x]])","A",9,7,31,0.2258,1,"{4397, 3817, 4022, 3920, 3774, 207, 3795}"
633,1,16,0,0.0868706,"\int \frac{\cot (x) \csc (x)}{\sqrt{\sin (2 x)}} \, dx","Int[(Cot[x]*Csc[x])/Sqrt[Sin[2*x]],x]","-\frac{2 \cos (x) \cot (x)}{3 \sqrt{\sin (2 x)}}","-\frac{2 \cos (x) \cot (x)}{3 \sqrt{\sin (2 x)}}",1,"(-2*Cos[x]*Cot[x])/(3*Sqrt[Sin[2*x]])","A",3,2,13,0.1538,1,"{4390, 30}"
634,1,69,0,0.3638891,"\int \frac{\csc ^2(x) \sec (x)}{\sqrt{\sin (2 x)} (-2+\tan (x))} \, dx","Int[(Csc[x]^2*Sec[x])/(Sqrt[Sin[2*x]]*(-2 + Tan[x])),x]","\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)}}","\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,"Cos[x]/(2*Sqrt[Sin[2*x]]) + (Cos[x]*Cot[x])/(3*Sqrt[Sin[2*x]]) - (5*ArcTanh[Sqrt[Tan[x]]/Sqrt[2]]*Sin[x])/(2*Sqrt[2]*Sqrt[Sin[2*x]]*Sqrt[Tan[x]])","A",6,4,21,0.1905,1,"{4390, 898, 1262, 207}"
635,1,79,0,0.5670158,"\int \frac{\cos ^2(x) \sin (x)}{\left(\sin ^2(x)-\sin (2 x)\right) \sin ^{\frac{5}{2}}(2 x)} \, dx","Int[(Cos[x]^2*Sin[x])/((Sin[x]^2 - Sin[2*x])*Sin[2*x]^(5/2)),x]","\frac{\sin ^2(x) \cos ^3(x)}{2 \sin ^{\frac{5}{2}}(2 x)}+\frac{\sin (x) \cos ^4(x)}{3 \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)}","\frac{\sin ^2(x) \cos ^3(x)}{2 \sin ^{\frac{5}{2}}(2 x)}+\frac{\sin (x) \cos ^4(x)}{3 \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,"(Cos[x]^4*Sin[x])/(3*Sin[2*x]^(5/2)) + (Cos[x]^3*Sin[x]^2)/(2*Sin[2*x]^(5/2)) - (5*ArcTanh[Sqrt[Tan[x]]/Sqrt[2]]*Sin[x]^5)/(2*Sqrt[2]*Sin[2*x]^(5/2)*Tan[x]^(5/2))","A",6,4,28,0.1429,1,"{4390, 898, 1262, 207}"
636,1,95,0,0.5757657,"\int \frac{\cos ^3(x) \cos (2 x)}{\left(\sin ^2(x)-\sin (2 x)\right) \sin ^{\frac{5}{2}}(2 x)} \, dx","Int[(Cos[x]^3*Cos[2*x])/((Sin[x]^2 - Sin[2*x])*Sin[2*x]^(5/2)),x]","\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)}","\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,"Cos[x]^5/(5*Sin[2*x]^(5/2)) + (Cos[x]^4*Sin[x])/(6*Sin[2*x]^(5/2)) - (3*Cos[x]^3*Sin[x]^2)/(4*Sin[2*x]^(5/2)) + (3*ArcTanh[Sqrt[Tan[x]]/Sqrt[2]]*Sin[x]^5)/(4*Sqrt[2]*Sin[2*x]^(5/2)*Tan[x]^(5/2))","A",6,4,30,0.1333,1,"{4390, 898, 1262, 207}"
637,1,30,0,0.0592225,"\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","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]),x]","\frac{(a \sin (c+d x)+b \sec (c+d x))^{n+1}}{d (n+1)}","\frac{(a \sin (c+d x)+b \sec (c+d x))^{n+1}}{d (n+1)}",1,"(b*Sec[c + d*x] + a*Sin[c + d*x])^(1 + n)/(d*(1 + n))","A",1,1,43,0.02326,1,"{4385}"
638,1,26,0,0.0441515,"\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","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]),x]","\frac{(a \sin (c+d x)+b \sec (c+d x))^4}{4 d}","\frac{(a \sin (c+d x)+b \sec (c+d x))^4}{4 d}",1,"(b*Sec[c + d*x] + a*Sin[c + d*x])^4/(4*d)","A",1,1,43,0.02326,1,"{4385}"
639,1,26,0,0.0429013,"\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","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]),x]","\frac{(a \sin (c+d x)+b \sec (c+d x))^3}{3 d}","\frac{(a \sin (c+d x)+b \sec (c+d x))^3}{3 d}",1,"(b*Sec[c + d*x] + a*Sin[c + d*x])^3/(3*d)","A",1,1,43,0.02326,1,"{4385}"
640,1,26,0,0.0280635,"\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","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]),x]","\frac{(a \sin (c+d x)+b \sec (c+d x))^2}{2 d}","\frac{(a \sin (c+d x)+b \sec (c+d x))^2}{2 d}",1,"(b*Sec[c + d*x] + a*Sin[c + d*x])^2/(2*d)","A",1,1,41,0.02439,1,"{4385}"
641,1,22,0,0.0484212,"\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","Int[(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 (c+d x)+b \sec (c+d x))}{d}","\frac{\log (a \sin (c+d x)+b \sec (c+d x))}{d}",1,"Log[b*Sec[c + d*x] + a*Sin[c + d*x]]/d","A",1,1,43,0.02326,1,"{4383}"
642,1,24,0,0.0441496,"\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","Int[(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{1}{d (a \sin (c+d x)+b \sec (c+d x))}","-\frac{1}{d (a \sin (c+d x)+b \sec (c+d x))}",1,"-(1/(d*(b*Sec[c + d*x] + a*Sin[c + d*x])))","A",1,1,43,0.02326,1,"{4385}"
643,1,26,0,0.045177,"\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","Int[(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{1}{2 d (a \sin (c+d x)+b \sec (c+d x))^2}","-\frac{1}{2 d (a \sin (c+d x)+b \sec (c+d x))^2}",1,"-1/(2*d*(b*Sec[c + d*x] + a*Sin[c + d*x])^2)","A",1,1,43,0.02326,1,"{4385}"
644,0,0,0,0.0128501,"\int F(c,d,\cos (a+b x),r,s) \sin (a+b x) \, dx","Int[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,"-(Defer[Subst][Defer[Int][F[c, d, x, r, s], x], x, Cos[a + b*x]]/b)","A",0,0,0,0,-1,"{}"
645,0,0,0,0.0127099,"\int \cos (a+b x) F(c,d,\sin (a+b x),r,s) \, dx","Int[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,"Defer[Subst][Defer[Int][F[c, d, x, r, s], x], x, Sin[a + b*x]]/b","A",0,0,0,0,-1,"{}"
646,0,0,0,0.0161296,"\int F(c,d,\tan (a+b x),r,s) \sec ^2(a+b x) \, dx","Int[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,"Defer[Subst][Defer[Int][F[c, d, x, r, s], x], x, Tan[a + b*x]]/b","A",0,0,0,0,-1,"{}"
647,0,0,0,0.0166877,"\int \csc ^2(a+b x) F(c,d,\cot (a+b x),r,s) \, dx","Int[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,"-(Defer[Subst][Defer[Int][F[c, d, x, r, s], x], x, Cot[a + b*x]]/b)","A",0,0,0,0,-1,"{}"
648,1,12,0,0.0223243,"\int \frac{\sin (x)}{a+b \cos (x)} \, dx","Int[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",2,2,11,0.1818,1,"{2668, 31}"
649,1,20,0,0.0245843,"\int (a+b \cos (x))^n \sin (x) \, dx","Int[(a + b*Cos[x])^n*Sin[x],x]","-\frac{(a+b \cos (x))^{n+1}}{b (n+1)}","-\frac{(a+b \cos (x))^{n+1}}{b (n+1)}",1,"-((a + b*Cos[x])^(1 + n)/(b*(1 + n)))","A",2,2,11,0.1818,1,"{2668, 32}"
650,1,5,0,0.022591,"\int \frac{\sin (x)}{\sqrt{1+\cos ^2(x)}} \, dx","Int[Sin[x]/Sqrt[1 + Cos[x]^2],x]","-\sinh ^{-1}(\cos (x))","-\sinh ^{-1}(\cos (x))",1,"-ArcSinh[Cos[x]]","A",2,2,13,0.1538,1,"{3190, 215}"
651,1,5,0,0.0090848,"\int \cos (\cos (x)) \sin (x) \, dx","Int[Cos[Cos[x]]*Sin[x],x]","-\sin (\cos (x))","-\sin (\cos (x))",1,"-Sin[Cos[x]]","A",2,2,6,0.3333,1,"{4335, 2637}"
652,1,28,0,0.0255663,"\int \cos (x) \cos (\cos (x)) \sin (x) \sin (\cos (x)) \, dx","Int[Cos[x]*Cos[Cos[x]]*Sin[x]*Sin[Cos[x]],x]","\frac{\cos (x)}{4}-\frac{1}{2} \cos (x) \sin ^2(\cos (x))-\frac{1}{4} \cos (\cos (x)) \sin (\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]/4 - (Cos[Cos[x]]*Sin[Cos[x]])/4 - (Cos[x]*Sin[Cos[x]]^2)/2","A",4,4,11,0.3636,1,"{4335, 3443, 2635, 8}"
653,1,26,0,0.0464201,"\int \cos (\cos (x)) \sin (x) \sin ^2(6 \cos (x)) \, dx","Int[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,"-Sin[Cos[x]]/2 + Sin[11*Cos[x]]/44 + Sin[13*Cos[x]]/52","A",6,3,13,0.2308,1,"{4335, 4354, 2637}"
654,1,36,0,0.0872477,"\int \cos ^3(x) \left(a+b \cos ^2(x)\right)^3 \sin (x) \, dx","Int[Cos[x]^3*(a + b*Cos[x]^2)^3*Sin[x],x]","\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}","\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,"(a*(a + b*Cos[x]^2)^4)/(8*b^2) - (a + b*Cos[x]^2)^5/(10*b^2)","A",4,3,17,0.1765,1,"{4335, 266, 43}"
655,1,9,0,0.0107249,"\int \sin (3 x) \sin (\cos (3 x)) \, dx","Int[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",2,2,10,0.2000,1,"{4335, 2638}"
656,1,31,0,0.0226498,"\int e^{\cos (1+3 x)} \cos (1+3 x) \sin (1+3 x) \, dx","Int[E^Cos[1 + 3*x]*Cos[1 + 3*x]*Sin[1 + 3*x],x]","\frac{1}{3} e^{\cos (3 x+1)}-\frac{1}{3} e^{\cos (3 x+1)} \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,"E^Cos[1 + 3*x]/3 - (E^Cos[1 + 3*x]*Cos[1 + 3*x])/3","A",3,3,21,0.1429,1,"{4335, 2176, 2194}"
657,1,9,0,0.0718241,"\int \frac{\cos ^2(x) \sin (x)}{\sqrt{1-\cos ^6(x)}} \, dx","Int[(Cos[x]^2*Sin[x])/Sqrt[1 - Cos[x]^6],x]","-\frac{1}{3} \sin ^{-1}\left(\cos ^3(x)\right)","-\frac{1}{3} \sin ^{-1}\left(\cos ^3(x)\right)",1,"-ArcSin[Cos[x]^3]/3","A",3,3,19,0.1579,1,"{4335, 275, 216}"
658,1,71,0,0.0658991,"\int \frac{\sin ^5(x)}{\sqrt{1-5 \cos (x)}} \, dx","Int[Sin[x]^5/Sqrt[1 - 5*Cos[x]],x]","\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}","\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,"(1152*Sqrt[1 - 5*Cos[x]])/3125 + (64*(1 - 5*Cos[x])^(3/2))/3125 - (88*(1 - 5*Cos[x])^(5/2))/15625 - (8*(1 - 5*Cos[x])^(7/2))/21875 + (2*(1 - 5*Cos[x])^(9/2))/28125","A",3,2,15,0.1333,1,"{2668, 697}"
659,1,18,0,0.0144725,"\int e^{n \cos (a+b x)} \sin (a+b x) \, dx","Int[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",2,2,17,0.1176,1,"{4335, 2194}"
660,1,23,0,0.0149893,"\int e^{n \cos (a c+b c x)} \sin (c (a+b x)) \, dx","Int[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",2,2,22,0.09091,1,"{4335, 2194}"
661,1,24,0,0.0144949,"\int e^{n \cos (c (a+b x))} \sin (a c+b c x) \, dx","Int[E^(n*Cos[c*(a + b*x)])*Sin[a*c + b*c*x],x]","-\frac{e^{n \cos (a c+b c x)}}{b c n}","-\frac{e^{n \cos (a c+b c x)}}{b c n}",1,"-(E^(n*Cos[a*c + b*c*x])/(b*c*n))","A",2,2,22,0.09091,1,"{4335, 2194}"
662,1,14,0,0.0215174,"\int e^{n \cos (a+b x)} \tan (a+b x) \, dx","Int[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",2,2,17,0.1176,1,"{4339, 2178}"
663,1,19,0,0.0219697,"\int e^{n \cos (a c+b c x)} \tan (c (a+b x)) \, dx","Int[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",2,2,22,0.09091,1,"{4339, 2178}"
664,1,20,0,0.0231347,"\int e^{n \cos (c (a+b x))} \tan (a c+b c x) \, dx","Int[E^(n*Cos[c*(a + b*x)])*Tan[a*c + b*c*x],x]","-\frac{\text{Ei}(n \cos (a c+b x c))}{b c}","-\frac{\text{Ei}(n \cos (a c+b x c))}{b c}",1,"-(ExpIntegralEi[n*Cos[a*c + b*c*x]]/(b*c))","A",2,2,22,0.09091,1,"{4339, 2178}"
665,1,11,0,0.0218965,"\int \frac{\cos (x)}{a+b \sin (x)} \, dx","Int[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",2,2,11,0.1818,1,"{2668, 31}"
666,1,19,0,0.0216845,"\int \cos (x) (a+b \sin (x))^n \, dx","Int[Cos[x]*(a + b*Sin[x])^n,x]","\frac{(a+b \sin (x))^{n+1}}{b (n+1)}","\frac{(a+b \sin (x))^{n+1}}{b (n+1)}",1,"(a + b*Sin[x])^(1 + n)/(b*(1 + n))","A",2,2,11,0.1818,1,"{2668, 32}"
667,1,3,0,0.0227818,"\int \frac{\cos (x)}{\sqrt{1+\sin ^2(x)}} \, dx","Int[Cos[x]/Sqrt[1 + Sin[x]^2],x]","\sinh ^{-1}(\sin (x))","\sinh ^{-1}(\sin (x))",1,"ArcSinh[Sin[x]]","A",2,2,13,0.1538,1,"{3190, 215}"
668,1,7,0,0.0249788,"\int \frac{\cos (x)}{\sqrt{4-\sin ^2(x)}} \, dx","Int[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",2,2,15,0.1333,1,"{3190, 216}"
669,1,13,0,0.0263306,"\int \frac{\cos (3 x)}{\sqrt{4-\sin ^2(3 x)}} \, dx","Int[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",2,2,19,0.1053,1,"{3190, 216}"
670,1,21,0,0.0239342,"\int \cos (x) \sqrt{1+\csc (x)} \, dx","Int[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",4,4,11,0.3636,1,"{3873, 47, 63, 207}"
671,1,28,0,0.026376,"\int \cos (x) \sqrt{4-\sin ^2(x)} \, dx","Int[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",3,3,15,0.2000,1,"{3190, 195, 216}"
672,1,14,0,0.0341567,"\int \cos (x) \sin (x) \sqrt{1+\sin ^2(x)} \, dx","Int[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",2,2,15,0.1333,1,"{3198, 261}"
673,1,19,0,0.0316316,"\int \frac{\cos (x)}{\sqrt{2 \sin (x)+\sin ^2(x)}} \, dx","Int[Cos[x]/Sqrt[2*Sin[x] + Sin[x]^2],x]","2 \tanh ^{-1}\left(\frac{\sin (x)}{\sqrt{\sin ^2(x)+2 \sin (x)}}\right)","2 \tanh ^{-1}\left(\frac{\sin (x)}{\sqrt{\sin ^2(x)+2 \sin (x)}}\right)",1,"2*ArcTanh[Sin[x]/Sqrt[2*Sin[x] + Sin[x]^2]]","A",3,3,16,0.1875,1,"{3258, 620, 206}"
674,1,3,0,0.0084907,"\int \cos (x) \cos (\sin (x)) \, dx","Int[Cos[x]*Cos[Sin[x]],x]","\sin (\sin (x))","\sin (\sin (x))",1,"Sin[Sin[x]]","A",2,2,6,0.3333,1,"{4334, 2637}"
675,1,4,0,0.0216749,"\int \cos (x) \cos (\sin (x)) \cos (\sin (\sin (x))) \, dx","Int[Cos[x]*Cos[Sin[x]]*Cos[Sin[Sin[x]]],x]","\sin (\sin (\sin (x)))","\sin (\sin (\sin (x)))",1,"Sin[Sin[Sin[x]]]","A",3,2,10,0.2000,1,"{4334, 2637}"
676,1,4,0,0.0069563,"\int \cos (x) \sec (\sin (x)) \, dx","Int[Cos[x]*Sec[Sin[x]],x]","\tanh ^{-1}(\sin (\sin (x)))","\tanh ^{-1}(\sin (\sin (x)))",1,"ArcTanh[Sin[Sin[x]]]","A",2,2,6,0.3333,1,"{4334, 3770}"
677,1,36,0,0.076912,"\int \cos (x) \sin ^3(x) \left(a+b \sin ^2(x)\right)^3 \, dx","Int[Cos[x]*Sin[x]^3*(a + b*Sin[x]^2)^3,x]","\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}","\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,"-(a*(a + b*Sin[x]^2)^4)/(8*b^2) + (a + b*Sin[x]^2)^5/(10*b^2)","A",4,3,17,0.1765,1,"{3198, 266, 43}"
678,1,14,0,0.0163624,"\int e^{\sin (x)} \cos (x) \sin (x) \, dx","Int[E^Sin[x]*Cos[x]*Sin[x],x]","e^{\sin (x)} \sin (x)-e^{\sin (x)}","e^{\sin (x)} \sin (x)-e^{\sin (x)}",1,"-E^Sin[x] + E^Sin[x]*Sin[x]","A",3,3,9,0.3333,1,"{4334, 2176, 2194}"
679,1,25,0,0.0488918,"\int \frac{\cos ^3(x)}{\sqrt{\sin ^3(x)}} \, dx","Int[Cos[x]^3/Sqrt[Sin[x]^3],x]","-\frac{2 \sin (x)}{\sqrt{\sin ^3(x)}}-\frac{2}{3} \sqrt{\sin ^3(x)}","-\frac{2 \sin (x)}{\sqrt{\sin ^3(x)}}-\frac{2}{3} \sqrt{\sin ^3(x)}",1,"(-2*Sin[x])/Sqrt[Sin[x]^3] - (2*Sqrt[Sin[x]^3])/3","A",4,3,13,0.2308,1,"{3207, 2564, 14}"
680,1,10,0,0.026561,"\int \frac{e^{\sqrt{\sin (x)}} \cos (x)}{\sqrt{\sin (x)}} \, dx","Int[(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",2,2,17,0.1176,1,"{4334, 2209}"
681,1,6,0,0.0095759,"\int e^{4+\sin (x)} \cos (x) \, dx","Int[E^(4 + Sin[x])*Cos[x],x]","e^{\sin (x)+4}","e^{\sin (x)+4}",1,"E^(4 + Sin[x])","A",2,2,9,0.2222,1,"{4334, 2194}"
