1,1,36,0,0.0486186,"\int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx","Int[Sin[x]^3*(a*Cos[x] + b*Sin[x]),x]","\frac{1}{4} a \sin ^4(x)+\frac{3 b x}{8}-\frac{1}{4} b \sin ^3(x) \cos (x)-\frac{3}{8} b \sin (x) \cos (x)","\frac{1}{4} a \sin ^4(x)+\frac{3 b x}{8}-\frac{1}{4} b \sin ^3(x) \cos (x)-\frac{3}{8} b \sin (x) \cos (x)",1,"(3*b*x)/8 - (3*b*Cos[x]*Sin[x])/8 - (b*Cos[x]*Sin[x]^3)/4 + (a*Sin[x]^4)/4","A",7,5,14,0.3571,1,"{3089, 2564, 30, 2635, 8}"
2,1,24,0,0.041309,"\int \sin ^2(x) (a \cos (x)+b \sin (x)) \, dx","Int[Sin[x]^2*(a*Cos[x] + b*Sin[x]),x]","\frac{1}{3} a \sin ^3(x)+\frac{1}{3} b \cos ^3(x)-b \cos (x)","\frac{1}{3} a \sin ^3(x)+\frac{1}{3} b \cos ^3(x)-b \cos (x)",1,"-(b*Cos[x]) + (b*Cos[x]^3)/3 + (a*Sin[x]^3)/3","A",6,4,14,0.2857,1,"{3089, 2564, 30, 2633}"
3,1,25,0,0.026968,"\int \sin (x) (a \cos (x)+b \sin (x)) \, dx","Int[Sin[x]*(a*Cos[x] + b*Sin[x]),x]","\frac{1}{2} a \sin ^2(x)+\frac{b x}{2}-\frac{1}{2} b \sin (x) \cos (x)","\frac{1}{2} a \sin ^2(x)+\frac{b x}{2}-\frac{1}{2} b \sin (x) \cos (x)",1,"(b*x)/2 - (b*Cos[x]*Sin[x])/2 + (a*Sin[x]^2)/2","A",6,5,12,0.4167,1,"{3089, 2564, 30, 2635, 8}"
4,1,10,0,0.00609,"\int (a \cos (x)+b \sin (x)) \, dx","Int[a*Cos[x] + b*Sin[x],x]","a \sin (x)-b \cos (x)","a \sin (x)-b \cos (x)",1,"-(b*Cos[x]) + a*Sin[x]","A",3,2,9,0.2222,1,"{2637, 2638}"
5,1,9,0,0.0190863,"\int \csc (x) (a \cos (x)+b \sin (x)) \, dx","Int[Csc[x]*(a*Cos[x] + b*Sin[x]),x]","a \log (\sin (x))+b x","a \log (\sin (x))+b x",1,"b*x + a*Log[Sin[x]]","A",3,2,12,0.1667,1,"{3085, 3475}"
6,1,12,0,0.0333175,"\int \csc ^2(x) (a \cos (x)+b \sin (x)) \, dx","Int[Csc[x]^2*(a*Cos[x] + b*Sin[x]),x]","-a \csc (x)-b \tanh ^{-1}(\cos (x))","-a \csc (x)-b \tanh ^{-1}(\cos (x))",1,"-(b*ArcTanh[Cos[x]]) - a*Csc[x]","A",5,4,14,0.2857,1,"{3089, 3770, 2606, 8}"
7,1,15,0,0.043457,"\int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx","Int[Csc[x]^3*(a*Cos[x] + b*Sin[x]),x]","-\frac{1}{2} a \csc ^2(x)-b \cot (x)","-\frac{1}{2} a \csc ^2(x)-b \cot (x)",1,"-(b*Cot[x]) - (a*Csc[x]^2)/2","A",6,5,14,0.3571,1,"{3089, 3767, 8, 2606, 30}"
8,1,91,0,0.1069422,"\int \frac{\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Int[Sin[x]^3/(a*Cos[x] + b*Sin[x]),x]","\frac{a^2 b x}{\left(a^2+b^2\right)^2}+\frac{b x}{2 \left(a^2+b^2\right)}-\frac{a \sin ^2(x)}{2 \left(a^2+b^2\right)}-\frac{b \sin (x) \cos (x)}{2 \left(a^2+b^2\right)}-\frac{a^3 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}","\frac{a^2 b x}{\left(a^2+b^2\right)^2}+\frac{b x}{2 \left(a^2+b^2\right)}-\frac{a \sin ^2(x)}{2 \left(a^2+b^2\right)}-\frac{b \sin (x) \cos (x)}{2 \left(a^2+b^2\right)}-\frac{a^3 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}",1,"(a^2*b*x)/(a^2 + b^2)^2 + (b*x)/(2*(a^2 + b^2)) - (a^3*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 - (b*Cos[x]*Sin[x])/(2*(a^2 + b^2)) - (a*Sin[x]^2)/(2*(a^2 + b^2))","A",5,5,16,0.3125,1,"{3099, 3097, 3133, 2635, 8}"
9,1,68,0,0.0778495,"\int \frac{\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Int[Sin[x]^2/(a*Cos[x] + b*Sin[x]),x]","-\frac{a \sin (x)}{a^2+b^2}-\frac{b \cos (x)}{a^2+b^2}-\frac{a^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}","-\frac{a \sin (x)}{a^2+b^2}-\frac{b \cos (x)}{a^2+b^2}-\frac{a^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}",1,"-((a^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) - (b*Cos[x])/(a^2 + b^2) - (a*Sin[x])/(a^2 + b^2)","A",4,4,16,0.2500,1,"{3099, 3074, 206, 2638}"
10,1,35,0,0.0566863,"\int \frac{\sin (x)}{a \cos (x)+b \sin (x)} \, dx","Int[Sin[x]/(a*Cos[x] + b*Sin[x]),x]","\frac{b x}{a^2+b^2}-\frac{a \log (a \cos (x)+b \sin (x))}{a^2+b^2}","\frac{b x}{a^2+b^2}-\frac{a \log (a \cos (x)+b \sin (x))}{a^2+b^2}",1,"(b*x)/(a^2 + b^2) - (a*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)","A",2,2,14,0.1429,1,"{3097, 3133}"
11,1,36,0,0.0181187,"\int \frac{1}{a \cos (x)+b \sin (x)} \, dx","Int[(a*Cos[x] + b*Sin[x])^(-1),x]","-\frac{\tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}","-\frac{\tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}",1,"-(ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/Sqrt[a^2 + b^2])","A",2,2,11,0.1818,1,"{3074, 206}"
12,1,23,0,0.0698094,"\int \frac{\csc (x)}{a \cos (x)+b \sin (x)} \, dx","Int[Csc[x]/(a*Cos[x] + b*Sin[x]),x]","\frac{\log (\sin (x))}{a}-\frac{\log (a \cos (x)+b \sin (x))}{a}","\frac{\log (\sin (x))}{a}-\frac{\log (a \cos (x)+b \sin (x))}{a}",1,"Log[Sin[x]]/a - Log[a*Cos[x] + b*Sin[x]]/a","A",3,3,14,0.2143,1,"{3101, 3475, 3133}"
13,1,55,0,0.0675421,"\int \frac{\csc ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Int[Csc[x]^2/(a*Cos[x] + b*Sin[x]),x]","-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^2}+\frac{b \tanh ^{-1}(\cos (x))}{a^2}-\frac{\csc (x)}{a}","-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^2}+\frac{b \tanh ^{-1}(\cos (x))}{a^2}-\frac{\csc (x)}{a}",1,"(b*ArcTanh[Cos[x]])/a^2 - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/a^2 - Csc[x]/a","A",4,4,16,0.2500,1,"{3103, 3770, 3074, 206}"
14,1,55,0,0.1207393,"\int \frac{\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Int[Csc[x]^3/(a*Cos[x] + b*Sin[x]),x]","\frac{\left(a^2+b^2\right) \log (\sin (x))}{a^3}-\frac{\left(a^2+b^2\right) \log (a \cos (x)+b \sin (x))}{a^3}+\frac{b \cot (x)}{a^2}-\frac{\csc ^2(x)}{2 a}","\frac{\left(a^2+b^2\right) \log (\sin (x))}{a^3}-\frac{\left(a^2+b^2\right) \log (a \cos (x)+b \sin (x))}{a^3}+\frac{b \cot (x)}{a^2}-\frac{\csc ^2(x)}{2 a}",1,"(b*Cot[x])/a^2 - Csc[x]^2/(2*a) + ((a^2 + b^2)*Log[Sin[x]])/a^3 - ((a^2 + b^2)*Log[a*Cos[x] + b*Sin[x]])/a^3","A",6,6,16,0.3750,1,"{3103, 3767, 8, 3101, 3475, 3133}"
15,1,283,0,1.1692239,"\int \frac{\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[Sin[x]^3/(a*Cos[x] + b*Sin[x])^2,x]","\frac{3 a^3 \sin (x)}{b^3 \left(a^2+b^2\right)}+\frac{3 a^2 \cos (x)}{b^2 \left(a^2+b^2\right)}+\frac{2 a^2 \left(a+b \tan \left(\frac{x}{2}\right)\right)}{\left(a^2+b^2\right)^2 \left(-a \tan ^2\left(\frac{x}{2}\right)+a+2 b \tan \left(\frac{x}{2}\right)\right)}-\frac{2 a^3 \cos ^2\left(\frac{x}{2}\right) \left(\left(a^2-b^2\right) \tan \left(\frac{x}{2}\right)+2 a b\right)}{b^3 \left(a^2+b^2\right)^2}+\frac{2 a^2 \left(3 a^2+b^2\right) \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2}}\right)}{b \left(a^2+b^2\right)^{5/2}}-\frac{2 a^2 b \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}-\frac{3 a^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{b \left(a^2+b^2\right)^{3/2}}-\frac{2 a \sin (x)}{b^3}-\frac{\cos (x)}{b^2}","\frac{-b \left(a^2+b^2\right) \sin (2 x)+a \left(a^2+b^2\right) \cos (2 x)+3 a \left(a^2-b^2\right)}{2 \left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}+\frac{6 a^2 b \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}",1,"(-3*a^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(b*(a^2 + b^2)^(3/2)) - (2*a^2*b*ArcTanh[(b - a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (2*a^2*(3*a^2 + b^2)*ArcTanh[(b - a*Tan[x/2])/Sqrt[a^2 + b^2]])/(b*(a^2 + b^2)^(5/2)) - Cos[x]/b^2 + (3*a^2*Cos[x])/(b^2*(a^2 + b^2)) - (2*a*Sin[x])/b^3 + (3*a^3*Sin[x])/(b^3*(a^2 + b^2)) - (2*a^3*Cos[x/2]^2*(2*a*b + (a^2 - b^2)*Tan[x/2]))/(b^3*(a^2 + b^2)^2) + (2*a^2*(a + b*Tan[x/2]))/((a^2 + b^2)^2*(a + 2*b*Tan[x/2] - a*Tan[x/2]^2))","B",19,11,16,0.6875,1,"{4401, 2637, 2638, 6742, 639, 203, 638, 618, 206, 3100, 3074}"
16,1,64,0,0.1183662,"\int \frac{\sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[Sin[x]^2/(a*Cos[x] + b*Sin[x])^2,x]","-\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}+\frac{a}{\left(a^2+b^2\right) (a \cot (x)+b)}-\frac{2 a b \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}","-\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}+\frac{a}{\left(a^2+b^2\right) (a \cot (x)+b)}-\frac{2 a b \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}",1,"-(((a^2 - b^2)*x)/(a^2 + b^2)^2) + a/((a^2 + b^2)*(b + a*Cot[x])) - (2*a*b*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2","A",4,4,16,0.2500,1,"{3085, 3483, 3531, 3530}"
17,1,60,0,0.0464099,"\int \frac{\sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[Sin[x]/(a*Cos[x] + b*Sin[x])^2,x]","\frac{a}{\left(a^2+b^2\right) (a \cos (x)+b \sin (x))}-\frac{b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}","\frac{a}{\left(a^2+b^2\right) (a \cos (x)+b \sin (x))}-\frac{b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}",1,"-((b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) + a/((a^2 + b^2)*(a*Cos[x] + b*Sin[x]))","A",3,3,14,0.2143,1,"{3154, 3074, 206}"
18,1,17,0,0.0127537,"\int \frac{1}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(a*Cos[x] + b*Sin[x])^(-2),x]","\frac{\sin (x)}{a (a \cos (x)+b \sin (x))}","\frac{\sin (x)}{a (a \cos (x)+b \sin (x))}",1,"Sin[x]/(a*(a*Cos[x] + b*Sin[x]))","A",1,1,11,0.09091,1,"{3075}"
19,1,63,0,0.0601325,"\int \frac{\csc (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[Csc[x]/(a*Cos[x] + b*Sin[x])^2,x]","\frac{b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^2 \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cos (x))}{a^2}+\frac{1}{a (a \cos (x)+b \sin (x))}","\frac{b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^2 \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cos (x))}{a^2}+\frac{1}{a (a \cos (x)+b \sin (x))}",1,"-(ArcTanh[Cos[x]]/a^2) + (b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) + 1/(a*(a*Cos[x] + b*Sin[x]))","A",4,4,14,0.2857,1,"{3093, 3770, 3074, 206}"
20,1,49,0,0.0761826,"\int \frac{\csc ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[Csc[x]^2/(a*Cos[x] + b*Sin[x])^2,x]","-\frac{\frac{b}{a^2}+\frac{1}{b}}{a+b \tan (x)}-\frac{2 b \log (\tan (x))}{a^3}+\frac{2 b \log (a+b \tan (x))}{a^3}-\frac{\cot (x)}{a^2}","-\frac{\frac{b}{a^2}+\frac{1}{b}}{a+b \tan (x)}-\frac{2 b \log (\tan (x))}{a^3}+\frac{2 b \log (a+b \tan (x))}{a^3}-\frac{\cot (x)}{a^2}",1,"-(Cot[x]/a^2) - (2*b*Log[Tan[x]])/a^3 + (2*b*Log[a + b*Tan[x]])/a^3 - (b^(-1) + b/a^2)/(a + b*Tan[x])","A",3,2,16,0.1250,1,"{3087, 894}"
21,1,118,0,0.1804626,"\int \frac{\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[Csc[x]^3/(a*Cos[x] + b*Sin[x])^2,x]","\frac{a^2+b^2}{a^3 (a \cos (x)+b \sin (x))}-\frac{2 b^2 \tanh ^{-1}(\cos (x))}{a^4}-\frac{\left(a^2+b^2\right) \tanh ^{-1}(\cos (x))}{a^4}+\frac{3 b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^4}+\frac{2 b \csc (x)}{a^3}-\frac{\tanh ^{-1}(\cos (x))}{2 a^2}-\frac{\cot (x) \csc (x)}{2 a^2}","\frac{a^2+b^2}{a^3 (a \cos (x)+b \sin (x))}-\frac{2 b^2 \tanh ^{-1}(\cos (x))}{a^4}-\frac{\left(a^2+b^2\right) \tanh ^{-1}(\cos (x))}{a^4}+\frac{3 b \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^4}+\frac{2 b \csc (x)}{a^3}-\frac{\tanh ^{-1}(\cos (x))}{2 a^2}-\frac{\cot (x) \csc (x)}{2 a^2}",1,"-ArcTanh[Cos[x]]/(2*a^2) - (2*b^2*ArcTanh[Cos[x]])/a^4 - ((a^2 + b^2)*ArcTanh[Cos[x]])/a^4 + (3*b*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/a^4 + (2*b*Csc[x])/a^3 - (Cot[x]*Csc[x])/(2*a^2) + (a^2 + b^2)/(a^3*(a*Cos[x] + b*Sin[x]))","A",11,7,16,0.4375,1,"{3105, 3093, 3770, 3074, 206, 3768, 3103}"
22,1,98,0,0.198367,"\int \frac{\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx","Int[Sin[x]^3/(a*Cos[x] + b*Sin[x])^3,x]","-\frac{b x \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3}+\frac{2 a b}{\left(a^2+b^2\right)^2 (a \cot (x)+b)}+\frac{a}{2 \left(a^2+b^2\right) (a \cot (x)+b)^2}+\frac{a \left(a^2-3 b^2\right) \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}","-\frac{b x \left(3 a^2-b^2\right)}{\left(a^2+b^2\right)^3}+\frac{2 a b}{\left(a^2+b^2\right)^2 (a \cot (x)+b)}+\frac{a}{2 \left(a^2+b^2\right) (a \cot (x)+b)^2}+\frac{a \left(a^2-3 b^2\right) \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}",1,"-((b*(3*a^2 - b^2)*x)/(a^2 + b^2)^3) + a/(2*(a^2 + b^2)*(b + a*Cot[x])^2) + (2*a*b)/((a^2 + b^2)^2*(b + a*Cot[x])) + (a*(a^2 - 3*b^2)*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3","A",5,5,16,0.3125,1,"{3085, 3483, 3529, 3531, 3530}"
23,1,300,0,0.6953633,"\int \frac{\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx","Int[Sin[x]^2/(a*Cos[x] + b*Sin[x])^3,x]","-\frac{a b \left(5 a^2+2 b^2\right) \tan \left(\frac{x}{2}\right)+3 a^2 b^2+4 a^4+2 b^4}{a b \left(a^2+b^2\right)^2 \left(-a \tan ^2\left(\frac{x}{2}\right)+a+2 b \tan \left(\frac{x}{2}\right)\right)}+\frac{2 \left(\left(a^2+2 b^2\right) \tan \left(\frac{x}{2}\right)+a b\right)}{a \left(a^2+b^2\right) \left(-a \tan ^2\left(\frac{x}{2}\right)+a+2 b \tan \left(\frac{x}{2}\right)\right)^2}+\frac{2 a}{b \left(a^2+b^2\right) (a \cos (x)+b \sin (x))}-\frac{a^2 \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{x}{2}\right)}{\sqrt{a^2+b^2}}\right)}{b^2 \left(a^2+b^2\right)^{5/2}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{b^2 \left(a^2+b^2\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{b^2 \sqrt{a^2+b^2}}","\frac{a \left(\left(a^2+4 b^2\right) \sin (x)+3 a b \cos (x)\right)}{2 \left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))^2}-\frac{\left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}",1,"(2*a^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(b^2*(a^2 + b^2)^(3/2)) - ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/(b^2*Sqrt[a^2 + b^2]) - (a^2*(2*a^2 - b^2)*ArcTanh[(b - a*Tan[x/2])/Sqrt[a^2 + b^2]])/(b^2*(a^2 + b^2)^(5/2)) + (2*a)/(b*(a^2 + b^2)*(a*Cos[x] + b*Sin[x])) + (2*(a*b + (a^2 + 2*b^2)*Tan[x/2]))/(a*(a^2 + b^2)*(a + 2*b*Tan[x/2] - a*Tan[x/2]^2)^2) - (4*a^4 + 3*a^2*b^2 + 2*b^4 + a*b*(5*a^2 + 2*b^2)*Tan[x/2])/(a*b*(a^2 + b^2)^2*(a + 2*b*Tan[x/2] - a*Tan[x/2]^2))","B",13,7,16,0.4375,1,"{4401, 1660, 12, 618, 206, 3155, 3074}"
24,1,19,0,0.0260682,"\int \frac{\sin (x)}{(a \cos (x)+b \sin (x))^3} \, dx","Int[Sin[x]/(a*Cos[x] + b*Sin[x])^3,x]","\frac{\tan ^2(x)}{2 a (a+b \tan (x))^2}","\frac{1}{2 a (a \cot (x)+b)^2}",1,"Tan[x]^2/(2*a*(a + b*Tan[x])^2)","A",2,2,14,0.1429,1,"{3087, 37}"
25,1,73,0,0.0351555,"\int \frac{1}{(a \cos (x)+b \sin (x))^3} \, dx","Int[(a*Cos[x] + b*Sin[x])^(-3),x]","-\frac{b \cos (x)-a \sin (x)}{2 \left(a^2+b^2\right) (a \cos (x)+b \sin (x))^2}-\frac{\tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{2 \left(a^2+b^2\right)^{3/2}}","-\frac{b \cos (x)-a \sin (x)}{2 \left(a^2+b^2\right) (a \cos (x)+b \sin (x))^2}-\frac{\tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{2 \left(a^2+b^2\right)^{3/2}}",1,"-ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/(2*(a^2 + b^2)^(3/2)) - (b*Cos[x] - a*Sin[x])/(2*(a^2 + b^2)*(a*Cos[x] + b*Sin[x])^2)","A",3,3,11,0.2727,1,"{3076, 3074, 206}"
26,1,59,0,0.0819996,"\int \frac{\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx","Int[Csc[x]/(a*Cos[x] + b*Sin[x])^3,x]","\frac{\frac{1}{a^2}-\frac{1}{b^2}}{a+b \tan (x)}-\frac{\log (a+b \tan (x))}{a^3}+\frac{\log (\tan (x))}{a^3}+\frac{\frac{a}{b^2}+\frac{1}{a}}{2 (a+b \tan (x))^2}","\frac{\frac{1}{a^2}-\frac{1}{b^2}}{a+b \tan (x)}-\frac{\log (a+b \tan (x))}{a^3}+\frac{\log (\tan (x))}{a^3}+\frac{\frac{a}{b^2}+\frac{1}{a}}{2 (a+b \tan (x))^2}",1,"Log[Tan[x]]/a^3 - Log[a + b*Tan[x]]/a^3 + (a^(-1) + a/b^2)/(2*(a + b*Tan[x])^2) + (a^(-2) - b^(-2))/(a + b*Tan[x])","A",3,2,14,0.1429,1,"{3087, 894}"
27,1,184,0,0.2243822,"\int \frac{\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx","Int[Csc[x]^2/(a*Cos[x] + b*Sin[x])^3,x]","-\frac{2 b^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^4 \sqrt{a^2+b^2}}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^4}-\frac{\tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{2 a^2 \sqrt{a^2+b^2}}-\frac{2 b}{a^3 (a \cos (x)+b \sin (x))}-\frac{b \cos (x)-a \sin (x)}{2 a^2 (a \cos (x)+b \sin (x))^2}+\frac{3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac{\csc (x)}{a^3}","-\frac{2 b^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^4 \sqrt{a^2+b^2}}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{a^4}-\frac{\tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{2 a^2 \sqrt{a^2+b^2}}-\frac{2 b}{a^3 (a \cos (x)+b \sin (x))}-\frac{b \cos (x)-a \sin (x)}{2 a^2 (a \cos (x)+b \sin (x))^2}+\frac{3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac{\csc (x)}{a^3}",1,"(3*b*ArcTanh[Cos[x]])/a^4 - ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]]/(2*a^2*Sqrt[a^2 + b^2]) - (2*b^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2 + b^2]) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/a^4 - Csc[x]/a^3 - (b*Cos[x] - a*Sin[x])/(2*a^2*(a*Cos[x] + b*Sin[x])^2) - (2*b)/(a^3*(a*Cos[x] + b*Sin[x]))","A",12,7,16,0.4375,1,"{3105, 3076, 3074, 206, 3103, 3770, 3093}"
28,1,117,0,0.1366473,"\int \frac{\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx","Int[Csc[x]^3/(a*Cos[x] + b*Sin[x])^3,x]","\frac{\left(a^2+b^2\right)^2}{2 a^3 b^2 (a+b \tan (x))^2}-\frac{\left(a^2-3 b^2\right) \left(a^2+b^2\right)}{a^4 b^2 (a+b \tan (x))}+\frac{2 \left(a^2+3 b^2\right) \log (\tan (x))}{a^5}-\frac{2 \left(a^2+3 b^2\right) \log (a+b \tan (x))}{a^5}+\frac{3 b \cot (x)}{a^4}-\frac{\cot ^2(x)}{2 a^3}","\frac{\left(a^2+b^2\right)^2}{2 a^3 b^2 (a+b \tan (x))^2}-\frac{\left(a^2-3 b^2\right) \left(a^2+b^2\right)}{a^4 b^2 (a+b \tan (x))}+\frac{2 \left(a^2+3 b^2\right) \log (\tan (x))}{a^5}-\frac{2 \left(a^2+3 b^2\right) \log (a+b \tan (x))}{a^5}+\frac{3 b \cot (x)}{a^4}-\frac{\cot ^2(x)}{2 a^3}",1,"(3*b*Cot[x])/a^4 - Cot[x]^2/(2*a^3) + (2*(a^2 + 3*b^2)*Log[Tan[x]])/a^5 - (2*(a^2 + 3*b^2)*Log[a + b*Tan[x]])/a^5 + (a^2 + b^2)^2/(2*a^3*b^2*(a + b*Tan[x])^2) - ((a^2 - 3*b^2)*(a^2 + b^2))/(a^4*b^2*(a + b*Tan[x]))","A",3,2,16,0.1250,1,"{3087, 894}"
29,1,66,0,0.0631324,"\int \sin ^{-n}(c+d x) (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/Sin[c + d*x]^n,x]","-\frac{i \sin ^{-n}(c+d x) \, _2F_1\left(1,n;n+1;-\frac{1}{2} i (\cot (c+d x)+i)\right) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n}","-\frac{i \sin ^{-n}(c+d x) \, _2F_1\left(1,n;n+1;-\frac{1}{2} i (\cot (c+d x)+i)\right) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n}",1,"((-I/2)*Hypergeometric2F1[1, n, 1 + n, (-I/2)*(I + Cot[c + d*x])]*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n)/(d*n*Sin[c + d*x]^n)","A",1,1,33,0.03030,1,"{3083}"
30,1,87,0,0.0907608,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \cos ^6(c+d x)}{6 d}","\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \cos ^6(c+d x)}{6 d}",1,"(5*a*x)/16 - (b*Cos[c + d*x]^6)/(6*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",8,5,26,0.1923,1,"{3090, 2635, 8, 2565, 30}"
31,1,60,0,0.0700506,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^5(c+d x)}{5 d}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^5(c+d x)}{5 d}",1,"-(b*Cos[c + d*x]^5)/(5*d) + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)","A",6,4,26,0.1538,1,"{3090, 2633, 2565, 30}"
32,1,65,0,0.0777632,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}-\frac{b \cos ^4(c+d x)}{4 d}","\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}-\frac{b \cos ^4(c+d x)}{4 d}",1,"(3*a*x)/8 - (b*Cos[c + d*x]^4)/(4*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",7,5,26,0.1923,1,"{3090, 2635, 8, 2565, 30}"
33,1,44,0,0.0646631,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^3(c+d x)}{3 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^3(c+d x)}{3 d}",1,"-(b*Cos[c + d*x]^3)/(3*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)","A",6,4,26,0.1538,1,"{3090, 2633, 2565, 30}"
34,1,43,0,0.0434389,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}+\frac{b \sin ^2(c+d x)}{2 d}","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}+\frac{b \sin ^2(c+d x)}{2 d}",1,"(a*x)/2 + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b*Sin[c + d*x]^2)/(2*d)","A",6,5,24,0.2083,1,"{3090, 2635, 8, 2564, 30}"
35,1,24,0,0.0133799,"\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}"
36,1,17,0,0.0257225,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","a x-\frac{b \log (\cos (c+d x))}{d}","a x-\frac{b \log (\cos (c+d x))}{d}",1,"a*x - (b*Log[Cos[c + d*x]])/d","A",3,2,24,0.08333,1,"{3086, 3475}"
37,1,24,0,0.0453008,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \sec (c+d x)}{d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \sec (c+d x)}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d + (b*Sec[c + d*x])/d","A",5,4,26,0.1538,1,"{3090, 3770, 2606, 8}"
38,1,28,0,0.0578926,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d}","\frac{a \tan (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d}",1,"(b*Sec[c + d*x]^2)/(2*d) + (a*Tan[c + d*x])/d","A",6,5,26,0.1923,1,"{3090, 3767, 8, 2606, 30}"
39,1,52,0,0.0666957,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]^3)/(3*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,5,26,0.1923,1,"{3090, 3768, 3770, 2606, 30}"
40,1,44,0,0.0631878,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^4(c+d x)}{4 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^4(c+d x)}{4 d}",1,"(b*Sec[c + d*x]^4)/(4*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",6,4,26,0.1538,1,"{3090, 3767, 2606, 30}"
41,1,74,0,0.0808342,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \sec ^5(c+d x)}{5 d}","\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \sec ^5(c+d x)}{5 d}",1,"(3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (b*Sec[c + d*x]^5)/(5*d) + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,5,26,0.1923,1,"{3090, 3768, 3770, 2606, 30}"
42,1,60,0,0.0706185,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^6(c+d x)}{6 d}","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^6(c+d x)}{6 d}",1,"(b*Sec[c + d*x]^6)/(6*d) + (a*Tan[c + d*x])/d + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)","A",6,4,26,0.1538,1,"{3090, 3767, 2606, 30}"
43,1,137,0,0.1382499,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","-\frac{a^2 \sin ^7(c+d x)}{7 d}+\frac{3 a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{b^2 \sin ^7(c+d x)}{7 d}-\frac{2 b^2 \sin ^5(c+d x)}{5 d}+\frac{b^2 \sin ^3(c+d x)}{3 d}","-\frac{a^2 \sin ^7(c+d x)}{7 d}+\frac{3 a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos ^7(c+d x)}{7 d}+\frac{b^2 \sin ^7(c+d x)}{7 d}-\frac{2 b^2 \sin ^5(c+d x)}{5 d}+\frac{b^2 \sin ^3(c+d x)}{3 d}",1,"(-2*a*b*Cos[c + d*x]^7)/(7*d) + (a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^3)/d + (b^2*Sin[c + d*x]^3)/(3*d) + (3*a^2*Sin[c + d*x]^5)/(5*d) - (2*b^2*Sin[c + d*x]^5)/(5*d) - (a^2*Sin[c + d*x]^7)/(7*d) + (b^2*Sin[c + d*x]^7)/(7*d)","A",9,6,28,0.2143,1,"{3090, 2633, 2565, 30, 2564, 270}"
44,1,174,0,0.1703584,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^2 x}{16}-\frac{a b \cos ^6(c+d x)}{3 d}-\frac{b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b^2 x}{16}","\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^2 x}{16}-\frac{a b \cos ^6(c+d x)}{3 d}-\frac{b^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b^2 x}{16}",1,"(5*a^2*x)/16 + (b^2*x)/16 - (a*b*Cos[c + d*x]^6)/(3*d) + (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (b^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (b^2*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",12,6,28,0.2143,1,"{3090, 2635, 8, 2565, 30, 2568}"
45,1,103,0,0.1219475,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos ^5(c+d x)}{5 d}-\frac{b^2 \sin ^5(c+d x)}{5 d}+\frac{b^2 \sin ^3(c+d x)}{3 d}","\frac{a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos ^5(c+d x)}{5 d}-\frac{b^2 \sin ^5(c+d x)}{5 d}+\frac{b^2 \sin ^3(c+d x)}{3 d}",1,"(-2*a*b*Cos[c + d*x]^5)/(5*d) + (a^2*Sin[c + d*x])/d - (2*a^2*Sin[c + d*x]^3)/(3*d) + (b^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^5)/(5*d) - (b^2*Sin[c + d*x]^5)/(5*d)","A",9,6,28,0.2143,1,"{3090, 2633, 2565, 30, 2564, 14}"
46,1,126,0,0.1345955,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a^2 x}{8}-\frac{a b \cos ^4(c+d x)}{2 d}-\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{b^2 x}{8}","\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a^2 x}{8}-\frac{a b \cos ^4(c+d x)}{2 d}-\frac{b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{b^2 x}{8}",1,"(3*a^2*x)/8 + (b^2*x)/8 - (a*b*Cos[c + d*x]^4)/(2*d) + (3*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (b^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",10,6,28,0.2143,1,"{3090, 2635, 8, 2565, 30, 2568}"
47,1,67,0,0.0908333,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","-\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos ^3(c+d x)}{3 d}+\frac{b^2 \sin ^3(c+d x)}{3 d}","-\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos ^3(c+d x)}{3 d}+\frac{b^2 \sin ^3(c+d x)}{3 d}",1,"(-2*a*b*Cos[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x])/d - (a^2*Sin[c + d*x]^3)/(3*d) + (b^2*Sin[c + d*x]^3)/(3*d)","A",8,5,26,0.1923,1,"{3090, 2633, 2565, 30, 2564}"
48,1,55,0,0.0200204,"\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}"
49,1,55,0,0.07073,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos (c+d x)}{d}-\frac{b^2 \sin (c+d x)}{d}+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a^2 \sin (c+d x)}{d}-\frac{2 a b \cos (c+d x)}{d}-\frac{b^2 \sin (c+d x)}{d}+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b^2*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Cos[c + d*x])/d + (a^2*Sin[c + d*x])/d - (b^2*Sin[c + d*x])/d","A",7,6,26,0.2308,1,"{3090, 2637, 2638, 2592, 321, 206}"
50,1,39,0,0.0564281,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","x \left(a^2-b^2\right)-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","x \left(a^2-b^2\right)-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"(a^2 - b^2)*x - (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d","A",3,3,28,0.1071,1,"{3086, 3477, 3475}"
51,1,67,0,0.0936005,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/d - (b^2*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a*b*Sec[c + d*x])/d + (b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,5,28,0.1786,1,"{3090, 3770, 2606, 8, 2611}"
52,1,30,0,0.0464568,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\tan ^3(c+d x) (a \cot (c+d x)+b)^3}{3 b d}","\frac{\tan ^3(c+d x) (a \cot (c+d x)+b)^3}{3 b d}",1,"((b + a*Cot[c + d*x])^3*Tan[c + d*x]^3)/(3*b*d)","A",2,2,28,0.07143,1,"{3088, 37}"
53,1,120,0,0.1419657,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{b^2 \tan (c+d x) \sec (c+d x)}{8 d}","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{2 a b \sec ^3(c+d x)}{3 d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{b^2 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/(2*d) - (b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (2*a*b*Sec[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",9,6,28,0.2143,1,"{3090, 3768, 3770, 2606, 30, 2611}"
54,1,85,0,0.0766075,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\left(a^2+b^2\right) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^4(c+d x)}{2 d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^5(c+d x)}{5 d}","\frac{\left(a^2+b^2\right) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^4(c+d x)}{2 d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^5(c+d x)}{5 d}",1,"(a^2*Tan[c + d*x])/d + (a*b*Tan[c + d*x]^2)/d + ((a^2 + b^2)*Tan[c + d*x]^3)/(3*d) + (a*b*Tan[c + d*x]^4)/(2*d) + (b^2*Tan[c + d*x]^5)/(5*d)","A",3,2,28,0.07143,1,"{3088, 894}"
55,1,168,0,0.1695573,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{2 a b \sec ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{b^2 \tan (c+d x) \sec ^5(c+d x)}{6 d}-\frac{b^2 \tan (c+d x) \sec ^3(c+d x)}{24 d}-\frac{b^2 \tan (c+d x) \sec (c+d x)}{16 d}","\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{2 a b \sec ^5(c+d x)}{5 d}-\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{b^2 \tan (c+d x) \sec ^5(c+d x)}{6 d}-\frac{b^2 \tan (c+d x) \sec ^3(c+d x)}{24 d}-\frac{b^2 \tan (c+d x) \sec (c+d x)}{16 d}",1,"(3*a^2*ArcTanh[Sin[c + d*x]])/(8*d) - (b^2*ArcTanh[Sin[c + d*x]])/(16*d) + (2*a*b*Sec[c + d*x]^5)/(5*d) + (3*a^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (b^2*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (b^2*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (b^2*Sec[c + d*x]^5*Tan[c + d*x])/(6*d)","A",11,6,28,0.2143,1,"{3090, 3768, 3770, 2606, 30, 2611}"
56,1,125,0,0.1044035,"\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\left(a^2+2 b^2\right) \tan ^5(c+d x)}{5 d}+\frac{\left(2 a^2+b^2\right) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^6(c+d x)}{3 d}+\frac{a b \tan ^4(c+d x)}{d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^7(c+d x)}{7 d}","\frac{\left(a^2+2 b^2\right) \tan ^5(c+d x)}{5 d}+\frac{\left(2 a^2+b^2\right) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^6(c+d x)}{3 d}+\frac{a b \tan ^4(c+d x)}{d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^7(c+d x)}{7 d}",1,"(a^2*Tan[c + d*x])/d + (a*b*Tan[c + d*x]^2)/d + ((2*a^2 + b^2)*Tan[c + d*x]^3)/(3*d) + (a*b*Tan[c + d*x]^4)/d + ((a^2 + 2*b^2)*Tan[c + d*x]^5)/(5*d) + (a*b*Tan[c + d*x]^6)/(3*d) + (b^2*Tan[c + d*x]^7)/(7*d)","A",3,2,28,0.07143,1,"{3088, 948}"
57,1,265,0,0.2456907,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{3 a^2 b \cos ^8(c+d x)}{8 d}+\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^3 x}{128}-\frac{3 a b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a b^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{5 a b^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{15 a b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{15}{128} a b^2 x+\frac{b^3 \cos ^8(c+d x)}{8 d}-\frac{b^3 \cos ^6(c+d x)}{6 d}","-\frac{3 a^2 b \cos ^8(c+d x)}{8 d}+\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^3 x}{128}-\frac{3 a b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{a b^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{5 a b^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{15 a b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{15}{128} a b^2 x+\frac{b^3 \cos ^8(c+d x)}{8 d}-\frac{b^3 \cos ^6(c+d x)}{6 d}",1,"(35*a^3*x)/128 + (15*a*b^2*x)/128 - (b^3*Cos[c + d*x]^6)/(6*d) - (3*a^2*b*Cos[c + d*x]^8)/(8*d) + (b^3*Cos[c + d*x]^8)/(8*d) + (35*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^3*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (3*a*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)","A",17,7,28,0.2500,1,"{3090, 2635, 8, 2565, 30, 2568, 14}"
58,1,175,0,0.1837036,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{3 a^2 b \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}-\frac{a^3 \sin ^3(c+d x)}{d}+\frac{a^3 \sin (c+d x)}{d}+\frac{3 a b^2 \sin ^7(c+d x)}{7 d}-\frac{6 a b^2 \sin ^5(c+d x)}{5 d}+\frac{a b^2 \sin ^3(c+d x)}{d}+\frac{b^3 \cos ^7(c+d x)}{7 d}-\frac{b^3 \cos ^5(c+d x)}{5 d}","-\frac{3 a^2 b \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^7(c+d x)}{7 d}+\frac{3 a^3 \sin ^5(c+d x)}{5 d}-\frac{a^3 \sin ^3(c+d x)}{d}+\frac{a^3 \sin (c+d x)}{d}+\frac{3 a b^2 \sin ^7(c+d x)}{7 d}-\frac{6 a b^2 \sin ^5(c+d x)}{5 d}+\frac{a b^2 \sin ^3(c+d x)}{d}+\frac{b^3 \cos ^7(c+d x)}{7 d}-\frac{b^3 \cos ^5(c+d x)}{5 d}",1,"-(b^3*Cos[c + d*x]^5)/(5*d) - (3*a^2*b*Cos[c + d*x]^7)/(7*d) + (b^3*Cos[c + d*x]^7)/(7*d) + (a^3*Sin[c + d*x])/d - (a^3*Sin[c + d*x]^3)/d + (a*b^2*Sin[c + d*x]^3)/d + (3*a^3*Sin[c + d*x]^5)/(5*d) - (6*a*b^2*Sin[c + d*x]^5)/(5*d) - (a^3*Sin[c + d*x]^7)/(7*d) + (3*a*b^2*Sin[c + d*x]^7)/(7*d)","A",12,7,28,0.2500,1,"{3090, 2633, 2565, 30, 2564, 270, 14}"
59,1,216,0,0.2126405,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{a^2 b \cos ^6(c+d x)}{2 d}+\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^3 x}{16}-\frac{a b^2 \sin (c+d x) \cos ^5(c+d x)}{2 d}+\frac{a b^2 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{3}{16} a b^2 x-\frac{b^3 \sin ^6(c+d x)}{6 d}+\frac{b^3 \sin ^4(c+d x)}{4 d}","-\frac{a^2 b \cos ^6(c+d x)}{2 d}+\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^3 x}{16}-\frac{a b^2 \sin (c+d x) \cos ^5(c+d x)}{2 d}+\frac{a b^2 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{3 a b^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{3}{16} a b^2 x-\frac{b^3 \sin ^6(c+d x)}{6 d}+\frac{b^3 \sin ^4(c+d x)}{4 d}",1,"(5*a^3*x)/16 + (3*a*b^2*x)/16 - (a^2*b*Cos[c + d*x]^6)/(2*d) + (5*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (3*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) + (a^3*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (a*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(2*d) + (b^3*Sin[c + d*x]^4)/(4*d) - (b^3*Sin[c + d*x]^6)/(6*d)","A",15,8,28,0.2857,1,"{3090, 2635, 8, 2565, 30, 2568, 2564, 14}"
60,1,140,0,0.1632647,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{3 a^2 b \cos ^5(c+d x)}{5 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{2 a^3 \sin ^3(c+d x)}{3 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{3 a b^2 \sin ^5(c+d x)}{5 d}+\frac{a b^2 \sin ^3(c+d x)}{d}+\frac{b^3 \cos ^5(c+d x)}{5 d}-\frac{b^3 \cos ^3(c+d x)}{3 d}","-\frac{3 a^2 b \cos ^5(c+d x)}{5 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{2 a^3 \sin ^3(c+d x)}{3 d}+\frac{a^3 \sin (c+d x)}{d}-\frac{3 a b^2 \sin ^5(c+d x)}{5 d}+\frac{a b^2 \sin ^3(c+d x)}{d}+\frac{b^3 \cos ^5(c+d x)}{5 d}-\frac{b^3 \cos ^3(c+d x)}{3 d}",1,"-(b^3*Cos[c + d*x]^3)/(3*d) - (3*a^2*b*Cos[c + d*x]^5)/(5*d) + (b^3*Cos[c + d*x]^5)/(5*d) + (a^3*Sin[c + d*x])/d - (2*a^3*Sin[c + d*x]^3)/(3*d) + (a*b^2*Sin[c + d*x]^3)/d + (a^3*Sin[c + d*x]^5)/(5*d) - (3*a*b^2*Sin[c + d*x]^5)/(5*d)","A",12,6,28,0.2143,1,"{3090, 2633, 2565, 30, 2564, 14}"
61,1,78,0,0.0635502,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{3}{8} a x \left(a^2+b^2\right)+\frac{\sin ^4(c+d x) (a \cot (c+d x)+b)^3}{4 d}+\frac{3 a \sin ^2(c+d x) (a \cot (c+d x)+b) (a-b \cot (c+d x))}{8 d}","\frac{3}{8} a x \left(a^2+b^2\right)+\frac{\sin ^4(c+d x) (a \cot (c+d x)+b)^3}{4 d}+\frac{3 a \sin ^2(c+d x) (a \cot (c+d x)+b) (a-b \cot (c+d x))}{8 d}",1,"(3*a*(a^2 + b^2)*x)/8 + (3*a*(b + a*Cot[c + d*x])*(a - b*Cot[c + d*x])*Sin[c + d*x]^2)/(8*d) + ((b + a*Cot[c + d*x])^3*Sin[c + d*x]^4)/(4*d)","A",4,4,26,0.1538,1,"{3088, 805, 723, 203}"
62,1,58,0,0.0232823,"\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}"
63,1,91,0,0.1193876,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\sin ^2(c+d x) \left(a \left(a^2-3 b^2\right) \cot (c+d x)+b \left(3 a^2-b^2\right)\right)}{2 d}+\frac{1}{2} a x \left(a^2+3 b^2\right)-\frac{b^3 \log (\sin (c+d x))}{d}+\frac{b^3 \log (\tan (c+d x))}{d}","\frac{\sin ^2(c+d x) \left(a \left(a^2-3 b^2\right) \cot (c+d x)+b \left(3 a^2-b^2\right)\right)}{2 d}+\frac{1}{2} a x \left(a^2+3 b^2\right)-\frac{b^3 \log (\sin (c+d x))}{d}+\frac{b^3 \log (\tan (c+d x))}{d}",1,"(a*(a^2 + 3*b^2)*x)/2 - (b^3*Log[Sin[c + d*x]])/d + (b^3*Log[Tan[c + d*x]])/d + ((b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Cot[c + d*x])*Sin[c + d*x]^2)/(2*d)","A",7,6,26,0.2308,1,"{3088, 1805, 801, 635, 203, 260}"
64,1,86,0,0.1116419,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{3 a^2 b \cos (c+d x)}{d}+\frac{a^3 \sin (c+d x)}{d}-\frac{3 a b^2 \sin (c+d x)}{d}+\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^3 \cos (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}","-\frac{3 a^2 b \cos (c+d x)}{d}+\frac{a^3 \sin (c+d x)}{d}-\frac{3 a b^2 \sin (c+d x)}{d}+\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^3 \cos (c+d x)}{d}+\frac{b^3 \sec (c+d x)}{d}",1,"(3*a*b^2*ArcTanh[Sin[c + d*x]])/d - (3*a^2*b*Cos[c + d*x])/d + (b^3*Cos[c + d*x])/d + (b^3*Sec[c + d*x])/d + (a^3*Sin[c + d*x])/d - (3*a*b^2*Sin[c + d*x])/d","A",10,8,28,0.2857,1,"{3090, 2637, 2638, 2592, 321, 206, 2590, 14}"
65,1,72,0,0.0925312,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+a x \left(a^2-3 b^2\right)+\frac{2 a b^2 \tan (c+d x)}{d}+\frac{b (a+b \tan (c+d x))^2}{2 d}","-\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+a x \left(a^2-3 b^2\right)+\frac{2 a b^2 \tan (c+d x)}{d}+\frac{b (a+b \tan (c+d x))^2}{2 d}",1,"a*(a^2 - 3*b^2)*x - (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (2*a*b^2*Tan[c + d*x])/d + (b*(a + b*Tan[c + d*x])^2)/(2*d)","A",4,4,28,0.1429,1,"{3086, 3482, 3525, 3475}"
66,1,103,0,0.1225247,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{3 a^2 b \sec (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b^3 \sec ^3(c+d x)}{3 d}-\frac{b^3 \sec (c+d x)}{d}","\frac{3 a^2 b \sec (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b^3 \sec ^3(c+d x)}{3 d}-\frac{b^3 \sec (c+d x)}{d}",1,"(a^3*ArcTanh[Sin[c + d*x]])/d - (3*a*b^2*ArcTanh[Sin[c + d*x]])/(2*d) + (3*a^2*b*Sec[c + d*x])/d - (b^3*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^3)/(3*d) + (3*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",9,5,28,0.1786,1,"{3090, 3770, 2606, 8, 2611}"
67,1,30,0,0.0474499,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\tan ^4(c+d x) (a \cot (c+d x)+b)^4}{4 b d}","\frac{\tan ^4(c+d x) (a \cot (c+d x)+b)^4}{4 b d}",1,"((b + a*Cot[c + d*x])^4*Tan[c + d*x]^4)/(4*b*d)","A",2,2,28,0.07143,1,"{3088, 37}"
68,1,158,0,0.1809349,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{a^2 b \sec ^3(c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b^3 \sec ^5(c+d x)}{5 d}-\frac{b^3 \sec ^3(c+d x)}{3 d}","\frac{a^2 b \sec ^3(c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b^3 \sec ^5(c+d x)}{5 d}-\frac{b^3 \sec ^3(c+d x)}{3 d}",1,"(a^3*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a*b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*b*Sec[c + d*x]^3)/d - (b^3*Sec[c + d*x]^3)/(3*d) + (b^3*Sec[c + d*x]^5)/(5*d) + (a^3*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (3*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (3*a*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",12,7,28,0.2500,1,"{3090, 3768, 3770, 2606, 30, 2611, 14}"
69,1,120,0,0.0979102,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{b \left(3 a^2+b^2\right) \tan ^4(c+d x)}{4 d}+\frac{a \left(a^2+3 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{3 a^2 b \tan ^2(c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}+\frac{b^3 \tan ^6(c+d x)}{6 d}","\frac{b \left(3 a^2+b^2\right) \tan ^4(c+d x)}{4 d}+\frac{a \left(a^2+3 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{3 a^2 b \tan ^2(c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}+\frac{b^3 \tan ^6(c+d x)}{6 d}",1,"(a^3*Tan[c + d*x])/d + (3*a^2*b*Tan[c + d*x]^2)/(2*d) + (a*(a^2 + 3*b^2)*Tan[c + d*x]^3)/(3*d) + (b*(3*a^2 + b^2)*Tan[c + d*x]^4)/(4*d) + (3*a*b^2*Tan[c + d*x]^5)/(5*d) + (b^3*Tan[c + d*x]^6)/(6*d)","A",3,2,28,0.07143,1,"{3088, 894}"
70,1,210,0,0.2197082,"\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{3 a^2 b \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b^2 \tan (c+d x) \sec ^5(c+d x)}{2 d}-\frac{a b^2 \tan (c+d x) \sec ^3(c+d x)}{8 d}-\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}-\frac{b^3 \sec ^5(c+d x)}{5 d}","\frac{3 a^2 b \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b^2 \tan (c+d x) \sec ^5(c+d x)}{2 d}-\frac{a b^2 \tan (c+d x) \sec ^3(c+d x)}{8 d}-\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}-\frac{b^3 \sec ^5(c+d x)}{5 d}",1,"(3*a^3*ArcTanh[Sin[c + d*x]])/(8*d) - (3*a*b^2*ArcTanh[Sin[c + d*x]])/(16*d) + (3*a^2*b*Sec[c + d*x]^5)/(5*d) - (b^3*Sec[c + d*x]^5)/(5*d) + (b^3*Sec[c + d*x]^7)/(7*d) + (3*a^3*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (3*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (a*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (a*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(2*d)","A",14,7,28,0.2500,1,"{3090, 3768, 3770, 2606, 30, 2611, 14}"
71,1,174,0,0.1395599,"\int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{b \left(3 a^2+2 b^2\right) \tan ^6(c+d x)}{6 d}+\frac{a \left(a^2+6 b^2\right) \tan ^5(c+d x)}{5 d}+\frac{b \left(6 a^2+b^2\right) \tan ^4(c+d x)}{4 d}+\frac{a \left(2 a^2+3 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{3 a^2 b \tan ^2(c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a b^2 \tan ^7(c+d x)}{7 d}+\frac{b^3 \tan ^8(c+d x)}{8 d}","\frac{b \left(3 a^2+2 b^2\right) \tan ^6(c+d x)}{6 d}+\frac{a \left(a^2+6 b^2\right) \tan ^5(c+d x)}{5 d}+\frac{b \left(6 a^2+b^2\right) \tan ^4(c+d x)}{4 d}+\frac{a \left(2 a^2+3 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{3 a^2 b \tan ^2(c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a b^2 \tan ^7(c+d x)}{7 d}+\frac{b^3 \tan ^8(c+d x)}{8 d}",1,"(a^3*Tan[c + d*x])/d + (3*a^2*b*Tan[c + d*x]^2)/(2*d) + (a*(2*a^2 + 3*b^2)*Tan[c + d*x]^3)/(3*d) + (b*(6*a^2 + b^2)*Tan[c + d*x]^4)/(4*d) + (a*(a^2 + 6*b^2)*Tan[c + d*x]^5)/(5*d) + (b*(3*a^2 + 2*b^2)*Tan[c + d*x]^6)/(6*d) + (3*a*b^2*Tan[c + d*x]^7)/(7*d) + (b^3*Tan[c + d*x]^8)/(8*d)","A",3,2,28,0.07143,1,"{3088, 948}"
72,1,259,0,0.2679577,"\int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{3 a^2 b \sec ^7(c+d x)}{7 d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^3 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^3 \tan (c+d x) \sec (c+d x)}{16 d}-\frac{15 a b^2 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{3 a b^2 \tan (c+d x) \sec ^7(c+d x)}{8 d}-\frac{a b^2 \tan (c+d x) \sec ^5(c+d x)}{16 d}-\frac{5 a b^2 \tan (c+d x) \sec ^3(c+d x)}{64 d}-\frac{15 a b^2 \tan (c+d x) \sec (c+d x)}{128 d}+\frac{b^3 \sec ^9(c+d x)}{9 d}-\frac{b^3 \sec ^7(c+d x)}{7 d}","\frac{3 a^2 b \sec ^7(c+d x)}{7 d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^3 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^3 \tan (c+d x) \sec (c+d x)}{16 d}-\frac{15 a b^2 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{3 a b^2 \tan (c+d x) \sec ^7(c+d x)}{8 d}-\frac{a b^2 \tan (c+d x) \sec ^5(c+d x)}{16 d}-\frac{5 a b^2 \tan (c+d x) \sec ^3(c+d x)}{64 d}-\frac{15 a b^2 \tan (c+d x) \sec (c+d x)}{128 d}+\frac{b^3 \sec ^9(c+d x)}{9 d}-\frac{b^3 \sec ^7(c+d x)}{7 d}",1,"(5*a^3*ArcTanh[Sin[c + d*x]])/(16*d) - (15*a*b^2*ArcTanh[Sin[c + d*x]])/(128*d) + (3*a^2*b*Sec[c + d*x]^7)/(7*d) - (b^3*Sec[c + d*x]^7)/(7*d) + (b^3*Sec[c + d*x]^9)/(9*d) + (5*a^3*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (15*a*b^2*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (5*a^3*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) - (5*a*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (a^3*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) - (a*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(16*d) + (3*a*b^2*Sec[c + d*x]^7*Tan[c + d*x])/(8*d)","A",16,7,28,0.2500,1,"{3090, 3768, 3770, 2606, 30, 2611, 14}"
73,1,213,0,0.1786427,"\int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{3 b \left(a^2+b^2\right) \tan ^8(c+d x)}{8 d}+\frac{a \left(a^2+9 b^2\right) \tan ^7(c+d x)}{7 d}+\frac{b \left(3 a^2+b^2\right) \tan ^6(c+d x)}{2 d}+\frac{3 a \left(a^2+3 b^2\right) \tan ^5(c+d x)}{5 d}+\frac{b \left(9 a^2+b^2\right) \tan ^4(c+d x)}{4 d}+\frac{a \left(a^2+b^2\right) \tan ^3(c+d x)}{d}+\frac{3 a^2 b \tan ^2(c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{a b^2 \tan ^9(c+d x)}{3 d}+\frac{b^3 \tan ^{10}(c+d x)}{10 d}","\frac{3 b \left(a^2+b^2\right) \tan ^8(c+d x)}{8 d}+\frac{a \left(a^2+9 b^2\right) \tan ^7(c+d x)}{7 d}+\frac{b \left(3 a^2+b^2\right) \tan ^6(c+d x)}{2 d}+\frac{3 a \left(a^2+3 b^2\right) \tan ^5(c+d x)}{5 d}+\frac{b \left(9 a^2+b^2\right) \tan ^4(c+d x)}{4 d}+\frac{a \left(a^2+b^2\right) \tan ^3(c+d x)}{d}+\frac{3 a^2 b \tan ^2(c+d x)}{2 d}+\frac{a^3 \tan (c+d x)}{d}+\frac{a b^2 \tan ^9(c+d x)}{3 d}+\frac{b^3 \tan ^{10}(c+d x)}{10 d}",1,"(a^3*Tan[c + d*x])/d + (3*a^2*b*Tan[c + d*x]^2)/(2*d) + (a*(a^2 + b^2)*Tan[c + d*x]^3)/d + (b*(9*a^2 + b^2)*Tan[c + d*x]^4)/(4*d) + (3*a*(a^2 + 3*b^2)*Tan[c + d*x]^5)/(5*d) + (b*(3*a^2 + b^2)*Tan[c + d*x]^6)/(2*d) + (a*(a^2 + 9*b^2)*Tan[c + d*x]^7)/(7*d) + (3*b*(a^2 + b^2)*Tan[c + d*x]^8)/(8*d) + (a*b^2*Tan[c + d*x]^9)/(3*d) + (b^3*Tan[c + d*x]^10)/(10*d)","A",3,2,28,0.07143,1,"{3088, 948}"
74,1,279,0,0.258005,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac{18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^9(c+d x)}{9 d}+\frac{a^4 \sin ^9(c+d x)}{9 d}-\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{6 a^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^9(c+d x)}{9 d}-\frac{4 a b^3 \cos ^7(c+d x)}{7 d}+\frac{b^4 \sin ^9(c+d x)}{9 d}-\frac{2 b^4 \sin ^7(c+d x)}{7 d}+\frac{b^4 \sin ^5(c+d x)}{5 d}","-\frac{2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac{18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^9(c+d x)}{9 d}+\frac{a^4 \sin ^9(c+d x)}{9 d}-\frac{4 a^4 \sin ^7(c+d x)}{7 d}+\frac{6 a^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^9(c+d x)}{9 d}-\frac{4 a b^3 \cos ^7(c+d x)}{7 d}+\frac{b^4 \sin ^9(c+d x)}{9 d}-\frac{2 b^4 \sin ^7(c+d x)}{7 d}+\frac{b^4 \sin ^5(c+d x)}{5 d}",1,"(-4*a*b^3*Cos[c + d*x]^7)/(7*d) - (4*a^3*b*Cos[c + d*x]^9)/(9*d) + (4*a*b^3*Cos[c + d*x]^9)/(9*d) + (a^4*Sin[c + d*x])/d - (4*a^4*Sin[c + d*x]^3)/(3*d) + (2*a^2*b^2*Sin[c + d*x]^3)/d + (6*a^4*Sin[c + d*x]^5)/(5*d) - (18*a^2*b^2*Sin[c + d*x]^5)/(5*d) + (b^4*Sin[c + d*x]^5)/(5*d) - (4*a^4*Sin[c + d*x]^7)/(7*d) + (18*a^2*b^2*Sin[c + d*x]^7)/(7*d) - (2*b^4*Sin[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x]^9)/(9*d) - (2*a^2*b^2*Sin[c + d*x]^9)/(3*d) + (b^4*Sin[c + d*x]^9)/(9*d)","A",15,7,28,0.2500,1,"{3090, 2633, 2565, 30, 2564, 270, 14}"
75,1,381,0,0.3889782,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{3 a^2 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac{a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac{5 a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac{15 a^2 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{15}{64} a^2 b^2 x-\frac{a^3 b \cos ^8(c+d x)}{2 d}+\frac{a^4 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^4 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^4 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^4 x}{128}+\frac{a b^3 \cos ^8(c+d x)}{2 d}-\frac{2 a b^3 \cos ^6(c+d x)}{3 d}-\frac{b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 b^4 x}{128}","-\frac{3 a^2 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac{a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{8 d}+\frac{5 a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{32 d}+\frac{15 a^2 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{15}{64} a^2 b^2 x-\frac{a^3 b \cos ^8(c+d x)}{2 d}+\frac{a^4 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^4 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^4 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^4 x}{128}+\frac{a b^3 \cos ^8(c+d x)}{2 d}-\frac{2 a b^3 \cos ^6(c+d x)}{3 d}-\frac{b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{3 b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{3 b^4 x}{128}",1,"(35*a^4*x)/128 + (15*a^2*b^2*x)/64 + (3*b^4*x)/128 - (2*a*b^3*Cos[c + d*x]^6)/(3*d) - (a^3*b*Cos[c + d*x]^8)/(2*d) + (a*b^3*Cos[c + d*x]^8)/(2*d) + (35*a^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a^2*b^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (3*b^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a^2*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(32*d) + (b^4*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a^2*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(8*d) - (b^4*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^4*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (3*a^2*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(4*d) - (b^4*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d)","A",22,7,28,0.2500,1,"{3090, 2635, 8, 2565, 30, 2568, 14}"
76,1,220,0,0.2344978,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{6 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{12 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^7(c+d x)}{7 d}-\frac{a^4 \sin ^7(c+d x)}{7 d}+\frac{3 a^4 \sin ^5(c+d x)}{5 d}-\frac{a^4 \sin ^3(c+d x)}{d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^7(c+d x)}{7 d}-\frac{4 a b^3 \cos ^5(c+d x)}{5 d}-\frac{b^4 \sin ^7(c+d x)}{7 d}+\frac{b^4 \sin ^5(c+d x)}{5 d}","\frac{6 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac{12 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^7(c+d x)}{7 d}-\frac{a^4 \sin ^7(c+d x)}{7 d}+\frac{3 a^4 \sin ^5(c+d x)}{5 d}-\frac{a^4 \sin ^3(c+d x)}{d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^7(c+d x)}{7 d}-\frac{4 a b^3 \cos ^5(c+d x)}{5 d}-\frac{b^4 \sin ^7(c+d x)}{7 d}+\frac{b^4 \sin ^5(c+d x)}{5 d}",1,"(-4*a*b^3*Cos[c + d*x]^5)/(5*d) - (4*a^3*b*Cos[c + d*x]^7)/(7*d) + (4*a*b^3*Cos[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x])/d - (a^4*Sin[c + d*x]^3)/d + (2*a^2*b^2*Sin[c + d*x]^3)/d + (3*a^4*Sin[c + d*x]^5)/(5*d) - (12*a^2*b^2*Sin[c + d*x]^5)/(5*d) + (b^4*Sin[c + d*x]^5)/(5*d) - (a^4*Sin[c + d*x]^7)/(7*d) + (6*a^2*b^2*Sin[c + d*x]^7)/(7*d) - (b^4*Sin[c + d*x]^7)/(7*d)","A",15,7,28,0.2500,1,"{3090, 2633, 2565, 30, 2564, 270, 14}"
77,1,301,0,0.3033317,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{d}+\frac{a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a^2 b^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a^2 b^2 x-\frac{2 a^3 b \cos ^6(c+d x)}{3 d}+\frac{a^4 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a^4 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^4 x}{16}-\frac{2 a b^3 \sin ^6(c+d x)}{3 d}+\frac{a b^3 \sin ^4(c+d x)}{d}-\frac{b^4 \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac{b^4 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{b^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b^4 x}{16}","-\frac{a^2 b^2 \sin (c+d x) \cos ^5(c+d x)}{d}+\frac{a^2 b^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a^2 b^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a^2 b^2 x-\frac{2 a^3 b \cos ^6(c+d x)}{3 d}+\frac{a^4 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a^4 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^4 x}{16}-\frac{2 a b^3 \sin ^6(c+d x)}{3 d}+\frac{a b^3 \sin ^4(c+d x)}{d}-\frac{b^4 \sin ^3(c+d x) \cos ^3(c+d x)}{6 d}-\frac{b^4 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{b^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{b^4 x}{16}",1,"(5*a^4*x)/16 + (3*a^2*b^2*x)/8 + (b^4*x)/16 - (2*a^3*b*Cos[c + d*x]^6)/(3*d) + (5*a^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (3*a^2*b^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b^4*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^2*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (b^4*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (a^2*b^2*Cos[c + d*x]^5*Sin[c + d*x])/d - (b^4*Cos[c + d*x]^3*Sin[c + d*x]^3)/(6*d) + (a*b^3*Sin[c + d*x]^4)/d - (2*a*b^3*Sin[c + d*x]^6)/(3*d)","A",19,8,28,0.2857,1,"{3090, 2635, 8, 2565, 30, 2568, 2564, 14}"
78,1,165,0,0.1813166,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{6 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^5(c+d x)}{5 d}+\frac{a^4 \sin ^5(c+d x)}{5 d}-\frac{2 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^5(c+d x)}{5 d}-\frac{4 a b^3 \cos ^3(c+d x)}{3 d}+\frac{b^4 \sin ^5(c+d x)}{5 d}","-\frac{6 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^5(c+d x)}{5 d}+\frac{a^4 \sin ^5(c+d x)}{5 d}-\frac{2 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^5(c+d x)}{5 d}-\frac{4 a b^3 \cos ^3(c+d x)}{3 d}+\frac{b^4 \sin ^5(c+d x)}{5 d}",1,"(-4*a*b^3*Cos[c + d*x]^3)/(3*d) - (4*a^3*b*Cos[c + d*x]^5)/(5*d) + (4*a*b^3*Cos[c + d*x]^5)/(5*d) + (a^4*Sin[c + d*x])/d - (2*a^4*Sin[c + d*x]^3)/(3*d) + (2*a^2*b^2*Sin[c + d*x]^3)/d + (a^4*Sin[c + d*x]^5)/(5*d) - (6*a^2*b^2*Sin[c + d*x]^5)/(5*d) + (b^4*Sin[c + d*x]^5)/(5*d)","A",14,6,26,0.2308,1,"{3090, 2633, 2565, 30, 2564, 14}"
79,1,108,0,0.0441804,"\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}"
80,1,150,0,0.1529596,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^3(c+d x)}{3 d}-\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^3(c+d x)}{3 d}-\frac{4 a b^3 \cos (c+d x)}{d}-\frac{b^4 \sin ^3(c+d x)}{3 d}-\frac{b^4 \sin (c+d x)}{d}+\frac{b^4 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{2 a^2 b^2 \sin ^3(c+d x)}{d}-\frac{4 a^3 b \cos ^3(c+d x)}{3 d}-\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos ^3(c+d x)}{3 d}-\frac{4 a b^3 \cos (c+d x)}{d}-\frac{b^4 \sin ^3(c+d x)}{3 d}-\frac{b^4 \sin (c+d x)}{d}+\frac{b^4 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b^4*ArcTanh[Sin[c + d*x]])/d - (4*a*b^3*Cos[c + d*x])/d - (4*a^3*b*Cos[c + d*x]^3)/(3*d) + (4*a*b^3*Cos[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x])/d - (b^4*Sin[c + d*x])/d - (a^4*Sin[c + d*x]^3)/(3*d) + (2*a^2*b^2*Sin[c + d*x]^3)/d - (b^4*Sin[c + d*x]^3)/(3*d)","A",14,8,26,0.3077,1,"{3090, 2633, 2565, 30, 2564, 2592, 302, 206}"