682,1,10,0,0.0117849,"\int e^{\cos (x) \sin (x)} \cos (2 x) \, dx","Int[E^(Cos[x]*Sin[x])*Cos[2*x],x]","e^{\frac{1}{2} \sin (2 x)}","e^{\frac{1}{2} \sin (2 x)}",1,"E^(Sin[2*x]/2)","A",2,2,12,0.1667,1,"{4356, 2194}"
683,1,10,0,0.0109039,"\int e^{\cos \left(\frac{x}{2}\right) \sin \left(\frac{x}{2}\right)} \cos (x) \, dx","Int[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",2,2,18,0.1111,1,"{4356, 2194}"
684,1,17,0,0.0126711,"\int e^{n \sin (a+b x)} \cos (a+b x) \, dx","Int[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",2,2,17,0.1176,1,"{4334, 2194}"
685,1,22,0,0.0132438,"\int e^{n \sin (a c+b c x)} \cos (c (a+b x)) \, dx","Int[E^(n*Sin[a*c + b*c*x])*Cos[c*(a + b*x)],x]","\frac{e^{n \sin (c (a+b x))}}{b c n}","\frac{e^{n \sin (c (a+b x))}}{b c n}",1,"E^(n*Sin[c*(a + b*x)])/(b*c*n)","A",2,2,22,0.09091,1,"{4334, 2194}"
686,1,23,0,0.0130337,"\int e^{n \sin (c (a+b x))} \cos (a c+b c x) \, dx","Int[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",2,2,22,0.09091,1,"{4334, 2194}"
687,1,13,0,0.0202975,"\int e^{n \sin (a+b x)} \cot (a+b x) \, dx","Int[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",2,2,17,0.1176,1,"{4338, 2178}"
688,1,18,0,0.0206306,"\int e^{n \sin (a c+b c x)} \cot (c (a+b x)) \, dx","Int[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",2,2,22,0.09091,1,"{4338, 2178}"
689,1,19,0,0.0203697,"\int e^{n \sin (c (a+b x))} \cot (a c+b c x) \, dx","Int[E^(n*Sin[c*(a + b*x)])*Cot[a*c + b*c*x],x]","\frac{\text{Ei}(n \sin (a c+b x c))}{b c}","\frac{\text{Ei}(n \sin (a c+b x c))}{b c}",1,"ExpIntegralEi[n*Sin[a*c + b*c*x]]/(b*c)","A",2,2,22,0.09091,1,"{4338, 2178}"
690,1,11,0,0.0341299,"\int \frac{\sec ^2(x)}{a+b \tan (x)} \, dx","Int[Sec[x]^2/(a + b*Tan[x]),x]","\frac{\log (a+b \tan (x))}{b}","\frac{\log (a+b \tan (x))}{b}",1,"Log[a + b*Tan[x]]/b","A",2,2,13,0.1538,1,"{3506, 31}"
691,1,11,0,0.0307602,"\int \frac{\sec ^2(x)}{1-\tan ^2(x)} \, dx","Int[Sec[x]^2/(1 - Tan[x]^2),x]","\frac{1}{2} \tanh ^{-1}(2 \sin (x) \cos (x))","\frac{1}{2} \tanh ^{-1}(2 \sin (x) \cos (x))",1,"ArcTanh[2*Cos[x]*Sin[x]]/2","A",2,2,15,0.1333,1,"{3675, 206}"
692,1,27,0,0.0292568,"\int \frac{\sec ^2(x)}{9+\tan ^2(x)} \, dx","Int[Sec[x]^2/(9 + Tan[x]^2),x]","\frac{x}{3}-\frac{1}{3} \tan ^{-1}\left(\frac{2 \sin (x) \cos (x)}{2 \cos ^2(x)+1}\right)","\frac{x}{3}-\frac{1}{3} \tan ^{-1}\left(\frac{2 \sin (x) \cos (x)}{2 \cos ^2(x)+1}\right)",1,"x/3 - ArcTan[(2*Cos[x]*Sin[x])/(1 + 2*Cos[x]^2)]/3","A",2,2,13,0.1538,1,"{3675, 203}"
693,1,19,0,0.0354784,"\int \sec ^2(x) (a+b \tan (x))^n \, dx","Int[Sec[x]^2*(a + b*Tan[x])^n,x]","\frac{(a+b \tan (x))^{n+1}}{b (n+1)}","\frac{(a+b \tan (x))^{n+1}}{b (n+1)}",1,"(a + b*Tan[x])^(1 + n)/(b*(1 + n))","A",2,2,13,0.1538,1,"{3506, 32}"
694,1,4,0,0.0426197,"\int \sec ^2(x) \left(1+\frac{1}{1+\tan ^2(x)}\right) \, dx","Int[Sec[x]^2*(1 + (1 + Tan[x]^2)^(-1)),x]","x+\tan (x)","x+\tan (x)",1,"x + Tan[x]","A",3,1,15,0.06667,1,"{203}"
695,1,4,0,0.0629814,"\int \frac{\sec ^2(x) \left(2+\tan ^2(x)\right)}{1+\tan ^2(x)} \, dx","Int[(Sec[x]^2*(2 + Tan[x]^2))/(1 + Tan[x]^2),x]","x+\tan (x)","x+\tan (x)",1,"x + Tan[x]","A",4,3,19,0.1579,1,"{3657, 3473, 8}"
696,1,33,0,0.0422412,"\int \frac{\sec ^2(x)}{2+2 \tan (x)+\tan ^2(x)} \, dx","Int[Sec[x]^2/(2 + 2*Tan[x] + Tan[x]^2),x]","x-\tan ^{-1}\left(\frac{-2 \cos ^2(x)+\sin (x) \cos (x)+1}{\cos ^2(x)+2 \sin (x) \cos (x)+2}\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,"x - ArcTan[(1 - 2*Cos[x]^2 + Cos[x]*Sin[x])/(2 + Cos[x]^2 + 2*Cos[x]*Sin[x])]","A",3,3,17,0.1765,1,"{4342, 617, 204}"
697,1,15,0,0.0479915,"\int \frac{\sec ^2(x)}{\tan ^2(x)+\tan ^3(x)} \, dx","Int[Sec[x]^2/(Tan[x]^2 + Tan[x]^3),x]","-\cot (x)-\log (\tan (x))+\log (\tan (x)+1)","\log (\cot (x)+1)-\cot (x)",1,"-Cot[x] - Log[Tan[x]] + Log[1 + Tan[x]]","A",3,2,16,0.1250,1,"{4342, 44}"
698,1,15,0,0.052689,"\int \frac{\sec ^2(x)}{-\tan ^2(x)+\tan ^3(x)} \, dx","Int[Sec[x]^2/(-Tan[x]^2 + Tan[x]^3),x]","\cot (x)+\log (1-\tan (x))-\log (\tan (x))","\cot (x)+\log (1-\cot (x))",1,"Cot[x] + Log[1 - Tan[x]] - Log[Tan[x]]","A",3,2,18,0.1111,1,"{4342, 44}"
699,1,176,0,0.1396702,"\int \frac{\sec ^2(x)}{3-4 \tan ^3(x)} \, dx","Int[Sec[x]^2/(3 - 4*Tan[x]^3),x]","\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}}","\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,"x/(3*2^(2/3)*3^(1/6)) - ArcTan[(6^(2/3) - 2*6^(2/3)*Cos[x]^2 + 2*(3 - 2*6^(1/3))*Cos[x]*Sin[x])/(3*2^(2/3)*3^(1/6) + 4*6^(1/3) + (6 - 4*6^(1/3))*Cos[x]^2 + 2*6^(2/3)*Cos[x]*Sin[x])]/(3*2^(2/3)*3^(1/6)) - Log[3^(1/3) - 2^(2/3)*Tan[x]]/(3*6^(2/3)) + Log[3^(2/3) + 2^(2/3)*3^(1/3)*Tan[x] + 2*2^(1/3)*Tan[x]^2]/(6*6^(2/3))","A",7,7,15,0.4667,1,"{3675, 200, 31, 634, 617, 204, 628}"
700,1,53,0,0.0656439,"\int \frac{\sec ^2(x)}{11-5 \tan (x)+5 \tan ^2(x)} \, dx","Int[Sec[x]^2/(11 - 5*Tan[x] + 5*Tan[x]^2),x]","\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}}","\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*x)/Sqrt[195] - (2*ArcTan[(-5 + 10*Cos[x]^2 + 12*Cos[x]*Sin[x])/(10 + Sqrt[195] + 12*Cos[x]^2 - 10*Cos[x]*Sin[x])])/Sqrt[195]","A",3,3,19,0.1579,1,"{4342, 618, 204}"
701,1,28,0,0.0873822,"\int \frac{\sec ^2(x) (a+b \tan (x))}{c+d \tan (x)} \, dx","Int[(Sec[x]^2*(a + b*Tan[x]))/(c + d*Tan[x]),x]","\frac{b \tan (x)}{d}-\frac{(b c-a d) \log (c+d \tan (x))}{d^2}","\frac{b \tan (x)}{d}-\frac{(b c-a d) \log (c+d \tan (x))}{d^2}",1,"-(((b*c - a*d)*Log[c + d*Tan[x]])/d^2) + (b*Tan[x])/d","A",3,2,19,0.1053,1,"{4342, 43}"
702,1,53,0,0.137769,"\int \frac{\sec ^2(x) (a+b \tan (x))^2}{c+d \tan (x)} \, dx","Int[(Sec[x]^2*(a + b*Tan[x])^2)/(c + d*Tan[x]),x]","-\frac{b \tan (x) (b c-a d)}{d^2}+\frac{(b c-a d)^2 \log (c+d \tan (x))}{d^3}+\frac{(a+b \tan (x))^2}{2 d}","-\frac{b \tan (x) (b c-a d)}{d^2}+\frac{(b c-a d)^2 \log (c+d \tan (x))}{d^3}+\frac{(a+b \tan (x))^2}{2 d}",1,"((b*c - a*d)^2*Log[c + d*Tan[x]])/d^3 - (b*(b*c - a*d)*Tan[x])/d^2 + (a + b*Tan[x])^2/(2*d)","A",3,2,21,0.09524,1,"{4342, 43}"
703,1,78,0,0.1496078,"\int \frac{\sec ^2(x) (a+b \tan (x))^3}{c+d \tan (x)} \, dx","Int[(Sec[x]^2*(a + b*Tan[x])^3)/(c + d*Tan[x]),x]","\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{(b c-a d)^3 \log (c+d \tan (x))}{d^4}+\frac{(a+b \tan (x))^3}{3 d}","\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{(b c-a d)^3 \log (c+d \tan (x))}{d^4}+\frac{(a+b \tan (x))^3}{3 d}",1,"-(((b*c - a*d)^3*Log[c + d*Tan[x]])/d^4) + (b*(b*c - a*d)^2*Tan[x])/d^3 - ((b*c - a*d)*(a + b*Tan[x])^2)/(2*d^2) + (a + b*Tan[x])^3/(3*d)","A",3,2,21,0.09524,1,"{4342, 43}"
704,1,12,0,0.0760194,"\int \frac{\sec ^2(x) \tan ^2(x)}{\left(2+\tan ^3(x)\right)^2} \, dx","Int[(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*(2 + Tan[x]^3))","A",2,2,17,0.1176,1,"{4342, 261}"
705,1,33,0,0.0920761,"\int \sec ^2(x) \tan ^6(x) \left(1+\tan ^2(x)\right)^3 \, dx","Int[Sec[x]^2*Tan[x]^6*(1 + Tan[x]^2)^3,x]","\frac{\tan ^{13}(x)}{13}+\frac{3 \tan ^{11}(x)}{11}+\frac{\tan ^9(x)}{3}+\frac{\tan ^7(x)}{7}","\frac{\tan ^{13}(x)}{13}+\frac{3 \tan ^{11}(x)}{11}+\frac{\tan ^9(x)}{3}+\frac{\tan ^7(x)}{7}",1,"Tan[x]^7/7 + Tan[x]^9/3 + (3*Tan[x]^11)/11 + Tan[x]^13/13","A",4,3,17,0.1765,1,"{3657, 2607, 270}"
706,1,46,0,0.0887961,"\int \frac{\sec ^2(x) \left(2+\tan ^2(x)\right)}{1+\tan ^3(x)} \, dx","Int[(Sec[x]^2*(2 + Tan[x]^2))/(1 + Tan[x]^3),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}}","\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*x)/Sqrt[3] + (2*ArcTan[(1 - 2*Cos[x]^2)/(2 + Sqrt[3] - 2*Cos[x]*Sin[x])])/Sqrt[3] + Log[1 + Tan[x]]","A",5,5,19,0.2632,1,"{4342, 1863, 31, 618, 204}"
707,1,4,0,0.0182728,"\int \left(1+\cos ^2(x)\right) \sec ^2(x) \, dx","Int[(1 + Cos[x]^2)*Sec[x]^2,x]","x+\tan (x)","x+\tan (x)",1,"x + Tan[x]","A",2,2,11,0.1818,1,"{3012, 8}"
708,1,21,0,0.1154769,"\int \frac{\sec ^2(x)}{1+\sec ^2(x)-3 \tan (x)} \, dx","Int[Sec[x]^2/(1 + Sec[x]^2 - 3*Tan[x]),x]","\log (2 \cos (x)-\sin (x))-\log (\cos (x)-\sin (x))","\log (2 \cos (x)-\sin (x))-\log (\cos (x)-\sin (x))",1,"-Log[Cos[x] - Sin[x]] + Log[2*Cos[x] - Sin[x]]","A",4,2,17,0.1176,1,"{616, 31}"
709,1,9,0,0.0467953,"\int \frac{\sec ^2(x)}{\sqrt{4-\sec ^2(x)}} \, dx","Int[Sec[x]^2/Sqrt[4 - Sec[x]^2],x]","\sin ^{-1}\left(\frac{\tan (x)}{\sqrt{3}}\right)","\sin ^{-1}\left(\frac{\tan (x)}{\sqrt{3}}\right)",1,"ArcSin[Tan[x]/Sqrt[3]]","A",2,2,17,0.1176,1,"{4146, 216}"
710,1,9,0,0.0468873,"\int \frac{\sec ^2(x)}{\sqrt{1-4 \tan ^2(x)}} \, dx","Int[Sec[x]^2/Sqrt[1 - 4*Tan[x]^2],x]","\frac{1}{2} \sin ^{-1}(2 \tan (x))","\frac{1}{2} \sin ^{-1}(2 \tan (x))",1,"ArcSin[2*Tan[x]]/2","A",2,2,17,0.1176,1,"{3675, 216}"
711,1,14,0,0.0444797,"\int \frac{\sec ^2(x)}{\sqrt{-4+\tan ^2(x)}} \, dx","Int[Sec[x]^2/Sqrt[-4 + Tan[x]^2],x]","\tanh ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)-4}}\right)","\tanh ^{-1}\left(\frac{\tan (x)}{\sqrt{\tan ^2(x)-4}}\right)",1,"ArcTanh[Tan[x]/Sqrt[-4 + Tan[x]^2]]","A",3,3,15,0.2000,1,"{3675, 217, 206}"
712,1,19,0,0.0494782,"\int \sqrt{1-\cot ^2(x)} \sec ^2(x) \, dx","Int[Sqrt[1 - Cot[x]^2]*Sec[x]^2,x]","\tan (x) \sqrt{1-\cot ^2(x)}+\sin ^{-1}(\cot (x))","\tan (x) \sqrt{1-\cot ^2(x)}+\sin ^{-1}(\cot (x))",1,"ArcSin[Cot[x]] + Sqrt[1 - Cot[x]^2]*Tan[x]","A",3,3,17,0.1765,1,"{3663, 277, 216}"
713,1,26,0,0.0458562,"\int \sec ^2(x) \sqrt{1-\tan ^2(x)} \, dx","Int[Sec[x]^2*Sqrt[1 - Tan[x]^2],x]","\frac{1}{2} \tan (x) \sqrt{1-\tan ^2(x)}+\frac{1}{2} \sin ^{-1}(\tan (x))","\frac{1}{2} \tan (x) \sqrt{1-\tan ^2(x)}+\frac{1}{2} \sin ^{-1}(\tan (x))",1,"ArcSin[Tan[x]]/2 + (Tan[x]*Sqrt[1 - Tan[x]^2])/2","A",3,3,17,0.1765,1,"{3675, 195, 216}"
714,1,4,0,0.0123224,"\int e^{\tan (x)} \sec ^2(x) \, dx","Int[E^Tan[x]*Sec[x]^2,x]","e^{\tan (x)}","e^{\tan (x)}",1,"E^Tan[x]","A",2,2,9,0.2222,1,"{4342, 2194}"
715,1,17,0,0.0666117,"\int \sec ^4(x) \left(-1+\sec ^2(x)\right)^2 \tan (x) \, dx","Int[Sec[x]^4*(-1 + Sec[x]^2)^2*Tan[x],x]","\frac{\tan ^8(x)}{8}+\frac{\tan ^6(x)}{6}","\frac{\tan ^8(x)}{8}+\frac{\tan ^6(x)}{6}",1,"Tan[x]^6/6 + Tan[x]^8/8","A",4,3,15,0.2000,1,"{4120, 2607, 14}"
716,1,12,0,0.0411302,"\int \frac{\csc ^2(x)}{a+b \cot (x)} \, dx","Int[Csc[x]^2/(a + b*Cot[x]),x]","-\frac{\log (a+b \cot (x))}{b}","-\frac{\log (a+b \cot (x))}{b}",1,"-(Log[a + b*Cot[x]]/b)","A",2,2,13,0.1538,1,"{3506, 31}"
717,1,20,0,0.0412348,"\int (a+b \cot (x))^n \csc ^2(x) \, dx","Int[(a + b*Cot[x])^n*Csc[x]^2,x]","-\frac{(a+b \cot (x))^{n+1}}{b (n+1)}","-\frac{(a+b \cot (x))^{n+1}}{b (n+1)}",1,"-((a + b*Cot[x])^(1 + n)/(b*(1 + n)))","A",2,2,13,0.1538,1,"{3506, 32}"
718,1,6,0,0.0159719,"\int \csc ^2(x) \left(1+\sin ^2(x)\right) \, dx","Int[Csc[x]^2*(1 + Sin[x]^2),x]","x-\cot (x)","x-\cot (x)",1,"x - Cot[x]","A",2,2,11,0.1818,1,"{3012, 8}"
719,1,6,0,0.0470889,"\int \left(1+\frac{1}{1+\cot ^2(x)}\right) \csc ^2(x) \, dx","Int[(1 + (1 + Cot[x]^2)^(-1))*Csc[x]^2,x]","x-\cot (x)","x-\cot (x)",1,"x - Cot[x]","A",4,2,15,0.1333,1,"{14, 203}"
720,1,28,0,0.0818833,"\int \frac{(a+b \cot (x)) \csc ^2(x)}{c+d \cot (x)} \, dx","Int[((a + b*Cot[x])*Csc[x]^2)/(c + d*Cot[x]),x]","\frac{(b c-a d) \log (c+d \cot (x))}{d^2}-\frac{b \cot (x)}{d}","\frac{(b c-a d) \log (c+d \cot (x))}{d^2}-\frac{b \cot (x)}{d}",1,"-((b*Cot[x])/d) + ((b*c - a*d)*Log[c + d*Cot[x]])/d^2","A",3,2,19,0.1053,1,"{4344, 43}"
721,1,53,0,0.1364347,"\int \frac{(a+b \cot (x))^2 \csc ^2(x)}{c+d \cot (x)} \, dx","Int[((a + b*Cot[x])^2*Csc[x]^2)/(c + d*Cot[x]),x]","\frac{b \cot (x) (b c-a d)}{d^2}-\frac{(b c-a d)^2 \log (c+d \cot (x))}{d^3}-\frac{(a+b \cot (x))^2}{2 d}","\frac{b \cot (x) (b c-a d)}{d^2}-\frac{(b c-a d)^2 \log (c+d \cot (x))}{d^3}-\frac{(a+b \cot (x))^2}{2 d}",1,"(b*(b*c - a*d)*Cot[x])/d^2 - (a + b*Cot[x])^2/(2*d) - ((b*c - a*d)^2*Log[c + d*Cot[x]])/d^3","A",3,2,21,0.09524,1,"{4344, 43}"
722,1,78,0,0.1389541,"\int \frac{(a+b \cot (x))^3 \csc ^2(x)}{c+d \cot (x)} \, dx","Int[((a + b*Cot[x])^3*Csc[x]^2)/(c + d*Cot[x]),x]","-\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{(b c-a d)^3 \log (c+d \cot (x))}{d^4}-\frac{(a+b \cot (x))^3}{3 d}","-\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{(b c-a d)^3 \log (c+d \cot (x))}{d^4}-\frac{(a+b \cot (x))^3}{3 d}",1,"-((b*(b*c - a*d)^2*Cot[x])/d^3) + ((b*c - a*d)*(a + b*Cot[x])^2)/(2*d^2) - (a + b*Cot[x])^3/(3*d) + ((b*c - a*d)^3*Log[c + d*Cot[x]])/d^4","A",3,2,21,0.09524,1,"{4344, 43}"
723,1,6,0,0.0147822,"\int e^{-\cot (x)} \csc ^2(x) \, dx","Int[Csc[x]^2/E^Cot[x],x]","e^{-\cot (x)}","e^{-\cot (x)}",1,"E^(-Cot[x])","A",2,2,11,0.1818,1,"{4344, 2194}"
724,1,20,0,0.0456383,"\int \frac{\sec (x) \tan (x)}{a+b \sec (x)} \, dx","Int[(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",4,4,13,0.3077,1,"{4339, 36, 29, 31}"
725,1,5,0,0.0332322,"\int \frac{\sec (x) \tan (x)}{1+\sec ^2(x)} \, dx","Int[(Sec[x]*Tan[x])/(1 + Sec[x]^2),x]","-\tan ^{-1}(\cos (x))","-\tan ^{-1}(\cos (x))",1,"-ArcTan[Cos[x]]","A",2,2,13,0.1538,1,"{4339, 203}"
726,1,11,0,0.0347168,"\int \frac{\sec (x) \tan (x)}{9+4 \sec ^2(x)} \, dx","Int[(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,"-ArcTan[(3*Cos[x])/2]/6","A",2,2,15,0.1333,1,"{4339, 203}"
727,1,7,0,0.031622,"\int \frac{\sec (x) \tan (x)}{\sec (x)+\sec ^2(x)} \, dx","Int[(Sec[x]*Tan[x])/(Sec[x] + Sec[x]^2),x]","-\log (\cos (x)+1)","-\log (\cos (x)+1)",1,"-Log[1 + Cos[x]]","A",2,2,14,0.1429,1,"{4339, 31}"
728,1,5,0,0.0448581,"\int \frac{\sec (x) \tan (x)}{\sqrt{4+\sec ^2(x)}} \, dx","Int[(Sec[x]*Tan[x])/Sqrt[4 + Sec[x]^2],x]","\text{csch}^{-1}(2 \cos (x))","\text{csch}^{-1}(2 \cos (x))",1,"ArcCsch[2*Cos[x]]","A",3,3,15,0.2000,1,"{4339, 335, 215}"
729,1,13,0,0.0792028,"\int \frac{\sec (x) \tan (x)}{\sqrt{1+\cos ^2(x)}} \, dx","Int[(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",2,1,15,0.06667,1,"{264}"
730,1,4,0,0.0217493,"\int e^{\sec (x)} \sec (x) \tan (x) \, dx","Int[E^Sec[x]*Sec[x]*Tan[x],x]","e^{\sec (x)}","e^{\sec (x)}",1,"E^Sec[x]","A",2,2,9,0.2222,1,"{4339, 2209}"
731,1,9,0,0.0216664,"\int 2^{\sec (x)} \sec (x) \tan (x) \, dx","Int[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",2,2,9,0.2222,1,"{4339, 2209}"
732,1,12,0,0.0545603,"\int \frac{\sec (2 x) \tan (2 x)}{(1+\sec (2 x))^{3/2}} \, dx","Int[(Sec[2*x]*Tan[2*x])/(1 + Sec[2*x])^(3/2),x]","-\frac{1}{\sqrt{\sec (2 x)+1}}","-\frac{1}{\sqrt{\sec (2 x)+1}}",1,"-(1/Sqrt[1 + Sec[2*x]])","A",2,2,19,0.1053,1,"{4339, 261}"
733,1,43,0,0.0949236,"\int \sqrt{1+5 \cos ^2(3 x)} \sec (3 x) \tan (3 x) \, dx","Int[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,"-(Sqrt[5]*ArcSinh[Sqrt[5]*Cos[3*x]])/3 + (Sqrt[1 + 5*Cos[3*x]^2]*Sec[3*x])/3","A",3,2,23,0.08696,1,"{277, 215}"
734,1,22,0,0.0909895,"\int \frac{\sec (3 x) \tan (3 x)}{\sqrt{1+5 \cos ^2(3 x)}} \, dx","Int[(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",2,1,23,0.04348,1,"{264}"
735,1,20,0,0.0423383,"\int \frac{\cot (x) \csc (x)}{a+b \csc (x)} \, dx","Int[(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",4,4,13,0.3077,1,"{4338, 36, 29, 31}"
736,1,14,0,0.0247173,"\int 5^{\csc (3 x)} \cot (3 x) \csc (3 x) \, dx","Int[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,"-5^Csc[3*x]/(3*Log[5])","A",2,2,15,0.1333,1,"{4338, 2209}"
737,1,3,0,0.0321606,"\int \frac{\cot (x) \csc (x)}{1+\csc ^2(x)} \, dx","Int[(Cot[x]*Csc[x])/(1 + Csc[x]^2),x]","\tan ^{-1}(\sin (x))","\tan ^{-1}(\sin (x))",1,"ArcTan[Sin[x]]","A",2,2,13,0.1538,1,"{4338, 203}"
738,1,43,0,0.0570532,"\int \frac{\cot (6 x) \csc (6 x)}{\left(5-11 \csc ^2(6 x)\right)^2} \, dx","Int[(Cot[6*x]*Csc[6*x])/(5 - 11*Csc[6*x]^2)^2,x]","\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}}","\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,"-ArcTanh[Sqrt[5/11]*Sin[6*x]]/(60*Sqrt[55]) + Sin[6*x]/(60*(11 - 5*Sin[6*x]^2))","A",3,3,21,0.1429,1,"{4338, 288, 206}"
739,1,14,0,0.0765542,"\int \frac{\cot (x) \csc (x)}{\sqrt{1+\sin ^2(x)}} \, dx","Int[(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",2,1,15,0.06667,1,"{264}"
740,1,43,0,0.1056295,"\int \frac{\cot (5 x) \csc ^3(5 x)}{\sqrt{1+\sin ^2(5 x)}} \, dx","Int[(Cot[5*x]*Csc[5*x]^3)/Sqrt[1 + Sin[5*x]^2],x]","\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)","\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,"(2*Csc[5*x]*Sqrt[1 + Sin[5*x]^2])/15 - (Csc[5*x]^3*Sqrt[1 + Sin[5*x]^2])/15","A",3,2,23,0.08696,1,"{271, 264}"
741,1,43,0,0.0366599,"\int e^{n \sin (a+b x)} \sin (2 a+2 b x) \, dx","Int[E^(n*Sin[a + b*x])*Sin[2*a + 2*b*x],x]","\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}","\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]))/(b*n^2) + (2*E^(n*Sin[a + b*x])*Sin[a + b*x])/(b*n)","A",4,3,20,0.1500,1,"{12, 2176, 2194}"
742,1,43,0,0.0347942,"\int e^{n \sin (a+b x)} \sin (2 (a+b x)) \, dx","Int[E^(n*Sin[a + b*x])*Sin[2*(a + b*x)],x]","\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}","\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]))/(b*n^2) + (2*E^(n*Sin[a + b*x])*Sin[a + b*x])/(b*n)","A",4,3,19,0.1579,1,"{12, 2176, 2194}"
743,1,64,0,0.0362031,"\int e^{n \sin \left(\frac{a}{2}+\frac{b x}{2}\right)} \sin (a+b x) \, dx","Int[E^(n*Sin[a/2 + (b*x)/2])*Sin[a + b*x],x]","\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}","\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/2 + (b*x)/2]))/(b*n^2) + (4*E^(n*Sin[a/2 + (b*x)/2])*Sin[a/2 + (b*x)/2])/(b*n)","A",4,3,24,0.1250,1,"{12, 2176, 2194}"
744,1,64,0,0.0384814,"\int e^{n \sin \left(\frac{1}{2} (a+b x)\right)} \sin (a+b x) \, dx","Int[E^(n*Sin[(a + b*x)/2])*Sin[a + b*x],x]","\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}","\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/2 + (b*x)/2]))/(b*n^2) + (4*E^(n*Sin[a/2 + (b*x)/2])*Sin[a/2 + (b*x)/2])/(b*n)","A",4,3,21,0.1429,1,"{12, 2176, 2194}"
745,1,43,0,0.0405534,"\int e^{n \cos (a+b x)} \sin (2 a+2 b x) \, dx","Int[E^(n*Cos[a + b*x])*Sin[2*a + 2*b*x],x]","\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}","\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]))/(b*n^2) - (2*E^(n*Cos[a + b*x])*Cos[a + b*x])/(b*n)","A",4,3,20,0.1500,1,"{12, 2176, 2194}"
746,1,43,0,0.0349637,"\int e^{n \cos (a+b x)} \sin (2 (a+b x)) \, dx","Int[E^(n*Cos[a + b*x])*Sin[2*(a + b*x)],x]","\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}","\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]))/(b*n^2) - (2*E^(n*Cos[a + b*x])*Cos[a + b*x])/(b*n)","A",4,3,19,0.1579,1,"{12, 2176, 2194}"
747,1,64,0,0.0374232,"\int e^{n \cos \left(\frac{a}{2}+\frac{b x}{2}\right)} \sin (a+b x) \, dx","Int[E^(n*Cos[a/2 + (b*x)/2])*Sin[a + b*x],x]","\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}","\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/2 + (b*x)/2]))/(b*n^2) - (4*E^(n*Cos[a/2 + (b*x)/2])*Cos[a/2 + (b*x)/2])/(b*n)","A",4,3,24,0.1250,1,"{12, 2176, 2194}"
748,1,64,0,0.0374376,"\int e^{n \cos \left(\frac{1}{2} (a+b x)\right)} \sin (a+b x) \, dx","Int[E^(n*Cos[(a + b*x)/2])*Sin[a + b*x],x]","\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}","\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/2 + (b*x)/2]))/(b*n^2) - (4*E^(n*Cos[a/2 + (b*x)/2])*Cos[a/2 + (b*x)/2])/(b*n)","A",4,3,21,0.1429,1,"{12, 2176, 2194}"
749,1,9,0,0.0219136,"\int \csc (x) \log (\tan (x)) \sec (x) \, dx","Int[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",1,3,8,0.3750,1,"{2620, 29, 6686}"
750,1,9,0,0.0195538,"\int \csc (2 x) \log (\tan (x)) \, dx","Int[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",1,2,8,0.2500,1,"{3770, 6686}"
751,1,3,0,0.0087911,"\int e^{\cos ^2(x)+\sin ^2(x)} \, dx","Int[E^(Cos[x]^2 + Sin[x]^2),x]","e x","e x",1,"E*x","A",3,2,11,0.1818,1,"{12, 203}"
752,1,8,0,0.0177246,"\int x \sec ^2(x) \, dx","Int[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",2,2,6,0.3333,1,"{4184, 3475}"
753,1,34,0,0.0224351,"\int x \cos ^4\left(x^2\right) \, dx","Int[x*Cos[x^2]^4,x]","\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)","\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 + (3*Cos[x^2]*Sin[x^2])/16 + (Cos[x^2]^3*Sin[x^2])/8","A",4,3,8,0.3750,1,"{3380, 2635, 8}"
754,1,10,0,0.0121782,"\int \sqrt{\cos (x)} \sin (x) \, dx","Int[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",2,2,9,0.2222,1,"{2565, 30}"
755,1,11,0,0.0117125,"\int e^{-2 x} \tan \left(e^{-2 x}\right) \, dx","Int[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",2,2,12,0.1667,1,"{2282, 3475}"
756,1,7,0,0.0443849,"\int \frac{\sec (x) \sin (2 x)}{1+\cos (x)} \, dx","Int[(Sec[x]*Sin[2*x])/(1 + Cos[x]),x]","-2 \log (\cos (x)+1)","-2 \log (\cos (x)+1)",1,"-2*Log[1 + Cos[x]]","A",3,2,13,0.1538,1,"{12, 31}"
757,1,19,0,0.0185204,"\int x \sec ^2(3 x) \, dx","Int[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",2,2,8,0.2500,1,"{4184, 3475}"
758,1,37,0,0.0135448,"\int e^{-2 \pi  x} \cos (2 \pi  x) \, dx","Int[Cos[2*Pi*x]/E^(2*Pi*x),x]","\frac{e^{-2 \pi  x} \sin (2 \pi  x)}{4 \pi }-\frac{e^{-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]/(4*E^(2*Pi*x)*Pi) + Sin[2*Pi*x]/(4*E^(2*Pi*x)*Pi)","A",1,1,12,0.08333,1,"{4433}"
759,1,129,0,0.3237995,"\int \left(\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right) \, dx","Int[Cos[x]^12*Sin[x]^10 - Cos[x]^10*Sin[x]^12,x]","-\frac{1}{22} \sin ^9(x) \cos ^{13}(x)-\frac{9}{440} \sin ^7(x) \cos ^{13}(x)-\frac{7}{880} \sin ^5(x) \cos ^{13}(x)-\frac{7 \sin ^3(x) \cos ^{13}(x)}{2816}+\frac{1}{22} \sin ^{11}(x) \cos ^{11}(x)+\frac{1}{40} \sin ^9(x) \cos ^{11}(x)+\frac{1}{80} \sin ^7(x) \cos ^{11}(x)+\frac{7 \sin ^5(x) \cos ^{11}(x)}{1280}+\frac{1}{512} \sin ^3(x) \cos ^{11}(x)-\frac{3 \sin (x) \cos ^{13}(x)}{5632}+\frac{3 \sin (x) \cos ^{11}(x)}{5632}","\frac{1}{11} \sin ^{11}(x) \cos ^{11}(x)",1,"(3*Cos[x]^11*Sin[x])/5632 - (3*Cos[x]^13*Sin[x])/5632 + (Cos[x]^11*Sin[x]^3)/512 - (7*Cos[x]^13*Sin[x]^3)/2816 + (7*Cos[x]^11*Sin[x]^5)/1280 - (7*Cos[x]^13*Sin[x]^5)/880 + (Cos[x]^11*Sin[x]^7)/80 - (9*Cos[x]^13*Sin[x]^7)/440 + (Cos[x]^11*Sin[x]^9)/40 - (Cos[x]^13*Sin[x]^9)/22 + (Cos[x]^11*Sin[x]^11)/22","B",25,3,20,0.1500,1,"{2568, 2635, 8}"
760,1,9,0,0.0066126,"\int x \cot \left(x^2\right) \, dx","Int[x*Cot[x^2],x]","\frac{1}{2} \log \left(\sin \left(x^2\right)\right)","\frac{1}{2} \log \left(\sin \left(x^2\right)\right)",1,"Log[Sin[x^2]]/2","A",2,2,6,0.3333,1,"{3748, 3475}"
761,1,8,0,0.0130906,"\int x \sec ^2\left(x^2\right) \, dx","Int[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",3,3,8,0.3750,1,"{4204, 3767, 8}"
762,1,15,0,0.0290622,"\int \frac{\sin (8 x)}{9+\sin ^4(4 x)} \, dx","Int[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",4,3,15,0.2000,1,"{12, 275, 203}"
763,1,23,0,0.0218966,"\int \frac{\cos (2 x)}{8+\sin ^2(2 x)} \, dx","Int[Cos[2*x]/(8 + Sin[2*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sin (2 x)}{2 \sqrt{2}}\right)}{4 \sqrt{2}}","\frac{\tan ^{-1}\left(\frac{\sin (2 x)}{2 \sqrt{2}}\right)}{4 \sqrt{2}}",1,"ArcTan[Sin[2*x]/(2*Sqrt[2])]/(4*Sqrt[2])","A",2,2,15,0.1333,1,"{3190, 203}"
764,1,37,0,0.0335579,"\int x \left(\cos ^3\left(x^2\right)-\sin ^3\left(x^2\right)\right) \, dx","Int[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{1}{6} \cos ^3\left(x^2\right)+\frac{\cos \left(x^2\right)}{2}","-\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,"Cos[x^2]/2 - Cos[x^2]^3/6 + Sin[x^2]/2 - Sin[x^2]^3/6","A",8,4,17,0.2353,1,"{14, 3380, 2633, 3379}"
765,1,10,0,0.0330559,"\int \frac{\cos (x) \sin (x)}{1-\cos (x)} \, dx","Int[(Cos[x]*Sin[x])/(1 - Cos[x]),x]","\cos (x)+\log (1-\cos (x))","\cos (x)+\log (1-\cos (x))",1,"Cos[x] + Log[1 - Cos[x]]","A",3,2,13,0.1538,1,"{2833, 43}"
766,1,8,0,0.0069327,"\int x \cos \left(x^2\right) \, dx","Int[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",2,2,6,0.3333,1,"{3380, 2637}"
767,1,10,0,0.0121073,"\int x^2 \cos \left(4 x^3\right) \, dx","Int[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",2,2,10,0.2000,1,"{3380, 2637}"
768,1,8,0,0.0094394,"\int x^3 \cos \left(x^4\right) \, dx","Int[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",2,2,8,0.2500,1,"{3380, 2637}"
769,1,10,0,0.0077483,"\int x \sin \left(\frac{x^2}{2}\right) \, dx","Int[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",2,2,10,0.2000,1,"{3379, 2638}"
770,1,8,0,0.063054,"\int x \sec \left(x^2\right) \tan \left(x^2\right) \, dx","Int[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",3,3,10,0.3000,1,"{6715, 2606, 8}"
771,1,10,0,0.0178687,"\int \frac{\tan ^2\left(\frac{1}{x}\right)}{x^2} \, dx","Int[Tan[x^(-1)]^2/x^2,x]","\frac{1}{x}-\tan \left(\frac{1}{x}\right)","\frac{1}{x}-\tan \left(\frac{1}{x}\right)",1,"x^(-1) - Tan[x^(-1)]","A",3,3,10,0.3000,1,"{3747, 3473, 8}"
772,1,11,0,0.010176,"\int x \tan \left(1+x^2\right) \, dx","Int[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,"-Log[Cos[1 + x^2]]/2","A",2,2,8,0.2500,1,"{3747, 3475}"
773,1,12,0,0.0045919,"\int \sin (\pi  (1+2 x)) \, dx","Int[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",1,1,8,0.1250,1,"{2638}"
774,1,21,0,0.0595257,"\int \frac{\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx","Int[(Cot[x] + Csc[x]^2)/(1 - Cos[x]^2),x]","-\frac{1}{3} \cot ^3(x)-\frac{\cot ^2(x)}{2}-\cot (x)","-\frac{1}{3} \cot ^3(x)-\frac{\cot ^2(x)}{2}-\cot (x)",1,"-Cot[x] - Cot[x]^2/2 - Cot[x]^3/3","A",3,1,18,0.05556,1,"{14}"
775,1,19,0,0.0372446,"\int x^2 \cos \left(4 x^3\right) \cos \left(5 x^3\right) \, dx","Int[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",6,3,16,0.1875,1,"{4572, 3380, 2637}"
776,1,47,0,0.0641899,"\int x^{14} \sin \left(x^3\right) \, dx","Int[x^14*Sin[x^3],x]","\frac{4}{3} x^9 \sin \left(x^3\right)-8 x^3 \sin \left(x^3\right)-\frac{1}{3} x^{12} \cos \left(x^3\right)+4 x^6 \cos \left(x^3\right)-8 \cos \left(x^3\right)","\frac{4}{3} x^9 \sin \left(x^3\right)-8 x^3 \sin \left(x^3\right)-\frac{1}{3} x^{12} \cos \left(x^3\right)+4 x^6 \cos \left(x^3\right)-8 \cos \left(x^3\right)",1,"-8*Cos[x^3] + 4*x^6*Cos[x^3] - (x^12*Cos[x^3])/3 - 8*x^3*Sin[x^3] + (4*x^9*Sin[x^3])/3","A",6,3,8,0.3750,1,"{3379, 3296, 2638}"
777,1,35,0,0.1578117,"\int e^{-3 x^3} x^2 \sin \left(2 x^3\right) \, dx","Int[(x^2*Sin[2*x^3])/E^(3*x^3),x]","-\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)","-\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,"(-2*Cos[2*x^3])/(39*E^(3*x^3)) - Sin[2*x^3]/(13*E^(3*x^3))","A",2,2,17,0.1176,1,"{6715, 4432}"