81,1,119,0,0.1832957,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\sin ^2(c+d x) \left(\left(-6 a^2 b^2+a^4+b^4\right) \cot (c+d x)+4 a b \left(a^2-b^2\right)\right)}{2 d}+\frac{1}{2} x \left(6 a^2 b^2+a^4-3 b^4\right)-\frac{4 a b^3 \log (\sin (c+d x))}{d}+\frac{4 a b^3 \log (\tan (c+d x))}{d}+\frac{b^4 \tan (c+d x)}{d}","\frac{\sin ^2(c+d x) \left(\left(-6 a^2 b^2+a^4+b^4\right) \cot (c+d x)+4 a b \left(a^2-b^2\right)\right)}{2 d}+\frac{1}{2} x \left(6 a^2 b^2+a^4-3 b^4\right)-\frac{4 a b^3 \log (\sin (c+d x))}{d}+\frac{4 a b^3 \log (\tan (c+d x))}{d}+\frac{b^4 \tan (c+d x)}{d}",1,"((a^4 + 6*a^2*b^2 - 3*b^4)*x)/2 - (4*a*b^3*Log[Sin[c + d*x]])/d + (4*a*b^3*Log[Tan[c + d*x]])/d + ((4*a*b*(a^2 - b^2) + (a^4 - 6*a^2*b^2 + b^4)*Cot[c + d*x])*Sin[c + d*x]^2)/(2*d) + (b^4*Tan[c + d*x])/d","A",7,6,28,0.2143,1,"{3088, 1805, 1802, 635, 203, 260}"
82,1,151,0,0.164174,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{6 a^2 b^2 \sin (c+d x)}{d}+\frac{6 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{4 a^3 b \cos (c+d x)}{d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos (c+d x)}{d}+\frac{4 a b^3 \sec (c+d x)}{d}+\frac{3 b^4 \sin (c+d x)}{2 d}+\frac{b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}","-\frac{6 a^2 b^2 \sin (c+d x)}{d}+\frac{6 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{4 a^3 b \cos (c+d x)}{d}+\frac{a^4 \sin (c+d x)}{d}+\frac{4 a b^3 \cos (c+d x)}{d}+\frac{4 a b^3 \sec (c+d x)}{d}+\frac{3 b^4 \sin (c+d x)}{2 d}+\frac{b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(6*a^2*b^2*ArcTanh[Sin[c + d*x]])/d - (3*b^4*ArcTanh[Sin[c + d*x]])/(2*d) - (4*a^3*b*Cos[c + d*x])/d + (4*a*b^3*Cos[c + d*x])/d + (4*a*b^3*Sec[c + d*x])/d + (a^4*Sin[c + d*x])/d - (6*a^2*b^2*Sin[c + d*x])/d + (3*b^4*Sin[c + d*x])/(2*d) + (b^4*Sin[c + d*x]*Tan[c + d*x]^2)/(2*d)","A",14,9,28,0.3214,1,"{3090, 2637, 2638, 2592, 321, 206, 2590, 14, 288}"
83,1,103,0,0.156476,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{b^2 \left(3 a^2-b^2\right) \tan (c+d x)}{d}-\frac{4 a b \left(a^2-b^2\right) \log (\cos (c+d x))}{d}+x \left(-6 a^2 b^2+a^4+b^4\right)+\frac{b (a+b \tan (c+d x))^3}{3 d}+\frac{a b (a+b \tan (c+d x))^2}{d}","\frac{b^2 \left(3 a^2-b^2\right) \tan (c+d x)}{d}-\frac{4 a b \left(a^2-b^2\right) \log (\cos (c+d x))}{d}+x \left(-6 a^2 b^2+a^4+b^4\right)+\frac{b (a+b \tan (c+d x))^3}{3 d}+\frac{a b (a+b \tan (c+d x))^2}{d}",1,"(a^4 - 6*a^2*b^2 + b^4)*x - (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/d + (b^2*(3*a^2 - b^2)*Tan[c + d*x])/d + (a*b*(a + b*Tan[c + d*x])^2)/d + (b*(a + b*Tan[c + d*x])^3)/(3*d)","A",5,5,28,0.1786,1,"{3086, 3482, 3528, 3525, 3475}"
84,1,168,0,0.1945259,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{3 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{d}+\frac{4 a^3 b \sec (c+d x)}{d}+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a b^3 \sec ^3(c+d x)}{3 d}-\frac{4 a b^3 \sec (c+d x)}{d}+\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^4 \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{3 b^4 \tan (c+d x) \sec (c+d x)}{8 d}","-\frac{3 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{d}+\frac{4 a^3 b \sec (c+d x)}{d}+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a b^3 \sec ^3(c+d x)}{3 d}-\frac{4 a b^3 \sec (c+d x)}{d}+\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^4 \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{3 b^4 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(a^4*ArcTanh[Sin[c + d*x]])/d - (3*a^2*b^2*ArcTanh[Sin[c + d*x]])/d + (3*b^4*ArcTanh[Sin[c + d*x]])/(8*d) + (4*a^3*b*Sec[c + d*x])/d - (4*a*b^3*Sec[c + d*x])/d + (4*a*b^3*Sec[c + d*x]^3)/(3*d) + (3*a^2*b^2*Sec[c + d*x]*Tan[c + d*x])/d - (3*b^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b^4*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d)","A",12,5,28,0.1786,1,"{3090, 3770, 2606, 8, 2611}"
85,1,30,0,0.0476237,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\tan ^5(c+d x) (a \cot (c+d x)+b)^5}{5 b d}","\frac{\tan ^5(c+d x) (a \cot (c+d x)+b)^5}{5 b d}",1,"((b + a*Cot[c + d*x])^5*Tan[c + d*x]^5)/(5*b*d)","A",2,2,28,0.07143,1,"{3088, 37}"
86,1,258,0,0.2941166,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{3 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{3 a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}-\frac{3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{4 d}+\frac{4 a^3 b \sec ^3(c+d x)}{3 d}+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{4 a b^3 \sec ^5(c+d x)}{5 d}-\frac{4 a b^3 \sec ^3(c+d x)}{3 d}+\frac{b^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{b^4 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac{b^4 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{b^4 \tan (c+d x) \sec (c+d x)}{16 d}","-\frac{3 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{3 a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}-\frac{3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{4 d}+\frac{4 a^3 b \sec ^3(c+d x)}{3 d}+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{4 a b^3 \sec ^5(c+d x)}{5 d}-\frac{4 a b^3 \sec ^3(c+d x)}{3 d}+\frac{b^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{b^4 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac{b^4 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{b^4 \tan (c+d x) \sec (c+d x)}{16 d}",1,"(a^4*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*b^2*ArcTanh[Sin[c + d*x]])/(4*d) + (b^4*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^3*b*Sec[c + d*x]^3)/(3*d) - (4*a*b^3*Sec[c + d*x]^3)/(3*d) + (4*a*b^3*Sec[c + d*x]^5)/(5*d) + (a^4*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (3*a^2*b^2*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (b^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (3*a^2*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) - (b^4*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (b^4*Sec[c + d*x]^3*Tan[c + d*x]^3)/(6*d)","A",16,7,28,0.2500,1,"{3090, 3768, 3770, 2606, 30, 2611, 14}"
87,1,143,0,0.1231621,"\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{b^2 \left(6 a^2+b^2\right) \tan ^5(c+d x)}{5 d}+\frac{a b \left(a^2+b^2\right) \tan ^4(c+d x)}{d}+\frac{a^2 \left(a^2+6 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{2 a^3 b \tan ^2(c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{2 a b^3 \tan ^6(c+d x)}{3 d}+\frac{b^4 \tan ^7(c+d x)}{7 d}","\frac{b^2 \left(6 a^2+b^2\right) \tan ^5(c+d x)}{5 d}+\frac{a b \left(a^2+b^2\right) \tan ^4(c+d x)}{d}+\frac{a^2 \left(a^2+6 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{2 a^3 b \tan ^2(c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{2 a b^3 \tan ^6(c+d x)}{3 d}+\frac{b^4 \tan ^7(c+d x)}{7 d}",1,"(a^4*Tan[c + d*x])/d + (2*a^3*b*Tan[c + d*x]^2)/d + (a^2*(a^2 + 6*b^2)*Tan[c + d*x]^3)/(3*d) + (a*b*(a^2 + b^2)*Tan[c + d*x]^4)/d + (b^2*(6*a^2 + b^2)*Tan[c + d*x]^5)/(5*d) + (2*a*b^3*Tan[c + d*x]^6)/(3*d) + (b^4*Tan[c + d*x]^7)/(7*d)","A",3,2,28,0.07143,1,"{3088, 894}"
88,1,330,0,0.3433499,"\int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{3 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{d}-\frac{a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{4 a^3 b \sec ^5(c+d x)}{5 d}+\frac{3 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^4 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{4 a b^3 \sec ^7(c+d x)}{7 d}-\frac{4 a b^3 \sec ^5(c+d x)}{5 d}+\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac{b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac{b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac{3 b^4 \tan (c+d x) \sec (c+d x)}{128 d}","-\frac{3 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{d}-\frac{a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}-\frac{3 a^2 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{4 a^3 b \sec ^5(c+d x)}{5 d}+\frac{3 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^4 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{4 a b^3 \sec ^7(c+d x)}{7 d}-\frac{4 a b^3 \sec ^5(c+d x)}{5 d}+\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac{b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac{b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac{3 b^4 \tan (c+d x) \sec (c+d x)}{128 d}",1,"(3*a^4*ArcTanh[Sin[c + d*x]])/(8*d) - (3*a^2*b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (3*b^4*ArcTanh[Sin[c + d*x]])/(128*d) + (4*a^3*b*Sec[c + d*x]^5)/(5*d) - (4*a*b^3*Sec[c + d*x]^5)/(5*d) + (4*a*b^3*Sec[c + d*x]^7)/(7*d) + (3*a^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (3*a^2*b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (3*b^4*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (a^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (a^2*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b^4*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (a^2*b^2*Sec[c + d*x]^5*Tan[c + d*x])/d - (b^4*Sec[c + d*x]^5*Tan[c + d*x])/(16*d) + (b^4*Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*d)","A",19,7,28,0.2500,1,"{3090, 3768, 3770, 2606, 30, 2611, 14}"
89,1,201,0,0.1713745,"\int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{2 b^2 \left(3 a^2+b^2\right) \tan ^7(c+d x)}{7 d}+\frac{2 a b \left(a^2+2 b^2\right) \tan ^6(c+d x)}{3 d}+\frac{\left(12 a^2 b^2+a^4+b^4\right) \tan ^5(c+d x)}{5 d}+\frac{a b \left(2 a^2+b^2\right) \tan ^4(c+d x)}{d}+\frac{2 a^2 \left(a^2+3 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{2 a^3 b \tan ^2(c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{a b^3 \tan ^8(c+d x)}{2 d}+\frac{b^4 \tan ^9(c+d x)}{9 d}","\frac{2 b^2 \left(3 a^2+b^2\right) \tan ^7(c+d x)}{7 d}+\frac{2 a b \left(a^2+2 b^2\right) \tan ^6(c+d x)}{3 d}+\frac{\left(12 a^2 b^2+a^4+b^4\right) \tan ^5(c+d x)}{5 d}+\frac{a b \left(2 a^2+b^2\right) \tan ^4(c+d x)}{d}+\frac{2 a^2 \left(a^2+3 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{2 a^3 b \tan ^2(c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{a b^3 \tan ^8(c+d x)}{2 d}+\frac{b^4 \tan ^9(c+d x)}{9 d}",1,"(a^4*Tan[c + d*x])/d + (2*a^3*b*Tan[c + d*x]^2)/d + (2*a^2*(a^2 + 3*b^2)*Tan[c + d*x]^3)/(3*d) + (a*b*(2*a^2 + b^2)*Tan[c + d*x]^4)/d + ((a^4 + 12*a^2*b^2 + b^4)*Tan[c + d*x]^5)/(5*d) + (2*a*b*(a^2 + 2*b^2)*Tan[c + d*x]^6)/(3*d) + (2*b^2*(3*a^2 + b^2)*Tan[c + d*x]^7)/(7*d) + (a*b^3*Tan[c + d*x]^8)/(2*d) + (b^4*Tan[c + d*x]^9)/(9*d)","A",3,2,28,0.07143,1,"{3088, 948}"
90,1,408,0,0.3979412,"\int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{15 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{3 a^2 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac{a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{8 d}-\frac{5 a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{32 d}-\frac{15 a^2 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac{4 a^3 b \sec ^7(c+d x)}{7 d}+\frac{5 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{4 a b^3 \sec ^9(c+d x)}{9 d}-\frac{4 a b^3 \sec ^7(c+d x)}{7 d}+\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{10 d}-\frac{3 b^4 \tan (c+d x) \sec ^7(c+d x)}{80 d}+\frac{b^4 \tan (c+d x) \sec ^5(c+d x)}{160 d}+\frac{b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac{3 b^4 \tan (c+d x) \sec (c+d x)}{256 d}","-\frac{15 a^2 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{3 a^2 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac{a^2 b^2 \tan (c+d x) \sec ^5(c+d x)}{8 d}-\frac{5 a^2 b^2 \tan (c+d x) \sec ^3(c+d x)}{32 d}-\frac{15 a^2 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac{4 a^3 b \sec ^7(c+d x)}{7 d}+\frac{5 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{4 a b^3 \sec ^9(c+d x)}{9 d}-\frac{4 a b^3 \sec ^7(c+d x)}{7 d}+\frac{3 b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{10 d}-\frac{3 b^4 \tan (c+d x) \sec ^7(c+d x)}{80 d}+\frac{b^4 \tan (c+d x) \sec ^5(c+d x)}{160 d}+\frac{b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac{3 b^4 \tan (c+d x) \sec (c+d x)}{256 d}",1,"(5*a^4*ArcTanh[Sin[c + d*x]])/(16*d) - (15*a^2*b^2*ArcTanh[Sin[c + d*x]])/(64*d) + (3*b^4*ArcTanh[Sin[c + d*x]])/(256*d) + (4*a^3*b*Sec[c + d*x]^7)/(7*d) - (4*a*b^3*Sec[c + d*x]^7)/(7*d) + (4*a*b^3*Sec[c + d*x]^9)/(9*d) + (5*a^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (15*a^2*b^2*Sec[c + d*x]*Tan[c + d*x])/(64*d) + (3*b^4*Sec[c + d*x]*Tan[c + d*x])/(256*d) + (5*a^4*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) - (5*a^2*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(32*d) + (b^4*Sec[c + d*x]^3*Tan[c + d*x])/(128*d) + (a^4*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) - (a^2*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(8*d) + (b^4*Sec[c + d*x]^5*Tan[c + d*x])/(160*d) + (3*a^2*b^2*Sec[c + d*x]^7*Tan[c + d*x])/(4*d) - (3*b^4*Sec[c + d*x]^7*Tan[c + d*x])/(80*d) + (b^4*Sec[c + d*x]^7*Tan[c + d*x]^3)/(10*d)","A",22,7,28,0.2500,1,"{3090, 3768, 3770, 2606, 30, 2611, 14}"
91,1,254,0,0.2191423,"\int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Int[Sec[c + d*x]^12*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{b^2 \left(2 a^2+b^2\right) \tan ^9(c+d x)}{3 d}+\frac{a b \left(a^2+3 b^2\right) \tan ^8(c+d x)}{2 d}+\frac{\left(18 a^2 b^2+a^4+3 b^4\right) \tan ^7(c+d x)}{7 d}+\frac{2 a b \left(a^2+b^2\right) \tan ^6(c+d x)}{d}+\frac{\left(18 a^2 b^2+3 a^4+b^4\right) \tan ^5(c+d x)}{5 d}+\frac{a b \left(3 a^2+b^2\right) \tan ^4(c+d x)}{d}+\frac{a^2 \left(a^2+2 b^2\right) \tan ^3(c+d x)}{d}+\frac{2 a^3 b \tan ^2(c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{2 a b^3 \tan ^{10}(c+d x)}{5 d}+\frac{b^4 \tan ^{11}(c+d x)}{11 d}","\frac{b^2 \left(2 a^2+b^2\right) \tan ^9(c+d x)}{3 d}+\frac{a b \left(a^2+3 b^2\right) \tan ^8(c+d x)}{2 d}+\frac{\left(18 a^2 b^2+a^4+3 b^4\right) \tan ^7(c+d x)}{7 d}+\frac{2 a b \left(a^2+b^2\right) \tan ^6(c+d x)}{d}+\frac{\left(18 a^2 b^2+3 a^4+b^4\right) \tan ^5(c+d x)}{5 d}+\frac{a b \left(3 a^2+b^2\right) \tan ^4(c+d x)}{d}+\frac{a^2 \left(a^2+2 b^2\right) \tan ^3(c+d x)}{d}+\frac{2 a^3 b \tan ^2(c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{2 a b^3 \tan ^{10}(c+d x)}{5 d}+\frac{b^4 \tan ^{11}(c+d x)}{11 d}",1,"(a^4*Tan[c + d*x])/d + (2*a^3*b*Tan[c + d*x]^2)/d + (a^2*(a^2 + 2*b^2)*Tan[c + d*x]^3)/d + (a*b*(3*a^2 + b^2)*Tan[c + d*x]^4)/d + ((3*a^4 + 18*a^2*b^2 + b^4)*Tan[c + d*x]^5)/(5*d) + (2*a*b*(a^2 + b^2)*Tan[c + d*x]^6)/d + ((a^4 + 18*a^2*b^2 + 3*b^4)*Tan[c + d*x]^7)/(7*d) + (a*b*(a^2 + 3*b^2)*Tan[c + d*x]^8)/(2*d) + (b^2*(2*a^2 + b^2)*Tan[c + d*x]^9)/(3*d) + (2*a*b^3*Tan[c + d*x]^10)/(5*d) + (b^4*Tan[c + d*x]^11)/(11*d)","A",3,2,28,0.07143,1,"{3088, 948}"
92,1,515,0,0.4845098,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^3 b^2 \sin (c+d x) \cos ^9(c+d x)}{d}+\frac{a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35}{128} a^3 b^2 x-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^5 \sin (c+d x) \cos ^9(c+d x)}{10 d}+\frac{9 a^5 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{21 a^5 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{21 a^5 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{63 a^5 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{63 a^5 x}{256}-\frac{a b^4 \sin ^3(c+d x) \cos ^7(c+d x)}{2 d}-\frac{3 a b^4 \sin (c+d x) \cos ^7(c+d x)}{16 d}+\frac{a b^4 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{15}{256} a b^4 x+\frac{b^5 \sin ^{10}(c+d x)}{10 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}","\frac{a^2 b^3 \cos ^{10}(c+d x)}{d}-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{a^3 b^2 \sin (c+d x) \cos ^9(c+d x)}{d}+\frac{a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^3 b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35}{128} a^3 b^2 x-\frac{a^4 b \cos ^{10}(c+d x)}{2 d}+\frac{a^5 \sin (c+d x) \cos ^9(c+d x)}{10 d}+\frac{9 a^5 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac{21 a^5 \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac{21 a^5 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{63 a^5 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{63 a^5 x}{256}-\frac{a b^4 \sin ^3(c+d x) \cos ^7(c+d x)}{2 d}-\frac{3 a b^4 \sin (c+d x) \cos ^7(c+d x)}{16 d}+\frac{a b^4 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{15}{256} a b^4 x+\frac{b^5 \sin ^{10}(c+d x)}{10 d}-\frac{b^5 \sin ^8(c+d x)}{4 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}",1,"(63*a^5*x)/256 + (35*a^3*b^2*x)/128 + (15*a*b^4*x)/256 - (5*a^2*b^3*Cos[c + d*x]^8)/(4*d) - (a^4*b*Cos[c + d*x]^10)/(2*d) + (a^2*b^3*Cos[c + d*x]^10)/d + (63*a^5*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (35*a^3*b^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a*b^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (21*a^5*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (35*a^3*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a*b^4*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (21*a^5*Cos[c + d*x]^5*Sin[c + d*x])/(160*d) + (7*a^3*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (a*b^4*Cos[c + d*x]^5*Sin[c + d*x])/(32*d) + (9*a^5*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) + (a^3*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (3*a*b^4*Cos[c + d*x]^7*Sin[c + d*x])/(16*d) + (a^5*Cos[c + d*x]^9*Sin[c + d*x])/(10*d) - (a^3*b^2*Cos[c + d*x]^9*Sin[c + d*x])/d - (a*b^4*Cos[c + d*x]^7*Sin[c + d*x]^3)/(2*d) + (b^5*Sin[c + d*x]^6)/(6*d) - (b^5*Sin[c + d*x]^8)/(4*d) + (b^5*Sin[c + d*x]^10)/(10*d)","A",29,10,28,0.3571,1,"{3090, 2635, 8, 2565, 30, 2568, 14, 2564, 266, 43}"
93,1,337,0,0.2999914,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","-\frac{10 a^3 b^2 \sin ^9(c+d x)}{9 d}+\frac{30 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac{6 a^3 b^2 \sin ^5(c+d x)}{d}+\frac{10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac{10 a^2 b^3 \cos ^9(c+d x)}{9 d}-\frac{10 a^2 b^3 \cos ^7(c+d x)}{7 d}-\frac{5 a^4 b \cos ^9(c+d x)}{9 d}+\frac{a^5 \sin ^9(c+d x)}{9 d}-\frac{4 a^5 \sin ^7(c+d x)}{7 d}+\frac{6 a^5 \sin ^5(c+d x)}{5 d}-\frac{4 a^5 \sin ^3(c+d x)}{3 d}+\frac{a^5 \sin (c+d x)}{d}+\frac{5 a b^4 \sin ^9(c+d x)}{9 d}-\frac{10 a b^4 \sin ^7(c+d x)}{7 d}+\frac{a b^4 \sin ^5(c+d x)}{d}-\frac{b^5 \cos ^9(c+d x)}{9 d}+\frac{2 b^5 \cos ^7(c+d x)}{7 d}-\frac{b^5 \cos ^5(c+d x)}{5 d}","-\frac{10 a^3 b^2 \sin ^9(c+d x)}{9 d}+\frac{30 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac{6 a^3 b^2 \sin ^5(c+d x)}{d}+\frac{10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac{10 a^2 b^3 \cos ^9(c+d x)}{9 d}-\frac{10 a^2 b^3 \cos ^7(c+d x)}{7 d}-\frac{5 a^4 b \cos ^9(c+d x)}{9 d}+\frac{a^5 \sin ^9(c+d x)}{9 d}-\frac{4 a^5 \sin ^7(c+d x)}{7 d}+\frac{6 a^5 \sin ^5(c+d x)}{5 d}-\frac{4 a^5 \sin ^3(c+d x)}{3 d}+\frac{a^5 \sin (c+d x)}{d}+\frac{5 a b^4 \sin ^9(c+d x)}{9 d}-\frac{10 a b^4 \sin ^7(c+d x)}{7 d}+\frac{a b^4 \sin ^5(c+d x)}{d}-\frac{b^5 \cos ^9(c+d x)}{9 d}+\frac{2 b^5 \cos ^7(c+d x)}{7 d}-\frac{b^5 \cos ^5(c+d x)}{5 d}",1,"-(b^5*Cos[c + d*x]^5)/(5*d) - (10*a^2*b^3*Cos[c + d*x]^7)/(7*d) + (2*b^5*Cos[c + d*x]^7)/(7*d) - (5*a^4*b*Cos[c + d*x]^9)/(9*d) + (10*a^2*b^3*Cos[c + d*x]^9)/(9*d) - (b^5*Cos[c + d*x]^9)/(9*d) + (a^5*Sin[c + d*x])/d - (4*a^5*Sin[c + d*x]^3)/(3*d) + (10*a^3*b^2*Sin[c + d*x]^3)/(3*d) + (6*a^5*Sin[c + d*x]^5)/(5*d) - (6*a^3*b^2*Sin[c + d*x]^5)/d + (a*b^4*Sin[c + d*x]^5)/d - (4*a^5*Sin[c + d*x]^7)/(7*d) + (30*a^3*b^2*Sin[c + d*x]^7)/(7*d) - (10*a*b^4*Sin[c + d*x]^7)/(7*d) + (a^5*Sin[c + d*x]^9)/(9*d) - (10*a^3*b^2*Sin[c + d*x]^9)/(9*d) + (5*a*b^4*Sin[c + d*x]^9)/(9*d)","A",18,7,28,0.2500,1,"{3090, 2633, 2565, 30, 2564, 270, 14}"
94,1,426,0,0.4125529,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac{5 a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac{5 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{25 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{25 a^3 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{25}{64} a^3 b^2 x-\frac{5 a^4 b \cos ^8(c+d x)}{8 d}+\frac{a^5 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^5 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^5 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^5 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^5 x}{128}-\frac{5 a b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{5 a b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{15}{128} a b^4 x-\frac{b^5 \sin ^8(c+d x)}{8 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}","\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac{5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac{5 a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac{5 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac{25 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac{25 a^3 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac{25}{64} a^3 b^2 x-\frac{5 a^4 b \cos ^8(c+d x)}{8 d}+\frac{a^5 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 a^5 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 a^5 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 a^5 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a^5 x}{128}-\frac{5 a b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac{5 a b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{15 a b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{15}{128} a b^4 x-\frac{b^5 \sin ^8(c+d x)}{8 d}+\frac{b^5 \sin ^6(c+d x)}{6 d}",1,"(35*a^5*x)/128 + (25*a^3*b^2*x)/64 + (15*a*b^4*x)/128 - (5*a^2*b^3*Cos[c + d*x]^6)/(3*d) - (5*a^4*b*Cos[c + d*x]^8)/(8*d) + (5*a^2*b^3*Cos[c + d*x]^8)/(4*d) + (35*a^5*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (25*a^3*b^2*Cos[c + d*x]*Sin[c + d*x])/(64*d) + (15*a*b^4*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*a^5*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (25*a^3*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(96*d) + (5*a*b^4*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^5*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (5*a^3*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(24*d) - (5*a*b^4*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^5*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (5*a^3*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(4*d) - (5*a*b^4*Cos[c + d*x]^5*Sin[c + d*x]^3)/(8*d) + (b^5*Sin[c + d*x]^6)/(6*d) - (b^5*Sin[c + d*x]^8)/(8*d)","A",25,8,28,0.2857,1,"{3090, 2635, 8, 2565, 30, 2568, 14, 2564}"
95,1,275,0,0.2813519,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{10 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac{4 a^3 b^2 \sin ^5(c+d x)}{d}+\frac{10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac{10 a^2 b^3 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 b^3 \cos ^5(c+d x)}{d}-\frac{5 a^4 b \cos ^7(c+d x)}{7 d}-\frac{a^5 \sin ^7(c+d x)}{7 d}+\frac{3 a^5 \sin ^5(c+d x)}{5 d}-\frac{a^5 \sin ^3(c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}-\frac{5 a b^4 \sin ^7(c+d x)}{7 d}+\frac{a b^4 \sin ^5(c+d x)}{d}-\frac{b^5 \cos ^7(c+d x)}{7 d}+\frac{2 b^5 \cos ^5(c+d x)}{5 d}-\frac{b^5 \cos ^3(c+d x)}{3 d}","\frac{10 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac{4 a^3 b^2 \sin ^5(c+d x)}{d}+\frac{10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac{10 a^2 b^3 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 b^3 \cos ^5(c+d x)}{d}-\frac{5 a^4 b \cos ^7(c+d x)}{7 d}-\frac{a^5 \sin ^7(c+d x)}{7 d}+\frac{3 a^5 \sin ^5(c+d x)}{5 d}-\frac{a^5 \sin ^3(c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}-\frac{5 a b^4 \sin ^7(c+d x)}{7 d}+\frac{a b^4 \sin ^5(c+d x)}{d}-\frac{b^5 \cos ^7(c+d x)}{7 d}+\frac{2 b^5 \cos ^5(c+d x)}{5 d}-\frac{b^5 \cos ^3(c+d x)}{3 d}",1,"-(b^5*Cos[c + d*x]^3)/(3*d) - (2*a^2*b^3*Cos[c + d*x]^5)/d + (2*b^5*Cos[c + d*x]^5)/(5*d) - (5*a^4*b*Cos[c + d*x]^7)/(7*d) + (10*a^2*b^3*Cos[c + d*x]^7)/(7*d) - (b^5*Cos[c + d*x]^7)/(7*d) + (a^5*Sin[c + d*x])/d - (a^5*Sin[c + d*x]^3)/d + (10*a^3*b^2*Sin[c + d*x]^3)/(3*d) + (3*a^5*Sin[c + d*x]^5)/(5*d) - (4*a^3*b^2*Sin[c + d*x]^5)/d + (a*b^4*Sin[c + d*x]^5)/d - (a^5*Sin[c + d*x]^7)/(7*d) + (10*a^3*b^2*Sin[c + d*x]^7)/(7*d) - (5*a*b^4*Sin[c + d*x]^7)/(7*d)","A",18,7,28,0.2500,1,"{3090, 2633, 2565, 30, 2564, 270, 14}"
96,1,126,0,0.0898664,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{5 a \left(a^2+b^2\right) \sin ^2(c+d x) (a \cot (c+d x)+b) (a-b \cot (c+d x))}{16 d}+\frac{5}{16} a x \left(a^2+b^2\right)^2+\frac{\sin ^6(c+d x) (a \cot (c+d x)+b)^5}{6 d}+\frac{5 a \sin ^4(c+d x) (a \cot (c+d x)+b)^3 (a-b \cot (c+d x))}{24 d}","\frac{5 a \left(a^2+b^2\right) \sin ^2(c+d x) (a \cot (c+d x)+b) (a-b \cot (c+d x))}{16 d}+\frac{5}{16} a x \left(a^2+b^2\right)^2+\frac{\sin ^6(c+d x) (a \cot (c+d x)+b)^5}{6 d}+\frac{5 a \sin ^4(c+d x) (a \cot (c+d x)+b)^3 (a-b \cot (c+d x))}{24 d}",1,"(5*a*(a^2 + b^2)^2*x)/16 + (5*a*(a^2 + b^2)*(b + a*Cot[c + d*x])*(a - b*Cot[c + d*x])*Sin[c + d*x]^2)/(16*d) + (5*a*(b + a*Cot[c + d*x])^3*(a - b*Cot[c + d*x])*Sin[c + d*x]^4)/(24*d) + ((b + a*Cot[c + d*x])^5*Sin[c + d*x]^6)/(6*d)","A",5,4,26,0.1538,1,"{3088, 805, 723, 203}"
97,1,94,0,0.0468233,"\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}"