778,1,4,0,0.0065855,"\int 2 x \cos \left(x^2\right) \, dx","Int[2*x*Cos[x^2],x]","\sin \left(x^2\right)","\sin \left(x^2\right)",1,"Sin[x^2]","A",3,3,7,0.4286,1,"{12, 3380, 2637}"
779,1,6,0,0.0128978,"\int 3 x^2 \cos \left(7+x^3\right) \, dx","Int[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",3,3,11,0.2727,1,"{12, 3380, 2637}"
780,1,7,0,0.0040017,"\int \left(\frac{1}{1+x^2}+\sin (x)\right) \, dx","Int[(1 + x^2)^(-1) + Sin[x],x]","\tan ^{-1}(x)-\cos (x)","\tan ^{-1}(x)-\cos (x)",1,"ArcTan[x] - Cos[x]","A",3,2,10,0.2000,1,"{203, 2638}"
781,1,10,0,0.0093647,"\int x \sin \left(1+x^2\right) \, dx","Int[x*Sin[1 + x^2],x]","-\frac{1}{2} \cos \left(x^2+1\right)","-\frac{1}{2} \cos \left(x^2+1\right)",1,"-Cos[1 + x^2]/2","A",2,2,8,0.2500,1,"{3379, 2638}"
782,1,10,0,0.0087427,"\int x \cos \left(1+x^2\right) \, dx","Int[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",2,2,8,0.2500,1,"{3380, 2637}"
783,1,10,0,0.0098915,"\int \left(1+x^2 \cos \left(x^3\right)\right) \, dx","Int[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",3,2,10,0.2000,1,"{3380, 2637}"
784,1,10,0,0.0119009,"\int x^2 \sin \left(1+x^3\right) \, dx","Int[x^2*Sin[1 + x^3],x]","-\frac{1}{3} \cos \left(x^3+1\right)","-\frac{1}{3} \cos \left(x^3+1\right)",1,"-Cos[1 + x^3]/3","A",2,2,10,0.2000,1,"{3379, 2638}"
785,1,6,0,0.0091064,"\int 12 x^2 \cos \left(x^3\right) \, dx","Int[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",3,3,9,0.3333,1,"{12, 3380, 2637}"
786,1,14,0,0.0111327,"\int (1+x) \sin (1+x) \, dx","Int[(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",2,2,8,0.2500,1,"{3296, 2637}"
787,1,20,0,0.0174761,"\int x^5 \cos \left(x^3\right) \, dx","Int[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",3,3,8,0.3750,1,"{3380, 3296, 2638}"
788,1,23,0,0.0089591,"\int e^{-3 x} \cos (x) \, dx","Int[Cos[x]/E^(3*x),x]","\frac{1}{10} e^{-3 x} \sin (x)-\frac{3}{10} e^{-3 x} \cos (x)","\frac{1}{10} e^{-3 x} \sin (x)-\frac{3}{10} e^{-3 x} \cos (x)",1,"(-3*Cos[x])/(10*E^(3*x)) + Sin[x]/(10*E^(3*x))","A",1,1,8,0.1250,1,"{4433}"
789,1,20,0,0.0166995,"\int x^3 \sin \left(x^2\right) \, dx","Int[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,"-(x^2*Cos[x^2])/2 + Sin[x^2]/2","A",3,3,8,0.3750,1,"{3379, 3296, 2637}"
790,1,20,0,0.0168783,"\int x^3 \cos \left(x^2\right) \, dx","Int[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",3,3,8,0.3750,1,"{3380, 3296, 2638}"
791,1,9,0,0.0104997,"\int \cos (x) \cos (2 \sin (x)) \, dx","Int[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",2,2,8,0.2500,1,"{4334, 2637}"
792,1,11,0,0.0317343,"\int \frac{\cos (x) \sin (x)}{1+\cos ^2(x)} \, dx","Int[(Cos[x]*Sin[x])/(1 + Cos[x]^2),x]","-\frac{1}{2} \log \left(\cos ^2(x)+1\right)","-\frac{1}{2} \log \left(\cos ^2(x)+1\right)",1,"-Log[1 + Cos[x]^2]/2","A",2,2,13,0.1538,1,"{4335, 260}"
793,1,10,0,0.0378183,"\int (1+\cos (x)) (x+\sin (x))^3 \, dx","Int[(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",1,1,11,0.09091,1,"{6686}"
794,1,9,0,0.0286004,"\int (1+\cos (x)) \csc ^2(x) \, dx","Int[(1 + Cos[x])*Csc[x]^2,x]","-\cot (x)-\csc (x)","-\cot (x)-\csc (x)",1,"-Cot[x] - Csc[x]","A",3,3,9,0.3333,1,"{2669, 3767, 8}"
795,1,5,0,0.0162149,"\int \sin (x) \tan ^2(x) \, dx","Int[Sin[x]*Tan[x]^2,x]","\cos (x)+\sec (x)","\cos (x)+\sec (x)",1,"Cos[x] + Sec[x]","A",3,2,7,0.2857,1,"{2590, 14}"
796,0,0,0,0.640077,"\int e^{\sin (x)} \sec ^2(x) \left(x \cos ^3(x)-\sin (x)\right) \, dx","Int[E^Sin[x]*Sec[x]^2*(x*Cos[x]^3 - Sin[x]),x]","\int e^{\sin (x)} \sec ^2(x) \left(x \cos ^3(x)-\sin (x)\right) \, dx","e^{\sin (x)} (x \cos (x)-1) \sec (x)",1,"Defer[Int][E^Sin[x]*x*Cos[x], x] - Defer[Int][E^Sin[x]*Sec[x]*Tan[x], x]","F",0,0,0,0,-1,"{}"
797,1,9,0,0.0164081,"\int x \csc ^2(x) \, dx","Int[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",2,2,6,0.3333,1,"{4184, 3475}"
798,1,20,0,0.0167845,"\int \cos (x) \sin \left(\frac{\pi }{6}+x\right) \, dx","Int[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",3,2,11,0.1818,1,"{4574, 2638}"
799,1,19,0,0.0149719,"\int x \sin ^3\left(x^2\right) \, dx","Int[x*Sin[x^2]^3,x]","\frac{1}{6} \cos ^3\left(x^2\right)-\frac{\cos \left(x^2\right)}{2}","\frac{1}{6} \cos ^3\left(x^2\right)-\frac{\cos \left(x^2\right)}{2}",1,"-Cos[x^2]/2 + Cos[x^2]^3/6","A",3,2,8,0.2500,1,"{3379, 2633}"
800,1,14,0,0.0146159,"\int \sin ^2(x) \tan (x) \, dx","Int[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",3,2,7,0.2857,1,"{2590, 14}"
801,1,22,0,0.0329977,"\int \cos ^2(x) \cot ^3(x) \, dx","Int[Cos[x]^2*Cot[x]^3,x]","\frac{\sin ^2(x)}{2}-\frac{1}{2} \csc ^2(x)-2 \log (\sin (x))","\frac{\sin ^2(x)}{2}-\frac{1}{2} \csc ^2(x)-2 \log (\sin (x))",1,"-Csc[x]^2/2 - 2*Log[Sin[x]] + Sin[x]^2/2","A",4,3,9,0.3333,1,"{2590, 266, 43}"
802,1,5,0,0.0147563,"\int \sec (x) (1-\sin (x)) \, dx","Int[Sec[x]*(1 - Sin[x]),x]","\log (\sin (x)+1)","\log (\sin (x)+1)",1,"Log[1 + Sin[x]]","A",2,2,9,0.2222,1,"{2667, 31}"
803,1,7,0,0.0157383,"\int (1+\cos (x)) \csc (x) \, dx","Int[(1 + Cos[x])*Csc[x],x]","\log (1-\cos (x))","\log (1-\cos (x))",1,"Log[1 - Cos[x]]","A",2,2,7,0.2857,1,"{2667, 31}"
804,1,5,0,0.0218715,"\int \cos ^2(x) \left(1-\tan ^2(x)\right) \, dx","Int[Cos[x]^2*(1 - Tan[x]^2),x]","\sin (x) \cos (x)","\sin (x) \cos (x)",1,"Cos[x]*Sin[x]","A",2,2,13,0.1538,1,"{3675, 383}"
805,1,15,0,0.0453018,"\int \csc (2 x) (\cos (x)+\sin (x)) \, dx","Int[Csc[2*x]*(Cos[x] + Sin[x]),x]","\frac{1}{2} \tanh ^{-1}(\sin (x))-\frac{1}{2} \tanh ^{-1}(\cos (x))","\frac{1}{2} \tanh ^{-1}(\sin (x))-\frac{1}{2} \tanh ^{-1}(\cos (x))",1,"-ArcTanh[Cos[x]]/2 + ArcTanh[Sin[x]]/2","A",6,4,10,0.4000,1,"{4401, 4287, 3770, 4288}"
806,1,11,0,0.0463967,"\int \frac{\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx","Int[(Cos[x]*(-3 + 2*Sin[x]))/(2 - 3*Sin[x] + Sin[x]^2),x]","\log \left(\sin ^2(x)-3 \sin (x)+2\right)","\log \left(\sin ^2(x)-3 \sin (x)+2\right)",1,"Log[2 - 3*Sin[x] + Sin[x]^2]","A",2,2,21,0.09524,1,"{4334, 628}"
807,1,20,0,0.0527477,"\int \frac{\cos ^2(x) \sin (x)}{5+\cos ^2(x)} \, dx","Int[(Cos[x]^2*Sin[x])/(5 + Cos[x]^2),x]","\sqrt{5} \tan ^{-1}\left(\frac{\cos (x)}{\sqrt{5}}\right)-\cos (x)","\sqrt{5} \tan ^{-1}\left(\frac{\cos (x)}{\sqrt{5}}\right)-\cos (x)",1,"Sqrt[5]*ArcTan[Cos[x]/Sqrt[5]] - Cos[x]","A",3,3,15,0.2000,1,"{4335, 321, 203}"
808,1,11,0,0.0214209,"\int \frac{\cos (x)}{\sin (x)+\sin ^2(x)} \, dx","Int[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",2,2,12,0.1667,1,"{3258, 615}"
809,1,26,0,0.0490014,"\int \frac{\cos (x)}{\sin (x)+\sin ^{\sqrt{2}}(x)} \, dx","Int[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",5,5,16,0.3125,1,"{4334, 266, 36, 29, 31}"
810,1,24,0,0.0282516,"\int \frac{1}{2 \sin (x)+\sin (2 x)} \, dx","Int[(2*Sin[x] + Sin[2*x])^(-1),x]","\frac{1}{8} \tan ^2\left(\frac{x}{2}\right)+\frac{1}{4} \log \left(\tan \left(\frac{x}{2}\right)\right)","\frac{1}{8} \tan ^2\left(\frac{x}{2}\right)+\frac{1}{4} \log \left(\tan \left(\frac{x}{2}\right)\right)",1,"Log[Tan[x/2]]/4 + Tan[x/2]^2/8","A",4,2,11,0.1818,1,"{12, 14}"
811,1,40,0,0.0650418,"\int \left(-3+4 x+x^2\right) \sin (2 x) \, dx","Int[(-3 + 4*x + x^2)*Sin[2*x],x]","-\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)","-\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*Cos[2*x])/4 - 2*x*Cos[2*x] - (x^2*Cos[2*x])/2 + Sin[2*x] + (x*Sin[2*x])/2","A",8,4,13,0.3077,1,"{6742, 2638, 3296, 2637}"
812,1,27,0,0.0101598,"\int e^{-3 x} \cos (4 x) \, dx","Int[Cos[4*x]/E^(3*x),x]","\frac{4}{25} e^{-3 x} \sin (4 x)-\frac{3}{25} e^{-3 x} \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])/(25*E^(3*x)) + (4*Sin[4*x])/(25*E^(3*x))","A",1,1,10,0.1000,1,"{4433}"
813,1,23,0,0.0397623,"\int \frac{\cos (x) \sin (x)}{\sqrt{1+\sin (x)}} \, dx","Int[(Cos[x]*Sin[x])/Sqrt[1 + Sin[x]],x]","\frac{2}{3} (\sin (x)+1)^{3/2}-2 \sqrt{\sin (x)+1}","\frac{2}{3} (\sin (x)+1)^{3/2}-2 \sqrt{\sin (x)+1}",1,"-2*Sqrt[1 + Sin[x]] + (2*(1 + Sin[x])^(3/2))/3","A",3,2,13,0.1538,1,"{2833, 43}"
814,1,30,0,0.0303572,"\int \left(x+60 \cos ^5(x) \sin ^4(x)\right) \, dx","Int[x + 60*Cos[x]^5*Sin[x]^4,x]","\frac{x^2}{2}+\frac{20 \sin ^9(x)}{3}-\frac{120 \sin ^7(x)}{7}+12 \sin ^5(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 + 12*Sin[x]^5 - (120*Sin[x]^7)/7 + (20*Sin[x]^9)/3","A",4,2,12,0.1667,1,"{2564, 270}"
815,1,6,0,0.0106472,"\int \cos (x) (\sec (x)+\tan (x)) \, dx","Int[Cos[x]*(Sec[x] + Tan[x]),x]","x-\cos (x)","x-\cos (x)",1,"x - Cos[x]","A",3,2,8,0.2500,1,"{3161, 2638}"
816,1,7,0,0.0376473,"\int \cos (x) \left(\sec ^3(x)+\tan (x)\right) \, dx","Int[Cos[x]*(Sec[x]^3 + Tan[x]),x]","\tan (x)-\cos (x)","\tan (x)-\cos (x)",1,"-Cos[x] + Tan[x]","A",5,4,10,0.4000,1,"{4401, 3767, 8, 2638}"
817,1,13,0,0.0135493,"\int \frac{1}{2} \left(-\cot (x) \csc (x)+\csc ^2(x)\right) \, dx","Int[(-(Cot[x]*Csc[x]) + Csc[x]^2)/2,x]","\frac{\csc (x)}{2}-\frac{\cot (x)}{2}","\frac{\csc (x)}{2}-\frac{\cot (x)}{2}",1,"-Cot[x]/2 + Csc[x]/2","A",6,4,15,0.2667,1,"{12, 2606, 8, 3767}"
818,1,11,0,0.0082446,"\int \left(-\csc ^2(x)+\sin (2 x)\right) \, dx","Int[-Csc[x]^2 + Sin[2*x],x]","\cot (x)-\frac{1}{2} \cos (2 x)","\cot (x)-\frac{1}{2} \cos (2 x)",1,"-Cos[2*x]/2 + Cot[x]","A",4,3,11,0.2727,1,"{3767, 8, 2638}"
819,1,10,0,0.0076479,"\int (2 \cot (2 x)-3 \sin (3 x)) \, dx","Int[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",3,2,13,0.1538,1,"{3475, 2638}"
820,1,10,0,0.0077968,"\int x \sin \left(2 x^2\right) \, dx","Int[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,"-Cos[2*x^2]/4","A",2,2,8,0.2500,1,"{3379, 2638}"
821,1,18,0,0.0411609,"\int -\cos (1-x) \sin (1-x) \sqrt{1+\sin ^2(1-x)} \, dx","Int[-(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",2,2,28,0.07143,1,"{3198, 261}"
822,1,10,0,0.0123809,"\int \frac{\cos \left(\frac{1}{x}\right) \sin \left(\frac{1}{x}\right)}{x^2} \, dx","Int[(Cos[x^(-1)]*Sin[x^(-1)])/x^2,x]","-\frac{1}{2} \sin ^2\left(\frac{1}{x}\right)","-\frac{1}{2} \sin ^2\left(\frac{1}{x}\right)",1,"-Sin[x^(-1)]^2/2","A",1,1,12,0.08333,1,"{3441}"
823,1,16,0,0.0161971,"\int \cos \left(\frac{1}{2} (1+3 x)\right) \sin ^3\left(\frac{1}{2} (1+3 x)\right) \, dx","Int[Cos[(1 + 3*x)/2]*Sin[(1 + 3*x)/2]^3,x]","\frac{1}{6} \sin ^4\left(\frac{3 x}{2}+\frac{1}{2}\right)","\frac{1}{6} \sin ^4\left(\frac{3 x}{2}+\frac{1}{2}\right)",1,"Sin[1/2 + (3*x)/2]^4/6","A",2,2,23,0.08696,1,"{2564, 30}"
824,1,7,0,0.0072001,"\int 4 x \tan \left(x^2\right) \, dx","Int[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",3,3,7,0.4286,1,"{12, 3747, 3475}"
825,1,13,0,0.0117344,"\int x \sec \left(5-x^2\right) \, dx","Int[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,"-ArcTanh[Sin[5 - x^2]]/2","A",2,2,10,0.2000,1,"{4204, 3770}"
826,1,5,0,0.0090749,"\int \frac{\csc \left(\frac{1}{x}\right)}{x^2} \, dx","Int[Csc[x^(-1)]/x^2,x]","\tanh ^{-1}\left(\cos \left(\frac{1}{x}\right)\right)","\tanh ^{-1}\left(\cos \left(\frac{1}{x}\right)\right)",1,"ArcTanh[Cos[x^(-1)]]","A",2,2,8,0.2500,1,"{4205, 3770}"
827,1,9,0,0.0454076,"\int (\csc (x)-\sec (x)) (\cos (x)+\sin (x)) \, dx","Int[(Csc[x] - Sec[x])*(Cos[x] + Sin[x]),x]","\log (\tan (x))+2 \log (\cos (x))","\log (\sin (x))+\log (\cos (x))",1,"2*Log[Cos[x]] + Log[Tan[x]]","A",4,2,13,0.1538,1,"{446, 72}"
828,1,4,0,0.0173933,"\int (-\cos (3 x) \sin (2 x)+\cos (2 x) \sin (3 x)) \, dx","Int[-(Cos[3*x]*Sin[2*x]) + Cos[2*x]*Sin[3*x],x]","-\cos (x)","-\cos (x)",1,"-Cos[x]","A",3,1,20,0.05000,1,"{4284}"
829,1,13,0,0.0200766,"\int 4 x \sec ^2(2 x) \, dx","Int[4*x*Sec[2*x]^2,x]","2 x \tan (2 x)+\log (\cos (2 x))","2 x \tan (2 x)+\log (\cos (2 x))",1,"Log[Cos[2*x]] + 2*x*Tan[2*x]","A",3,3,9,0.3333,1,"{12, 4184, 3475}"
830,1,16,0,0.0290192,"\int 4 \sin ^2(x) \tan ^2(x) \, dx","Int[4*Sin[x]^2*Tan[x]^2,x]","-6 x+6 \tan (x)-2 \sin ^2(x) \tan (x)","-6 x+6 \tan (x)-2 \sin ^2(x) \tan (x)",1,"-6*x + 6*Tan[x] - 2*Sin[x]^2*Tan[x]","A",5,5,10,0.5000,1,"{12, 2591, 288, 321, 203}"
831,1,32,0,0.0336789,"\int \cos ^4(x) \cot ^2(x) \, dx","Int[Cos[x]^4*Cot[x]^2,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)","-\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 - (15*Cot[x])/8 + (5*Cos[x]^2*Cot[x])/8 + (Cos[x]^4*Cot[x])/4","A",5,4,9,0.4444,1,"{2591, 288, 321, 203}"
832,1,18,0,0.0282526,"\int 16 \cos ^2(x) \sin ^2(x) \, dx","Int[16*Cos[x]^2*Sin[x]^2,x]","2 x-4 \sin (x) \cos ^3(x)+2 \sin (x) \cos (x)","2 x-4 \sin (x) \cos ^3(x)+2 \sin (x) \cos (x)",1,"2*x + 2*Cos[x]*Sin[x] - 4*Cos[x]^3*Sin[x]","A",4,4,10,0.4000,1,"{12, 2568, 2635, 8}"
833,1,34,0,0.0502599,"\int 8 \cos ^2(x) \sin ^4(x) \, dx","Int[8*Cos[x]^2*Sin[x]^4,x]","\frac{x}{2}-\frac{4}{3} \sin ^3(x) \cos ^3(x)-\sin (x) \cos ^3(x)+\frac{1}{2} \sin (x) \cos (x)","\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,"x/2 + (Cos[x]*Sin[x])/2 - Cos[x]^3*Sin[x] - (4*Cos[x]^3*Sin[x]^3)/3","A",5,4,10,0.4000,1,"{12, 2568, 2635, 8}"
834,1,13,0,0.0241216,"\int 35 \cos ^3(x) \sin ^4(x) \, dx","Int[35*Cos[x]^3*Sin[x]^4,x]","7 \sin ^5(x)-5 \sin ^7(x)","7 \sin ^5(x)-5 \sin ^7(x)",1,"7*Sin[x]^5 - 5*Sin[x]^7","A",4,3,10,0.3000,1,"{12, 2564, 14}"
835,1,46,0,0.056109,"\int 4 \cos ^4(x) \sin ^4(x) \, dx","Int[4*Cos[x]^4*Sin[x]^4,x]","\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)","\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,"(3*x)/32 + (3*Cos[x]*Sin[x])/32 + (Cos[x]^3*Sin[x])/16 - (Cos[x]^5*Sin[x])/4 - (Cos[x]^5*Sin[x]^3)/2","A",6,4,10,0.4000,1,"{12, 2568, 2635, 8}"
836,1,9,0,0.028245,"\int \frac{\cos (x)}{-\sin (x)+\sin ^3(x)} \, dx","Int[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",5,5,14,0.3571,1,"{4334, 266, 36, 31, 29}"
837,1,14,0,0.0166557,"\int \left(-1+2 \cos ^2(x)+\cos (x) \sin (x)\right) \, dx","Int[-1 + 2*Cos[x]^2 + Cos[x]*Sin[x],x]","\frac{\sin ^2(x)}{2}+\sin (x) \cos (x)","\frac{\sin ^2(x)}{2}+\sin (x) \cos (x)",1,"Cos[x]*Sin[x] + Sin[x]^2/2","A",5,4,13,0.3077,1,"{2635, 8, 2564, 30}"
838,1,1,0,0.0103859,"\int \left(\cos ^2(x)+\sin ^2(x)\right) \, dx","Int[Cos[x]^2 + Sin[x]^2,x]","x","x",1,"x","A",5,2,9,0.2222,1,"{2635, 8}"
839,1,6,0,0.0110818,"\int \left(-\cos ^2(x)+\sin ^2(x)\right) \, dx","Int[-Cos[x]^2 + Sin[x]^2,x]","\sin (x) (-\cos (x))","\sin (x) (-\cos (x))",1,"-(Cos[x]*Sin[x])","A",5,2,11,0.1818,1,"{2635, 8}"
840,1,9,0,0.0089618,"\int 2^{\sin (x)} \cos (x) \, dx","Int[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",2,2,7,0.2857,1,"{4334, 2194}"
841,1,8,0,0.0167934,"\int \left(\tan ^3(x)+\tan ^5(x)\right) \, dx","Int[Tan[x]^3 + Tan[x]^5,x]","\frac{\tan ^4(x)}{4}","\frac{\tan ^4(x)}{4}",1,"Tan[x]^4/4","A",6,2,9,0.2222,1,"{3473, 3475}"
842,1,6,0,0.1780284,"\int x \sec (x) (2+x \tan (x)) \, dx","Int[x*Sec[x]*(2 + x*Tan[x]),x]","x^2 \sec (x)","x^2 \sec (x)",1,"x^2*Sec[x]","A",13,5,10,0.5000,1,"{6742, 4181, 2279, 2391, 3757}"
843,1,8,0,0.1966438,"\int \frac{\cot \left(\sqrt{x}\right) \csc \left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Int[(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",3,3,18,0.1667,1,"{6715, 2606, 8}"
844,1,8,0,0.0110285,"\int \frac{\cos \left(\sqrt{x}\right) \sin \left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Int[(Cos[Sqrt[x]]*Sin[Sqrt[x]])/Sqrt[x],x]","\sin ^2\left(\sqrt{x}\right)","\sin ^2\left(\sqrt{x}\right)",1,"Sin[Sqrt[x]]^2","A",1,1,18,0.05556,1,"{3441}"
845,1,8,0,0.1838755,"\int \frac{\sec \left(\sqrt{x}\right) \tan \left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Int[(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",3,3,18,0.1667,1,"{6715, 2606, 8}"
846,1,70,0,0.1699043,"\int \frac{\sin ^2(x)}{a+b \sin (2 x)} \, dx","Int[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 \left(a \tan ^2(x)+a+2 b \tan (x)\right)}{4 b}-\frac{\log (\cos (x))}{2 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[Cos[x]]/(2*b) - Log[a + 2*b*Tan[x] + a*Tan[x]^2]/(4*b)","A",9,7,15,0.4667,1,"{1075, 12, 634, 618, 204, 628, 260}"
847,1,70,0,0.1328852,"\int \frac{\cos ^2(x)}{a+b \sin (2 x)} \, dx","Int[Cos[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 \left(a \tan ^2(x)+a+2 b \tan (x)\right)}{4 b}+\frac{\log (\cos (x))}{2 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[Cos[x]]/(2*b) + Log[a + 2*b*Tan[x] + a*Tan[x]^2]/(4*b)","A",8,7,15,0.4667,1,"{981, 634, 618, 204, 628, 12, 260}"
848,1,52,0,0.1249841,"\int \frac{\sin ^2(x)}{a+b \cos (2 x)} \, dx","Int[Sin[x]^2/(a + b*Cos[2*x]),x]","\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}","\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,"-x/(2*b) + (Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[x])/Sqrt[a + b]])/(2*Sqrt[a - b]*b)","A",4,2,15,0.1333,1,"{1130, 205}"
849,1,52,0,0.0936759,"\int \frac{\cos ^2(x)}{a+b \cos (2 x)} \, dx","Int[Cos[x]^2/(a + b*Cos[2*x]),x]","\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}}","\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/(2*b) - (Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[x])/Sqrt[a + b]])/(2*b*Sqrt[a + b])","A",4,2,15,0.1333,1,"{1093, 205}"
850,1,30,0,0.0351015,"\int \frac{\tan (c+d x)}{\sqrt{a \sin ^2(c+d x)}} \, dx","Int[Tan[c + d*x]/Sqrt[a*Sin[c + d*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin ^2(c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin ^2(c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"ArcTanh[Sqrt[a*Sin[c + d*x]^2]/Sqrt[a]]/(Sqrt[a]*d)","A",3,3,21,0.1429,1,"{3205, 63, 206}"
851,1,31,0,0.0319034,"\int \frac{\cot (c+d x)}{\sqrt{a \cos ^2(c+d x)}} \, dx","Int[Cot[c + d*x]/Sqrt[a*Cos[c + d*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \cos ^2(c+d x)}}{\sqrt{a}}\right)}{\sqrt{a} d}",1,"-(ArcTanh[Sqrt[a*Cos[c + d*x]^2]/Sqrt[a]]/(Sqrt[a]*d))","A",3,3,21,0.1429,1,"{3205, 63, 206}"
852,1,8,0,0.0127035,"\int \frac{x \cos \left(x^2\right)}{\sqrt{\sin \left(x^2\right)}} \, dx","Int[(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",1,1,14,0.07143,1,"{3441}"
853,1,19,0,0.0303875,"\int \frac{\cos (x)}{\sqrt{1-\cos (2 x)}} \, dx","Int[Cos[x]/Sqrt[1 - Cos[2*x]],x]","\frac{\sin (x) \log (\sin (x))}{\sqrt{2} \sqrt{\sin ^2(x)}}","\frac{\sin (x) \log (\sin (x))}{\sqrt{2} \sqrt{\sin ^2(x)}}",1,"(Log[Sin[x]]*Sin[x])/(Sqrt[2]*Sqrt[Sin[x]^2])","A",4,4,15,0.2667,1,"{4356, 12, 15, 29}"
854,1,29,0,0.0563327,"\int \frac{\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx","Int[(Cos[Log[x]]^2*Sin[Log[x]]^2)/x,x]","\frac{\log (x)}{8}-\frac{1}{4} \sin (\log (x)) \cos ^3(\log (x))+\frac{1}{8} \sin (\log (x)) \cos (\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 + (Cos[Log[x]]*Sin[Log[x]])/8 - (Cos[Log[x]]^3*Sin[Log[x]])/4","A",4,3,14,0.2143,1,"{2568, 2635, 8}"
855,1,37,0,0.1328451,"\int \frac{\sin ^3(x)}{\cos ^3(x)+\sin ^3(x)} \, dx","Int[Sin[x]^3/(Cos[x]^3 + Sin[x]^3),x]","\frac{x}{2}+\frac{1}{3} \log \left(\tan ^2(x)-\tan (x)+1\right)-\frac{1}{6} \log (\tan (x)+1)+\frac{1}{2} \log (\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]]/2 - Log[1 + Tan[x]]/6 + Log[1 - Tan[x] + Tan[x]^2]/3","A",7,5,16,0.3125,1,"{2074, 635, 203, 260, 628}"
856,1,37,0,0.0906246,"\int \frac{\cos ^3(x)}{\cos ^3(x)+\sin ^3(x)} \, dx","Int[Cos[x]^3/(Cos[x]^3 + Sin[x]^3),x]","\frac{x}{2}-\frac{1}{3} \log \left(\tan ^2(x)-\tan (x)+1\right)+\frac{1}{6} \log (\tan (x)+1)-\frac{1}{2} \log (\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]]/2 + Log[1 + Tan[x]]/6 - Log[1 - Tan[x] + Tan[x]^2]/3","A",7,5,16,0.3125,1,"{2058, 635, 203, 260, 628}"
857,1,44,0,0.0556506,"\int \frac{\sec (x)}{-5+\cos ^2(x)+4 \sin (x)} \, dx","Int[Sec[x]/(-5 + Cos[x]^2 + 4*Sin[x]),x]","\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)","\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,"Log[1 - Sin[x]]/2 - (4*Log[2 - Sin[x]])/9 - Log[1 + Sin[x]]/18 + 1/(3*(2 - Sin[x]))","A",4,2,15,0.1333,1,"{710, 801}"
858,1,88,0,2.2426298,"\int \frac{1}{\cos ^{\frac{3}{2}}(x) \sqrt{3 \cos (x)+\sin (x)}} \, dx","Int[1/(Cos[x]^(3/2)*Sqrt[3*Cos[x] + Sin[x]]),x]","\frac{2 \cos ^2\left(\frac{x}{2}\right) \left(-3 \tan ^2\left(\frac{x}{2}\right)+2 \tan \left(\frac{x}{2}\right)+3\right)}{\sqrt{\cos ^2\left(\frac{x}{2}\right) \left(-3 \tan ^2\left(\frac{x}{2}\right)+2 \tan \left(\frac{x}{2}\right)+3\right)} \sqrt{\cos ^2\left(\frac{x}{2}\right) \left(1-\tan ^2\left(\frac{x}{2}\right)\right)}}","\frac{2 \sqrt{\sin (x)+3 \cos (x)}}{\sqrt{\cos (x)}}",1,"(2*Cos[x/2]^2*(3 + 2*Tan[x/2] - 3*Tan[x/2]^2))/(Sqrt[Cos[x/2]^2*(3 + 2*Tan[x/2] - 3*Tan[x/2]^2)]*Sqrt[Cos[x/2]^2*(1 - Tan[x/2]^2)])","B",5,3,18,0.1667,1,"{6719, 1063, 8}"
859,0,0,0,2.568918,"\int \frac{\csc (x) \sqrt{\cos (x)+\sin (x)}}{\cos ^{\frac{3}{2}}(x)} \, dx","Int[(Csc[x]*Sqrt[Cos[x] + Sin[x]])/Cos[x]^(3/2),x]","\int \frac{\csc (x) \sqrt{\cos (x)+\sin (x)}}{\cos ^{\frac{3}{2}}(x)} \, dx","-\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,"Defer[Int][(Csc[x]*Sqrt[Cos[x] + Sin[x]])/Cos[x]^(3/2), x]","F",0,0,0,0,-1,"{}"
860,1,72,0,1.7071632,"\int \frac{\cos (x)+\sin (x)}{\sqrt{1+\sin (2 x)}} \, dx","Int[(Cos[x] + Sin[x])/Sqrt[1 + Sin[2*x]],x]","\frac{2 \cos ^2\left(\frac{x}{2}\right) \tan ^{-1}\left(\tan \left(\frac{x}{2}\right)\right) \left(-\tan ^2\left(\frac{x}{2}\right)+2 \tan \left(\frac{x}{2}\right)+1\right)}{\sqrt{\cos ^4\left(\frac{x}{2}\right) \left(-\tan ^2\left(\frac{x}{2}\right)+2 \tan \left(\frac{x}{2}\right)+1\right)^2}}","\frac{x \sqrt{\sin (2 x)+1}}{\sin (x)+\cos (x)}",1,"(2*ArcTan[Tan[x/2]]*Cos[x/2]^2*(1 + 2*Tan[x/2] - Tan[x/2]^2))/Sqrt[Cos[x/2]^4*(1 + 2*Tan[x/2] - Tan[x/2]^2)^2]","B",17,9,16,0.5625,1,"{4401, 6719, 1075, 628, 635, 203, 260, 12, 1023}"
861,1,13,0,0.145602,"\int \sec (x) \sqrt{\sec (x)+\tan (x)} \, dx","Int[Sec[x]*Sqrt[Sec[x] + Tan[x]],x]","2 \sqrt{(\sin (x)+1) \sec (x)}","2 \sqrt{(\sin (x)+1) \sec (x)}",1,"2*Sqrt[Sec[x]*(1 + Sin[x])]","A",4,4,12,0.3333,1,"{4397, 4400, 2705, 2671}"
862,1,14,0,0.0438361,"\int \sec (x) \sqrt{4+3 \sec (x)} \tan (x) \, dx","Int[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",2,2,15,0.1333,1,"{4339, 261}"
863,1,25,0,0.0853067,"\int \sec (x) \sqrt{1+\sec (x)} \tan ^3(x) \, dx","Int[Sec[x]*Sqrt[1 + Sec[x]]*Tan[x]^3,x]","\frac{2}{7} (\sec (x)+1)^{7/2}-\frac{4}{5} (\sec (x)+1)^{5/2}","\frac{2}{7} (\sec (x)+1)^{7/2}-\frac{4}{5} (\sec (x)+1)^{5/2}",1,"(-4*(1 + Sec[x])^(5/2))/5 + (2*(1 + Sec[x])^(7/2))/7","A",6,5,15,0.3333,1,"{4373, 1570, 1469, 627, 43}"
864,1,25,0,0.0810866,"\int \cot ^3(x) \csc (x) \sqrt{1+\csc (x)} \, dx","Int[Cot[x]^3*Csc[x]*Sqrt[1 + Csc[x]],x]","\frac{4}{5} (\csc (x)+1)^{5/2}-\frac{2}{7} (\csc (x)+1)^{7/2}","\frac{4}{5} (\csc (x)+1)^{5/2}-\frac{2}{7} (\csc (x)+1)^{7/2}",1,"(4*(1 + Csc[x])^(5/2))/5 - (2*(1 + Csc[x])^(7/2))/7","A",6,5,15,0.3333,1,"{4372, 1570, 1469, 627, 43}"
865,1,20,0,0.151274,"\int \sqrt{\csc (x)} (x \cos (x)-4 \sec (x) \tan (x)) \, dx","Int[Sqrt[Csc[x]]*(x*Cos[x] - 4*Sec[x]*Tan[x]),x]","\frac{2 x}{\sqrt{\csc (x)}}-\frac{4 \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)/Sqrt[Csc[x]] - (4*Sec[x])/Csc[x]^(3/2)","A",8,5,18,0.2778,1,"{6742, 4213, 3771, 2639, 2626}"
866,1,84,0,0.1622325,"\int \cot (x) \sqrt{-1+\csc ^2(x)} \left(1-\sin ^2(x)\right)^3 \, dx","Int[Cot[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3,x]","-\frac{35}{16} \sqrt{\csc ^2(x)-1}+\frac{1}{6} \sin ^6(x) \left(\csc ^2(x)-1\right)^{7/2}+\frac{7}{24} \sin ^4(x) \left(\csc ^2(x)-1\right)^{5/2}+\frac{35}{48} \sin ^2(x) \left(\csc ^2(x)-1\right)^{3/2}+\frac{35}{16} \tan ^{-1}\left(\sqrt{\csc ^2(x)-1}\right)","-\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,"(35*ArcTan[Sqrt[-1 + Csc[x]^2]])/16 - (35*Sqrt[-1 + Csc[x]^2])/16 + (35*(-1 + Csc[x]^2)^(3/2)*Sin[x]^2)/48 + (7*(-1 + Csc[x]^2)^(5/2)*Sin[x]^4)/24 + ((-1 + Csc[x]^2)^(7/2)*Sin[x]^6)/6","A",10,8,23,0.3478,1,"{3175, 4360, 25, 266, 47, 50, 63, 203}"
867,1,81,0,0.158987,"\int \cos (x) \sqrt{-1+\csc ^2(x)} \left(1-\sin ^2(x)\right)^3 \, dx","Int[Cos[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3,x]","\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))","\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]*Sin[x] + (Cos[x]^2*Sqrt[Cot[x]^2]*Sin[x])/3 + (Cos[x]^4*Sqrt[Cot[x]^2]*Sin[x])/5 + (Cos[x]^6*Sqrt[Cot[x]^2]*Sin[x])/7 - ArcTanh[Cos[x]]*Sqrt[Cot[x]^2]*Tan[x]","A",7,6,23,0.2609,1,"{3175, 4121, 3658, 2592, 302, 206}"
868,1,76,0,0.5348548,"\int \frac{x \csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}} \, dx","Int[(x*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2],x]","\frac{i \sec (x) \text{PolyLog}\left(2,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{i \sec (x) \text{PolyLog}\left(2,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 x \sec (x) \tanh ^{-1}\left(e^{i x}\right)}{\sqrt{a \sec ^2(x)}}","\frac{i \sec (x) \text{PolyLog}\left(2,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{i \sec (x) \text{PolyLog}\left(2,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 x \sec (x) \tanh ^{-1}\left(e^{i x}\right)}{\sqrt{a \sec ^2(x)}}",1,"(-2*x*ArcTanh[E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] + (I*PolyLog[2, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - (I*PolyLog[2, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2]","A",6,4,16,0.2500,1,"{6720, 4183, 2279, 2391}"
869,1,128,0,0.5918563,"\int \frac{x^2 \csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}} \, dx","Int[(x^2*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2],x]","\frac{2 i x \sec (x) \text{PolyLog}\left(2,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 i x \sec (x) \text{PolyLog}\left(2,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 \sec (x) \text{PolyLog}\left(3,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}+\frac{2 \sec (x) \text{PolyLog}\left(3,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 x^2 \sec (x) \tanh ^{-1}\left(e^{i x}\right)}{\sqrt{a \sec ^2(x)}}","\frac{2 i x \sec (x) \text{PolyLog}\left(2,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 i x \sec (x) \text{PolyLog}\left(2,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 \sec (x) \text{PolyLog}\left(3,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}+\frac{2 \sec (x) \text{PolyLog}\left(3,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 x^2 \sec (x) \tanh ^{-1}\left(e^{i x}\right)}{\sqrt{a \sec ^2(x)}}",1,"(-2*x^2*ArcTanh[E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] + ((2*I)*x*PolyLog[2, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - ((2*I)*x*PolyLog[2, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - (2*PolyLog[3, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] + (2*PolyLog[3, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2]","A",8,5,18,0.2778,1,"{6720, 4183, 2531, 2282, 6589}"