98,1,170,0,0.221735,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","-\frac{\sin ^4(c+d x) \left(a \left(-10 a^2 b^2+a^4+5 b^4\right) \cot (c+d x)+b \left(-10 a^2 b^2+5 a^4+b^4\right)\right)}{4 d}+\frac{\sin ^2(c+d x) \left(5 a \left(a^2-3 b^2\right) \left(a^2+b^2\right) \cot (c+d x)+4 b \left(5 a^4-b^4\right)\right)}{8 d}+\frac{1}{8} a x \left(10 a^2 b^2+3 a^4+15 b^4\right)-\frac{b^5 \log (\sin (c+d x))}{d}+\frac{b^5 \log (\tan (c+d x))}{d}","-\frac{\sin ^4(c+d x) \left(a \left(-10 a^2 b^2+a^4+5 b^4\right) \cot (c+d x)+b \left(-10 a^2 b^2+5 a^4+b^4\right)\right)}{4 d}+\frac{\sin ^2(c+d x) \left(5 a \left(a^2-3 b^2\right) \left(a^2+b^2\right) \cot (c+d x)+4 b \left(5 a^4-b^4\right)\right)}{8 d}+\frac{1}{8} a x \left(10 a^2 b^2+3 a^4+15 b^4\right)-\frac{b^5 \log (\sin (c+d x))}{d}+\frac{b^5 \log (\tan (c+d x))}{d}",1,"(a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*x)/8 - (b^5*Log[Sin[c + d*x]])/d + (b^5*Log[Tan[c + d*x]])/d + ((4*b*(5*a^4 - b^4) + 5*a*(a^2 - 3*b^2)*(a^2 + b^2)*Cot[c + d*x])*Sin[c + d*x]^2)/(8*d) - ((b*(5*a^4 - 10*a^2*b^2 + b^4) + a*(a^4 - 10*a^2*b^2 + 5*b^4)*Cot[c + d*x])*Sin[c + d*x]^4)/(4*d)","A",8,6,26,0.2308,1,"{3088, 1805, 801, 635, 203, 260}"
99,1,205,0,0.2159126,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac{10 a^2 b^3 \cos ^3(c+d x)}{3 d}-\frac{10 a^2 b^3 \cos (c+d x)}{d}-\frac{5 a^4 b \cos ^3(c+d x)}{3 d}-\frac{a^5 \sin ^3(c+d x)}{3 d}+\frac{a^5 \sin (c+d x)}{d}-\frac{5 a b^4 \sin ^3(c+d x)}{3 d}-\frac{5 a b^4 \sin (c+d x)}{d}+\frac{5 a b^4 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^5 \cos ^3(c+d x)}{3 d}+\frac{2 b^5 \cos (c+d x)}{d}+\frac{b^5 \sec (c+d x)}{d}","\frac{10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac{10 a^2 b^3 \cos ^3(c+d x)}{3 d}-\frac{10 a^2 b^3 \cos (c+d x)}{d}-\frac{5 a^4 b \cos ^3(c+d x)}{3 d}-\frac{a^5 \sin ^3(c+d x)}{3 d}+\frac{a^5 \sin (c+d x)}{d}-\frac{5 a b^4 \sin ^3(c+d x)}{3 d}-\frac{5 a b^4 \sin (c+d x)}{d}+\frac{5 a b^4 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^5 \cos ^3(c+d x)}{3 d}+\frac{2 b^5 \cos (c+d x)}{d}+\frac{b^5 \sec (c+d x)}{d}",1,"(5*a*b^4*ArcTanh[Sin[c + d*x]])/d - (10*a^2*b^3*Cos[c + d*x])/d + (2*b^5*Cos[c + d*x])/d - (5*a^4*b*Cos[c + d*x]^3)/(3*d) + (10*a^2*b^3*Cos[c + d*x]^3)/(3*d) - (b^5*Cos[c + d*x]^3)/(3*d) + (b^5*Sec[c + d*x])/d + (a^5*Sin[c + d*x])/d - (5*a*b^4*Sin[c + d*x])/d - (a^5*Sin[c + d*x]^3)/(3*d) + (10*a^3*b^2*Sin[c + d*x]^3)/(3*d) - (5*a*b^4*Sin[c + d*x]^3)/(3*d)","A",17,10,28,0.3571,1,"{3090, 2633, 2565, 30, 2564, 2592, 302, 206, 2590, 270}"
100,1,169,0,0.2312625,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","-\frac{2 b^3 \left(5 a^2-b^2\right) \log (\sin (c+d x))}{d}+\frac{2 b^3 \left(5 a^2-b^2\right) \log (\tan (c+d x))}{d}+\frac{\sin ^2(c+d x) \left(a \left(-10 a^2 b^2+a^4+5 b^4\right) \cot (c+d x)+b \left(-10 a^2 b^2+5 a^4+b^4\right)\right)}{2 d}+\frac{1}{2} a x \left(10 a^2 b^2+a^4-15 b^4\right)+\frac{5 a b^4 \tan (c+d x)}{d}+\frac{b^5 \tan ^2(c+d x)}{2 d}","-\frac{2 b^3 \left(5 a^2-b^2\right) \log (\sin (c+d x))}{d}+\frac{2 b^3 \left(5 a^2-b^2\right) \log (\tan (c+d x))}{d}+\frac{\sin ^2(c+d x) \left(a \left(-10 a^2 b^2+a^4+5 b^4\right) \cot (c+d x)+b \left(-10 a^2 b^2+5 a^4+b^4\right)\right)}{2 d}+\frac{1}{2} a x \left(10 a^2 b^2+a^4-15 b^4\right)+\frac{5 a b^4 \tan (c+d x)}{d}+\frac{b^5 \tan ^2(c+d x)}{2 d}",1,"(a*(a^4 + 10*a^2*b^2 - 15*b^4)*x)/2 - (2*b^3*(5*a^2 - b^2)*Log[Sin[c + d*x]])/d + (2*b^3*(5*a^2 - b^2)*Log[Tan[c + d*x]])/d + ((b*(5*a^4 - 10*a^2*b^2 + b^4) + a*(a^4 - 10*a^2*b^2 + 5*b^4)*Cot[c + d*x])*Sin[c + d*x]^2)/(2*d) + (5*a*b^4*Tan[c + d*x])/d + (b^5*Tan[c + d*x]^2)/(2*d)","A",7,6,28,0.2143,1,"{3088, 1805, 1802, 635, 203, 260}"
101,1,204,0,0.2080006,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","-\frac{10 a^3 b^2 \sin (c+d x)}{d}+\frac{10 a^2 b^3 \cos (c+d x)}{d}+\frac{10 a^2 b^3 \sec (c+d x)}{d}+\frac{10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{5 a^4 b \cos (c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}+\frac{15 a b^4 \sin (c+d x)}{2 d}+\frac{5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^5 \cos (c+d x)}{d}+\frac{b^5 \sec ^3(c+d x)}{3 d}-\frac{2 b^5 \sec (c+d x)}{d}","-\frac{10 a^3 b^2 \sin (c+d x)}{d}+\frac{10 a^2 b^3 \cos (c+d x)}{d}+\frac{10 a^2 b^3 \sec (c+d x)}{d}+\frac{10 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{5 a^4 b \cos (c+d x)}{d}+\frac{a^5 \sin (c+d x)}{d}+\frac{15 a b^4 \sin (c+d x)}{2 d}+\frac{5 a b^4 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^5 \cos (c+d x)}{d}+\frac{b^5 \sec ^3(c+d x)}{3 d}-\frac{2 b^5 \sec (c+d x)}{d}",1,"(10*a^3*b^2*ArcTanh[Sin[c + d*x]])/d - (15*a*b^4*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^4*b*Cos[c + d*x])/d + (10*a^2*b^3*Cos[c + d*x])/d - (b^5*Cos[c + d*x])/d + (10*a^2*b^3*Sec[c + d*x])/d - (2*b^5*Sec[c + d*x])/d + (b^5*Sec[c + d*x]^3)/(3*d) + (a^5*Sin[c + d*x])/d - (10*a^3*b^2*Sin[c + d*x])/d + (15*a*b^4*Sin[c + d*x])/(2*d) + (5*a*b^4*Sin[c + d*x]*Tan[c + d*x]^2)/(2*d)","A",17,10,28,0.3571,1,"{3090, 2637, 2638, 2592, 321, 206, 2590, 14, 288, 270}"
102,1,147,0,0.2301645,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{b \left(3 a^2-b^2\right) (a+b \tan (c+d x))^2}{2 d}+\frac{4 a b^2 \left(a^2-b^2\right) \tan (c+d x)}{d}-\frac{b \left(-10 a^2 b^2+5 a^4+b^4\right) \log (\cos (c+d x))}{d}+a x \left(-10 a^2 b^2+a^4+5 b^4\right)+\frac{b (a+b \tan (c+d x))^4}{4 d}+\frac{2 a b (a+b \tan (c+d x))^3}{3 d}","\frac{b \left(3 a^2-b^2\right) (a+b \tan (c+d x))^2}{2 d}+\frac{4 a b^2 \left(a^2-b^2\right) \tan (c+d x)}{d}-\frac{b \left(-10 a^2 b^2+5 a^4+b^4\right) \log (\cos (c+d x))}{d}+a x \left(-10 a^2 b^2+a^4+5 b^4\right)+\frac{b (a+b \tan (c+d x))^4}{4 d}+\frac{2 a b (a+b \tan (c+d x))^3}{3 d}",1,"a*(a^4 - 10*a^2*b^2 + 5*b^4)*x - (b*(5*a^4 - 10*a^2*b^2 + b^4)*Log[Cos[c + d*x]])/d + (4*a*b^2*(a^2 - b^2)*Tan[c + d*x])/d + (b*(3*a^2 - b^2)*(a + b*Tan[c + d*x])^2)/(2*d) + (2*a*b*(a + b*Tan[c + d*x])^3)/(3*d) + (b*(a + b*Tan[c + d*x])^4)/(4*d)","A",6,5,28,0.1786,1,"{3086, 3482, 3528, 3525, 3475}"
103,1,224,0,0.2331134,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{10 a^2 b^3 \sec ^3(c+d x)}{3 d}-\frac{10 a^2 b^3 \sec (c+d x)}{d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{d}+\frac{5 a^4 b \sec (c+d x)}{d}+\frac{a^5 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{5 a b^4 \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b^5 \sec ^5(c+d x)}{5 d}-\frac{2 b^5 \sec ^3(c+d x)}{3 d}+\frac{b^5 \sec (c+d x)}{d}","\frac{10 a^2 b^3 \sec ^3(c+d x)}{3 d}-\frac{10 a^2 b^3 \sec (c+d x)}{d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{d}+\frac{5 a^4 b \sec (c+d x)}{d}+\frac{a^5 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{5 a b^4 \tan ^3(c+d x) \sec (c+d x)}{4 d}-\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b^5 \sec ^5(c+d x)}{5 d}-\frac{2 b^5 \sec ^3(c+d x)}{3 d}+\frac{b^5 \sec (c+d x)}{d}",1,"(a^5*ArcTanh[Sin[c + d*x]])/d - (5*a^3*b^2*ArcTanh[Sin[c + d*x]])/d + (15*a*b^4*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^4*b*Sec[c + d*x])/d - (10*a^2*b^3*Sec[c + d*x])/d + (b^5*Sec[c + d*x])/d + (10*a^2*b^3*Sec[c + d*x]^3)/(3*d) - (2*b^5*Sec[c + d*x]^3)/(3*d) + (b^5*Sec[c + d*x]^5)/(5*d) + (5*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/d - (15*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (5*a*b^4*Sec[c + d*x]*Tan[c + d*x]^3)/(4*d)","A",15,6,28,0.2143,1,"{3090, 3770, 2606, 8, 2611, 194}"
104,1,30,0,0.0477584,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{\tan ^6(c+d x) (a \cot (c+d x)+b)^6}{6 b d}","\frac{\tan ^6(c+d x) (a \cot (c+d x)+b)^6}{6 b d}",1,"((b + a*Cot[c + d*x])^6*Tan[c + d*x]^6)/(6*b*d)","A",2,2,28,0.07143,1,"{3088, 37}"
105,1,318,0,0.3369031,"\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac{10 a^2 b^3 \sec ^3(c+d x)}{3 d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{4 d}+\frac{5 a^4 b \sec ^3(c+d x)}{3 d}+\frac{a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^5 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{5 a b^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a b^4 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{5 a b^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b^5 \sec ^7(c+d x)}{7 d}-\frac{2 b^5 \sec ^5(c+d x)}{5 d}+\frac{b^5 \sec ^3(c+d x)}{3 d}","\frac{2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac{10 a^2 b^3 \sec ^3(c+d x)}{3 d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{4 d}+\frac{5 a^4 b \sec ^3(c+d x)}{3 d}+\frac{a^5 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^5 \tan (c+d x) \sec (c+d x)}{2 d}+\frac{5 a b^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a b^4 \tan ^3(c+d x) \sec ^3(c+d x)}{6 d}-\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{8 d}+\frac{5 a b^4 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b^5 \sec ^7(c+d x)}{7 d}-\frac{2 b^5 \sec ^5(c+d x)}{5 d}+\frac{b^5 \sec ^3(c+d x)}{3 d}",1,"(a^5*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*b^2*ArcTanh[Sin[c + d*x]])/(4*d) + (5*a*b^4*ArcTanh[Sin[c + d*x]])/(16*d) + (5*a^4*b*Sec[c + d*x]^3)/(3*d) - (10*a^2*b^3*Sec[c + d*x]^3)/(3*d) + (b^5*Sec[c + d*x]^3)/(3*d) + (2*a^2*b^3*Sec[c + d*x]^5)/d - (2*b^5*Sec[c + d*x]^5)/(5*d) + (b^5*Sec[c + d*x]^7)/(7*d) + (a^5*Sec[c + d*x]*Tan[c + d*x])/(2*d) - (5*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/(4*d) + (5*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (5*a^3*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) - (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x])/(8*d) + (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x]^3)/(6*d)","A",19,8,28,0.2857,1,"{3090, 3768, 3770, 2606, 30, 2611, 14, 270}"
106,1,177,0,0.1516523,"\int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{b^3 \left(10 a^2+b^2\right) \tan ^6(c+d x)}{6 d}+\frac{a b^2 \left(2 a^2+b^2\right) \tan ^5(c+d x)}{d}+\frac{5 a^2 b \left(a^2+2 b^2\right) \tan ^4(c+d x)}{4 d}+\frac{a^3 \left(a^2+10 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{a^5 \tan (c+d x)}{d}+\frac{5 a b^4 \tan ^7(c+d x)}{7 d}+\frac{b^5 \tan ^8(c+d x)}{8 d}","\frac{b^3 \left(10 a^2+b^2\right) \tan ^6(c+d x)}{6 d}+\frac{a b^2 \left(2 a^2+b^2\right) \tan ^5(c+d x)}{d}+\frac{5 a^2 b \left(a^2+2 b^2\right) \tan ^4(c+d x)}{4 d}+\frac{a^3 \left(a^2+10 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{a^5 \tan (c+d x)}{d}+\frac{5 a b^4 \tan ^7(c+d x)}{7 d}+\frac{b^5 \tan ^8(c+d x)}{8 d}",1,"(a^5*Tan[c + d*x])/d + (5*a^4*b*Tan[c + d*x]^2)/(2*d) + (a^3*(a^2 + 10*b^2)*Tan[c + d*x]^3)/(3*d) + (5*a^2*b*(a^2 + 2*b^2)*Tan[c + d*x]^4)/(4*d) + (a*b^2*(2*a^2 + b^2)*Tan[c + d*x]^5)/d + (b^3*(10*a^2 + b^2)*Tan[c + d*x]^6)/(6*d) + (5*a*b^4*Tan[c + d*x]^7)/(7*d) + (b^5*Tan[c + d*x]^8)/(8*d)","A",3,2,28,0.07143,1,"{3088, 894}"
107,1,391,0,0.3891824,"\int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{3 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{12 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^4 b \sec ^5(c+d x)}{d}+\frac{3 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^5 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^5 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{5 a b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac{5 a b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{128 d}+\frac{b^5 \sec ^9(c+d x)}{9 d}-\frac{2 b^5 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^5(c+d x)}{5 d}","\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{2 a^2 b^3 \sec ^5(c+d x)}{d}-\frac{5 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{3 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{12 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^4 b \sec ^5(c+d x)}{d}+\frac{3 a^5 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^5 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^5 \tan (c+d x) \sec (c+d x)}{8 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{128 d}+\frac{5 a b^4 \tan ^3(c+d x) \sec ^5(c+d x)}{8 d}-\frac{5 a b^4 \tan (c+d x) \sec ^5(c+d x)}{16 d}+\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{64 d}+\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{128 d}+\frac{b^5 \sec ^9(c+d x)}{9 d}-\frac{2 b^5 \sec ^7(c+d x)}{7 d}+\frac{b^5 \sec ^5(c+d x)}{5 d}",1,"(3*a^5*ArcTanh[Sin[c + d*x]])/(8*d) - (5*a^3*b^2*ArcTanh[Sin[c + d*x]])/(8*d) + (15*a*b^4*ArcTanh[Sin[c + d*x]])/(128*d) + (a^4*b*Sec[c + d*x]^5)/d - (2*a^2*b^3*Sec[c + d*x]^5)/d + (b^5*Sec[c + d*x]^5)/(5*d) + (10*a^2*b^3*Sec[c + d*x]^7)/(7*d) - (2*b^5*Sec[c + d*x]^7)/(7*d) + (b^5*Sec[c + d*x]^9)/(9*d) + (3*a^5*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (5*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (15*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(128*d) + (a^5*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) - (5*a^3*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(12*d) + (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x])/(64*d) + (5*a^3*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(3*d) - (5*a*b^4*Sec[c + d*x]^5*Tan[c + d*x])/(16*d) + (5*a*b^4*Sec[c + d*x]^5*Tan[c + d*x]^3)/(8*d)","A",22,8,28,0.2857,1,"{3090, 3768, 3770, 2606, 30, 2611, 14, 270}"
108,1,242,0,0.2215953,"\int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{b^3 \left(5 a^2+b^2\right) \tan ^8(c+d x)}{4 d}+\frac{10 a b^2 \left(a^2+b^2\right) \tan ^7(c+d x)}{7 d}+\frac{b \left(20 a^2 b^2+5 a^4+b^4\right) \tan ^6(c+d x)}{6 d}+\frac{a \left(20 a^2 b^2+a^4+5 b^4\right) \tan ^5(c+d x)}{5 d}+\frac{5 a^2 b \left(a^2+b^2\right) \tan ^4(c+d x)}{2 d}+\frac{2 a^3 \left(a^2+5 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{a^5 \tan (c+d x)}{d}+\frac{5 a b^4 \tan ^9(c+d x)}{9 d}+\frac{b^5 \tan ^{10}(c+d x)}{10 d}","\frac{b^3 \left(5 a^2+b^2\right) \tan ^8(c+d x)}{4 d}+\frac{10 a b^2 \left(a^2+b^2\right) \tan ^7(c+d x)}{7 d}+\frac{b \left(20 a^2 b^2+5 a^4+b^4\right) \tan ^6(c+d x)}{6 d}+\frac{a \left(20 a^2 b^2+a^4+5 b^4\right) \tan ^5(c+d x)}{5 d}+\frac{5 a^2 b \left(a^2+b^2\right) \tan ^4(c+d x)}{2 d}+\frac{2 a^3 \left(a^2+5 b^2\right) \tan ^3(c+d x)}{3 d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{a^5 \tan (c+d x)}{d}+\frac{5 a b^4 \tan ^9(c+d x)}{9 d}+\frac{b^5 \tan ^{10}(c+d x)}{10 d}",1,"(a^5*Tan[c + d*x])/d + (5*a^4*b*Tan[c + d*x]^2)/(2*d) + (2*a^3*(a^2 + 5*b^2)*Tan[c + d*x]^3)/(3*d) + (5*a^2*b*(a^2 + b^2)*Tan[c + d*x]^4)/(2*d) + (a*(a^4 + 20*a^2*b^2 + 5*b^4)*Tan[c + d*x]^5)/(5*d) + (b*(5*a^4 + 20*a^2*b^2 + b^4)*Tan[c + d*x]^6)/(6*d) + (10*a*b^2*(a^2 + b^2)*Tan[c + d*x]^7)/(7*d) + (b^3*(5*a^2 + b^2)*Tan[c + d*x]^8)/(4*d) + (5*a*b^4*Tan[c + d*x]^9)/(9*d) + (b^5*Tan[c + d*x]^10)/(10*d)","A",3,2,28,0.07143,1,"{3088, 948}"
109,1,472,0,0.4667431,"\int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Int[Sec[c + d*x]^12*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{25 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{24 d}-\frac{25 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{96 d}-\frac{25 a^3 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac{5 a^4 b \sec ^7(c+d x)}{7 d}+\frac{5 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^5 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{a b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{2 d}-\frac{3 a b^4 \tan (c+d x) \sec ^7(c+d x)}{16 d}+\frac{a b^4 \tan (c+d x) \sec ^5(c+d x)}{32 d}+\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{256 d}+\frac{b^5 \sec ^{11}(c+d x)}{11 d}-\frac{2 b^5 \sec ^9(c+d x)}{9 d}+\frac{b^5 \sec ^7(c+d x)}{7 d}","\frac{10 a^2 b^3 \sec ^9(c+d x)}{9 d}-\frac{10 a^2 b^3 \sec ^7(c+d x)}{7 d}-\frac{25 a^3 b^2 \tanh ^{-1}(\sin (c+d x))}{64 d}+\frac{5 a^3 b^2 \tan (c+d x) \sec ^7(c+d x)}{4 d}-\frac{5 a^3 b^2 \tan (c+d x) \sec ^5(c+d x)}{24 d}-\frac{25 a^3 b^2 \tan (c+d x) \sec ^3(c+d x)}{96 d}-\frac{25 a^3 b^2 \tan (c+d x) \sec (c+d x)}{64 d}+\frac{5 a^4 b \sec ^7(c+d x)}{7 d}+\frac{5 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{5 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{5 a^5 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{15 a b^4 \tanh ^{-1}(\sin (c+d x))}{256 d}+\frac{a b^4 \tan ^3(c+d x) \sec ^7(c+d x)}{2 d}-\frac{3 a b^4 \tan (c+d x) \sec ^7(c+d x)}{16 d}+\frac{a b^4 \tan (c+d x) \sec ^5(c+d x)}{32 d}+\frac{5 a b^4 \tan (c+d x) \sec ^3(c+d x)}{128 d}+\frac{15 a b^4 \tan (c+d x) \sec (c+d x)}{256 d}+\frac{b^5 \sec ^{11}(c+d x)}{11 d}-\frac{2 b^5 \sec ^9(c+d x)}{9 d}+\frac{b^5 \sec ^7(c+d x)}{7 d}",1,"(5*a^5*ArcTanh[Sin[c + d*x]])/(16*d) - (25*a^3*b^2*ArcTanh[Sin[c + d*x]])/(64*d) + (15*a*b^4*ArcTanh[Sin[c + d*x]])/(256*d) + (5*a^4*b*Sec[c + d*x]^7)/(7*d) - (10*a^2*b^3*Sec[c + d*x]^7)/(7*d) + (b^5*Sec[c + d*x]^7)/(7*d) + (10*a^2*b^3*Sec[c + d*x]^9)/(9*d) - (2*b^5*Sec[c + d*x]^9)/(9*d) + (b^5*Sec[c + d*x]^11)/(11*d) + (5*a^5*Sec[c + d*x]*Tan[c + d*x])/(16*d) - (25*a^3*b^2*Sec[c + d*x]*Tan[c + d*x])/(64*d) + (15*a*b^4*Sec[c + d*x]*Tan[c + d*x])/(256*d) + (5*a^5*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) - (25*a^3*b^2*Sec[c + d*x]^3*Tan[c + d*x])/(96*d) + (5*a*b^4*Sec[c + d*x]^3*Tan[c + d*x])/(128*d) + (a^5*Sec[c + d*x]^5*Tan[c + d*x])/(6*d) - (5*a^3*b^2*Sec[c + d*x]^5*Tan[c + d*x])/(24*d) + (a*b^4*Sec[c + d*x]^5*Tan[c + d*x])/(32*d) + (5*a^3*b^2*Sec[c + d*x]^7*Tan[c + d*x])/(4*d) - (3*a*b^4*Sec[c + d*x]^7*Tan[c + d*x])/(16*d) + (a*b^4*Sec[c + d*x]^7*Tan[c + d*x]^3)/(2*d)","A",25,8,28,0.2857,1,"{3090, 3768, 3770, 2606, 30, 2611, 14, 270}"
110,1,227,0,0.2142514,"\int \frac{\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{b^3 \cos ^2(c+d x)}{2 d \left(a^2+b^2\right)^2}+\frac{b \cos ^4(c+d x)}{4 d \left(a^2+b^2\right)}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d \left(a^2+b^2\right)}+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{2 d \left(a^2+b^2\right)^2}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d \left(a^2+b^2\right)}+\frac{b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{a b^4 x}{\left(a^2+b^2\right)^3}+\frac{a b^2 x}{2 \left(a^2+b^2\right)^2}+\frac{3 a x}{8 \left(a^2+b^2\right)}","\frac{b^3 \cos ^2(c+d x)}{2 d \left(a^2+b^2\right)^2}+\frac{b \cos ^4(c+d x)}{4 d \left(a^2+b^2\right)}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d \left(a^2+b^2\right)}+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{2 d \left(a^2+b^2\right)^2}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d \left(a^2+b^2\right)}+\frac{b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{a b^4 x}{\left(a^2+b^2\right)^3}+\frac{a b^2 x}{2 \left(a^2+b^2\right)^2}+\frac{3 a x}{8 \left(a^2+b^2\right)}",1,"(a*b^4*x)/(a^2 + b^2)^3 + (a*b^2*x)/(2*(a^2 + b^2)^2) + (3*a*x)/(8*(a^2 + b^2)) + (b^3*Cos[c + d*x]^2)/(2*(a^2 + b^2)^2*d) + (b*Cos[c + d*x]^4)/(4*(a^2 + b^2)*d) + (b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/(2*(a^2 + b^2)^2*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*(a^2 + b^2)*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*(a^2 + b^2)*d)","A",9,5,28,0.1786,1,"{3100, 2635, 8, 3098, 3133}"
111,1,166,0,0.1749495,"\int \frac{\cos ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d \left(a^2+b^2\right)}+\frac{a b^2 \sin (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a \sin (c+d x)}{d \left(a^2+b^2\right)}+\frac{b \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)}+\frac{b^3 \cos (c+d x)}{d \left(a^2+b^2\right)^2}-\frac{b^4 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{5/2}}","-\frac{a \sin ^3(c+d x)}{3 d \left(a^2+b^2\right)}+\frac{a b^2 \sin (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a \sin (c+d x)}{d \left(a^2+b^2\right)}+\frac{b \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)}+\frac{b^3 \cos (c+d x)}{d \left(a^2+b^2\right)^2}-\frac{b^4 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{5/2}}",1,"-((b^4*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d)) + (b^3*Cos[c + d*x])/((a^2 + b^2)^2*d) + (b*Cos[c + d*x]^3)/(3*(a^2 + b^2)*d) + (a*b^2*Sin[c + d*x])/((a^2 + b^2)^2*d) + (a*Sin[c + d*x])/((a^2 + b^2)*d) - (a*Sin[c + d*x]^3)/(3*(a^2 + b^2)*d)","A",7,5,28,0.1786,1,"{3100, 2633, 2637, 3074, 206}"
112,1,119,0,0.129146,"\int \frac{\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{b \cos ^2(c+d x)}{2 d \left(a^2+b^2\right)}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d \left(a^2+b^2\right)}+\frac{b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{a b^2 x}{\left(a^2+b^2\right)^2}+\frac{a x}{2 \left(a^2+b^2\right)}","\frac{b \cos ^2(c+d x)}{2 d \left(a^2+b^2\right)}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d \left(a^2+b^2\right)}+\frac{b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{a b^2 x}{\left(a^2+b^2\right)^2}+\frac{a x}{2 \left(a^2+b^2\right)}",1,"(a*b^2*x)/(a^2 + b^2)^2 + (a*x)/(2*(a^2 + b^2)) + (b*Cos[c + d*x]^2)/(2*(a^2 + b^2)*d) + (b^3*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*(a^2 + b^2)*d)","A",5,5,28,0.1786,1,"{3100, 2635, 8, 3098, 3133}"
113,1,91,0,0.0817828,"\int \frac{\cos ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \sin (c+d x)}{d \left(a^2+b^2\right)}+\frac{b \cos (c+d x)}{d \left(a^2+b^2\right)}-\frac{b^2 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{3/2}}","\frac{a \sin (c+d x)}{d \left(a^2+b^2\right)}+\frac{b \cos (c+d x)}{d \left(a^2+b^2\right)}-\frac{b^2 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{3/2}}",1,"-((b^2*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d)) + (b*Cos[c + d*x])/((a^2 + b^2)*d) + (a*Sin[c + d*x])/((a^2 + b^2)*d)","A",4,4,28,0.1429,1,"{3100, 2637, 3074, 206}"
114,1,45,0,0.0657701,"\int \frac{\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)}+\frac{a x}{a^2+b^2}","\frac{b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)}+\frac{a x}{a^2+b^2}",1,"(a*x)/(a^2 + b^2) + (b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)","A",2,2,26,0.07692,1,"{3098, 3133}"
115,1,47,0,0.022887,"\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}"
116,1,41,0,0.0816556,"\int \frac{\sec (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{\log (a \cos (c+d x)+b \sin (c+d x))}{b d}-\frac{\log (\cos (c+d x))}{b d}","\frac{\log (a \cos (c+d x)+b \sin (c+d x))}{b d}-\frac{\log (\cos (c+d x))}{b d}",1,"-(Log[Cos[c + d*x]]/(b*d)) + Log[a*Cos[c + d*x] + b*Sin[c + d*x]]/(b*d)","A",3,3,26,0.1154,1,"{3102, 3475, 3133}"
117,1,80,0,0.0835684,"\int \frac{\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^2 d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{\sec (c+d x)}{b d}","-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^2 d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{\sec (c+d x)}{b d}",1,"-((a*ArcTanh[Sin[c + d*x]])/(b^2*d)) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*d) + Sec[c + d*x]/(b*d)","A",4,4,28,0.1429,1,"{3104, 3770, 3074, 206}"
118,1,88,0,0.1409723,"\int \frac{\sec ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","-\frac{\left(a^2+b^2\right) \log (\cos (c+d x))}{b^3 d}+\frac{\left(a^2+b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec ^2(c+d x)}{2 b d}","-\frac{\left(a^2+b^2\right) \log (\cos (c+d x))}{b^3 d}+\frac{\left(a^2+b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec ^2(c+d x)}{2 b d}",1,"-(((a^2 + b^2)*Log[Cos[c + d*x]])/(b^3*d)) + ((a^2 + b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(b^3*d) + Sec[c + d*x]^2/(2*b*d) - (a*Tan[c + d*x])/(b^2*d)","A",6,6,28,0.2143,1,"{3104, 3767, 8, 3102, 3475, 3133}"
119,1,153,0,0.1572371,"\int \frac{\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{\left(a^2+b^2\right) \sec (c+d x)}{b^3 d}-\frac{a \left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac{\left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}-\frac{a \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{\sec ^3(c+d x)}{3 b d}","\frac{\left(a^2+b^2\right) \sec (c+d x)}{b^3 d}-\frac{a \left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac{\left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d}-\frac{a \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}-\frac{a \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{\sec ^3(c+d x)}{3 b d}",1,"-(a*ArcTanh[Sin[c + d*x]])/(2*b^2*d) - (a*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(b^4*d) - ((a^2 + b^2)^(3/2)*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*d) + ((a^2 + b^2)*Sec[c + d*x])/(b^3*d) + Sec[c + d*x]^3/(3*b*d) - (a*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d)","A",7,5,28,0.1786,1,"{3104, 3768, 3770, 3074, 206}"
120,1,158,0,0.222416,"\int \frac{\sec ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","-\frac{a \left(a^2+b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(a^2+b^2\right) \sec ^2(c+d x)}{2 b^3 d}-\frac{\left(a^2+b^2\right)^2 \log (\cos (c+d x))}{b^5 d}+\frac{\left(a^2+b^2\right)^2 \log (a \cos (c+d x)+b \sin (c+d x))}{b^5 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec ^4(c+d x)}{4 b d}","-\frac{a \left(a^2+b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(a^2+b^2\right) \sec ^2(c+d x)}{2 b^3 d}-\frac{\left(a^2+b^2\right)^2 \log (\cos (c+d x))}{b^5 d}+\frac{\left(a^2+b^2\right)^2 \log (a \cos (c+d x)+b \sin (c+d x))}{b^5 d}-\frac{a \tan ^3(c+d x)}{3 b^2 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\sec ^4(c+d x)}{4 b d}",1,"-(((a^2 + b^2)^2*Log[Cos[c + d*x]])/(b^5*d)) + ((a^2 + b^2)^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(b^5*d) + ((a^2 + b^2)*Sec[c + d*x]^2)/(2*b^3*d) + Sec[c + d*x]^4/(4*b*d) - (a*Tan[c + d*x])/(b^2*d) - (a*(a^2 + b^2)*Tan[c + d*x])/(b^4*d) - (a*Tan[c + d*x]^3)/(3*b^2*d)","A",9,6,28,0.2143,1,"{3104, 3767, 8, 3102, 3475, 3133}"
121,1,262,0,0.2561621,"\int \frac{\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^6/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{\left(a^2+b^2\right) \sec ^3(c+d x)}{3 b^3 d}+\frac{\left(a^2+b^2\right)^2 \sec (c+d x)}{b^5 d}-\frac{a \left(a^2+b^2\right)^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac{a \left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{a \left(a^2+b^2\right) \tan (c+d x) \sec (c+d x)}{2 b^4 d}-\frac{\left(a^2+b^2\right)^{5/2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 b^2 d}-\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 b^2 d}-\frac{3 a \tan (c+d x) \sec (c+d x)}{8 b^2 d}+\frac{\sec ^5(c+d x)}{5 b d}","\frac{\left(a^2+b^2\right) \sec ^3(c+d x)}{3 b^3 d}+\frac{\left(a^2+b^2\right)^2 \sec (c+d x)}{b^5 d}-\frac{a \left(a^2+b^2\right)^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac{a \left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{a \left(a^2+b^2\right) \tan (c+d x) \sec (c+d x)}{2 b^4 d}-\frac{\left(a^2+b^2\right)^{5/2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 b^2 d}-\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 b^2 d}-\frac{3 a \tan (c+d x) \sec (c+d x)}{8 b^2 d}+\frac{\sec ^5(c+d x)}{5 b d}",1,"(-3*a*ArcTanh[Sin[c + d*x]])/(8*b^2*d) - (a*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (a*(a^2 + b^2)^2*ArcTanh[Sin[c + d*x]])/(b^6*d) - ((a^2 + b^2)^(5/2)*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*d) + ((a^2 + b^2)^2*Sec[c + d*x])/(b^5*d) + ((a^2 + b^2)*Sec[c + d*x]^3)/(3*b^3*d) + Sec[c + d*x]^5/(5*b*d) - (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*b^2*d) - (a*(a^2 + b^2)*Sec[c + d*x]*Tan[c + d*x])/(2*b^4*d) - (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*b^2*d)","A",11,5,28,0.1786,1,"{3104, 3768, 3770, 3074, 206}"