870,1,186,0,0.5695976,"\int \frac{x^3 \csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}} \, dx","Int[(x^3*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2],x]","\frac{3 i x^2 \sec (x) \text{PolyLog}\left(2,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{3 i x^2 \sec (x) \text{PolyLog}\left(2,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{6 x \sec (x) \text{PolyLog}\left(3,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}+\frac{6 x \sec (x) \text{PolyLog}\left(3,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{6 i \sec (x) \text{PolyLog}\left(4,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}+\frac{6 i \sec (x) \text{PolyLog}\left(4,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 x^3 \sec (x) \tanh ^{-1}\left(e^{i x}\right)}{\sqrt{a \sec ^2(x)}}","\frac{3 i x^2 \sec (x) \text{PolyLog}\left(2,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{3 i x^2 \sec (x) \text{PolyLog}\left(2,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{6 x \sec (x) \text{PolyLog}\left(3,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}+\frac{6 x \sec (x) \text{PolyLog}\left(3,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{6 i \sec (x) \text{PolyLog}\left(4,-e^{i x}\right)}{\sqrt{a \sec ^2(x)}}+\frac{6 i \sec (x) \text{PolyLog}\left(4,e^{i x}\right)}{\sqrt{a \sec ^2(x)}}-\frac{2 x^3 \sec (x) \tanh ^{-1}\left(e^{i x}\right)}{\sqrt{a \sec ^2(x)}}",1,"(-2*x^3*ArcTanh[E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] + ((3*I)*x^2*PolyLog[2, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - ((3*I)*x^2*PolyLog[2, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - (6*x*PolyLog[3, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] + (6*x*PolyLog[3, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] - ((6*I)*PolyLog[4, -E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2] + ((6*I)*PolyLog[4, E^(I*x)]*Sec[x])/Sqrt[a*Sec[x]^2]","A",10,6,18,0.3333,1,"{6720, 4183, 2531, 6609, 2282, 6589}"
871,1,81,0,0.4871431,"\int \frac{x \csc (x) \sec (x)}{\sqrt{a \sec ^4(x)}} \, dx","Int[(x*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^4],x]","-\frac{i \sec ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right)}{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)}}","-\frac{i \sec ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right)}{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,"((-I/2)*x^2*Sec[x]^2)/Sqrt[a*Sec[x]^4] + (x*Log[1 - E^((2*I)*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4] - ((I/2)*PolyLog[2, E^((2*I)*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4]","A",5,5,16,0.3125,1,"{6720, 3717, 2190, 2279, 2391}"
872,1,109,0,0.5742103,"\int \frac{x^2 \csc (x) \sec (x)}{\sqrt{a \sec ^4(x)}} \, dx","Int[(x^2*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^4],x]","-\frac{i x \sec ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right)}{\sqrt{a \sec ^4(x)}}+\frac{\sec ^2(x) \text{PolyLog}\left(3,e^{2 i x}\right)}{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)}}","-\frac{i x \sec ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right)}{\sqrt{a \sec ^4(x)}}+\frac{\sec ^2(x) \text{PolyLog}\left(3,e^{2 i x}\right)}{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/3)*x^3*Sec[x]^2)/Sqrt[a*Sec[x]^4] + (x^2*Log[1 - E^((2*I)*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4] - (I*x*PolyLog[2, E^((2*I)*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4] + (PolyLog[3, E^((2*I)*x)]*Sec[x]^2)/(2*Sqrt[a*Sec[x]^4])","A",6,6,18,0.3333,1,"{6720, 3717, 2190, 2531, 2282, 6589}"
873,1,143,0,0.6101255,"\int \frac{x^3 \csc (x) \sec (x)}{\sqrt{a \sec ^4(x)}} \, dx","Int[(x^3*Csc[x]*Sec[x])/Sqrt[a*Sec[x]^4],x]","-\frac{3 i x^2 \sec ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right)}{2 \sqrt{a \sec ^4(x)}}+\frac{3 x \sec ^2(x) \text{PolyLog}\left(3,e^{2 i x}\right)}{2 \sqrt{a \sec ^4(x)}}+\frac{3 i \sec ^2(x) \text{PolyLog}\left(4,e^{2 i x}\right)}{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)}}","-\frac{3 i x^2 \sec ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right)}{2 \sqrt{a \sec ^4(x)}}+\frac{3 x \sec ^2(x) \text{PolyLog}\left(3,e^{2 i x}\right)}{2 \sqrt{a \sec ^4(x)}}+\frac{3 i \sec ^2(x) \text{PolyLog}\left(4,e^{2 i x}\right)}{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,"((-I/4)*x^4*Sec[x]^2)/Sqrt[a*Sec[x]^4] + (x^3*Log[1 - E^((2*I)*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4] - (((3*I)/2)*x^2*PolyLog[2, E^((2*I)*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4] + (3*x*PolyLog[3, E^((2*I)*x)]*Sec[x]^2)/(2*Sqrt[a*Sec[x]^4]) + (((3*I)/4)*PolyLog[4, E^((2*I)*x)]*Sec[x]^2)/Sqrt[a*Sec[x]^4]","A",7,7,18,0.3889,1,"{6720, 3717, 2190, 2531, 6609, 2282, 6589}"
874,1,105,0,0.3433697,"\int x \csc (x) \sec (x) \sqrt{a \sec ^2(x)} \, dx","Int[x*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^2],x]","i \cos (x) \text{PolyLog}\left(2,-e^{i x}\right) \sqrt{a \sec ^2(x)}-i \cos (x) \text{PolyLog}\left(2,e^{i x}\right) \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))","i \cos (x) \text{PolyLog}\left(2,-e^{i x}\right) \sqrt{a \sec ^2(x)}-i \cos (x) \text{PolyLog}\left(2,e^{i x}\right) \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*Sqrt[a*Sec[x]^2] - 2*x*ArcTanh[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] - ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2] + I*Cos[x]*PolyLog[2, -E^(I*x)]*Sqrt[a*Sec[x]^2] - I*Cos[x]*PolyLog[2, E^(I*x)]*Sqrt[a*Sec[x]^2]","A",10,10,16,0.6250,1,"{6720, 2622, 321, 207, 4420, 6271, 4183, 2279, 2391, 3770}"
875,1,225,0,0.5313862,"\int x^2 \csc (x) \sec (x) \sqrt{a \sec ^2(x)} \, dx","Int[x^2*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^2],x]","2 i x \cos (x) \text{PolyLog}\left(2,-e^{i x}\right) \sqrt{a \sec ^2(x)}-2 i x \cos (x) \text{PolyLog}\left(2,e^{i x}\right) \sqrt{a \sec ^2(x)}-2 i \cos (x) \text{PolyLog}\left(2,-i e^{i x}\right) \sqrt{a \sec ^2(x)}+2 i \cos (x) \text{PolyLog}\left(2,i e^{i x}\right) \sqrt{a \sec ^2(x)}-2 \cos (x) \text{PolyLog}\left(3,-e^{i x}\right) \sqrt{a \sec ^2(x)}+2 \cos (x) \text{PolyLog}\left(3,e^{i x}\right) \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)}","2 i x \cos (x) \text{PolyLog}\left(2,-e^{i x}\right) \sqrt{a \sec ^2(x)}-2 i x \cos (x) \text{PolyLog}\left(2,e^{i x}\right) \sqrt{a \sec ^2(x)}-2 i \cos (x) \text{PolyLog}\left(2,-i e^{i x}\right) \sqrt{a \sec ^2(x)}+2 i \cos (x) \text{PolyLog}\left(2,i e^{i x}\right) \sqrt{a \sec ^2(x)}-2 \cos (x) \text{PolyLog}\left(3,-e^{i x}\right) \sqrt{a \sec ^2(x)}+2 \cos (x) \text{PolyLog}\left(3,e^{i x}\right) \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*Sqrt[a*Sec[x]^2] + (4*I)*x*ArcTan[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] - 2*x^2*ArcTanh[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] + (2*I)*x*Cos[x]*PolyLog[2, -E^(I*x)]*Sqrt[a*Sec[x]^2] - (2*I)*Cos[x]*PolyLog[2, (-I)*E^(I*x)]*Sqrt[a*Sec[x]^2] + (2*I)*Cos[x]*PolyLog[2, I*E^(I*x)]*Sqrt[a*Sec[x]^2] - (2*I)*x*Cos[x]*PolyLog[2, E^(I*x)]*Sqrt[a*Sec[x]^2] - 2*Cos[x]*PolyLog[3, -E^(I*x)]*Sqrt[a*Sec[x]^2] + 2*Cos[x]*PolyLog[3, E^(I*x)]*Sqrt[a*Sec[x]^2]","A",17,14,18,0.7778,1,"{6720, 2622, 321, 207, 4420, 14, 6273, 4183, 2531, 2282, 6589, 4181, 2279, 2391}"
876,1,341,0,0.6285303,"\int x^3 \csc (x) \sec (x) \sqrt{a \sec ^2(x)} \, dx","Int[x^3*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^2],x]","3 i x^2 \cos (x) \text{PolyLog}\left(2,-e^{i x}\right) \sqrt{a \sec ^2(x)}-3 i x^2 \cos (x) \text{PolyLog}\left(2,e^{i x}\right) \sqrt{a \sec ^2(x)}-6 i x \cos (x) \text{PolyLog}\left(2,-i e^{i x}\right) \sqrt{a \sec ^2(x)}+6 i x \cos (x) \text{PolyLog}\left(2,i e^{i x}\right) \sqrt{a \sec ^2(x)}-6 x \cos (x) \text{PolyLog}\left(3,-e^{i x}\right) \sqrt{a \sec ^2(x)}+6 x \cos (x) \text{PolyLog}\left(3,e^{i x}\right) \sqrt{a \sec ^2(x)}+6 \cos (x) \text{PolyLog}\left(3,-i e^{i x}\right) \sqrt{a \sec ^2(x)}-6 \cos (x) \text{PolyLog}\left(3,i e^{i x}\right) \sqrt{a \sec ^2(x)}-6 i \cos (x) \text{PolyLog}\left(4,-e^{i x}\right) \sqrt{a \sec ^2(x)}+6 i \cos (x) \text{PolyLog}\left(4,e^{i x}\right) \sqrt{a \sec ^2(x)}+x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \cos (x) \tan ^{-1}\left(e^{i x}\right) \sqrt{a \sec ^2(x)}-2 x^3 \cos (x) \tanh ^{-1}\left(e^{i x}\right) \sqrt{a \sec ^2(x)}","3 i x^2 \cos (x) \text{PolyLog}\left(2,-e^{i x}\right) \sqrt{a \sec ^2(x)}-3 i x^2 \cos (x) \text{PolyLog}\left(2,e^{i x}\right) \sqrt{a \sec ^2(x)}-6 i x \cos (x) \text{PolyLog}\left(2,-i e^{i x}\right) \sqrt{a \sec ^2(x)}+6 i x \cos (x) \text{PolyLog}\left(2,i e^{i x}\right) \sqrt{a \sec ^2(x)}-6 x \cos (x) \text{PolyLog}\left(3,-e^{i x}\right) \sqrt{a \sec ^2(x)}+6 x \cos (x) \text{PolyLog}\left(3,e^{i x}\right) \sqrt{a \sec ^2(x)}+6 \cos (x) \text{PolyLog}\left(3,-i e^{i x}\right) \sqrt{a \sec ^2(x)}-6 \cos (x) \text{PolyLog}\left(3,i e^{i x}\right) \sqrt{a \sec ^2(x)}-6 i \cos (x) \text{PolyLog}\left(4,-e^{i x}\right) \sqrt{a \sec ^2(x)}+6 i \cos (x) \text{PolyLog}\left(4,e^{i x}\right) \sqrt{a \sec ^2(x)}+x^3 \sqrt{a \sec ^2(x)}+6 i x^2 \cos (x) \tan ^{-1}\left(e^{i x}\right) \sqrt{a \sec ^2(x)}-2 x^3 \cos (x) \tanh ^{-1}\left(e^{i x}\right) \sqrt{a \sec ^2(x)}",1,"x^3*Sqrt[a*Sec[x]^2] + (6*I)*x^2*ArcTan[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] - 2*x^3*ArcTanh[E^(I*x)]*Cos[x]*Sqrt[a*Sec[x]^2] + (3*I)*x^2*Cos[x]*PolyLog[2, -E^(I*x)]*Sqrt[a*Sec[x]^2] - (6*I)*x*Cos[x]*PolyLog[2, (-I)*E^(I*x)]*Sqrt[a*Sec[x]^2] + (6*I)*x*Cos[x]*PolyLog[2, I*E^(I*x)]*Sqrt[a*Sec[x]^2] - (3*I)*x^2*Cos[x]*PolyLog[2, E^(I*x)]*Sqrt[a*Sec[x]^2] - 6*x*Cos[x]*PolyLog[3, -E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*Cos[x]*PolyLog[3, (-I)*E^(I*x)]*Sqrt[a*Sec[x]^2] - 6*Cos[x]*PolyLog[3, I*E^(I*x)]*Sqrt[a*Sec[x]^2] + 6*x*Cos[x]*PolyLog[3, E^(I*x)]*Sqrt[a*Sec[x]^2] - (6*I)*Cos[x]*PolyLog[4, -E^(I*x)]*Sqrt[a*Sec[x]^2] + (6*I)*Cos[x]*PolyLog[4, E^(I*x)]*Sqrt[a*Sec[x]^2]","A",21,13,18,0.7222,1,"{6720, 2622, 321, 207, 4420, 14, 6273, 4183, 2531, 6609, 2282, 6589, 4181}"
877,1,142,0,0.3994371,"\int x \csc (x) \sec (x) \sqrt{a \sec ^4(x)} \, dx","Int[x*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]","\frac{1}{2} i \cos ^2(x) \text{PolyLog}\left(2,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{1}{2} i \cos ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right) \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)}","\frac{1}{2} i \cos ^2(x) \text{PolyLog}\left(2,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{1}{2} i \cos ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right) \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,"(x*Cos[x]^2*Sqrt[a*Sec[x]^4])/2 - 2*x*ArcTanh[E^((2*I)*x)]*Cos[x]^2*Sqrt[a*Sec[x]^4] + (I/2)*Cos[x]^2*PolyLog[2, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - (I/2)*Cos[x]^2*PolyLog[2, E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - (Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x])/2 + (x*Sqrt[a*Sec[x]^4]*Sin[x]^2)/2","A",12,11,16,0.6875,1,"{6720, 2620, 14, 4420, 2548, 4419, 4183, 2279, 2391, 3473, 8}"
878,1,220,0,0.5372659,"\int x^2 \csc (x) \sec (x) \sqrt{a \sec ^4(x)} \, dx","Int[x^2*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]","i x \cos ^2(x) \text{PolyLog}\left(2,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}-i x \cos ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{1}{2} \cos ^2(x) \text{PolyLog}\left(3,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{1}{2} \cos ^2(x) \text{PolyLog}\left(3,e^{2 i x}\right) \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)}","i x \cos ^2(x) \text{PolyLog}\left(2,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}-i x \cos ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{1}{2} \cos ^2(x) \text{PolyLog}\left(3,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{1}{2} \cos ^2(x) \text{PolyLog}\left(3,e^{2 i x}\right) \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,"(x^2*Cos[x]^2*Sqrt[a*Sec[x]^4])/2 - 2*x^2*ArcTanh[E^((2*I)*x)]*Cos[x]^2*Sqrt[a*Sec[x]^4] - Cos[x]^2*Log[Cos[x]]*Sqrt[a*Sec[x]^4] + I*x*Cos[x]^2*PolyLog[2, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - I*x*Cos[x]^2*PolyLog[2, E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - (Cos[x]^2*PolyLog[3, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4])/2 + (Cos[x]^2*PolyLog[3, E^((2*I)*x)]*Sqrt[a*Sec[x]^4])/2 - x*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x] + (x^2*Sqrt[a*Sec[x]^4]*Sin[x]^2)/2","A",16,13,18,0.7222,1,"{6720, 2620, 14, 4420, 2551, 4419, 4183, 2531, 2282, 6589, 3720, 3475, 30}"
879,1,356,0,0.6369571,"\int x^3 \csc (x) \sec (x) \sqrt{a \sec ^4(x)} \, dx","Int[x^3*Csc[x]*Sec[x]*Sqrt[a*Sec[x]^4],x]","\frac{3}{2} i x^2 \cos ^2(x) \text{PolyLog}\left(2,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{3}{2} i x^2 \cos ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{3}{2} x \cos ^2(x) \text{PolyLog}\left(3,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{3}{2} x \cos ^2(x) \text{PolyLog}\left(3,e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{3}{2} i \cos ^2(x) \text{PolyLog}\left(2,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{3}{4} i \cos ^2(x) \text{PolyLog}\left(4,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{3}{4} i \cos ^2(x) \text{PolyLog}\left(4,e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{3}{2} i x^2 \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} 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)}","\frac{3}{2} i x^2 \cos ^2(x) \text{PolyLog}\left(2,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{3}{2} i x^2 \cos ^2(x) \text{PolyLog}\left(2,e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{3}{2} x \cos ^2(x) \text{PolyLog}\left(3,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{3}{2} x \cos ^2(x) \text{PolyLog}\left(3,e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{3}{2} i \cos ^2(x) \text{PolyLog}\left(2,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}-\frac{3}{4} i \cos ^2(x) \text{PolyLog}\left(4,-e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{3}{4} i \cos ^2(x) \text{PolyLog}\left(4,e^{2 i x}\right) \sqrt{a \sec ^4(x)}+\frac{1}{2} x^3 \cos ^2(x) \sqrt{a \sec ^4(x)}+\frac{3}{2} i x^2 \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} 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,"((3*I)/2)*x^2*Cos[x]^2*Sqrt[a*Sec[x]^4] + (x^3*Cos[x]^2*Sqrt[a*Sec[x]^4])/2 - 2*x^3*ArcTanh[E^((2*I)*x)]*Cos[x]^2*Sqrt[a*Sec[x]^4] - 3*x*Cos[x]^2*Log[1 + E^((2*I)*x)]*Sqrt[a*Sec[x]^4] + ((3*I)/2)*Cos[x]^2*PolyLog[2, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4] + ((3*I)/2)*x^2*Cos[x]^2*PolyLog[2, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - ((3*I)/2)*x^2*Cos[x]^2*PolyLog[2, E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - (3*x*Cos[x]^2*PolyLog[3, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4])/2 + (3*x*Cos[x]^2*PolyLog[3, E^((2*I)*x)]*Sqrt[a*Sec[x]^4])/2 - ((3*I)/4)*Cos[x]^2*PolyLog[4, -E^((2*I)*x)]*Sqrt[a*Sec[x]^4] + ((3*I)/4)*Cos[x]^2*PolyLog[4, E^((2*I)*x)]*Sqrt[a*Sec[x]^4] - (3*x^2*Cos[x]*Sqrt[a*Sec[x]^4]*Sin[x])/2 + (x^3*Sqrt[a*Sec[x]^4]*Sin[x]^2)/2","A",21,17,18,0.9444,1,"{6720, 2620, 14, 4420, 2551, 4419, 4183, 2531, 6609, 2282, 6589, 3720, 3719, 2190, 2279, 2391, 30}"
880,1,25,0,0.031031,"\int \sin (x) \sin (2 x) \sin (3 x) \, dx","Int[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,"-Cos[2*x]/8 - Cos[4*x]/16 + Cos[6*x]/24","A",5,2,11,0.1818,1,"{4355, 2638}"
881,1,30,0,0.0326408,"\int \cos (x) \cos (2 x) \cos (3 x) \, dx","Int[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",5,2,11,0.1818,1,"{4355, 2637}"
882,1,30,0,0.0334716,"\int \cos (x) \sin (2 x) \sin (3 x) \, dx","Int[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",5,2,11,0.1818,1,"{4355, 2637}"
883,1,25,0,0.0310978,"\int \cos (2 x) \cos (3 x) \sin (x) \, dx","Int[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,"-Cos[2*x]/8 + Cos[4*x]/16 - Cos[6*x]/24","A",5,2,11,0.1818,1,"{4355, 2638}"
884,1,8,0,0.0062987,"\int x \sin \left(x^2\right) \, dx","Int[x*Sin[x^2],x]","-\frac{1}{2} \cos \left(x^2\right)","-\frac{1}{2} \cos \left(x^2\right)",1,"-Cos[x^2]/2","A",2,2,6,0.3333,1,"{3379, 2638}"
885,1,11,0,0.0207024,"\int (-\cos (x)+\sin (x)) (\cos (x)+\sin (x))^5 \, dx","Int[(-Cos[x] + Sin[x])*(Cos[x] + Sin[x])^5,x]","-\frac{1}{6} (\sin (x)+\cos (x))^6","-\frac{1}{6} (\sin (x)+\cos (x))^6",1,"-(Cos[x] + Sin[x])^6/6","A",1,1,15,0.06667,1,"{3145}"
886,1,11,0,0.0187379,"\int 2 x \sec ^2(x) \tan (x) \, dx","Int[2*x*Sec[x]^2*Tan[x],x]","x \sec ^2(x)-\tan (x)","x \sec ^2(x)-\tan (x)",1,"x*Sec[x]^2 - Tan[x]","A",4,4,9,0.4444,1,"{12, 3757, 3767, 8}"
887,1,12,0,0.0468899,"\int \frac{1+\cos ^2(x)}{1+\cos (2 x)} \, dx","Int[(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",3,2,15,0.1333,1,"{388, 203}"
888,1,17,0,0.0390701,"\int \frac{\sin (x)}{\cos ^3(x)-\cos ^5(x)} \, dx","Int[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",4,3,16,0.1875,1,"{4335, 266, 44}"
889,1,19,0,0.0396507,"\int \sec (x) \left(5-11 \sec ^5(x)\right)^2 \tan (x) \, dx","Int[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",3,2,15,0.1333,1,"{4339, 270}"
890,1,44,0,0.0380145,"\int \sin ^3(5 x) \tan ^3(5 x) \, dx","Int[Sin[5*x]^3*Tan[5*x]^3,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))","\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,"-ArcTanh[Sin[5*x]]/2 + Sin[5*x]/2 + Sin[5*x]^3/6 + (Sin[5*x]^3*Tan[5*x]^2)/10","A",5,4,13,0.3077,1,"{2592, 288, 302, 206}"
891,1,37,0,0.0328969,"\int \sin ^3(5 x) \tan ^4(5 x) \, dx","Int[Sin[5*x]^3*Tan[5*x]^4,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)","\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,"(-3*Cos[5*x])/5 + Cos[5*x]^3/15 - (3*Sec[5*x])/5 + Sec[5*x]^3/15","A",3,2,13,0.1538,1,"{2590, 270}"
892,1,54,0,0.0411704,"\int \sin ^5(6 x) \tan ^3(6 x) \, dx","Int[Sin[6*x]^5*Tan[6*x]^3,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))","\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*Sin[6*x])/12 + (7*Sin[6*x]^3)/36 + (7*Sin[6*x]^5)/60 + (Sin[6*x]^5*Tan[6*x]^2)/12","A",5,4,13,0.3077,1,"{2592, 288, 302, 206}"
893,1,37,0,0.0391507,"\int \left(-1+\sec ^2(2 x)\right)^3 \sin (2 x) \, dx","Int[(-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",4,3,15,0.2000,1,"{4120, 2590, 270}"
894,1,34,0,0.0249877,"\int \sin (x) \tan ^5(x) \, dx","Int[Sin[x]*Tan[x]^5,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))","-\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*Sin[x])/8 - (5*Sin[x]*Tan[x]^2)/8 + (Sin[x]*Tan[x]^4)/4","A",5,4,7,0.5714,1,"{2592, 288, 321, 206}"
895,1,43,0,0.0360787,"\int \cos ^5(2 x) \cot ^4(2 x) \, dx","Int[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",3,2,13,0.1538,1,"{2590, 270}"
896,1,87,0,0.1301784,"\int \cos (3 x) \left(-1+\csc ^2(3 x)\right)^3 \left(1-\sin ^2(3 x)\right)^5 \, dx","Int[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",5,4,27,0.1481,1,"{3175, 4120, 2590, 270}"
897,1,42,0,0.1183091,"\int \cot (2 x) \left(-1+\csc ^2(2 x)\right)^2 \left(1-\sin ^2(2 x)\right)^2 \, dx","Int[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",5,4,27,0.1481,1,"{3175, 4360, 266, 43}"
898,1,63,0,0.123567,"\int \cos (2 x) \left(-1+\csc ^2(2 x)\right)^4 \left(1-\sin ^2(2 x)\right)^2 \, dx","Int[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",5,4,27,0.1481,1,"{3175, 4120, 2590, 270}"
899,1,60,0,0.1260588,"\int \cot (3 x) \left(-1+\csc ^2(3 x)\right)^3 \left(1-\sin ^2(3 x)\right)^2 \, dx","Int[Cot[3*x]*(-1 + Csc[3*x]^2)^3*(1 - Sin[3*x]^2)^2,x]","-\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))","-\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,"(-5*Csc[3*x]^2)/3 + (5*Csc[3*x]^4)/12 - Csc[3*x]^6/18 - (10*Log[Sin[3*x]])/3 + (5*Sin[3*x]^2)/6 - Sin[3*x]^4/12","A",5,4,27,0.1481,1,"{3175, 4360, 266, 43}"
900,1,47,0,0.1023267,"\int \left(1+\cot ^2(9 x)\right)^2 \left(1+\tan ^2(9 x)\right)^3 \, dx","Int[(1 + Cot[9*x]^2)^2*(1 + Tan[9*x]^2)^3,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)","\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,"(-4*Cot[9*x])/9 - Cot[9*x]^3/27 + (2*Tan[9*x])/3 + (4*Tan[9*x]^3)/27 + Tan[9*x]^5/45","A",5,3,21,0.1429,1,"{3657, 2620, 270}"
901,1,43,0,0.1157409,"\int \frac{\cos (x) \left(9-7 \sin ^3(x)\right)^2}{1-\sin ^2(x)} \, dx","Int[(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}+63 \sin ^2(x)-49 \sin (x)-2 \log (1-\sin (x))+128 \log (\sin (x)+1)","-\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,"-2*Log[1 - Sin[x]] + 128*Log[1 + Sin[x]] - 49*Sin[x] + 63*Sin[x]^2 - (49*Sin[x]^3)/3 - (49*Sin[x]^5)/5","A",7,5,23,0.2174,1,"{3175, 3223, 1810, 633, 31}"
902,1,42,0,0.0406121,"\int \cos ^4(2 x) \cot ^5(2 x) \, dx","Int[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",4,3,13,0.2308,1,"{2590, 266, 43}"
903,1,74,0,0.2460077,"\int \frac{\sec (x) \tan ^2(x)}{4+3 \sec (x)} \, dx","Int[(Sec[x]*Tan[x]^2)/(4 + 3*Sec[x]),x]","\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)","\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,"(-4*ArcTanh[Sin[x]])/9 - (Sqrt[7]*Log[Sqrt[7]*Cos[x/2] - Sin[x/2]])/9 + (Sqrt[7]*Log[Sqrt[7]*Cos[x/2] + Sin[x/2]])/9 + Tan[x]/3","A",7,7,15,0.4667,1,"{4397, 2723, 3056, 3001, 3770, 2659, 206}"
904,1,14,0,0.0106727,"\int x \sec (1+x) \tan (1+x) \, dx","Int[x*Sec[1 + x]*Tan[1 + x],x]","x \sec (x+1)-\tanh ^{-1}(\sin (x+1))","x \sec (x+1)-\tanh ^{-1}(\sin (x+1))",1,"-ArcTanh[Sin[1 + x]] + x*Sec[1 + x]","A",2,2,10,0.2000,1,"{3757, 3770}"
905,1,14,0,0.0379825,"\int \frac{\sin (2 x)}{\sqrt{9-\sin ^2(x)}} \, dx","Int[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",3,2,17,0.1176,1,"{12, 261}"
906,1,11,0,0.0538037,"\int \frac{\sin (2 x)}{\sqrt{9-\cos ^4(x)}} \, dx","Int[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",5,4,17,0.2353,1,"{12, 1107, 619, 216}"
907,1,34,0,0.0485441,"\int \frac{\cos \left(\frac{1}{x}\right)}{x^5} \, dx","Int[Cos[x^(-1)]/x^5,x]","-\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)","-\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,"6*Cos[x^(-1)] - (3*Cos[x^(-1)])/x^2 - Sin[x^(-1)]/x^3 + (6*Sin[x^(-1)])/x","A",5,3,8,0.3750,1,"{3380, 3296, 2638}"
908,1,21,0,0.0298582,"\int \cos ^3(1+x) \sin ^3(1+x) \, dx","Int[Cos[1 + x]^3*Sin[1 + x]^3,x]","\frac{1}{4} \sin ^4(x+1)-\frac{1}{6} \sin ^6(x+1)","\frac{1}{4} \sin ^4(x+1)-\frac{1}{6} \sin ^6(x+1)",1,"Sin[1 + x]^4/4 - Sin[1 + x]^6/6","A",3,2,13,0.1538,1,"{2564, 14}"
909,1,99,0,0.0595834,"\int (1+2 x)^3 \sin ^2(1+2 x) \, dx","Int[(1 + 2*x)^3*Sin[1 + 2*x]^2,x]","-\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)","-\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*x)/4 - (3*x^2)/4 + (1 + 2*x)^4/16 + (3*(1 + 2*x)*Cos[1 + 2*x]*Sin[1 + 2*x])/8 - ((1 + 2*x)^3*Cos[1 + 2*x]*Sin[1 + 2*x])/4 - (3*Sin[1 + 2*x]^2)/16 + (3*(1 + 2*x)^2*Sin[1 + 2*x]^2)/8","A",4,3,16,0.1875,1,"{3311, 32, 3310}"
910,1,37,0,0.0895854,"\int \frac{-1+\sec (x)}{1-\tan (x)} \, dx","Int[(-1 + Sec[x])/(1 - Tan[x]),x]","-\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}}","-\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 + ArcTanh[(Cos[x]*(1 + Tan[x]))/Sqrt[2]]/Sqrt[2] + Log[Cos[x] - Sin[x]]/2","A",6,5,13,0.3846,1,"{4401, 3484, 3530, 3509, 206}"
911,1,57,0,0.0734621,"\int x^2 \cos (3 x) \cos (5 x) \, dx","Int[x^2*Cos[3*x]*Cos[5*x],x]","\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)","\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,"(x*Cos[2*x])/4 + (x*Cos[8*x])/64 - Sin[2*x]/8 + (x^2*Sin[2*x])/4 - Sin[8*x]/512 + (x^2*Sin[8*x])/16","A",8,3,12,0.2500,1,"{4429, 3296, 2637}"
912,1,243,0,0.2113685,"\int \frac{\cos (x)+\sin (x)}{\sqrt{\cos (x)} \sqrt{\sin (x)}} \, dx","Int[(Cos[x] + Sin[x])/(Sqrt[Cos[x]]*Sqrt[Sin[x]]),x]","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\cos (x)}}{\sqrt{\sin (x)}}\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\cos (x)}}{\sqrt{\sin (x)}}+1\right)}{\sqrt{2}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)}{\sqrt{2}}+\frac{\log \left(\tan (x)-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)}{2 \sqrt{2}}-\frac{\log \left(\tan (x)+\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)}{2 \sqrt{2}}-\frac{\log \left(\cot (x)-\frac{\sqrt{2} \sqrt{\cos (x)}}{\sqrt{\sin (x)}}+1\right)}{2 \sqrt{2}}+\frac{\log \left(\cot (x)+\frac{\sqrt{2} \sqrt{\cos (x)}}{\sqrt{\sin (x)}}+1\right)}{2 \sqrt{2}}","\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,"ArcTan[1 - (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/Sqrt[2] - ArcTan[1 + (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/Sqrt[2] - ArcTan[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/Sqrt[2] + ArcTan[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]]]/Sqrt[2] - Log[1 + Cot[x] - (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/(2*Sqrt[2]) + Log[1 + Cot[x] + (Sqrt[2]*Sqrt[Cos[x]])/Sqrt[Sin[x]]]/(2*Sqrt[2]) + Log[1 - (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2]) - Log[1 + (Sqrt[2]*Sqrt[Sin[x]])/Sqrt[Cos[x]] + Tan[x]]/(2*Sqrt[2])","B",22,9,18,0.5000,1,"{3107, 2575, 297, 1162, 617, 204, 1165, 628, 2574}"
913,1,5,0,0.0231152,"\int \sec ^2(x) (1+\sin (x)) \, dx","Int[Sec[x]^2*(1 + Sin[x]),x]","\tan (x)+\sec (x)","\tan (x)+\sec (x)",1,"Sec[x] + Tan[x]","A",3,3,9,0.3333,1,"{2669, 3767, 8}"
914,0,0,0,0.2759106,"\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","Int[10*x^9*Cos[x^5*Log[x]] - x^10*(x^4 + 5*x^4*Log[x])*Sin[x^5*Log[x]],x]","\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","x^{10} \cos \left(x^5 \log (x)\right)",1,"10*Defer[Int][x^9*Cos[x^5*Log[x]], x] - Defer[Int][x^14*Sin[x^5*Log[x]], x] - 5*Defer[Int][x^14*Log[x]*Sin[x^5*Log[x]], x]","F",0,0,0,0,-1,"{}"
915,0,0,0,0.0625198,"\int \cos ^2\left(\frac{x}{2}\right) \tan \left(\frac{\pi }{4}+\frac{x}{2}\right) \, dx","Int[Cos[x/2]^2*Tan[Pi/4 + x/2],x]","\int \cos ^2\left(\frac{x}{2}\right) \tan \left(\frac{\pi }{4}+\frac{x}{2}\right) \, dx","\frac{x}{2}-\frac{\cos (x)}{2}-\log \left(\cos \left(\frac{x}{2}+\frac{\pi }{4}\right)\right)",1,"Defer[Int][Cos[x/2]^2*Tan[Pi/4 + x/2], x]","F",0,0,0,0,-1,"{}"