122,1,145,0,0.2926132,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{b^4}{a d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}-\frac{\sin ^2(c+d x) \left(2 a b-\left(a^2-b^2\right) \cot (c+d x)\right)}{2 d \left(a^2+b^2\right)^2}+\frac{4 a b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(6 a^2 b^2+a^4-3 b^4\right)}{2 \left(a^2+b^2\right)^3}","\frac{b^4}{a d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}-\frac{\sin ^2(c+d x) \left(2 a b-\left(a^2-b^2\right) \cot (c+d x)\right)}{2 d \left(a^2+b^2\right)^2}+\frac{4 a b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{x \left(6 a^2 b^2+a^4-3 b^4\right)}{2 \left(a^2+b^2\right)^3}",1,"((a^4 + 6*a^2*b^2 - 3*b^4)*x)/(2*(a^2 + b^2)^3) + b^4/(a*(a^2 + b^2)^2*d*(b + a*Cot[c + d*x])) + (4*a*b^3*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - ((2*a*b - (a^2 - b^2)*Cot[c + d*x])*Sin[c + d*x]^2)/(2*(a^2 + b^2)^2*d)","A",7,6,28,0.2143,1,"{3088, 1647, 1629, 635, 203, 260}"
123,1,231,0,1.0483348,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","-\frac{2 b^3 \left(a+b \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a d \left(a^2+b^2\right)^2 \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+2 b \tan \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 \left(\left(a^2-b^2\right) \tan \left(\frac{1}{2} (c+d x)\right)+2 a b\right)}{d \left(a^2+b^2\right)^2 \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)}+\frac{2 b^4 \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2+b^2}}\right)}{a d \left(a^2+b^2\right)^{5/2}}-\frac{2 b^2 \left(3 a^2+b^2\right) \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2+b^2}}\right)}{a d \left(a^2+b^2\right)^{5/2}}","\frac{\left(a^2-b^2\right) \sin (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{2 a b \cos (c+d x)}{d \left(a^2+b^2\right)^2}-\frac{b^3}{d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))}-\frac{3 a b^2 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{5/2}}",1,"(2*b^4*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a*(a^2 + b^2)^(5/2)*d) - (2*b^2*(3*a^2 + b^2)*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a*(a^2 + b^2)^(5/2)*d) + (2*(2*a*b + (a^2 - b^2)*Tan[(c + d*x)/2]))/((a^2 + b^2)^2*d*(1 + Tan[(c + d*x)/2]^2)) - (2*b^3*(a + b*Tan[(c + d*x)/2]))/(a*(a^2 + b^2)^2*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2))","A",11,6,28,0.2143,1,"{6742, 639, 203, 638, 618, 206}"
124,1,82,0,0.1365274,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","-\frac{b}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}","-\frac{b}{d \left(a^2+b^2\right) (a+b \tan (c+d x))}+\frac{2 a b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}",1,"((a^2 - b^2)*x)/(a^2 + b^2)^2 + (2*a*b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^2*d) - b/((a^2 + b^2)*d*(a + b*Tan[c + d*x]))","A",4,4,28,0.1429,1,"{3086, 3483, 3531, 3530}"
125,1,83,0,0.0665358,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","-\frac{b}{d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))}-\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{3/2}}","-\frac{b}{d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))}-\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{3/2}}",1,"-((a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(3/2)*d)) - b/((a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","A",3,3,26,0.1154,1,"{3155, 3074, 206}"
126,1,32,0,0.0167772,"\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}"
127,1,92,0,0.0794156,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^2 d \sqrt{a^2+b^2}}-\frac{1}{b d (a \cos (c+d x)+b \sin (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}","\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^2 d \sqrt{a^2+b^2}}-\frac{1}{b d (a \cos (c+d x)+b \sin (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"ArcTanh[Sin[c + d*x]]/(b^2*d) + (a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]*d) - 1/(b*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","A",4,4,26,0.1538,1,"{3094, 3770, 3074, 206}"
128,1,75,0,0.0962912,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\frac{a}{b^2}+\frac{1}{a}}{d (a \cot (c+d x)+b)}-\frac{2 a \log (\tan (c+d x))}{b^3 d}-\frac{2 a \log (a \cot (c+d x)+b)}{b^3 d}+\frac{\tan (c+d x)}{b^2 d}","\frac{\frac{a}{b^2}+\frac{1}{a}}{d (a \cot (c+d x)+b)}-\frac{2 a \log (\tan (c+d x))}{b^3 d}-\frac{2 a \log (a \cot (c+d x)+b)}{b^3 d}+\frac{\tan (c+d x)}{b^2 d}",1,"(a^(-1) + a/b^2)/(d*(b + a*Cot[c + d*x])) - (2*a*Log[b + a*Cot[c + d*x]])/(b^3*d) - (2*a*Log[Tan[c + d*x]])/(b^3*d) + Tan[c + d*x]/(b^2*d)","A",3,2,28,0.07143,1,"{3088, 894}"
129,1,179,0,0.2396096,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac{\left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac{a^2+b^2}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}+\frac{3 a \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d}-\frac{2 a \sec (c+d x)}{b^3 d}+\frac{\tanh ^{-1}(\sin (c+d x))}{2 b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b^2 d}","\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac{\left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac{a^2+b^2}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}+\frac{3 a \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d}-\frac{2 a \sec (c+d x)}{b^3 d}+\frac{\tanh ^{-1}(\sin (c+d x))}{2 b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b^2 d}",1,"(2*a^2*ArcTanh[Sin[c + d*x]])/(b^4*d) + ArcTanh[Sin[c + d*x]]/(2*b^2*d) + ((a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(b^4*d) + (3*a*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*d) - (2*a*Sec[c + d*x])/(b^3*d) - (a^2 + b^2)/(b^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) + (Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d)","A",11,7,28,0.2500,1,"{3106, 3094, 3770, 3074, 206, 3768, 3104}"
130,1,141,0,0.1498993,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\left(3 a^2+2 b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(a^2+b^2\right)^2}{a b^4 d (a \cot (c+d x)+b)}-\frac{4 a \left(a^2+b^2\right) \log (\tan (c+d x))}{b^5 d}-\frac{4 a \left(a^2+b^2\right) \log (a \cot (c+d x)+b)}{b^5 d}-\frac{a \tan ^2(c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}","\frac{\left(3 a^2+2 b^2\right) \tan (c+d x)}{b^4 d}+\frac{\left(a^2+b^2\right)^2}{a b^4 d (a \cot (c+d x)+b)}-\frac{4 a \left(a^2+b^2\right) \log (\tan (c+d x))}{b^5 d}-\frac{4 a \left(a^2+b^2\right) \log (a \cot (c+d x)+b)}{b^5 d}-\frac{a \tan ^2(c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}",1,"(a^2 + b^2)^2/(a*b^4*d*(b + a*Cot[c + d*x])) - (4*a*(a^2 + b^2)*Log[b + a*Cot[c + d*x]])/(b^5*d) - (4*a*(a^2 + b^2)*Log[Tan[c + d*x]])/(b^5*d) + ((3*a^2 + 2*b^2)*Tan[c + d*x])/(b^4*d) - (a*Tan[c + d*x]^2)/(b^3*d) + Tan[c + d*x]^3/(3*b^2*d)","A",3,2,28,0.07143,1,"{3088, 894}"
131,1,492,0,1.7422679,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{3 b^4 \left(a^2+2 b^2\right) \left(b-a \tan \left(\frac{1}{2} (c+d x)\right)\right)}{a^3 d \left(a^2+b^2\right)^3 \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+2 b \tan \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 b^4 \left(\left(a^2+2 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right)+a b\right)}{a^3 d \left(a^2+b^2\right)^2 \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+2 b \tan \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{4 b^3 \left(a b \left(3 a^2+2 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right)+2 a^4-b^4\right)}{a^3 d \left(a^2+b^2\right)^3 \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+2 b \tan \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 \left(a \left(a^2-3 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right)+b \left(3 a^2-b^2\right)\right)}{d \left(a^2+b^2\right)^3 \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)}-\frac{3 b^4 \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2+b^2}}\right)}{a^2 d \left(a^2+b^2\right)^{7/2}}+\frac{4 b^4 \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2+b^2}}\right)}{a^2 d \left(a^2+b^2\right)^{7/2}}-\frac{2 b^2 \left(3 a^2 b^2+6 a^4+b^4\right) \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2+b^2}}\right)}{a^2 d \left(a^2+b^2\right)^{7/2}}","\frac{a \left(a^2-3 b^2\right) \sin (c+d x)}{d \left(a^2+b^2\right)^3}+\frac{b \left(3 a^2-b^2\right) \cos (c+d x)}{d \left(a^2+b^2\right)^3}+\frac{b^4 \sin (c+d x)}{2 a d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{b^3 \left(8 a^2+b^2\right)}{2 a d \left(a^2+b^2\right)^3 (a \cos (c+d x)+b \sin (c+d x))}-\frac{3 b^2 \left(4 a^2-b^2\right) \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{7/2}}",1,"(-3*b^4*(a^2 + 2*b^2)*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(7/2)*d) + (4*b^4*(3*a^2 + 2*b^2)*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(7/2)*d) - (2*b^2*(6*a^4 + 3*a^2*b^2 + b^4)*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(7/2)*d) + (2*(b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Tan[(c + d*x)/2]))/((a^2 + b^2)^3*d*(1 + Tan[(c + d*x)/2]^2)) + (2*b^4*(a*b + (a^2 + 2*b^2)*Tan[(c + d*x)/2]))/(a^3*(a^2 + b^2)^2*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2)^2) - (3*b^4*(a^2 + 2*b^2)*(b - a*Tan[(c + d*x)/2]))/(a^3*(a^2 + b^2)^3*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2)) - (4*b^3*(2*a^4 - b^4 + a*b*(3*a^2 + 2*b^2)*Tan[(c + d*x)/2]))/(a^3*(a^2 + b^2)^3*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2))","B",15,7,28,0.2500,1,"{6742, 639, 203, 638, 618, 206, 614}"
132,1,122,0,0.2122987,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{2 a b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(3 a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{a x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^3}","-\frac{2 a b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{b}{2 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}+\frac{b \left(3 a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{a x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^3}",1,"(a*(a^2 - 3*b^2)*x)/(a^2 + b^2)^3 + (b*(3*a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^3*d) - b/(2*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) - (2*a*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))","A",5,5,28,0.1786,1,"{3086, 3483, 3529, 3531, 3530}"
133,1,225,0,0.5877046,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{2 b^2 \left(\left(a^2+2 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right)+a b\right)}{a^3 d \left(a^2+b^2\right) \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+2 b \tan \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b \left(a b \left(5 a^2+2 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right)+3 a^2 b^2+4 a^4+2 b^4\right)}{a^3 d \left(a^2+b^2\right)^2 \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+2 b \tan \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{\left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{5/2}}","\frac{\left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{5/2}}-\frac{b \left(\left(4 a^2+b^2\right) \cos (c+d x)+3 a b \sin (c+d x)\right)}{2 d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^2}",1,"-(((2*a^2 - b^2)*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(5/2)*d)) + (2*b^2*(a*b + (a^2 + 2*b^2)*Tan[(c + d*x)/2]))/(a^3*(a^2 + b^2)*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2)^2) - (b*(4*a^4 + 3*a^2*b^2 + 2*b^4 + a*b*(5*a^2 + 2*b^2)*Tan[(c + d*x)/2]))/(a^3*(a^2 + b^2)^2*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2))","A",6,4,28,0.1429,1,"{1660, 12, 618, 206}"
134,1,30,0,0.0312509,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{\cot ^2(c+d x)}{2 b d (a \cot (c+d x)+b)^2}","-\frac{1}{2 b d (a+b \tan (c+d x))^2}",1,"-Cot[c + d*x]^2/(2*b*d*(b + a*Cot[c + d*x])^2)","A",2,2,26,0.07692,1,"{3088, 37}"
135,1,103,0,0.0494178,"\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}"
136,1,86,0,0.1031371,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\frac{1}{a^2}-\frac{1}{b^2}}{d (a \cot (c+d x)+b)}-\frac{\frac{b}{a^2}+\frac{1}{b}}{2 d (a \cot (c+d x)+b)^2}+\frac{\log (a \cot (c+d x)+b)}{b^3 d}+\frac{\log (\tan (c+d x))}{b^3 d}","\frac{\frac{1}{a^2}-\frac{1}{b^2}}{d (a \cot (c+d x)+b)}-\frac{\frac{b}{a^2}+\frac{1}{b}}{2 d (a \cot (c+d x)+b)^2}+\frac{\log (a \cot (c+d x)+b)}{b^3 d}+\frac{\log (\tan (c+d x))}{b^3 d}",1,"-(b^(-1) + b/a^2)/(2*d*(b + a*Cot[c + d*x])^2) + (a^(-2) - b^(-2))/(d*(b + a*Cot[c + d*x])) + Log[b + a*Cot[c + d*x]]/(b^3*d) + Log[Tan[c + d*x]]/(b^3*d)","A",3,2,26,0.07692,1,"{3088, 894}"
137,1,260,0,0.2861495,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{2 a^2 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d \sqrt{a^2+b^2}}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d}-\frac{\tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 b^2 d \sqrt{a^2+b^2}}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac{2 a}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{b \cos (c+d x)-a \sin (c+d x)}{2 b^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\sec (c+d x)}{b^3 d}","-\frac{2 a^2 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d \sqrt{a^2+b^2}}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d}-\frac{\tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 b^2 d \sqrt{a^2+b^2}}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac{2 a}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{b \cos (c+d x)-a \sin (c+d x)}{2 b^2 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\sec (c+d x)}{b^3 d}",1,"(-3*a*ArcTanh[Sin[c + d*x]])/(b^4*d) - (2*a^2*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*Sqrt[a^2 + b^2]*d) - ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]]/(2*b^2*Sqrt[a^2 + b^2]*d) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*d) + Sec[c + d*x]/(b^3*d) - (b*Cos[c + d*x] - a*Sin[c + d*x])/(2*b^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (2*a)/(b^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","A",12,7,28,0.2500,1,"{3106, 3076, 3074, 206, 3104, 3770, 3094}"
138,1,161,0,0.1678245,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","-\frac{\left(a^2+b^2\right)^2}{2 a^2 b^3 d (a \cot (c+d x)+b)^2}-\frac{\left(3 a^2-b^2\right) \left(a^2+b^2\right)}{a^2 b^4 d (a \cot (c+d x)+b)}+\frac{2 \left(3 a^2+b^2\right) \log (\tan (c+d x))}{b^5 d}+\frac{2 \left(3 a^2+b^2\right) \log (a \cot (c+d x)+b)}{b^5 d}-\frac{3 a \tan (c+d x)}{b^4 d}+\frac{\tan ^2(c+d x)}{2 b^3 d}","-\frac{\left(a^2+b^2\right)^2}{2 a^2 b^3 d (a \cot (c+d x)+b)^2}-\frac{\left(3 a^2-b^2\right) \left(a^2+b^2\right)}{a^2 b^4 d (a \cot (c+d x)+b)}+\frac{2 \left(3 a^2+b^2\right) \log (\tan (c+d x))}{b^5 d}+\frac{2 \left(3 a^2+b^2\right) \log (a \cot (c+d x)+b)}{b^5 d}-\frac{3 a \tan (c+d x)}{b^4 d}+\frac{\tan ^2(c+d x)}{2 b^3 d}",1,"-(a^2 + b^2)^2/(2*a^2*b^3*d*(b + a*Cot[c + d*x])^2) - ((3*a^2 - b^2)*(a^2 + b^2))/(a^2*b^4*d*(b + a*Cot[c + d*x])) + (2*(3*a^2 + b^2)*Log[b + a*Cot[c + d*x]])/(b^5*d) + (2*(3*a^2 + b^2)*Log[Tan[c + d*x]])/(b^5*d) - (3*a*Tan[c + d*x])/(b^4*d) + Tan[c + d*x]^2/(2*b^3*d)","A",3,2,28,0.07143,1,"{3088, 894}"
139,1,383,0,0.7854876,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{4 a^2 \sec (c+d x)}{b^5 d}+\frac{2 \left(a^2+b^2\right) \sec (c+d x)}{b^5 d}-\frac{4 a^3 \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac{6 a \left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac{4 a \left(a^2+b^2\right)}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{\left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{8 a^2 \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d}-\frac{2 \left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 b^4 d}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{3 a \tan (c+d x) \sec (c+d x)}{2 b^4 d}+\frac{\sec ^3(c+d x)}{3 b^3 d}","\frac{4 a^2 \sec (c+d x)}{b^5 d}+\frac{2 \left(a^2+b^2\right) \sec (c+d x)}{b^5 d}-\frac{4 a^3 \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac{6 a \left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac{4 a \left(a^2+b^2\right)}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{\left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{8 a^2 \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d}-\frac{2 \left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d}-\frac{\sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 b^4 d}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{3 a \tan (c+d x) \sec (c+d x)}{2 b^4 d}+\frac{\sec ^3(c+d x)}{3 b^3 d}",1,"(-4*a^3*ArcTanh[Sin[c + d*x]])/(b^6*d) - (3*a*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (6*a*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(b^6*d) - (8*a^2*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*d) - (Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*b^4*d) - (2*(a^2 + b^2)^(3/2)*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*d) + (4*a^2*Sec[c + d*x])/(b^5*d) + (2*(a^2 + b^2)*Sec[c + d*x])/(b^5*d) + Sec[c + d*x]^3/(3*b^3*d) - ((a^2 + b^2)*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*b^4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (4*a*(a^2 + b^2))/(b^5*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) - (3*a*Sec[c + d*x]*Tan[c + d*x])/(2*b^4*d)","A",31,8,28,0.2857,1,"{3106, 3076, 3074, 206, 3104, 3770, 3094, 3768}"
140,1,232,0,0.2435825,"\int \frac{\sec ^5(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{3 \left(2 a^2+b^2\right) \tan ^2(c+d x)}{2 b^5 d}-\frac{a \left(10 a^2+9 b^2\right) \tan (c+d x)}{b^6 d}-\frac{\left(5 a^2-b^2\right) \left(a^2+b^2\right)^2}{a^2 b^6 d (a \cot (c+d x)+b)}-\frac{\left(a^2+b^2\right)^3}{2 a^2 b^5 d (a \cot (c+d x)+b)^2}+\frac{3 \left(a^2+b^2\right) \left(5 a^2+b^2\right) \log (\tan (c+d x))}{b^7 d}+\frac{3 \left(a^2+b^2\right) \left(5 a^2+b^2\right) \log (a \cot (c+d x)+b)}{b^7 d}-\frac{a \tan ^3(c+d x)}{b^4 d}+\frac{\tan ^4(c+d x)}{4 b^3 d}","\frac{3 \left(2 a^2+b^2\right) \tan ^2(c+d x)}{2 b^5 d}-\frac{a \left(10 a^2+9 b^2\right) \tan (c+d x)}{b^6 d}-\frac{\left(5 a^2-b^2\right) \left(a^2+b^2\right)^2}{a^2 b^6 d (a \cot (c+d x)+b)}-\frac{\left(a^2+b^2\right)^3}{2 a^2 b^5 d (a \cot (c+d x)+b)^2}+\frac{3 \left(a^2+b^2\right) \left(5 a^2+b^2\right) \log (\tan (c+d x))}{b^7 d}+\frac{3 \left(a^2+b^2\right) \left(5 a^2+b^2\right) \log (a \cot (c+d x)+b)}{b^7 d}-\frac{a \tan ^3(c+d x)}{b^4 d}+\frac{\tan ^4(c+d x)}{4 b^3 d}",1,"-(a^2 + b^2)^3/(2*a^2*b^5*d*(b + a*Cot[c + d*x])^2) - ((5*a^2 - b^2)*(a^2 + b^2)^2)/(a^2*b^6*d*(b + a*Cot[c + d*x])) + (3*(a^2 + b^2)*(5*a^2 + b^2)*Log[b + a*Cot[c + d*x]])/(b^7*d) + (3*(a^2 + b^2)*(5*a^2 + b^2)*Log[Tan[c + d*x]])/(b^7*d) - (a*(10*a^2 + 9*b^2)*Tan[c + d*x])/(b^6*d) + (3*(2*a^2 + b^2)*Tan[c + d*x]^2)/(2*b^5*d) - (a*Tan[c + d*x]^3)/(b^4*d) + Tan[c + d*x]^4/(4*b^3*d)","A",3,2,28,0.07143,1,"{3088, 894}"
141,1,165,0,0.3032297,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Int[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{b \left(3 a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{a b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{b}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{4 a b \left(a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{x \left(-6 a^2 b^2+a^4+b^4\right)}{\left(a^2+b^2\right)^4}","-\frac{b \left(3 a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{a b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{b}{3 d \left(a^2+b^2\right) (a+b \tan (c+d x))^3}+\frac{4 a b \left(a^2-b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{x \left(-6 a^2 b^2+a^4+b^4\right)}{\left(a^2+b^2\right)^4}",1,"((a^4 - 6*a^2*b^2 + b^4)*x)/(a^2 + b^2)^4 + (4*a*b*(a^2 - b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^4*d) - b/(3*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a*b)/((a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^2 - b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))","A",6,5,28,0.1786,1,"{3086, 3483, 3529, 3531, 3530}"
142,1,362,0,1.1730946,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Int[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{8 b^3 \left(b \left(3 a^2+4 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right)+a \left(a^2+2 b^2\right)\right)}{3 a^5 d \left(a^2+b^2\right) \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+2 b \tan \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 b^2 \left(a \left(30 a^2 b^2+9 a^4+16 b^4\right) \tan \left(\frac{1}{2} (c+d x)\right)+b \left(18 a^2 b^2+15 a^4+8 b^4\right)\right)}{3 a^5 d \left(a^2+b^2\right)^2 \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+2 b \tan \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b \left(a b \left(6 a^2 b^2+9 a^4+2 b^4\right) \tan \left(\frac{1}{2} (c+d x)\right)+9 a^4 b^2+12 a^2 b^4+6 a^6+4 b^6\right)}{a^4 d \left(a^2+b^2\right)^3 \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+2 b \tan \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{a \left(2 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{b-a \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{7/2}}","\frac{\frac{1}{2} b \left(b^2-9 a^2\right) \left(2 \left(a^2+b^2\right)+3 a b \sin (2 (c+d x))\right)-3 \left(-a^2 b^3+3 a^4 b+b^5\right) \cos (2 (c+d x))}{6 d \left(a^2+b^2\right)^3 (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{a \left(2 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{7/2}}",1,"-((a*(2*a^2 - 3*b^2)*ArcTanh[(b - a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(7/2)*d)) - (8*b^3*(a*(a^2 + 2*b^2) + b*(3*a^2 + 4*b^2)*Tan[(c + d*x)/2]))/(3*a^5*(a^2 + b^2)*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2)^3) + (2*b^2*(b*(15*a^4 + 18*a^2*b^2 + 8*b^4) + a*(9*a^4 + 30*a^2*b^2 + 16*b^4)*Tan[(c + d*x)/2]))/(3*a^5*(a^2 + b^2)^2*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2)^2) - (b*(6*a^6 + 9*a^4*b^2 + 12*a^2*b^4 + 4*b^6 + a*b*(9*a^4 + 6*a^2*b^2 + 2*b^4)*Tan[(c + d*x)/2]))/(a^4*(a^2 + b^2)^3*d*(a + 2*b*Tan[(c + d*x)/2] - a*Tan[(c + d*x)/2]^2))","B",7,4,28,0.1429,1,"{1660, 12, 618, 206}"
143,1,30,0,0.0528595,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Int[Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{\cot ^3(c+d x)}{3 b d (a \cot (c+d x)+b)^3}","-\frac{\cot ^3(c+d x)}{3 b d (a \cot (c+d x)+b)^3}",1,"-Cot[c + d*x]^3/(3*b*d*(b + a*Cot[c + d*x])^3)","A",2,2,28,0.07143,1,"{3088, 37}"
144,1,141,0,0.1109759,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Int[Cos[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{b}{3 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a (b \cos (c+d x)-a \sin (c+d x))}{2 d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{a \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)^{5/2}}","-\frac{b}{3 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a (b \cos (c+d x)-a \sin (c+d x))}{2 d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{a \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)^{5/2}}",1,"-(a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*(a^2 + b^2)^(5/2)*d) - b/(3*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (a*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)","A",5,5,26,0.1923,1,"{3158, 12, 3076, 3074, 206}"
145,1,98,0,0.0404252,"\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}"
146,1,231,0,0.1695936,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Int[Sec[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{a (b \cos (c+d x)-a \sin (c+d x))}{2 b^2 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d \sqrt{a^2+b^2}}+\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 b^2 d \left(a^2+b^2\right)^{3/2}}-\frac{1}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^4 d}","\frac{a (b \cos (c+d x)-a \sin (c+d x))}{2 b^2 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^4 d \sqrt{a^2+b^2}}+\frac{a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 b^2 d \left(a^2+b^2\right)^{3/2}}-\frac{1}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{1}{3 b d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"ArcTanh[Sin[c + d*x]]/(b^4*d) + (a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*b^2*(a^2 + b^2)^(3/2)*d) + (a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^4*Sqrt[a^2 + b^2]*d) - 1/(3*b*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (a*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*b^2*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - 1/(b^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","A",8,5,26,0.1923,1,"{3094, 3770, 3074, 206, 3076}"
147,1,138,0,0.1600196,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Int[Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\left(a^2+b^2\right)^2}{3 a^3 b^2 d (a \cot (c+d x)+b)^3}+\frac{\frac{1}{a^3}+\frac{3 a}{b^4}}{d (a \cot (c+d x)+b)}+\frac{\frac{a}{b^3}-\frac{b}{a^3}}{d (a \cot (c+d x)+b)^2}-\frac{4 a \log (\tan (c+d x))}{b^5 d}-\frac{4 a \log (a \cot (c+d x)+b)}{b^5 d}+\frac{\tan (c+d x)}{b^4 d}","\frac{\left(a^2+b^2\right)^2}{3 a^3 b^2 d (a \cot (c+d x)+b)^3}+\frac{\frac{1}{a^3}+\frac{3 a}{b^4}}{d (a \cot (c+d x)+b)}+\frac{\frac{a}{b^3}-\frac{b}{a^3}}{d (a \cot (c+d x)+b)^2}-\frac{4 a \log (\tan (c+d x))}{b^5 d}-\frac{4 a \log (a \cot (c+d x)+b)}{b^5 d}+\frac{\tan (c+d x)}{b^4 d}",1,"(a^2 + b^2)^2/(3*a^3*b^2*d*(b + a*Cot[c + d*x])^3) + (a/b^3 - b/a^3)/(d*(b + a*Cot[c + d*x])^2) + (a^(-3) + (3*a)/b^4)/(d*(b + a*Cot[c + d*x])) - (4*a*Log[b + a*Cot[c + d*x]])/(b^5*d) - (4*a*Log[Tan[c + d*x]])/(b^5*d) + Tan[c + d*x]/(b^4*d)","A",3,2,28,0.07143,1,"{3088, 894}"
148,1,400,0,0.7964257,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Int[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{8 a^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac{2 \left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac{4 a^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{2 \left(a^2+b^2\right)}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{a^2+b^2}{3 b^3 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{4 a^3 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d \sqrt{a^2+b^2}}+\frac{6 a \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d}+\frac{3 a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 b^4 d \sqrt{a^2+b^2}}-\frac{4 a \sec (c+d x)}{b^5 d}+\frac{3 a (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b^4 d}","\frac{8 a^2 \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac{2 \left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\frac{4 a^2}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{2 \left(a^2+b^2\right)}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}-\frac{a^2+b^2}{3 b^3 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{4 a^3 \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d \sqrt{a^2+b^2}}+\frac{6 a \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{b^6 d}+\frac{3 a \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 b^4 d \sqrt{a^2+b^2}}-\frac{4 a \sec (c+d x)}{b^5 d}+\frac{3 a (b \cos (c+d x)-a \sin (c+d x))}{2 b^4 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b^4 d}",1,"(8*a^2*ArcTanh[Sin[c + d*x]])/(b^6*d) + ArcTanh[Sin[c + d*x]]/(2*b^4*d) + (2*(a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(b^6*d) + (4*a^3*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*Sqrt[a^2 + b^2]*d) + (3*a*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(2*b^4*Sqrt[a^2 + b^2]*d) + (6*a*Sqrt[a^2 + b^2]*ArcTanh[(b*Cos[c + d*x] - a*Sin[c + d*x])/Sqrt[a^2 + b^2]])/(b^6*d) - (4*a*Sec[c + d*x])/(b^5*d) - (a^2 + b^2)/(3*b^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*a*(b*Cos[c + d*x] - a*Sin[c + d*x]))/(2*b^4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (4*a^2)/(b^5*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) - (2*(a^2 + b^2))/(b^5*d*(a*Cos[c + d*x] + b*Sin[c + d*x])) + (Sec[c + d*x]*Tan[c + d*x])/(2*b^4*d)","A",32,8,28,0.2857,1,"{3106, 3094, 3770, 3074, 206, 3076, 3768, 3104}"
149,1,232,0,0.2489322,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Int[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\left(10 a^2+3 b^2\right) \tan (c+d x)}{b^6 d}+\frac{\left(a^2+b^2\right)^3}{3 a^3 b^4 d (a \cot (c+d x)+b)^3}+\frac{9 a^4 b^2+10 a^6+b^6}{a^3 b^6 d (a \cot (c+d x)+b)}+\frac{3 a^4 b^2+2 a^6-b^6}{a^3 b^5 d (a \cot (c+d x)+b)^2}-\frac{4 a \left(5 a^2+3 b^2\right) \log (\tan (c+d x))}{b^7 d}-\frac{4 a \left(5 a^2+3 b^2\right) \log (a \cot (c+d x)+b)}{b^7 d}-\frac{2 a \tan ^2(c+d x)}{b^5 d}+\frac{\tan ^3(c+d x)}{3 b^4 d}","\frac{\left(10 a^2+3 b^2\right) \tan (c+d x)}{b^6 d}+\frac{\left(a^2+b^2\right)^3}{3 a^3 b^4 d (a \cot (c+d x)+b)^3}+\frac{9 a^4 b^2+10 a^6+b^6}{a^3 b^6 d (a \cot (c+d x)+b)}+\frac{3 a^4 b^2+2 a^6-b^6}{a^3 b^5 d (a \cot (c+d x)+b)^2}-\frac{4 a \left(5 a^2+3 b^2\right) \log (\tan (c+d x))}{b^7 d}-\frac{4 a \left(5 a^2+3 b^2\right) \log (a \cot (c+d x)+b)}{b^7 d}-\frac{2 a \tan ^2(c+d x)}{b^5 d}+\frac{\tan ^3(c+d x)}{3 b^4 d}",1,"(a^2 + b^2)^3/(3*a^3*b^4*d*(b + a*Cot[c + d*x])^3) + (2*a^6 + 3*a^4*b^2 - b^6)/(a^3*b^5*d*(b + a*Cot[c + d*x])^2) + (10*a^6 + 9*a^4*b^2 + b^6)/(a^3*b^6*d*(b + a*Cot[c + d*x])) - (4*a*(5*a^2 + 3*b^2)*Log[b + a*Cot[c + d*x]])/(b^7*d) - (4*a*(5*a^2 + 3*b^2)*Log[Tan[c + d*x]])/(b^7*d) + ((10*a^2 + 3*b^2)*Tan[c + d*x])/(b^6*d) - (2*a*Tan[c + d*x]^2)/(b^5*d) + Tan[c + d*x]^3/(3*b^4*d)","A",3,2,28,0.07143,1,"{3088, 894}"
150,1,99,0,0.1507339,"\int \frac{\cos ^5(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{i \cos ^6(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{24 a d}+\frac{5 \sin (c+d x) \cos (c+d x)}{16 a d}+\frac{5 x}{16 a}","\frac{i \cos ^6(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{24 a d}+\frac{5 \sin (c+d x) \cos (c+d x)}{16 a d}+\frac{5 x}{16 a}",1,"(5*x)/(16*a) + ((I/6)*Cos[c + d*x]^6)/(a*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(16*a*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(24*a*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(6*a*d)","A",9,6,31,0.1935,1,"{3092, 3090, 2635, 8, 2565, 30}"
151,1,70,0,0.1278159,"\int \frac{\cos ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{\sin ^5(c+d x)}{5 a d}-\frac{2 \sin ^3(c+d x)}{3 a d}+\frac{\sin (c+d x)}{a d}+\frac{i \cos ^5(c+d x)}{5 a d}","\frac{\sin ^5(c+d x)}{5 a d}-\frac{2 \sin ^3(c+d x)}{3 a d}+\frac{\sin (c+d x)}{a d}+\frac{i \cos ^5(c+d x)}{5 a d}",1,"((I/5)*Cos[c + d*x]^5)/(a*d) + Sin[c + d*x]/(a*d) - (2*Sin[c + d*x]^3)/(3*a*d) + Sin[c + d*x]^5/(5*a*d)","A",7,5,31,0.1613,1,"{3092, 3090, 2633, 2565, 30}"
152,1,75,0,0.1281293,"\int \frac{\cos ^3(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{i \cos ^4(c+d x)}{4 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x}{8 a}","\frac{i \cos ^4(c+d x)}{4 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x}{8 a}",1,"(3*x)/(8*a) + ((I/4)*Cos[c + d*x]^4)/(a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)","A",8,6,31,0.1935,1,"{3092, 3090, 2635, 8, 2565, 30}"
153,1,52,0,0.1203339,"\int \frac{\cos ^2(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","-\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin (c+d x)}{a d}+\frac{i \cos ^3(c+d x)}{3 a d}","-\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin (c+d x)}{a d}+\frac{i \cos ^3(c+d x)}{3 a d}",1,"((I/3)*Cos[c + d*x]^3)/(a*d) + Sin[c + d*x]/(a*d) - Sin[c + d*x]^3/(3*a*d)","A",7,5,31,0.1613,1,"{3092, 3090, 2633, 2565, 30}"
154,1,46,0,0.0294502,"\int \frac{\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{x}{2 a}+\frac{i \cos (c+d x)}{2 d (a \cos (c+d x)+i a \sin (c+d x))}","\frac{x}{2 a}+\frac{i \cos (c+d x)}{2 d (a \cos (c+d x)+i a \sin (c+d x))}",1,"x/(2*a) + ((I/2)*Cos[c + d*x])/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x]))","A",2,2,29,0.06897,1,"{3082, 8}"
155,1,29,0,0.015677,"\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}"
156,1,23,0,0.0701351,"\int \frac{\sec (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Sec[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{x}{a}+\frac{i \log (\cos (c+d x))}{a d}","\frac{x}{a}+\frac{i \log (\cos (c+d x))}{a d}",1,"x/a + (I*Log[Cos[c + d*x]])/(a*d)","A",4,3,29,0.1034,1,"{3092, 3090, 3475}"
157,1,31,0,0.0943114,"\int \frac{\sec ^2(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{i \sec (c+d x)}{a d}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{i \sec (c+d x)}{a d}",1,"ArcTanh[Sin[c + d*x]]/(a*d) - (I*Sec[c + d*x])/(a*d)","A",6,5,31,0.1613,1,"{3092, 3090, 3770, 2606, 8}"
158,1,34,0,0.1072435,"\int \frac{\sec ^3(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{\tan (c+d x)}{a d}-\frac{i \sec ^2(c+d x)}{2 a d}","\frac{\tan (c+d x)}{a d}-\frac{i \sec ^2(c+d x)}{2 a d}",1,"((-I/2)*Sec[c + d*x]^2)/(a*d) + Tan[c + d*x]/(a*d)","A",7,6,31,0.1935,1,"{3092, 3090, 3767, 8, 2606, 30}"
159,1,60,0,0.1171741,"\int \frac{\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","-\frac{i \sec ^3(c+d x)}{3 a d}+\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{\tan (c+d x) \sec (c+d x)}{2 a d}","-\frac{i \sec ^3(c+d x)}{3 a d}+\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{\tan (c+d x) \sec (c+d x)}{2 a d}",1,"ArcTanh[Sin[c + d*x]]/(2*a*d) - ((I/3)*Sec[c + d*x]^3)/(a*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*a*d)","A",7,6,31,0.1935,1,"{3092, 3090, 3768, 3770, 2606, 30}"
160,1,52,0,0.1155843,"\int \frac{\sec ^5(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{\tan ^3(c+d x)}{3 a d}+\frac{\tan (c+d x)}{a d}-\frac{i \sec ^4(c+d x)}{4 a d}","\frac{\tan ^3(c+d x)}{3 a d}+\frac{\tan (c+d x)}{a d}-\frac{i \sec ^4(c+d x)}{4 a d}",1,"((-I/4)*Sec[c + d*x]^4)/(a*d) + Tan[c + d*x]/(a*d) + Tan[c + d*x]^3/(3*a*d)","A",7,5,31,0.1613,1,"{3092, 3090, 3767, 2606, 30}"
161,1,84,0,0.1327695,"\int \frac{\sec ^6(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","-\frac{i \sec ^5(c+d x)}{5 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{8 a d}","-\frac{i \sec ^5(c+d x)}{5 a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}+\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 \tan (c+d x) \sec (c+d x)}{8 a d}",1,"(3*ArcTanh[Sin[c + d*x]])/(8*a*d) - ((I/5)*Sec[c + d*x]^5)/(a*d) + (3*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + (Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d)","A",8,6,31,0.1935,1,"{3092, 3090, 3768, 3770, 2606, 30}"
162,1,70,0,0.1221415,"\int \frac{\sec ^7(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^7/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{\tan ^5(c+d x)}{5 a d}+\frac{2 \tan ^3(c+d x)}{3 a d}+\frac{\tan (c+d x)}{a d}-\frac{i \sec ^6(c+d x)}{6 a d}","\frac{\tan ^5(c+d x)}{5 a d}+\frac{2 \tan ^3(c+d x)}{3 a d}+\frac{\tan (c+d x)}{a d}-\frac{i \sec ^6(c+d x)}{6 a d}",1,"((-I/6)*Sec[c + d*x]^6)/(a*d) + Tan[c + d*x]/(a*d) + (2*Tan[c + d*x]^3)/(3*a*d) + Tan[c + d*x]^5/(5*a*d)","A",7,5,31,0.1613,1,"{3092, 3090, 3767, 2606, 30}"
163,1,85,0,0.1857306,"\int \frac{\cos ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{2 \sin ^7(c+d x)}{7 a^2 d}+\frac{\sin ^5(c+d x)}{a^2 d}-\frac{4 \sin ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x)}{a^2 d}+\frac{2 i \cos ^7(c+d x)}{7 a^2 d}","-\frac{2 \sin ^7(c+d x)}{7 a^2 d}+\frac{\sin ^5(c+d x)}{a^2 d}-\frac{4 \sin ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x)}{a^2 d}+\frac{2 i \cos ^7(c+d x)}{7 a^2 d}",1,"(((2*I)/7)*Cos[c + d*x]^7)/(a^2*d) + Sin[c + d*x]/(a^2*d) - (4*Sin[c + d*x]^3)/(3*a^2*d) + Sin[c + d*x]^5/(a^2*d) - (2*Sin[c + d*x]^7)/(7*a^2*d)","A",10,7,31,0.2258,1,"{3092, 3090, 2633, 2565, 30, 2564, 270}"
164,1,101,0,0.1005256,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{1}{16 a^2 d (-\cot (c+d x)+i)}+\frac{11}{16 a^2 d (\cot (c+d x)+i)}-\frac{3 i}{8 a^2 d (\cot (c+d x)+i)^2}-\frac{1}{12 a^2 d (\cot (c+d x)+i)^3}+\frac{x}{4 a^2}","-\frac{1}{16 a^2 d (-\cot (c+d x)+i)}+\frac{11}{16 a^2 d (\cot (c+d x)+i)}-\frac{3 i}{8 a^2 d (\cot (c+d x)+i)^2}-\frac{1}{12 a^2 d (\cot (c+d x)+i)^3}+\frac{x}{4 a^2}",1,"x/(4*a^2) - 1/(16*a^2*d*(I - Cot[c + d*x])) - 1/(12*a^2*d*(I + Cot[c + d*x])^3) - ((3*I)/8)/(a^2*d*(I + Cot[c + d*x])^2) + 11/(16*a^2*d*(I + Cot[c + d*x]))","A",5,4,31,0.1290,1,"{3088, 848, 88, 203}"
165,1,68,0,0.1758876,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{2 \sin ^5(c+d x)}{5 a^2 d}-\frac{\sin ^3(c+d x)}{a^2 d}+\frac{\sin (c+d x)}{a^2 d}+\frac{2 i \cos ^5(c+d x)}{5 a^2 d}","\frac{2 \sin ^5(c+d x)}{5 a^2 d}-\frac{\sin ^3(c+d x)}{a^2 d}+\frac{\sin (c+d x)}{a^2 d}+\frac{2 i \cos ^5(c+d x)}{5 a^2 d}",1,"(((2*I)/5)*Cos[c + d*x]^5)/(a^2*d) + Sin[c + d*x]/(a^2*d) - Sin[c + d*x]^3/(a^2*d) + (2*Sin[c + d*x]^5)/(5*a^2*d)","A",10,7,31,0.2258,1,"{3092, 3090, 2633, 2565, 30, 2564, 14}"
166,1,89,0,0.0820806,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{i \cos (c+d x)}{4 d \left(a^2 \cos (c+d x)+i a^2 \sin (c+d x)\right)}+\frac{x}{4 a^2}+\frac{i \cos ^2(c+d x)}{4 d (a \cos (c+d x)+i a \sin (c+d x))^2}","\frac{i \cos (c+d x)}{4 d \left(a^2 \cos (c+d x)+i a^2 \sin (c+d x)\right)}+\frac{x}{4 a^2}+\frac{i \cos ^2(c+d x)}{4 d (a \cos (c+d x)+i a \sin (c+d x))^2}",1,"x/(4*a^2) + ((I/4)*Cos[c + d*x]^2)/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2) + ((I/4)*Cos[c + d*x])/(d*(a^2*Cos[c + d*x] + I*a^2*Sin[c + d*x]))","A",3,2,31,0.06452,1,"{3082, 8}"
167,1,52,0,0.1137706,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{2 \sin ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x)}{a^2 d}+\frac{2 i \cos ^3(c+d x)}{3 a^2 d}","-\frac{2 \sin ^3(c+d x)}{3 a^2 d}+\frac{\sin (c+d x)}{a^2 d}+\frac{2 i \cos ^3(c+d x)}{3 a^2 d}",1,"(((2*I)/3)*Cos[c + d*x]^3)/(a^2*d) + Sin[c + d*x]/(a^2*d) - (2*Sin[c + d*x]^3)/(3*a^2*d)","A",9,6,29,0.2069,1,"{3092, 3090, 2633, 2565, 30, 2564}"
168,1,31,0,0.0155068,"\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}"
169,1,46,0,0.1145899,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{2 \sin (c+d x)}{a^2 d}+\frac{2 i \cos (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}","\frac{2 \sin (c+d x)}{a^2 d}+\frac{2 i \cos (c+d x)}{a^2 d}-\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}",1,"-(ArcTanh[Sin[c + d*x]]/(a^2*d)) + ((2*I)*Cos[c + d*x])/(a^2*d) + (2*Sin[c + d*x])/(a^2*d)","A",8,7,29,0.2414,1,"{3092, 3090, 2637, 2638, 2592, 321, 206}"
170,1,55,0,0.0706444,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{\tan (c+d x)}{a^2 d}+\frac{2 i \log (\sin (c+d x))}{a^2 d}-\frac{2 i \log (\tan (c+d x))}{a^2 d}+\frac{2 x}{a^2}","-\frac{\tan (c+d x)}{a^2 d}+\frac{2 i \log (\sin (c+d x))}{a^2 d}-\frac{2 i \log (\tan (c+d x))}{a^2 d}+\frac{2 x}{a^2}",1,"(2*x)/a^2 + ((2*I)*Log[Sin[c + d*x]])/(a^2*d) - ((2*I)*Log[Tan[c + d*x]])/(a^2*d) - Tan[c + d*x]/(a^2*d)","A",4,3,31,0.09677,1,"{3088, 848, 77}"
171,1,56,0,0.1456357,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{2 i \sec (c+d x)}{a^2 d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{\tan (c+d x) \sec (c+d x)}{2 a^2 d}","-\frac{2 i \sec (c+d x)}{a^2 d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{\tan (c+d x) \sec (c+d x)}{2 a^2 d}",1,"(3*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - ((2*I)*Sec[c + d*x])/(a^2*d) - (Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d)","A",8,6,31,0.1935,1,"{3092, 3090, 3770, 2606, 8, 2611}"
172,1,34,0,0.0636908,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{i \tan ^3(c+d x) (-\cot (c+d x)+i)^3}{3 a^2 d}","-\frac{i \tan ^3(c+d x) (-\cot (c+d x)+i)^3}{3 a^2 d}",1,"((-I/3)*(I - Cot[c + d*x])^3*Tan[c + d*x]^3)/(a^2*d)","A",3,3,31,0.09677,1,"{3088, 848, 37}"
173,1,84,0,0.192213,"\int \frac{\sec ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{2 i \sec ^3(c+d x)}{3 a^2 d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a^2 d}+\frac{5 \tan (c+d x) \sec (c+d x)}{8 a^2 d}","-\frac{2 i \sec ^3(c+d x)}{3 a^2 d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{8 a^2 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{4 a^2 d}+\frac{5 \tan (c+d x) \sec (c+d x)}{8 a^2 d}",1,"(5*ArcTanh[Sin[c + d*x]])/(8*a^2*d) - (((2*I)/3)*Sec[c + d*x]^3)/(a^2*d) + (5*Sec[c + d*x]*Tan[c + d*x])/(8*a^2*d) - (Sec[c + d*x]^3*Tan[c + d*x])/(4*a^2*d)","A",10,7,31,0.2258,1,"{3092, 3090, 3768, 3770, 2606, 30, 2611}"
174,1,70,0,0.0784553,"\int \frac{\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{\tan ^5(c+d x)}{5 a^2 d}-\frac{i \tan ^4(c+d x)}{2 a^2 d}-\frac{i \tan ^2(c+d x)}{a^2 d}+\frac{\tan (c+d x)}{a^2 d}","-\frac{\tan ^5(c+d x)}{5 a^2 d}-\frac{i \tan ^4(c+d x)}{2 a^2 d}-\frac{i \tan ^2(c+d x)}{a^2 d}+\frac{\tan (c+d x)}{a^2 d}",1,"Tan[c + d*x]/(a^2*d) - (I*Tan[c + d*x]^2)/(a^2*d) - ((I/2)*Tan[c + d*x]^4)/(a^2*d) - Tan[c + d*x]^5/(5*a^2*d)","A",4,3,31,0.09677,1,"{3088, 848, 75}"
175,1,125,0,0.111441,"\int \frac{\cos ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","-\frac{1}{32 a^3 d (-\cot (c+d x)+i)}+\frac{13}{16 a^3 d (\cot (c+d x)+i)}-\frac{23 i}{32 a^3 d (\cot (c+d x)+i)^2}-\frac{1}{3 a^3 d (\cot (c+d x)+i)^3}+\frac{i}{16 a^3 d (\cot (c+d x)+i)^4}+\frac{5 x}{32 a^3}","-\frac{1}{32 a^3 d (-\cot (c+d x)+i)}+\frac{13}{16 a^3 d (\cot (c+d x)+i)}-\frac{23 i}{32 a^3 d (\cot (c+d x)+i)^2}-\frac{1}{3 a^3 d (\cot (c+d x)+i)^3}+\frac{i}{16 a^3 d (\cot (c+d x)+i)^4}+\frac{5 x}{32 a^3}",1,"(5*x)/(32*a^3) - 1/(32*a^3*d*(I - Cot[c + d*x])) + (I/16)/(a^3*d*(I + Cot[c + d*x])^4) - 1/(3*a^3*d*(I + Cot[c + d*x])^3) - ((23*I)/32)/(a^3*d*(I + Cot[c + d*x])^2) + 13/(16*a^3*d*(I + Cot[c + d*x]))","A",5,4,31,0.1290,1,"{3088, 848, 88, 203}"
176,1,106,0,0.2344476,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","-\frac{4 \sin ^7(c+d x)}{7 a^3 d}+\frac{9 \sin ^5(c+d x)}{5 a^3 d}-\frac{2 \sin ^3(c+d x)}{a^3 d}+\frac{\sin (c+d x)}{a^3 d}+\frac{4 i \cos ^7(c+d x)}{7 a^3 d}-\frac{i \cos ^5(c+d x)}{5 a^3 d}","-\frac{4 \sin ^7(c+d x)}{7 a^3 d}+\frac{9 \sin ^5(c+d x)}{5 a^3 d}-\frac{2 \sin ^3(c+d x)}{a^3 d}+\frac{\sin (c+d x)}{a^3 d}+\frac{4 i \cos ^7(c+d x)}{7 a^3 d}-\frac{i \cos ^5(c+d x)}{5 a^3 d}",1,"((-I/5)*Cos[c + d*x]^5)/(a^3*d) + (((4*I)/7)*Cos[c + d*x]^7)/(a^3*d) + Sin[c + d*x]/(a^3*d) - (2*Sin[c + d*x]^3)/(a^3*d) + (9*Sin[c + d*x]^5)/(5*a^3*d) - (4*Sin[c + d*x]^7)/(7*a^3*d)","A",13,8,31,0.2581,1,"{3092, 3090, 2633, 2565, 30, 2564, 270, 14}"
177,1,131,0,0.140537,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \cos (c+d x)}{8 d \left(a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right)}+\frac{x}{8 a^3}+\frac{i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac{i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2}","\frac{i \cos (c+d x)}{8 d \left(a^3 \cos (c+d x)+i a^3 \sin (c+d x)\right)}+\frac{x}{8 a^3}+\frac{i \cos ^3(c+d x)}{6 d (a \cos (c+d x)+i a \sin (c+d x))^3}+\frac{i \cos ^2(c+d x)}{8 a d (a \cos (c+d x)+i a \sin (c+d x))^2}",1,"x/(8*a^3) + ((I/6)*Cos[c + d*x]^3)/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3) + ((I/8)*Cos[c + d*x]^2)/(a*d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2) + ((I/8)*Cos[c + d*x])/(d*(a^3*Cos[c + d*x] + I*a^3*Sin[c + d*x]))","A",4,2,31,0.06452,1,"{3082, 8}"
178,1,90,0,0.2205014,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{4 \sin ^5(c+d x)}{5 a^3 d}-\frac{5 \sin ^3(c+d x)}{3 a^3 d}+\frac{\sin (c+d x)}{a^3 d}+\frac{4 i \cos ^5(c+d x)}{5 a^3 d}-\frac{i \cos ^3(c+d x)}{3 a^3 d}","\frac{4 \sin ^5(c+d x)}{5 a^3 d}-\frac{5 \sin ^3(c+d x)}{3 a^3 d}+\frac{\sin (c+d x)}{a^3 d}+\frac{4 i \cos ^5(c+d x)}{5 a^3 d}-\frac{i \cos ^3(c+d x)}{3 a^3 d}",1,"((-I/3)*Cos[c + d*x]^3)/(a^3*d) + (((4*I)/5)*Cos[c + d*x]^5)/(a^3*d) + Sin[c + d*x]/(a^3*d) - (5*Sin[c + d*x]^3)/(3*a^3*d) + (4*Sin[c + d*x]^5)/(5*a^3*d)","A",13,7,31,0.2258,1,"{3092, 3090, 2633, 2565, 30, 2564, 14}"
179,1,32,0,0.0318248,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \cot ^2(c+d x)}{2 a^3 d (\cot (c+d x)+i)^2}","\frac{i \cot ^2(c+d x)}{2 a^3 d (\cot (c+d x)+i)^2}",1,"((I/2)*Cot[c + d*x]^2)/(a^3*d*(I + Cot[c + d*x])^2)","A",2,2,29,0.06897,1,"{3088, 37}"
180,1,31,0,0.0164085,"\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}"
181,1,61,0,0.0643177,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{2}{a^3 d (\cot (c+d x)+i)}-\frac{i \log (\sin (c+d x))}{a^3 d}+\frac{i \log (\tan (c+d x))}{a^3 d}-\frac{x}{a^3}","\frac{2}{a^3 d (\cot (c+d x)+i)}-\frac{i \log (\sin (c+d x))}{a^3 d}+\frac{i \log (\tan (c+d x))}{a^3 d}-\frac{x}{a^3}",1,"-(x/a^3) + 2/(a^3*d*(I + Cot[c + d*x])) - (I*Log[Sin[c + d*x]])/(a^3*d) + (I*Log[Tan[c + d*x]])/(a^3*d)","A",4,3,29,0.1034,1,"{3088, 848, 77}"
182,1,62,0,0.1619563,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{4 \sin (c+d x)}{a^3 d}+\frac{4 i \cos (c+d x)}{a^3 d}+\frac{i \sec (c+d x)}{a^3 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{a^3 d}","\frac{4 \sin (c+d x)}{a^3 d}+\frac{4 i \cos (c+d x)}{a^3 d}+\frac{i \sec (c+d x)}{a^3 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{a^3 d}",1,"(-3*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((4*I)*Cos[c + d*x])/(a^3*d) + (I*Sec[c + d*x])/(a^3*d) + (4*Sin[c + d*x])/(a^3*d)","A",11,9,31,0.2903,1,"{3092, 3090, 2637, 2638, 2592, 321, 206, 2590, 14}"
183,1,75,0,0.0805796,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \tan ^2(c+d x)}{2 a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}+\frac{4 i \log (\sin (c+d x))}{a^3 d}-\frac{4 i \log (\tan (c+d x))}{a^3 d}+\frac{4 x}{a^3}","\frac{i \tan ^2(c+d x)}{2 a^3 d}-\frac{3 \tan (c+d x)}{a^3 d}+\frac{4 i \log (\sin (c+d x))}{a^3 d}-\frac{4 i \log (\tan (c+d x))}{a^3 d}+\frac{4 x}{a^3}",1,"(4*x)/a^3 + ((4*I)*Log[Sin[c + d*x]])/(a^3*d) - ((4*I)*Log[Tan[c + d*x]])/(a^3*d) - (3*Tan[c + d*x])/(a^3*d) + ((I/2)*Tan[c + d*x]^2)/(a^3*d)","A",4,3,31,0.09677,1,"{3088, 848, 88}"
184,1,76,0,0.1818944,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \sec ^3(c+d x)}{3 a^3 d}-\frac{4 i \sec (c+d x)}{a^3 d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{3 \tan (c+d x) \sec (c+d x)}{2 a^3 d}","\frac{i \sec ^3(c+d x)}{3 a^3 d}-\frac{4 i \sec (c+d x)}{a^3 d}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{3 \tan (c+d x) \sec (c+d x)}{2 a^3 d}",1,"(5*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - ((4*I)*Sec[c + d*x])/(a^3*d) + ((I/3)*Sec[c + d*x]^3)/(a^3*d) - (3*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d)","A",10,6,31,0.1935,1,"{3092, 3090, 3770, 2606, 8, 2611}"
185,1,34,0,0.062903,"\int \frac{\sec ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \tan ^4(c+d x) (-\cot (c+d x)+i)^4}{4 a^3 d}","\frac{i \tan ^4(c+d x) (-\cot (c+d x)+i)^4}{4 a^3 d}",1,"((I/4)*(I - Cot[c + d*x])^4*Tan[c + d*x]^4)/(a^3*d)","A",3,3,31,0.09677,1,"{3088, 848, 37}"
186,1,104,0,0.2299471,"\int \frac{\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \sec ^5(c+d x)}{5 a^3 d}-\frac{4 i \sec ^3(c+d x)}{3 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{3 \tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{8 a^3 d}","\frac{i \sec ^5(c+d x)}{5 a^3 d}-\frac{4 i \sec ^3(c+d x)}{3 a^3 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{3 \tan (c+d x) \sec ^3(c+d x)}{4 a^3 d}+\frac{7 \tan (c+d x) \sec (c+d x)}{8 a^3 d}",1,"(7*ArcTanh[Sin[c + d*x]])/(8*a^3*d) - (((4*I)/3)*Sec[c + d*x]^3)/(a^3*d) + ((I/5)*Sec[c + d*x]^5)/(a^3*d) + (7*Sec[c + d*x]*Tan[c + d*x])/(8*a^3*d) - (3*Sec[c + d*x]^3*Tan[c + d*x])/(4*a^3*d)","A",13,8,31,0.2581,1,"{3092, 3090, 3768, 3770, 2606, 30, 2611, 14}"
187,1,66,0,0.0611622,"\int \cos ^{-n}(c+d x) (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/Cos[c + d*x]^n,x]","-\frac{i \cos ^{-n}(c+d x) \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n}","-\frac{i \cos ^{-n}(c+d x) \, _2F_1\left(1,n;n+1;\frac{1}{2} (i \tan (c+d x)+1)\right) (a \cos (c+d x)+i a \sin (c+d x))^n}{2 d n}",1,"((-I/2)*Hypergeometric2F1[1, n, 1 + n, (1 + I*Tan[c + d*x])/2]*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n)/(d*n*Cos[c + d*x]^n)","A",1,1,33,0.03030,1,"{3084}"
188,1,5,0,0.0231376,"\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}"
189,1,10,0,0.0678942,"\int \frac{\sin (x)}{\sec (x)+\tan (x)} \, dx","Int[Sin[x]/(Sec[x] + Tan[x]),x]","\sin (x)-\log (\sin (x)+1)","\sin (x)-\log (\sin (x)+1)",1,"-Log[1 + Sin[x]] + Sin[x]","A",4,3,10,0.3000,1,"{4391, 2833, 43}"
190,1,4,0,0.0601193,"\int \frac{\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",3,3,10,0.3000,1,"{4391, 2682, 8}"
191,1,11,0,0.0528004,"\int \frac{\tan (x)}{\sec (x)+\tan (x)} \, dx","Int[Tan[x]/(Sec[x] + Tan[x]),x]","x+\frac{\cos (x)}{\sin (x)+1}","x+\frac{\cos (x)}{\sin (x)+1}",1,"x + Cos[x]/(1 + Sin[x])","A",3,3,10,0.3000,1,"{4391, 2735, 2648}"
192,1,9,0,0.0784214,"\int \frac{\cot (x)}{\sec (x)+\tan (x)} \, dx","Int[Cot[x]/(Sec[x] + Tan[x]),x]","-x-\tanh ^{-1}(\cos (x))","-x-\tanh ^{-1}(\cos (x))",1,"-x - ArcTanh[Cos[x]]","A",4,4,10,0.4000,1,"{4391, 2839, 3770, 8}"
193,1,10,0,0.0244206,"\int \frac{\sec (x)}{\sec (x)+\tan (x)} \, dx","Int[Sec[x]/(Sec[x] + Tan[x]),x]","-\frac{\cos (x)}{\sin (x)+1}","-\frac{\cos (x)}{\sin (x)+1}",1,"-(Cos[x]/(1 + Sin[x]))","A",2,2,10,0.2000,1,"{3165, 2648}"
194,1,11,0,0.0546386,"\int \frac{\csc (x)}{\sec (x)+\tan (x)} \, dx","Int[Csc[x]/(Sec[x] + Tan[x]),x]","\log (\sin (x))-\log (\sin (x)+1)","\log (\sin (x))-\log (\sin (x)+1)",1,"Log[Sin[x]] - Log[1 + Sin[x]]","A",5,5,10,0.5000,1,"{4391, 2707, 36, 29, 31}"
195,1,9,0,0.0270939,"\int \frac{1}{\sec (x)-\tan (x)} \, dx","Int[(Sec[x] - Tan[x])^(-1),x]","-\log (1-\sin (x))","-\log (1-\sin (x))",1,"-Log[1 - Sin[x]]","A",3,3,9,0.3333,1,"{3159, 2667, 31}"
196,1,14,0,0.0819025,"\int \frac{\sin (x)}{\sec (x)-\tan (x)} \, dx","Int[Sin[x]/(Sec[x] - Tan[x]),x]","-\sin (x)-\log (1-\sin (x))","-\sin (x)-\log (1-\sin (x))",1,"-Log[1 - Sin[x]] - Sin[x]","A",4,3,12,0.2500,1,"{4391, 2833, 43}"
197,1,6,0,0.0600477,"\int \frac{\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",3,3,12,0.2500,1,"{4391, 2682, 8}"
198,1,15,0,0.0604004,"\int \frac{\tan (x)}{\sec (x)-\tan (x)} \, dx","Int[Tan[x]/(Sec[x] - Tan[x]),x]","\frac{\cos (x)}{1-\sin (x)}-x","\frac{\cos (x)}{1-\sin (x)}-x",1,"-x + Cos[x]/(1 - Sin[x])","A",3,3,12,0.2500,1,"{4391, 2735, 2648}"
199,1,7,0,0.0888162,"\int \frac{\cot (x)}{\sec (x)-\tan (x)} \, dx","Int[Cot[x]/(Sec[x] - Tan[x]),x]","x-\tanh ^{-1}(\cos (x))","x-\tanh ^{-1}(\cos (x))",1,"x - ArcTanh[Cos[x]]","A",4,4,12,0.3333,1,"{4391, 2839, 3770, 8}"
200,1,11,0,0.0287537,"\int \frac{\sec (x)}{\sec (x)-\tan (x)} \, dx","Int[Sec[x]/(Sec[x] - Tan[x]),x]","\frac{\cos (x)}{1-\sin (x)}","\frac{\cos (x)}{1-\sin (x)}",1,"Cos[x]/(1 - Sin[x])","A",2,2,12,0.1667,1,"{3165, 2648}"
201,1,13,0,0.0620936,"\int \frac{\csc (x)}{\sec (x)-\tan (x)} \, dx","Int[Csc[x]/(Sec[x] - Tan[x]),x]","\log (\sin (x))-\log (1-\sin (x))","\log (\sin (x))-\log (1-\sin (x))",1,"-Log[1 - Sin[x]] + Log[Sin[x]]","A",5,5,12,0.4167,1,"{4391, 2707, 36, 29, 31}"
202,1,23,0,0.0941638,"\int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx","Int[Csc[c + d*x]*(Cot[c + d*x] + Csc[c + d*x]),x]","-\frac{\cot (c+d x)}{d}-\frac{\csc (c+d x)}{d}","-\frac{\cot (c+d x)}{d}-\frac{\csc (c+d x)}{d}",1,"-(Cot[c + d*x]/d) - Csc[c + d*x]/d","A",4,4,20,0.2000,1,"{4397, 2669, 3767, 8}"
203,1,6,0,0.070243,"\int \frac{\sin (x)}{\cot (x)+\csc (x)} \, dx","Int[Sin[x]/(Cot[x] + Csc[x]),x]","x-\sin (x)","x-\sin (x)",1,"x - Sin[x]","A",3,3,10,0.3000,1,"{4392, 2682, 8}"