916,1,65,0,0.0683732,"\int (2+3 x)^2 \sin ^3(x) \, dx","Int[(2 + 3*x)^2*Sin[x]^3,x]","\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)","\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,"14*Cos[x] - (2*(2 + 3*x)^2*Cos[x])/3 - (2*Cos[x]^3)/3 + 4*(2 + 3*x)*Sin[x] - ((2 + 3*x)^2*Cos[x]*Sin[x]^2)/3 + (2*(2 + 3*x)*Sin[x]^3)/3","A",6,4,12,0.3333,1,"{3311, 3296, 2638, 2633}"
917,1,8,0,0.0232947,"\int \sec ^{1+m}(x) \sin (x) \, dx","Int[Sec[x]^(1 + m)*Sin[x],x]","\frac{\sec ^m(x)}{m}","\frac{\sec ^m(x)}{m}",1,"Sec[x]^m/m","A",2,2,9,0.2222,1,"{2622, 30}"
918,1,32,0,0.0400978,"\int \cos ^n(a+b x) \sin ^{-2-n}(a+b x) \, dx","Int[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",1,1,21,0.04762,1,"{2563}"
919,1,3,0,0.0301497,"\int \frac{1}{\sec (x)+\sin (x) \tan (x)} \, dx","Int[(Sec[x] + Sin[x]*Tan[x])^(-1),x]","\tan ^{-1}(\sin (x))","\tan ^{-1}(\sin (x))",1,"ArcTan[Sin[x]]","A",3,3,10,0.3000,1,"{4397, 3190, 203}"
920,1,35,0,0.0654437,"\int \left(a+b x+c x^2\right) \sin (x) \, dx","Int[(a + b*x + c*x^2)*Sin[x],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)","-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]) + 2*c*Cos[x] - b*x*Cos[x] - c*x^2*Cos[x] + b*Sin[x] + 2*c*x*Sin[x]","A",8,4,13,0.3077,1,"{6742, 2638, 3296, 2637}"
921,1,8,0,0.0065648,"\int \frac{\sin \left(x^5\right)}{x} \, dx","Int[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",1,1,8,0.1250,1,"{3375}"
922,1,37,0,0.1732052,"\int \frac{\sin \left(2^x\right)}{1+2^x} \, dx","Int[Sin[2^x]/(1 + 2^x),x]","\frac{\sin (1) \text{CosIntegral}\left(2^x+1\right)}{\log (2)}+\frac{\text{Si}\left(2^x\right)}{\log (2)}-\frac{\cos (1) \text{Si}\left(1+2^x\right)}{\log (2)}","\frac{\sin (1) \text{CosIntegral}\left(2^x+1\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])/Log[2] + SinIntegral[2^x]/Log[2] - (Cos[1]*SinIntegral[1 + 2^x])/Log[2]","A",7,5,12,0.4167,1,"{2282, 6742, 3299, 3303, 3302}"
923,1,14,0,0.0131242,"\int x \cos \left(2 x^2\right) \sin ^{\frac{3}{4}}\left(2 x^2\right) \, dx","Int[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",1,1,18,0.05556,1,"{3441}"
924,1,10,0,0.0369431,"\int x \sec ^2\left(x^2\right) \tan ^2\left(x^2\right) \, dx","Int[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",1,1,14,0.07143,1,"{6686}"
925,1,17,0,0.0241569,"\int x^2 \cos ^7\left(a+b x^3\right) \sin \left(a+b x^3\right) \, dx","Int[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,"-Cos[a + b*x^3]^8/(24*b)","A",1,1,22,0.04545,1,"{3442}"
926,1,129,0,0.1444397,"\int x^5 \cos ^7\left(a+b x^3\right) \sin \left(a+b x^3\right) \, dx","Int[x^5*Cos[a + b*x^3]^7*Sin[a + b*x^3],x]","\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}","\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,"(35*x^3)/(3072*b) - (x^3*Cos[a + b*x^3]^8)/(24*b) + (35*Cos[a + b*x^3]*Sin[a + b*x^3])/(3072*b^2) + (35*Cos[a + b*x^3]^3*Sin[a + b*x^3])/(4608*b^2) + (7*Cos[a + b*x^3]^5*Sin[a + b*x^3])/(1152*b^2) + (Cos[a + b*x^3]^7*Sin[a + b*x^3])/(192*b^2)","A",7,4,22,0.1818,1,"{3444, 3380, 2635, 8}"
927,1,110,0,0.1124439,"\int x^5 \sec ^7\left(a+b x^3\right) \tan \left(a+b x^3\right) \, dx","Int[x^5*Sec[a + b*x^3]^7*Tan[a + b*x^3],x]","-\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}","-\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,"(-5*ArcTanh[Sin[a + b*x^3]])/(336*b^2) + (x^3*Sec[a + b*x^3]^7)/(21*b) - (5*Sec[a + b*x^3]*Tan[a + b*x^3])/(336*b^2) - (5*Sec[a + b*x^3]^3*Tan[a + b*x^3])/(504*b^2) - (Sec[a + b*x^3]^5*Tan[a + b*x^3])/(126*b^2)","A",6,4,22,0.1818,1,"{3757, 4204, 3768, 3770}"
928,1,6,0,0.0204549,"\int \frac{\sec ^2\left(\frac{1}{x}\right)}{x^2} \, dx","Int[Sec[x^(-1)]^2/x^2,x]","-\tan \left(\frac{1}{x}\right)","-\tan \left(\frac{1}{x}\right)",1,"-Tan[x^(-1)]","A",3,3,10,0.3000,1,"{4204, 3767, 8}"
929,1,4,0,0.0097691,"\int 3 x^2 \cos \left(x^3\right) \, dx","Int[3*x^2*Cos[x^3],x]","\sin \left(x^3\right)","\sin \left(x^3\right)",1,"Sin[x^3]","A",3,3,9,0.3333,1,"{12, 3380, 2637}"
930,1,27,0,0.0234298,"\int (1+2 x) \sec ^2(1+2 x) \, dx","Int[(1 + 2*x)*Sec[1 + 2*x]^2,x]","\frac{1}{2} (2 x+1) \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 + ((1 + 2*x)*Tan[1 + 2*x])/2","A",2,2,14,0.1429,1,"{4184, 3475}"
931,0,0,0,0.8107394,"\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","Int[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]","\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","\frac{2 x^2 \sqrt{3 \sin (a+b x)+x^3}}{3 b}",1,"Defer[Int][x^4/Sqrt[x^3 + 3*Sin[a + b*x]], x]/b + Defer[Int][(x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]], x] + (4*Defer[Int][x*Sqrt[x^3 + 3*Sin[a + b*x]], x])/(3*b)","F",0,0,0,0,-1,"{}"
932,0,0,0,0.1107889,"\int \frac{x^2 \cos (a+b x)}{\sqrt{x^3+3 \sin (a+b x)}} \, dx","Int[(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,"Defer[Int][(x^2*Cos[a + b*x])/Sqrt[x^3 + 3*Sin[a + b*x]], x]","A",0,0,0,0,-1,"{}"
933,0,0,0,0.3807739,"\int \frac{\cos (x)+\sin (x)}{e^{-x}+\sin (x)} \, dx","Int[(Cos[x] + Sin[x])/(E^(-x) + Sin[x]),x]","\int \frac{\cos (x)+\sin (x)}{e^{-x}+\sin (x)} \, dx","\log \left(e^x \sin (x)+1\right)",1,"x + Log[Sin[x]] - Defer[Int][(1 + E^x*Sin[x])^(-1), x] - Defer[Int][Cot[x]/(1 + E^x*Sin[x]), x]","F",0,0,0,0,-1,"{}"
934,1,77,0,0.1259697,"\int \sin (c+d x) \left(a \sin ^2(c+d x)+b \sin ^3(c+d x)\right) \, dx","Int[Sin[c + d*x]*(a*Sin[c + d*x]^2 + b*Sin[c + d*x]^3),x]","\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}","\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*x)/8 - (a*Cos[c + d*x])/d + (a*Cos[c + d*x]^3)/(3*d) - (3*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",7,5,28,0.1786,1,"{4393, 2748, 2633, 2635, 8}"
935,1,161,0,0.2699482,"\int \sin (c+d x) \left(a \sin ^2(c+d x)+b \sin ^3(c+d x)\right)^2 \, dx","Int[Sin[c + d*x]*(a*Sin[c + d*x]^2 + b*Sin[c + d*x]^3)^2,x]","-\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}","-\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,"(5*a*b*x)/8 - ((a^2 + b^2)*Cos[c + d*x])/d + ((2*a^2 + 3*b^2)*Cos[c + d*x]^3)/(3*d) - ((a^2 + 3*b^2)*Cos[c + d*x]^5)/(5*d) + (b^2*Cos[c + d*x]^7)/(7*d) - (5*a*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (5*a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(12*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^5)/(3*d)","A",9,6,30,0.2000,1,"{4393, 2789, 2635, 8, 3013, 373}"
936,1,89,0,0.108528,"\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","Int[Sin[c + d*x]*(a*Sin[c + d*x] + b*Sin[c + d*x]^2 + c*Sin[c + d*x]^3),x]","-\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}","-\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,"((4*a + 3*c)*x)/8 - (b*Cos[c + d*x])/d + (b*Cos[c + d*x]^3)/(3*d) - ((4*a + 3*c)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (c*Cos[c + d*x]*Sin[c + d*x]^3)/(4*d)","A",7,6,36,0.1667,1,"{4237, 3023, 2748, 2635, 8, 2633}"
937,1,288,0,0.4001138,"\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","Int[Sin[c + d*x]*(a*Sin[c + d*x] + b*Sin[c + d*x]^2 + c*Sin[c + d*x]^3)^2,x]","\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}","\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,"(3*a*b*x)/4 + (5*b*c*x)/8 - (a^2*Cos[c + d*x])/d - (c^2*Cos[c + d*x])/d - ((b^2 + 2*a*c)*Cos[c + d*x])/d + (a^2*Cos[c + d*x]^3)/(3*d) + (c^2*Cos[c + d*x]^3)/d + (2*(b^2 + 2*a*c)*Cos[c + d*x]^3)/(3*d) - (3*c^2*Cos[c + d*x]^5)/(5*d) - ((b^2 + 2*a*c)*Cos[c + d*x]^5)/(5*d) + (c^2*Cos[c + d*x]^7)/(7*d) - (3*a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) - (5*b*c*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*b*Cos[c + d*x]*Sin[c + d*x]^3)/(2*d) - (5*b*c*Cos[c + d*x]*Sin[c + d*x]^3)/(12*d) - (b*c*Cos[c + d*x]*Sin[c + d*x]^5)/(3*d)","A",16,5,38,0.1316,1,"{4394, 3256, 2633, 2635, 8}"
938,1,61,0,0.2894994,"\int \sin (c+d x) \left(a+\frac{b}{\sqrt{\sin (c+d x)}}+c \sin (c+d x)\right) \, dx","Int[Sin[c + d*x]*(a + b/Sqrt[Sin[c + d*x]] + c*Sin[c + d*x]),x]","-\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}","-\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,"(c*x)/2 - (a*Cos[c + d*x])/d + (2*b*EllipticE[(c - Pi/2 + d*x)/2, 2])/d - (c*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",7,6,29,0.2069,1,"{4395, 4401, 2639, 2638, 2635, 8}"
939,1,148,0,0.2394632,"\int \sin (c+d x) \left(a+\frac{b}{\sqrt{\sin (c+d x)}}+c \sin (c+d x)\right)^2 \, dx","Int[Sin[c + d*x]*(a + b/Sqrt[Sin[c + d*x]] + c*Sin[c + d*x])^2,x]","-\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}","-\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,"b^2*x + a*c*x - (a^2*Cos[c + d*x])/d - (c^2*Cos[c + d*x])/d + (c^2*Cos[c + d*x]^3)/(3*d) + (4*a*b*EllipticE[(c - Pi/2 + d*x)/2, 2])/d + (4*b*c*EllipticF[(c - Pi/2 + d*x)/2, 2])/(3*d) - (4*b*c*Cos[c + d*x]*Sqrt[Sin[c + d*x]])/(3*d) - (a*c*Cos[c + d*x]*Sin[c + d*x])/d","A",11,8,31,0.2581,1,"{4395, 4401, 2639, 2638, 2635, 2641, 8, 2633}"
940,1,34,0,0.0970483,"\int f^{a+b x} (\cos (c+d x)+i \sin (c+d x))^n \, dx","Int[f^(a + b*x)*(Cos[c + d*x] + I*Sin[c + d*x])^n,x]","\frac{f^{a+b x} \left(e^{i (c+d x)}\right)^n}{b \log (f)+i d n}","\frac{f^{a+b x} \left(e^{i (c+d x)}\right)^n}{b \log (f)+i d n}",1,"((E^(I*(c + d*x)))^n*f^(a + b*x))/(I*d*n + b*Log[f])","A",4,4,27,0.1481,1,"{4614, 2281, 2287, 2194}"
941,1,36,0,0.0973396,"\int f^{a+b x} (\cos (c+d x)-i \sin (c+d x))^n \, dx","Int[f^(a + b*x)*(Cos[c + d*x] - I*Sin[c + d*x])^n,x]","-\frac{f^{a+b x} \left(e^{-i (c+d x)}\right)^n}{-b \log (f)+i d n}","-\frac{f^{a+b x} \left(e^{-i (c+d x)}\right)^n}{-b \log (f)+i d n}",1,"-(((E^((-I)*(c + d*x)))^n*f^(a + b*x))/(I*d*n - b*Log[f]))","A",4,4,27,0.1481,1,"{4614, 2281, 2287, 2194}"
942,1,120,0,0.7011476,"\int \frac{\cos ^5(a+b x)-\sin ^5(a+b x)}{\cos ^5(a+b x)+\sin ^5(a+b x)} \, dx","Int[(Cos[a + b*x]^5 - Sin[a + b*x]^5)/(Cos[a + b*x]^5 + Sin[a + b*x]^5),x]","-\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}","-\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]]/b + Log[1 + Tan[a + b*x]]/(5*b) - (4*Log[2 - (1 - Sqrt[5])*Tan[a + b*x] + 2*Tan[a + b*x]^2])/(5*(1 - Sqrt[5])*b) - (4*Log[2 - (1 + Sqrt[5])*Tan[a + b*x] + 2*Tan[a + b*x]^2])/(5*(1 + Sqrt[5])*b)","A",7,4,39,0.1026,1,"{2074, 260, 2086, 628}"
943,1,72,0,0.1513791,"\int \frac{\cos ^4(a+b x)-\sin ^4(a+b x)}{\cos ^4(a+b x)+\sin ^4(a+b x)} \, dx","Int[(Cos[a + b*x]^4 - Sin[a + b*x]^4)/(Cos[a + b*x]^4 + Sin[a + b*x]^4),x]","\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}","\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,"-Log[1 - Sqrt[2]*Tan[a + b*x] + Tan[a + b*x]^2]/(2*Sqrt[2]*b) + Log[1 + Sqrt[2]*Tan[a + b*x] + Tan[a + b*x]^2]/(2*Sqrt[2]*b)","A",4,2,39,0.05128,1,"{1165, 628}"
944,1,55,0,0.4086126,"\int \frac{\cos ^3(a+b x)-\sin ^3(a+b x)}{\cos ^3(a+b x)+\sin ^3(a+b x)} \, dx","Int[(Cos[a + b*x]^3 - Sin[a + b*x]^3)/(Cos[a + b*x]^3 + Sin[a + b*x]^3),x]","-\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}","-\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]]/b) + Log[1 + Tan[a + b*x]]/(3*b) - (2*Log[1 - Tan[a + b*x] + Tan[a + b*x]^2])/(3*b)","A",5,3,39,0.07692,1,"{2074, 260, 628}"
945,1,16,0,0.0541188,"\int \frac{\cos ^2(a+b x)-\sin ^2(a+b x)}{\cos ^2(a+b x)+\sin ^2(a+b x)} \, dx","Int[(Cos[a + b*x]^2 - Sin[a + b*x]^2)/(Cos[a + b*x]^2 + Sin[a + b*x]^2),x]","\frac{\sin (a+b x) \cos (a+b x)}{b}","\frac{\sin (a+b x) \cos (a+b x)}{b}",1,"(Cos[a + b*x]*Sin[a + b*x])/b","A",6,3,39,0.07692,1,"{4380, 2635, 8}"
946,1,18,0,0.0281176,"\int \frac{\cos (a+b x)-\sin (a+b x)}{\cos (a+b x)+\sin (a+b x)} \, dx","Int[(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",1,1,31,0.03226,1,"{3133}"
947,1,19,0,0.3140896,"\int \frac{-\csc (a+b x)+\sec (a+b x)}{\csc (a+b x)+\sec (a+b x)} \, dx","Int[(-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",4,2,31,0.06452,1,"{801, 260}"
948,1,17,0,0.1745179,"\int \frac{-\csc ^2(a+b x)+\sec ^2(a+b x)}{\csc ^2(a+b x)+\sec ^2(a+b x)} \, dx","Int[(-Csc[a + b*x]^2 + Sec[a + b*x]^2)/(Csc[a + b*x]^2 + Sec[a + b*x]^2),x]","-\frac{\sin (a+b x) \cos (a+b x)}{b}","-\frac{\sin (a+b x) \cos (a+b x)}{b}",1,"-((Cos[a + b*x]*Sin[a + b*x])/b)","A",2,1,39,0.02564,1,"{383}"
949,1,54,0,0.5316026,"\int \frac{-\csc ^3(a+b x)+\sec ^3(a+b x)}{\csc ^3(a+b x)+\sec ^3(a+b x)} \, dx","Int[(-Csc[a + b*x]^3 + Sec[a + b*x]^3)/(Csc[a + b*x]^3 + Sec[a + b*x]^3),x]","\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}","\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]]/b - Log[1 + Tan[a + b*x]]/(3*b) + (2*Log[1 - Tan[a + b*x] + Tan[a + b*x]^2])/(3*b)","A",5,3,39,0.07692,1,"{6725, 260, 628}"
950,1,72,0,1.3959905,"\int \frac{-\csc ^4(a+b x)+\sec ^4(a+b x)}{\csc ^4(a+b x)+\sec ^4(a+b x)} \, dx","Int[(-Csc[a + b*x]^4 + Sec[a + b*x]^4)/(Csc[a + b*x]^4 + Sec[a + b*x]^4),x]","\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}","\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,"Log[1 - Sqrt[2]*Tan[a + b*x] + Tan[a + b*x]^2]/(2*Sqrt[2]*b) - Log[1 + Sqrt[2]*Tan[a + b*x] + Tan[a + b*x]^2]/(2*Sqrt[2]*b)","A",4,2,39,0.05128,1,"{1165, 628}"