204,1,10,0,0.0673322,"\int \frac{\cos (x)}{\cot (x)+\csc (x)} \, dx","Int[Cos[x]/(Cot[x] + Csc[x]),x]","\log (\cos (x)+1)-\cos (x)","\log (\cos (x)+1)-\cos (x)",1,"-Cos[x] + Log[1 + Cos[x]]","A",4,3,10,0.3000,1,"{4392, 2833, 43}"
205,1,7,0,0.0845227,"\int \frac{\tan (x)}{\cot (x)+\csc (x)} \, dx","Int[Tan[x]/(Cot[x] + Csc[x]),x]","\tanh ^{-1}(\sin (x))-x","\tanh ^{-1}(\sin (x))-x",1,"-x + ArcTanh[Sin[x]]","A",4,4,10,0.4000,1,"{4392, 2839, 3770, 8}"
206,1,12,0,0.0535888,"\int \frac{\cot (x)}{\cot (x)+\csc (x)} \, dx","Int[Cot[x]/(Cot[x] + Csc[x]),x]","x-\frac{\sin (x)}{\cos (x)+1}","x-\frac{\sin (x)}{\cos (x)+1}",1,"x - Sin[x]/(1 + Cos[x])","A",3,3,10,0.3000,1,"{4392, 2735, 2648}"
207,1,11,0,0.0595973,"\int \frac{\sec (x)}{\cot (x)+\csc (x)} \, dx","Int[Sec[x]/(Cot[x] + Csc[x]),x]","\log (\cos (x)+1)-\log (\cos (x))","\log (\cos (x)+1)-\log (\cos (x))",1,"-Log[Cos[x]] + Log[1 + Cos[x]]","A",5,5,10,0.5000,1,"{4392, 2707, 36, 29, 31}"
208,1,9,0,0.0255203,"\int \frac{\csc (x)}{\cot (x)+\csc (x)} \, dx","Int[Csc[x]/(Cot[x] + Csc[x]),x]","\frac{\sin (x)}{\cos (x)+1}","\frac{\sin (x)}{\cos (x)+1}",1,"Sin[x]/(1 + Cos[x])","A",2,2,10,0.2000,1,"{3166, 2648}"
209,1,4,0,0.0697868,"\int \frac{\sin (x)}{-\cot (x)+\csc (x)} \, dx","Int[Sin[x]/(-Cot[x] + Csc[x]),x]","x+\sin (x)","x+\sin (x)",1,"x + Sin[x]","A",3,3,12,0.2500,1,"{4392, 2682, 8}"
210,1,10,0,0.0684733,"\int \frac{\cos (x)}{-\cot (x)+\csc (x)} \, dx","Int[Cos[x]/(-Cot[x] + Csc[x]),x]","\cos (x)+\log (1-\cos (x))","\cos (x)+\log (1-\cos (x))",1,"Cos[x] + Log[1 - Cos[x]]","A",4,3,12,0.2500,1,"{4392, 2833, 43}"
211,1,5,0,0.0944992,"\int \frac{\tan (x)}{-\cot (x)+\csc (x)} \, dx","Int[Tan[x]/(-Cot[x] + Csc[x]),x]","x+\tanh ^{-1}(\sin (x))","x+\tanh ^{-1}(\sin (x))",1,"x + ArcTanh[Sin[x]]","A",4,4,12,0.3333,1,"{4392, 2839, 3770, 8}"
212,1,16,0,0.0598725,"\int \frac{\cot (x)}{-\cot (x)+\csc (x)} \, dx","Int[Cot[x]/(-Cot[x] + Csc[x]),x]","-x-\frac{\sin (x)}{1-\cos (x)}","-x-\frac{\sin (x)}{1-\cos (x)}",1,"-x - Sin[x]/(1 - Cos[x])","A",3,3,12,0.2500,1,"{4392, 2735, 2648}"
213,1,13,0,0.0605882,"\int \frac{\sec (x)}{-\cot (x)+\csc (x)} \, dx","Int[Sec[x]/(-Cot[x] + Csc[x]),x]","\log (1-\cos (x))-\log (\cos (x))","\log (1-\cos (x))-\log (\cos (x))",1,"Log[1 - Cos[x]] - Log[Cos[x]]","A",5,5,12,0.4167,1,"{4392, 2707, 36, 29, 31}"
214,1,12,0,0.029823,"\int \frac{\csc (x)}{-\cot (x)+\csc (x)} \, dx","Int[Csc[x]/(-Cot[x] + Csc[x]),x]","-\frac{\sin (x)}{1-\cos (x)}","-\frac{\sin (x)}{1-\cos (x)}",1,"-(Sin[x]/(1 - Cos[x]))","A",2,2,12,0.1667,1,"{3166, 2648}"
215,1,23,0,0.0386754,"\int \frac{1}{\csc (c+d x)+\sin (c+d x)} \, dx","Int[(Csc[c + d*x] + Sin[c + d*x])^(-1),x]","-\frac{\tanh ^{-1}\left(\frac{\cos (c+d x)}{\sqrt{2}}\right)}{\sqrt{2} d}","-\frac{\tanh ^{-1}\left(\frac{\cos (c+d x)}{\sqrt{2}}\right)}{\sqrt{2} d}",1,"-(ArcTanh[Cos[c + d*x]/Sqrt[2]]/(Sqrt[2]*d))","A",3,3,15,0.2000,1,"{4397, 3186, 206}"
216,1,51,0,0.1721219,"\int \frac{\sin (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Int[Sin[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","-\frac{\tan ^{-1}\left(\frac{\sin (c+d x) \cos (c+d x)}{\sin ^2(c+d x)+\sqrt{2}+1}\right)}{\sqrt{2} d}-\frac{x}{\sqrt{2}}+x","-\frac{\tan ^{-1}\left(\frac{\sin (c+d x) \cos (c+d x)}{\sin ^2(c+d x)+\sqrt{2}+1}\right)}{\sqrt{2} d}-\frac{x}{\sqrt{2}}+x",1,"x - x/Sqrt[2] - ArcTan[(Cos[c + d*x]*Sin[c + d*x])/(1 + Sqrt[2] + Sin[c + d*x]^2)]/(Sqrt[2]*d)","A",4,2,22,0.09091,1,"{1130, 203}"
217,1,18,0,0.0311779,"\int \frac{\cos (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Int[Cos[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","\frac{\log \left(\sin ^2(c+d x)+1\right)}{2 d}","\frac{\log \left(\sin ^2(c+d x)+1\right)}{2 d}",1,"Log[1 + Sin[c + d*x]^2]/(2*d)","A",2,2,22,0.09091,1,"{4334, 260}"
218,1,29,0,0.0375861,"\int \frac{\tan (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Int[Tan[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{\tan ^{-1}(\sin (c+d x))}{2 d}","\frac{\tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{\tan ^{-1}(\sin (c+d x))}{2 d}",1,"-ArcTan[Sin[c + d*x]]/(2*d) + ArcTanh[Sin[c + d*x]]/(2*d)","A",4,3,22,0.1364,1,"{298, 203, 206}"
219,1,11,0,0.0253427,"\int \frac{\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Int[Cot[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","\frac{\tan ^{-1}(\sin (c+d x))}{d}","\frac{\tan ^{-1}(\sin (c+d x))}{d}",1,"ArcTan[Sin[c + d*x]]/d","A",2,2,22,0.09091,1,"{4338, 203}"
220,1,16,0,0.0317371,"\int \frac{\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Int[Sec[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","\frac{\tanh ^{-1}\left(\sin ^2(c+d x)\right)}{2 d}","\frac{\tanh ^{-1}\left(\sin ^2(c+d x)\right)}{2 d}",1,"ArcTanh[Sin[c + d*x]^2]/(2*d)","A",3,2,22,0.09091,1,"{275, 206}"
221,1,48,0,0.10813,"\int \frac{\csc (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Int[Csc[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","\frac{\tan ^{-1}\left(\frac{\sin (c+d x) \cos (c+d x)}{\sin ^2(c+d x)+\sqrt{2}+1}\right)}{\sqrt{2} d}+\frac{x}{\sqrt{2}}","\frac{\tan ^{-1}\left(\frac{\sin (c+d x) \cos (c+d x)}{\sin ^2(c+d x)+\sqrt{2}+1}\right)}{\sqrt{2} d}+\frac{x}{\sqrt{2}}",1,"x/Sqrt[2] + ArcTan[(Cos[c + d*x]*Sin[c + d*x])/(1 + Sqrt[2] + Sin[c + d*x]^2)]/(Sqrt[2]*d)","A",2,1,22,0.04545,1,"{203}"
222,1,10,0,0.0250854,"\int \frac{1}{\csc (c+d x)-\sin (c+d x)} \, dx","Int[(Csc[c + d*x] - Sin[c + d*x])^(-1),x]","\frac{\sec (c+d x)}{d}","\frac{\sec (c+d x)}{d}",1,"Sec[c + d*x]/d","A",3,3,17,0.1765,1,"{4397, 2606, 8}"
223,1,14,0,0.1466817,"\int \frac{\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Int[Sin[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]","\frac{\tan (c+d x)}{d}-x","\frac{\tan (c+d x)}{d}-x",1,"-x + Tan[c + d*x]/d","A",3,2,24,0.08333,1,"{321, 203}"
224,1,12,0,0.0306763,"\int \frac{\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Int[Cos[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]","-\frac{\log (\cos (c+d x))}{d}","-\frac{\log (\cos (c+d x))}{d}",1,"-(Log[Cos[c + d*x]]/d)","A",2,2,24,0.08333,1,"{4334, 260}"
225,1,34,0,0.0391738,"\int \frac{\tan (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Int[Tan[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]","\frac{\tan (c+d x) \sec (c+d x)}{2 d}-\frac{\tanh ^{-1}(\sin (c+d x))}{2 d}","\frac{\tan (c+d x) \sec (c+d x)}{2 d}-\frac{\tanh ^{-1}(\sin (c+d x))}{2 d}",1,"-ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",3,2,24,0.08333,1,"{288, 206}"
226,1,11,0,0.0264199,"\int \frac{\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Int[Cot[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{d}","\frac{\tanh ^{-1}(\sin (c+d x))}{d}",1,"ArcTanh[Sin[c + d*x]]/d","A",2,2,24,0.08333,1,"{4338, 206}"
227,1,15,0,0.0317387,"\int \frac{\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Int[Sec[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]","\frac{\sec ^2(c+d x)}{2 d}","\frac{\sec ^2(c+d x)}{2 d}",1,"Sec[c + d*x]^2/(2*d)","A",2,1,24,0.04167,1,"{261}"
228,1,10,0,0.0862574,"\int \frac{\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Int[Csc[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]","\frac{\tan (c+d x)}{d}","\frac{\tan (c+d x)}{d}",1,"Tan[c + d*x]/d","A",2,1,24,0.04167,1,"{8}"
229,1,33,0,0.0619245,"\int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{a \cos ^4(c+d x)}{4 d}-\frac{b \cos ^3(c+d x)}{3 d}","-\frac{a \cos ^4(c+d x)}{4 d}-\frac{b \cos ^3(c+d x)}{3 d}",1,"-(b*Cos[c + d*x]^3)/(3*d) - (a*Cos[c + d*x]^4)/(4*d)","A",6,4,26,0.1538,1,"{4377, 12, 2565, 30}"
230,1,33,0,0.0538762,"\int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{b \sin ^2(c+d x)}{2 d}-\frac{a \cos ^3(c+d x)}{3 d}","\frac{b \sin ^2(c+d x)}{2 d}-\frac{a \cos ^3(c+d x)}{3 d}",1,"-(a*Cos[c + d*x]^3)/(3*d) + (b*Sin[c + d*x]^2)/(2*d)","A",6,5,26,0.1923,1,"{4377, 12, 2564, 30, 2565}"
231,1,29,0,0.0302323,"\int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Int[Cos[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{a \sin ^2(c+d x)}{2 d}-\frac{b \cos (c+d x)}{d}","-\frac{(a \cos (c+d x)+b)^2}{2 a d}",1,"-((b*Cos[c + d*x])/d) + (a*Sin[c + d*x]^2)/(2*d)","A",5,5,24,0.2083,1,"{4377, 12, 2638, 2564, 30}"
232,1,26,0,0.0130983,"\int (a \sin (c+d x)+b \tan (c+d x)) \, dx","Int[a*Sin[c + d*x] + b*Tan[c + d*x],x]","-\frac{a \cos (c+d x)}{d}-\frac{b \log (\cos (c+d x))}{d}","-\frac{a \cos (c+d x)}{d}-\frac{b \log (\cos (c+d x))}{d}",1,"-((a*Cos[c + d*x])/d) - (b*Log[Cos[c + d*x]])/d","A",3,2,17,0.1176,1,"{2638, 3475}"
233,1,25,0,0.0280599,"\int \sec (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Int[Sec[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","\frac{b \sec (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Log[Cos[c + d*x]])/d) + (b*Sec[c + d*x])/d","A",5,5,24,0.2083,1,"{4377, 12, 2606, 8, 3475}"
234,1,28,0,0.0501234,"\int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{a \sec (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d}","\frac{a \sec (c+d x)}{d}+\frac{b \sec ^2(c+d x)}{2 d}",1,"(a*Sec[c + d*x])/d + (b*Sec[c + d*x]^2)/(2*d)","A",6,5,26,0.1923,1,"{4377, 12, 2606, 30, 8}"
235,1,33,0,0.0569483,"\int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{a \sec ^2(c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d}","\frac{a \sec ^2(c+d x)}{2 d}+\frac{b \sec ^3(c+d x)}{3 d}",1,"(a*Sec[c + d*x]^2)/(2*d) + (b*Sec[c + d*x]^3)/(3*d)","A",6,4,26,0.1538,1,"{4377, 12, 2606, 30}"
236,1,106,0,0.3487891,"\int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{\left(4 a^2+b^2\right) \sin ^3(c+d x)}{30 d}+\frac{\sin ^3(c+d x) (a \cos (c+d x)+b)^2}{5 d}+\frac{b \sin ^3(c+d x) (a \cos (c+d x)+b)}{10 d}-\frac{a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a b x}{4}","\frac{\left(4 a^2+b^2\right) \sin ^3(c+d x)}{30 d}+\frac{\sin ^3(c+d x) (a \cos (c+d x)+b)^2}{5 d}+\frac{b \sin ^3(c+d x) (a \cos (c+d x)+b)}{10 d}-\frac{a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a b x}{4}",1,"(a*b*x)/4 - (a*b*Cos[c + d*x]*Sin[c + d*x])/(4*d) + ((4*a^2 + b^2)*Sin[c + d*x]^3)/(30*d) + (b*(b + a*Cos[c + d*x])*Sin[c + d*x]^3)/(10*d) + ((b + a*Cos[c + d*x])^2*Sin[c + d*x]^3)/(5*d)","A",6,5,28,0.1786,1,"{4397, 2862, 2669, 2635, 8}"
237,1,86,0,0.1895863,"\int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{\left(a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(a^2+4 b^2\right)+\frac{5 a b \sin ^3(c+d x)}{12 d}+\frac{a \sin ^3(c+d x) (a \cos (c+d x)+b)}{4 d}","-\frac{\left(a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(a^2+4 b^2\right)+\frac{5 a b \sin ^3(c+d x)}{12 d}+\frac{a \sin ^3(c+d x) (a \cos (c+d x)+b)}{4 d}",1,"((a^2 + 4*b^2)*x)/8 - ((a^2 + 4*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (5*a*b*Sin[c + d*x]^3)/(12*d) + (a*(b + a*Cos[c + d*x])*Sin[c + d*x]^3)/(4*d)","A",5,5,28,0.1786,1,"{4397, 2692, 2669, 2635, 8}"
238,1,87,0,0.3213835,"\int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Int[Cos[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{3 d}-\frac{a b \sin (c+d x) \cos (c+d x)}{3 d}-\frac{\sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}+a b x+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{3 d}-\frac{a b \sin (c+d x) \cos (c+d x)}{3 d}-\frac{\sin (c+d x) (a \cos (c+d x)+b)^2}{3 d}+a b x+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"a*b*x + (b^2*ArcTanh[Sin[c + d*x]])/d + ((a^2 - 2*b^2)*Sin[c + d*x])/(3*d) - (a*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) - ((b + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d)","A",7,7,26,0.2692,1,"{4397, 2889, 3050, 3033, 3023, 2735, 3770}"
239,1,77,0,0.1158768,"\int (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Int[(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a^2 x}{2}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}-b^2 x","-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a^2 x}{2}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}-b^2 x",1,"(a^2*x)/2 - b^2*x + (2*a*b*ArcTanh[Sin[c + d*x]])/d - (2*a*b*Sin[c + d*x])/d - (a^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b^2*Tan[c + d*x])/d","A",10,8,19,0.4211,1,"{4397, 2722, 2635, 8, 2592, 321, 206, 3473}"
240,1,90,0,0.4264198,"\int \sec (c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Int[Sec[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{3 a^2 \sin (c+d x)}{2 d}+\frac{a b \tan (c+d x)}{d}+\frac{\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^2}{2 d}-2 a b x","\frac{\left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{3 a^2 \sin (c+d x)}{2 d}+\frac{a b \tan (c+d x)}{d}+\frac{\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^2}{2 d}-2 a b x",1,"-2*a*b*x + ((2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (3*a^2*Sin[c + d*x])/(2*d) + (a*b*Tan[c + d*x])/d + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,7,26,0.2692,1,"{4397, 2889, 3048, 3031, 3023, 2735, 3770}"
241,1,99,0,0.4667966,"\int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Int[Sec[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \tan (c+d x)}{3 d}+a^2 (-x)-\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{\tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b)^2}{3 d}","\frac{\left(2 a^2-b^2\right) \tan (c+d x)}{3 d}+a^2 (-x)-\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{3 d}+\frac{\tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b)^2}{3 d}",1,"-(a^2*x) - (a*b*ArcTanh[Sin[c + d*x]])/d + ((2*a^2 - b^2)*Tan[c + d*x])/(3*d) + (a*b*Sec[c + d*x]*Tan[c + d*x])/(3*d) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,7,28,0.2500,1,"{4397, 2889, 3048, 3031, 3021, 2735, 3770}"
242,1,125,0,0.4396637,"\int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Int[Sec[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{\left(4 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(2 a^2-b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}-\frac{2 a b \tan (c+d x)}{3 d}+\frac{a b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b)^2}{4 d}","-\frac{\left(4 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(2 a^2-b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}-\frac{2 a b \tan (c+d x)}{3 d}+\frac{a b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b)^2}{4 d}",1,"-((4*a^2 + b^2)*ArcTanh[Sin[c + d*x]])/(8*d) - (2*a*b*Tan[c + d*x])/(3*d) + ((2*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*b*Sec[c + d*x]^2*Tan[c + d*x])/(6*d) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",9,9,28,0.3214,1,"{4397, 2889, 3048, 3031, 3021, 2748, 3767, 8, 3770}"
243,1,77,0,0.1887034,"\int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","-\frac{\left(a^2-b^2\right) (a \cos (c+d x)+b)^4}{4 a^3 d}+\frac{(a \cos (c+d x)+b)^6}{6 a^3 d}-\frac{2 b (a \cos (c+d x)+b)^5}{5 a^3 d}","-\frac{\left(a^2-b^2\right) (a \cos (c+d x)+b)^4}{4 a^3 d}+\frac{(a \cos (c+d x)+b)^6}{6 a^3 d}-\frac{2 b (a \cos (c+d x)+b)^5}{5 a^3 d}",1,"-((a^2 - b^2)*(b + a*Cos[c + d*x])^4)/(4*a^3*d) - (2*b*(b + a*Cos[c + d*x])^5)/(5*a^3*d) + (b + a*Cos[c + d*x])^6/(6*a^3*d)","A",4,3,28,0.1071,1,"{4397, 2668, 697}"
244,1,120,0,0.1942657,"\int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","-\frac{a \left(a^2-3 b^2\right) \cos ^3(c+d x)}{3 d}-\frac{b \left(3 a^2-b^2\right) \cos ^2(c+d x)}{2 d}+\frac{3 a^2 b \cos ^4(c+d x)}{4 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a b^2 \cos (c+d x)}{d}-\frac{b^3 \log (\cos (c+d x))}{d}","-\frac{a \left(a^2-3 b^2\right) \cos ^3(c+d x)}{3 d}-\frac{b \left(3 a^2-b^2\right) \cos ^2(c+d x)}{2 d}+\frac{3 a^2 b \cos ^4(c+d x)}{4 d}+\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a b^2 \cos (c+d x)}{d}-\frac{b^3 \log (\cos (c+d x))}{d}",1,"(-3*a*b^2*Cos[c + d*x])/d - (b*(3*a^2 - b^2)*Cos[c + d*x]^2)/(2*d) - (a*(a^2 - 3*b^2)*Cos[c + d*x]^3)/(3*d) + (3*a^2*b*Cos[c + d*x]^4)/(4*d) + (a^3*Cos[c + d*x]^5)/(5*d) - (b^3*Log[Cos[c + d*x]])/d","A",5,4,28,0.1429,1,"{4397, 2837, 12, 894}"
245,1,112,0,0.1772737,"\int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Int[Cos[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","-\frac{a \left(a^2-3 b^2\right) \cos ^2(c+d x)}{2 d}-\frac{b \left(3 a^2-b^2\right) \cos (c+d x)}{d}+\frac{a^2 b \cos ^3(c+d x)}{d}+\frac{a^3 \cos ^4(c+d x)}{4 d}-\frac{3 a b^2 \log (\cos (c+d x))}{d}+\frac{b^3 \sec (c+d x)}{d}","-\frac{a \left(a^2-3 b^2\right) \cos ^2(c+d x)}{2 d}-\frac{b \left(3 a^2-b^2\right) \cos (c+d x)}{d}+\frac{a^2 b \cos ^3(c+d x)}{d}+\frac{a^3 \cos ^4(c+d x)}{4 d}-\frac{3 a b^2 \log (\cos (c+d x))}{d}+\frac{b^3 \sec (c+d x)}{d}",1,"-((b*(3*a^2 - b^2)*Cos[c + d*x])/d) - (a*(a^2 - 3*b^2)*Cos[c + d*x]^2)/(2*d) + (a^2*b*Cos[c + d*x]^3)/d + (a^3*Cos[c + d*x]^4)/(4*d) - (3*a*b^2*Log[Cos[c + d*x]])/d + (b^3*Sec[c + d*x])/d","A",5,4,26,0.1538,1,"{4397, 2837, 12, 894}"
246,1,116,0,0.1039422,"\int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Int[(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","-\frac{a \left(a^2-3 b^2\right) \cos (c+d x)}{d}-\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a^2 b \cos ^2(c+d x)}{2 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}","-\frac{a \left(a^2-3 b^2\right) \cos (c+d x)}{d}-\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a^2 b \cos ^2(c+d x)}{2 d}+\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"-((a*(a^2 - 3*b^2)*Cos[c + d*x])/d) + (3*a^2*b*Cos[c + d*x]^2)/(2*d) + (a^3*Cos[c + d*x]^3)/(3*d) - (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Sec[c + d*x])/d + (b^3*Sec[c + d*x]^2)/(2*d)","A",4,3,19,0.1579,1,"{4397, 2721, 894}"
247,1,115,0,0.2415897,"\int \sec (c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Int[Sec[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{b \left(3 a^2-b^2\right) \sec (c+d x)}{d}-\frac{a \left(a^2-3 b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a^2 b \cos (c+d x)}{d}+\frac{a^3 \cos ^2(c+d x)}{2 d}+\frac{3 a b^2 \sec ^2(c+d x)}{2 d}+\frac{b^3 \sec ^3(c+d x)}{3 d}","\frac{b \left(3 a^2-b^2\right) \sec (c+d x)}{d}-\frac{a \left(a^2-3 b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a^2 b \cos (c+d x)}{d}+\frac{a^3 \cos ^2(c+d x)}{2 d}+\frac{3 a b^2 \sec ^2(c+d x)}{2 d}+\frac{b^3 \sec ^3(c+d x)}{3 d}",1,"(3*a^2*b*Cos[c + d*x])/d + (a^3*Cos[c + d*x]^2)/(2*d) - (a*(a^2 - 3*b^2)*Log[Cos[c + d*x]])/d + (b*(3*a^2 - b^2)*Sec[c + d*x])/d + (3*a*b^2*Sec[c + d*x]^2)/(2*d) + (b^3*Sec[c + d*x]^3)/(3*d)","A",5,4,26,0.1538,1,"{4397, 2837, 12, 894}"
248,1,111,0,0.2540235,"\int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Int[Sec[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{b \left(3 a^2-b^2\right) \sec ^2(c+d x)}{2 d}+\frac{a \left(a^2-3 b^2\right) \sec (c+d x)}{d}+\frac{3 a^2 b \log (\cos (c+d x))}{d}+\frac{a^3 \cos (c+d x)}{d}+\frac{a b^2 \sec ^3(c+d x)}{d}+\frac{b^3 \sec ^4(c+d x)}{4 d}","\frac{b \left(3 a^2-b^2\right) \sec ^2(c+d x)}{2 d}+\frac{a \left(a^2-3 b^2\right) \sec (c+d x)}{d}+\frac{3 a^2 b \log (\cos (c+d x))}{d}+\frac{a^3 \cos (c+d x)}{d}+\frac{a b^2 \sec ^3(c+d x)}{d}+\frac{b^3 \sec ^4(c+d x)}{4 d}",1,"(a^3*Cos[c + d*x])/d + (3*a^2*b*Log[Cos[c + d*x]])/d + (a*(a^2 - 3*b^2)*Sec[c + d*x])/d + (b*(3*a^2 - b^2)*Sec[c + d*x]^2)/(2*d) + (a*b^2*Sec[c + d*x]^3)/d + (b^3*Sec[c + d*x]^4)/(4*d)","A",5,4,28,0.1429,1,"{4397, 2837, 12, 894}"
249,1,119,0,0.2199062,"\int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Int[Sec[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{b \left(3 a^2-b^2\right) \sec ^3(c+d x)}{3 d}+\frac{a \left(a^2-3 b^2\right) \sec ^2(c+d x)}{2 d}-\frac{3 a^2 b \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}+\frac{3 a b^2 \sec ^4(c+d x)}{4 d}+\frac{b^3 \sec ^5(c+d x)}{5 d}","\frac{b \left(3 a^2-b^2\right) \sec ^3(c+d x)}{3 d}+\frac{a \left(a^2-3 b^2\right) \sec ^2(c+d x)}{2 d}-\frac{3 a^2 b \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}+\frac{3 a b^2 \sec ^4(c+d x)}{4 d}+\frac{b^3 \sec ^5(c+d x)}{5 d}",1,"(a^3*Log[Cos[c + d*x]])/d - (3*a^2*b*Sec[c + d*x])/d + (a*(a^2 - 3*b^2)*Sec[c + d*x]^2)/(2*d) + (b*(3*a^2 - b^2)*Sec[c + d*x]^3)/(3*d) + (3*a*b^2*Sec[c + d*x]^4)/(4*d) + (b^3*Sec[c + d*x]^5)/(5*d)","A",5,4,28,0.1429,1,"{4397, 2837, 12, 894}"
250,1,113,0,0.341056,"\int \frac{\cos ^3(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{b^4 \log (a \cos (c+d x)+b)}{a^3 d \left(a^2-b^2\right)}-\frac{b \cos (c+d x)}{a^2 d}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}+\frac{\cos ^2(c+d x)}{2 a d}","-\frac{b^4 \log (a \cos (c+d x)+b)}{a^3 d \left(a^2-b^2\right)}-\frac{b \cos (c+d x)}{a^2 d}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}+\frac{\cos ^2(c+d x)}{2 a d}",1,"-((b*Cos[c + d*x])/(a^2*d)) + Cos[c + d*x]^2/(2*a*d) + Log[1 - Cos[c + d*x]]/(2*(a + b)*d) + Log[1 + Cos[c + d*x]]/(2*(a - b)*d) - (b^4*Log[b + a*Cos[c + d*x]])/(a^3*(a^2 - b^2)*d)","A",5,4,28,0.1429,1,"{4397, 2837, 12, 1629}"
251,1,92,0,0.2789221,"\int \frac{\cos ^2(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Int[Cos[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{b^3 \log (a \cos (c+d x)+b)}{a^2 d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}+\frac{\cos (c+d x)}{a d}","\frac{b^3 \log (a \cos (c+d x)+b)}{a^2 d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}+\frac{\cos (c+d x)}{a d}",1,"Cos[c + d*x]/(a*d) + Log[1 - Cos[c + d*x]]/(2*(a + b)*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)*d) + (b^3*Log[b + a*Cos[c + d*x]])/(a^2*(a^2 - b^2)*d)","A",5,4,28,0.1429,1,"{4397, 2837, 12, 1629}"
252,1,80,0,0.2368988,"\int \frac{\cos (c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Int[Cos[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{b^2 \log (a \cos (c+d x)+b)}{a d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}","-\frac{b^2 \log (a \cos (c+d x)+b)}{a d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}",1,"Log[1 - Cos[c + d*x]]/(2*(a + b)*d) + Log[1 + Cos[c + d*x]]/(2*(a - b)*d) - (b^2*Log[b + a*Cos[c + d*x]])/(a*(a^2 - b^2)*d)","A",5,4,26,0.1538,1,"{4397, 2837, 12, 1629}"
253,1,74,0,0.0798118,"\int \frac{1}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Int[(a*Sin[c + d*x] + b*Tan[c + d*x])^(-1),x]","\frac{b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}","\frac{b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}",1,"Log[1 - Cos[c + d*x]]/(2*(a + b)*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)*d) + (b*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)*d)","A",4,3,19,0.1579,1,"{4397, 2721, 801}"
254,1,75,0,0.1855843,"\int \frac{\sec (c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Int[Sec[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{a \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}","-\frac{a \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}",1,"Log[1 - Cos[c + d*x]]/(2*(a + b)*d) + Log[1 + Cos[c + d*x]]/(2*(a - b)*d) - (a*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)*d)","A",7,5,26,0.1923,1,"{4397, 2668, 706, 31, 633}"
255,1,94,0,0.2702162,"\int \frac{\sec ^2(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Int[Sec[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{a^2 \log (a \cos (c+d x)+b)}{b d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}-\frac{\log (\cos (c+d x))}{b d}","\frac{a^2 \log (a \cos (c+d x)+b)}{b d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}-\frac{\log (\cos (c+d x))}{b d}",1,"Log[1 - Cos[c + d*x]]/(2*(a + b)*d) - Log[Cos[c + d*x]]/(b*d) - Log[1 + Cos[c + d*x]]/(2*(a - b)*d) + (a^2*Log[b + a*Cos[c + d*x]])/(b*(a^2 - b^2)*d)","A",5,4,28,0.1429,1,"{4397, 2837, 12, 894}"
256,1,108,0,0.2841284,"\int \frac{\sec ^3(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Int[Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{a^3 \log (a \cos (c+d x)+b)}{b^2 d \left(a^2-b^2\right)}+\frac{a \log (\cos (c+d x))}{b^2 d}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}+\frac{\sec (c+d x)}{b d}","-\frac{a^3 \log (a \cos (c+d x)+b)}{b^2 d \left(a^2-b^2\right)}+\frac{a \log (\cos (c+d x))}{b^2 d}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}+\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}+\frac{\sec (c+d x)}{b d}",1,"Log[1 - Cos[c + d*x]]/(2*(a + b)*d) + (a*Log[Cos[c + d*x]])/(b^2*d) + Log[1 + Cos[c + d*x]]/(2*(a - b)*d) - (a^3*Log[b + a*Cos[c + d*x]])/(b^2*(a^2 - b^2)*d) + Sec[c + d*x]/(b*d)","A",5,4,28,0.1429,1,"{4397, 2837, 12, 894}"
257,1,243,0,0.6107123,"\int \frac{\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Int[Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{b^5 \sin (c+d x)}{a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 b^6 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{2 b^4 \left(5 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{2 b x}{a^3}-\frac{\sin (c+d x)}{a^2 d}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}","-\frac{b^5 \sin (c+d x)}{a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 b^6 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{2 b^4 \left(5 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{2 b x}{a^3}-\frac{\sin (c+d x)}{a^2 d}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}",1,"(2*b*x)/a^3 + (2*b^6*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (2*b^4*(5*a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(a^2*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) - (b^5*Sin[c + d*x])/(a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",12,8,28,0.2857,1,"{4397, 2897, 2648, 2637, 2664, 12, 2659, 208}"
258,1,227,0,0.4911709,"\int \frac{\cos ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Int[Cos[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{b^4 \sin (c+d x)}{a d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{4 b^3 \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{x}{a^2}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}","\frac{b^4 \sin (c+d x)}{a d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{4 b^3 \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{x}{a^2}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}",1,"-(x/a^2) - (2*b^5*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) - (4*b^3*(2*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) + (b^4*Sin[c + d*x])/(a*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",11,7,28,0.2500,1,"{4397, 2897, 2648, 2664, 12, 2659, 208}"
259,1,219,0,0.3848744,"\int \frac{\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Int[Cos[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{b^3 \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 b^2 \left(3 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d (a-b)^{5/2} (a+b)^{5/2}}+\frac{2 b^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}","-\frac{b^3 \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 b^2 \left(3 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d (a-b)^{5/2} (a+b)^{5/2}}+\frac{2 b^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}",1,"(2*b^4*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*(a - b)^(5/2)*(a + b)^(5/2)*d) + (2*b^2*(3*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*(a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) - (b^3*Sin[c + d*x])/((a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",11,7,26,0.2692,1,"{4397, 2897, 2648, 2659, 208, 2664, 12}"
260,1,203,0,0.3992406,"\int \frac{1}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Int[(a*Sin[c + d*x] + b*Tan[c + d*x])^(-2),x]","\frac{a b^2 \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{4 a^2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{2 b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}","\frac{a b^2 \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{4 a^2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{2 b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}",1,"(-4*a^2*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (2*b^3*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) + Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) + (a*b^2*Sin[c + d*x])/((a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",11,7,19,0.3684,1,"{4397, 2731, 2648, 2664, 12, 2659, 208}"
261,1,136,0,0.3175271,"\int \frac{\sec (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Int[Sec[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{\csc (c+d x) \left(a^2-3 a b \cos (c+d x)+2 b^2\right)}{d \left(a^2-b^2\right)^2}-\frac{b \csc (c+d x)}{d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{2 a \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}","-\frac{\csc (c+d x) \left(a^2-3 a b \cos (c+d x)+2 b^2\right)}{d \left(a^2-b^2\right)^2}-\frac{b \csc (c+d x)}{d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{2 a \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}",1,"(2*a*(a^2 + 2*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*Csc[c + d*x])/((a^2 - b^2)*d*(b + a*Cos[c + d*x])) - ((a^2 + 2*b^2 - 3*a*b*Cos[c + d*x])*Csc[c + d*x])/((a^2 - b^2)^2*d)","A",6,6,26,0.2308,1,"{4397, 2864, 2866, 12, 2659, 208}"
262,1,131,0,0.3340152,"\int \frac{\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Int[Sec[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{a \csc (c+d x)}{d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{\csc (c+d x) \left(3 a b-\left(2 a^2+b^2\right) \cos (c+d x)\right)}{d \left(a^2-b^2\right)^2}-\frac{6 a^2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}","\frac{a \csc (c+d x)}{d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{\csc (c+d x) \left(3 a b-\left(2 a^2+b^2\right) \cos (c+d x)\right)}{d \left(a^2-b^2\right)^2}-\frac{6 a^2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}",1,"(-6*a^2*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) + (a*Csc[c + d*x])/((a^2 - b^2)*d*(b + a*Cos[c + d*x])) + ((3*a*b - (2*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x])/((a^2 - b^2)^2*d)","A",6,6,28,0.2143,1,"{4397, 2694, 2866, 12, 2659, 208}"
263,1,231,0,0.4369501,"\int \frac{\sec ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Int[Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{a^4 \sin (c+d x)}{b d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{2 a^3 \left(a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{2 a^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}","-\frac{a^4 \sin (c+d x)}{b d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{2 a^3 \left(a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{2 a^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}-\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"ArcTanh[Sin[c + d*x]]/(b^2*d) + (2*a^3*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - (2*a^3*(a^2 - 3*b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^2*(a + b)^(5/2)*d) - Sin[c + d*x]/(2*(a + b)^2*d*(1 - Cos[c + d*x])) - Sin[c + d*x]/(2*(a - b)^2*d*(1 + Cos[c + d*x])) - (a^4*Sin[c + d*x])/(b*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",12,8,28,0.2857,1,"{4397, 2897, 2648, 2664, 12, 2659, 208, 3770}"
264,1,248,0,0.9016387,"\int \frac{\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Int[Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{b^6}{2 a^3 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{2 b^5 \left(3 a^2-b^2\right)}{a^3 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{b^4 \left(-4 a^2 b^2+15 a^4+b^4\right) \log (a \cos (c+d x)+b)}{a^3 d \left(a^2-b^2\right)^4}-\frac{\csc ^2(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}-\frac{(2 a+5 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(2 a-5 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}","\frac{b^6}{2 a^3 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{2 b^5 \left(3 a^2-b^2\right)}{a^3 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{b^4 \left(-4 a^2 b^2+15 a^4+b^4\right) \log (a \cos (c+d x)+b)}{a^3 d \left(a^2-b^2\right)^4}-\frac{\csc ^2(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}-\frac{(2 a+5 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(2 a-5 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}",1,"b^6/(2*a^3*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) - (2*b^5*(3*a^2 - b^2))/(a^3*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) - ((a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) - ((2*a + 5*b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) - ((2*a - 5*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) - (b^4*(15*a^4 - 4*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]])/(a^3*(a^2 - b^2)^4*d)","A",6,5,28,0.1786,1,"{4397, 2837, 12, 1647, 1629}"
265,1,232,0,0.7617746,"\int \frac{\cos ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Int[Cos[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","-\frac{b^5}{2 a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b^4 \left(5 a^2-b^2\right)}{a^2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{2 b^3 \left(5 a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^2(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}-\frac{(a+4 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(a-4 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}","-\frac{b^5}{2 a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b^4 \left(5 a^2-b^2\right)}{a^2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{2 b^3 \left(5 a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^2(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}-\frac{(a+4 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(a-4 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}",1,"-b^5/(2*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (b^4*(5*a^2 - b^2))/(a^2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) - ((a + 4*b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) + ((a - 4*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (2*b^3*(5*a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)","A",6,5,28,0.1786,1,"{4397, 2837, 12, 1647, 1629}"
266,1,211,0,0.6679584,"\int \frac{\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Int[Cos[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{b^4}{2 a d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{4 a b^3}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{6 a b^2 \left(a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}-\frac{\csc ^2(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}-\frac{3 b \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{3 b \log (\cos (c+d x)+1)}{4 d (a-b)^4}","\frac{b^4}{2 a d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{4 a b^3}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{6 a b^2 \left(a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}-\frac{\csc ^2(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}-\frac{3 b \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{3 b \log (\cos (c+d x)+1)}{4 d (a-b)^4}",1,"b^4/(2*a*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) - (4*a*b^3)/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) - ((a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) - (3*b*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) + (3*b*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) - (6*a*b^2*(a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)","A",6,5,26,0.1923,1,"{4397, 2837, 12, 1647, 1629}"
267,1,229,0,0.4804229,"\int \frac{1}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Int[(a*Sin[c + d*x] + b*Tan[c + d*x])^(-3),x]","-\frac{b^3}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2+b^2\right)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{b \left(8 a^2 b^2+3 a^4+b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^2(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}","-\frac{b^3}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b^2 \left(3 a^2+b^2\right)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{b \left(8 a^2 b^2+3 a^4+b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^2(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}",1,"-b^3/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (b^2*(3*a^2 + b^2))/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((b*(3*a^2 + b^2) - a*(a^2 + 3*b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) + ((a - 2*b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) - ((a + 2*b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (b*(3*a^4 + 8*a^2*b^2 + b^4)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)","A",5,4,19,0.2105,1,"{4397, 2721, 1647, 1629}"
268,1,231,0,0.620968,"\int \frac{\sec (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Int[Sec[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{a b^2}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{2 a b \left(a^2+b^2\right)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{a \left(8 a^2 b^2+a^4+3 b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}-\frac{\csc ^2(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}+\frac{(2 a-b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(2 a+b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}","\frac{a b^2}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{2 a b \left(a^2+b^2\right)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{a \left(8 a^2 b^2+a^4+3 b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}-\frac{\csc ^2(c+d x) \left(a \left(a^2+3 b^2\right)-b \left(3 a^2+b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}+\frac{(2 a-b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(2 a+b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}",1,"(a*b^2)/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) - (2*a*b*(a^2 + b^2))/((a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) - ((a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)^3*d) + ((2*a - b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) + ((2*a + b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) - (a*(a^4 + 8*a^2*b^2 + 3*b^4)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)","A",6,5,26,0.1923,1,"{4397, 2837, 12, 1647, 1629}"
269,1,212,0,0.3639854,"\int \frac{\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Int[Sec[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{3 a^2 \left(a^2+3 b^2\right)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{3 a^2 b}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{6 a^2 b \left(a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^2(c+d x) (b-a \cos (c+d x))}{2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{3 a \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{3 a \log (\cos (c+d x)+1)}{4 d (a-b)^4}","\frac{3 a^2 \left(a^2+3 b^2\right)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{3 a^2 b}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{6 a^2 b \left(a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{\csc ^2(c+d x) (b-a \cos (c+d x))}{2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{3 a \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{3 a \log (\cos (c+d x)+1)}{4 d (a-b)^4}",1,"(-3*a^2*b)/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (3*a^2*(a^2 + 3*b^2))/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + ((b - a*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + (3*a*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) - (3*a*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) + (6*a^2*b*(a^2 + b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)","A",6,5,28,0.1786,1,"{4397, 2837, 12, 823, 801}"
270,1,228,0,0.4079301,"\int \frac{\sec ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Int[Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","-\frac{a b \left(11 a^2+b^2\right)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{a \left(2 a^2+b^2\right)}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{2 a^3 \left(a^2+5 b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}-\frac{\csc ^2(c+d x) (a-b \cos (c+d x))}{2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{(4 a+b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(4 a-b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}","-\frac{a b \left(11 a^2+b^2\right)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{a \left(2 a^2+b^2\right)}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{2 a^3 \left(a^2+5 b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}-\frac{\csc ^2(c+d x) (a-b \cos (c+d x))}{2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{(4 a+b) \log (1-\cos (c+d x))}{4 d (a+b)^4}+\frac{(4 a-b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}",1,"(a*(2*a^2 + b^2))/(2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) - (a*b*(11*a^2 + b^2))/(2*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) - ((a - b*Cos[c + d*x])*Csc[c + d*x]^2)/(2*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + ((4*a + b)*Log[1 - Cos[c + d*x]])/(4*(a + b)^4*d) + ((4*a - b)*Log[1 + Cos[c + d*x]])/(4*(a - b)^4*d) - (2*a^3*(a^2 + 5*b^2)*Log[b + a*Cos[c + d*x]])/((a^2 - b^2)^4*d)","A",5,4,28,0.1429,1,"{4397, 2668, 741, 801}"
271,1,155,0,0.3848046,"\int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Int[Cos[c + d*x]^m*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","-\frac{b \left(3 a^2-b^2\right) \cos ^m(c+d x)}{d m}-\frac{a \left(a^2-3 b^2\right) \cos ^{m+1}(c+d x)}{d (m+1)}+\frac{3 a^2 b \cos ^{m+2}(c+d x)}{d (m+2)}+\frac{a^3 \cos ^{m+3}(c+d x)}{d (m+3)}+\frac{3 a b^2 \cos ^{m-1}(c+d x)}{d (1-m)}+\frac{b^3 \cos ^{m-2}(c+d x)}{d (2-m)}","-\frac{b \left(3 a^2-b^2\right) \cos ^m(c+d x)}{d m}-\frac{a \left(a^2-3 b^2\right) \cos ^{m+1}(c+d x)}{d (m+1)}+\frac{3 a^2 b \cos ^{m+2}(c+d x)}{d (m+2)}+\frac{a^3 \cos ^{m+3}(c+d x)}{d (m+3)}+\frac{3 a b^2 \cos ^{m-1}(c+d x)}{d (1-m)}+\frac{b^3 \cos ^{m-2}(c+d x)}{d (2-m)}",1,"(b^3*Cos[c + d*x]^(-2 + m))/(d*(2 - m)) + (3*a*b^2*Cos[c + d*x]^(-1 + m))/(d*(1 - m)) - (b*(3*a^2 - b^2)*Cos[c + d*x]^m)/(d*m) - (a*(a^2 - 3*b^2)*Cos[c + d*x]^(1 + m))/(d*(1 + m)) + (3*a^2*b*Cos[c + d*x]^(2 + m))/(d*(2 + m)) + (a^3*Cos[c + d*x]^(3 + m))/(d*(3 + m))","A",4,3,28,0.1071,1,"{4397, 2837, 948}"
272,1,264,0,0.7641083,"\int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Int[Cos[c + d*x]^m*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{\left(a^2 (1-m)-b^2 (m+2)\right) \sin (c+d x) \cos ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\cos ^2(c+d x)\right)}{d (1-m) m (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{\left(a^2-2 b^2\right) \sin (c+d x) \cos ^{m-1}(c+d x)}{d m (m+2)}-\frac{2 a b \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\cos ^2(c+d x)\right)}{d m (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 a b \sin (c+d x) \cos ^m(c+d x)}{d \left(m^2+3 m+2\right)}-\frac{\sin (c+d x) \cos ^{m-1}(c+d x) (a \cos (c+d x)+b)^2}{d (m+2)}","-\frac{\left(a^2 (1-m)-b^2 (m+2)\right) \sin (c+d x) \cos ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\cos ^2(c+d x)\right)}{d (1-m) m (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{\left(a^2-2 b^2\right) \sin (c+d x) \cos ^{m-1}(c+d x)}{d m (m+2)}-\frac{2 a b \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\cos ^2(c+d x)\right)}{d m (m+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 a b \sin (c+d x) \cos ^m(c+d x)}{d \left(m^2+3 m+2\right)}-\frac{\sin (c+d x) \cos ^{m-1}(c+d x) (a \cos (c+d x)+b)^2}{d (m+2)}",1,"((a^2 - 2*b^2)*Cos[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*m*(2 + m)) - (2*a*b*Cos[c + d*x]^m*Sin[c + d*x])/(d*(2 + 3*m + m^2)) - (Cos[c + d*x]^(-1 + m)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*(2 + m)) - ((a^2*(1 - m) - b^2*(2 + m))*Cos[c + d*x]^(-1 + m)*Hypergeometric2F1[1/2, (-1 + m)/2, (1 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(1 - m)*m*(2 + m)*Sqrt[Sin[c + d*x]^2]) - (2*a*b*Cos[c + d*x]^m*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*m*(1 + m)*Sqrt[Sin[c + d*x]^2])","A",8,7,28,0.2500,1,"{4397, 2889, 3050, 3033, 3023, 2748, 2643}"
273,1,39,0,0.073123,"\int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Int[Cos[c + d*x]^m*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{a \cos ^{m+1}(c+d x)}{d (m+1)}-\frac{b \cos ^m(c+d x)}{d m}","-\frac{a \cos ^{m+1}(c+d x)}{d (m+1)}-\frac{b \cos ^m(c+d x)}{d m}",1,"-((b*Cos[c + d*x]^m)/(d*m)) - (a*Cos[c + d*x]^(1 + m))/(d*(1 + m))","A",6,4,26,0.1538,1,"{4377, 12, 2565, 30}"
274,1,144,0,0.4315726,"\int \frac{\cos ^m(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Int[Cos[c + d*x]^m/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{a^2 \cos ^{m+2}(c+d x) \, _2F_1\left(1,m+2;m+3;-\frac{a \cos (c+d x)}{b}\right)}{b d (m+2) \left(a^2-b^2\right)}+\frac{\cos ^{m+2}(c+d x) \, _2F_1(1,m+2;m+3;-\cos (c+d x))}{2 d (m+2) (a-b)}-\frac{\cos ^{m+2}(c+d x) \, _2F_1(1,m+2;m+3;\cos (c+d x))}{2 d (m+2) (a+b)}","-\frac{a^2 \cos ^{m+2}(c+d x) \, _2F_1\left(1,m+2;m+3;-\frac{a \cos (c+d x)}{b}\right)}{b d (m+2) \left(a^2-b^2\right)}+\frac{\cos ^{m+2}(c+d x) \, _2F_1(1,m+2;m+3;-\cos (c+d x))}{2 d (m+2) (a-b)}-\frac{\cos ^{m+2}(c+d x) \, _2F_1(1,m+2;m+3;\cos (c+d x))}{2 d (m+2) (a+b)}",1,"(Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, -Cos[c + d*x]])/(2*(a - b)*d*(2 + m)) - (Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, Cos[c + d*x]])/(2*(a + b)*d*(2 + m)) - (a^2*Cos[c + d*x]^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, -((a*Cos[c + d*x])/b)])/(b*(a^2 - b^2)*d*(2 + m))","A",7,4,28,0.1429,1,"{4397, 2837, 961, 64}"
275,1,65,0,0.0652865,"\int \frac{\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx","Int[(Cos[x]*Sin[x])/(a*Cos[x] + b*Sin[x]),x]","\frac{b \sin (x)}{a^2+b^2}-\frac{a \cos (x)}{a^2+b^2}+\frac{a b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}","\frac{b \sin (x)}{a^2+b^2}-\frac{a \cos (x)}{a^2+b^2}+\frac{a b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}",1,"(a*b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (a*Cos[x])/(a^2 + b^2) + (b*Sin[x])/(a^2 + b^2)","A",5,5,16,0.3125,1,"{3109, 2637, 2638, 3074, 206}"
276,1,92,0,0.1389015,"\int \frac{\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Int[(Cos[x]*Sin[x]^2)/(a*Cos[x] + b*Sin[x]),x]","\frac{a x}{2 \left(a^2+b^2\right)}-\frac{a b^2 x}{\left(a^2+b^2\right)^2}+\frac{b \sin ^2(x)}{2 \left(a^2+b^2\right)}-\frac{a \sin (x) \cos (x)}{2 \left(a^2+b^2\right)}+\frac{a^2 b \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}","\frac{a x}{2 \left(a^2+b^2\right)}-\frac{a b^2 x}{\left(a^2+b^2\right)^2}+\frac{b \sin ^2(x)}{2 \left(a^2+b^2\right)}-\frac{a \sin (x) \cos (x)}{2 \left(a^2+b^2\right)}+\frac{a^2 b \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}",1,"-((a*b^2*x)/(a^2 + b^2)^2) + (a*x)/(2*(a^2 + b^2)) + (a^2*b*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 - (a*Cos[x]*Sin[x])/(2*(a^2 + b^2)) + (b*Sin[x]^2)/(2*(a^2 + b^2))","A",7,7,18,0.3889,1,"{3109, 2564, 30, 2635, 8, 3097, 3133}"
277,1,122,0,0.1664398,"\int \frac{\cos (x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Int[(Cos[x]*Sin[x]^3)/(a*Cos[x] + b*Sin[x]),x]","\frac{b \sin ^3(x)}{3 \left(a^2+b^2\right)}+\frac{a^2 b \sin (x)}{\left(a^2+b^2\right)^2}+\frac{a \cos ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a \cos (x)}{a^2+b^2}+\frac{a b^2 \cos (x)}{\left(a^2+b^2\right)^2}+\frac{a^3 b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}","\frac{b \sin ^3(x)}{3 \left(a^2+b^2\right)}+\frac{a^2 b \sin (x)}{\left(a^2+b^2\right)^2}+\frac{a \cos ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a \cos (x)}{a^2+b^2}+\frac{a b^2 \cos (x)}{\left(a^2+b^2\right)^2}+\frac{a^3 b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}",1,"(a^3*b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (a*b^2*Cos[x])/(a^2 + b^2)^2 - (a*Cos[x])/(a^2 + b^2) + (a*Cos[x]^3)/(3*(a^2 + b^2)) + (a^2*b*Sin[x])/(a^2 + b^2)^2 + (b*Sin[x]^3)/(3*(a^2 + b^2))","A",9,8,18,0.4444,1,"{3109, 2564, 30, 2633, 3099, 3074, 206, 2638}"
278,1,93,0,0.1334335,"\int \frac{\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx","Int[(Cos[x]^2*Sin[x])/(a*Cos[x] + b*Sin[x]),x]","-\frac{a^2 b x}{\left(a^2+b^2\right)^2}+\frac{b x}{2 \left(a^2+b^2\right)}+\frac{a \sin ^2(x)}{2 \left(a^2+b^2\right)}+\frac{b \sin (x) \cos (x)}{2 \left(a^2+b^2\right)}-\frac{a b^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}","-\frac{a^2 b x}{\left(a^2+b^2\right)^2}+\frac{b x}{2 \left(a^2+b^2\right)}+\frac{a \sin ^2(x)}{2 \left(a^2+b^2\right)}+\frac{b \sin (x) \cos (x)}{2 \left(a^2+b^2\right)}-\frac{a b^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}",1,"-((a^2*b*x)/(a^2 + b^2)^2) + (b*x)/(2*(a^2 + b^2)) - (a*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 + (b*Cos[x]*Sin[x])/(2*(a^2 + b^2)) + (a*Sin[x]^2)/(2*(a^2 + b^2))","A",7,7,18,0.3889,1,"{3109, 2635, 8, 2564, 30, 3098, 3133}"
279,1,112,0,0.1960088,"\int \frac{\cos ^2(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Int[(Cos[x]^2*Sin[x]^2)/(a*Cos[x] + b*Sin[x]),x]","\frac{a \sin ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a b^2 \sin (x)}{\left(a^2+b^2\right)^2}-\frac{b \cos ^3(x)}{3 \left(a^2+b^2\right)}+\frac{a^2 b \cos (x)}{\left(a^2+b^2\right)^2}-\frac{a^2 b^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}","\frac{a \sin ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a b^2 \sin (x)}{\left(a^2+b^2\right)^2}-\frac{b \cos ^3(x)}{3 \left(a^2+b^2\right)}+\frac{a^2 b \cos (x)}{\left(a^2+b^2\right)^2}-\frac{a^2 b^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}",1,"-((a^2*b^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (a^2*b*Cos[x])/(a^2 + b^2)^2 - (b*Cos[x]^3)/(3*(a^2 + b^2)) - (a*b^2*Sin[x])/(a^2 + b^2)^2 + (a*Sin[x]^3)/(3*(a^2 + b^2))","A",10,8,20,0.4000,1,"{3109, 2565, 30, 2564, 2637, 2638, 3074, 206}"
280,1,176,0,0.2784531,"\int \frac{\cos ^2(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Int[(Cos[x]^2*Sin[x]^3)/(a*Cos[x] + b*Sin[x]),x]","\frac{b x}{8 \left(a^2+b^2\right)}-\frac{a^2 b x}{2 \left(a^2+b^2\right)^2}+\frac{a^2 b^3 x}{\left(a^2+b^2\right)^3}+\frac{a \sin ^4(x)}{4 \left(a^2+b^2\right)}-\frac{a b^2 \sin ^2(x)}{2 \left(a^2+b^2\right)^2}-\frac{b \sin (x) \cos ^3(x)}{4 \left(a^2+b^2\right)}+\frac{b \sin (x) \cos (x)}{8 \left(a^2+b^2\right)}+\frac{a^2 b \sin (x) \cos (x)}{2 \left(a^2+b^2\right)^2}-\frac{a^3 b^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}","\frac{b x}{8 \left(a^2+b^2\right)}-\frac{a^2 b x}{2 \left(a^2+b^2\right)^2}+\frac{a^2 b^3 x}{\left(a^2+b^2\right)^3}+\frac{a \sin ^4(x)}{4 \left(a^2+b^2\right)}-\frac{a b^2 \sin ^2(x)}{2 \left(a^2+b^2\right)^2}-\frac{b \sin (x) \cos ^3(x)}{4 \left(a^2+b^2\right)}+\frac{b \sin (x) \cos (x)}{8 \left(a^2+b^2\right)}+\frac{a^2 b \sin (x) \cos (x)}{2 \left(a^2+b^2\right)^2}-\frac{a^3 b^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}",1,"(a^2*b^3*x)/(a^2 + b^2)^3 - (a^2*b*x)/(2*(a^2 + b^2)^2) + (b*x)/(8*(a^2 + b^2)) - (a^3*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (a^2*b*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (b*Cos[x]*Sin[x])/(8*(a^2 + b^2)) - (b*Cos[x]^3*Sin[x])/(4*(a^2 + b^2)) - (a*b^2*Sin[x]^2)/(2*(a^2 + b^2)^2) + (a*Sin[x]^4)/(4*(a^2 + b^2))","A",13,8,20,0.4000,1,"{3109, 2568, 2635, 8, 2564, 30, 3097, 3133}"
281,1,123,0,0.1576305,"\int \frac{\cos ^3(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx","Int[(Cos[x]^3*Sin[x])/(a*Cos[x] + b*Sin[x]),x]","-\frac{b \sin ^3(x)}{3 \left(a^2+b^2\right)}+\frac{b \sin (x)}{a^2+b^2}-\frac{a^2 b \sin (x)}{\left(a^2+b^2\right)^2}-\frac{a \cos ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a b^2 \cos (x)}{\left(a^2+b^2\right)^2}+\frac{a b^3 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}","-\frac{b \sin ^3(x)}{3 \left(a^2+b^2\right)}+\frac{b \sin (x)}{a^2+b^2}-\frac{a^2 b \sin (x)}{\left(a^2+b^2\right)^2}-\frac{a \cos ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a b^2 \cos (x)}{\left(a^2+b^2\right)^2}+\frac{a b^3 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}",1,"(a*b^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a*b^2*Cos[x])/(a^2 + b^2)^2 - (a*Cos[x]^3)/(3*(a^2 + b^2)) - (a^2*b*Sin[x])/(a^2 + b^2)^2 + (b*Sin[x])/(a^2 + b^2) - (b*Sin[x]^3)/(3*(a^2 + b^2))","A",9,8,18,0.4444,1,"{3109, 2633, 2565, 30, 3100, 2637, 3074, 206}"
282,1,175,0,0.2782931,"\int \frac{\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Int[(Cos[x]^3*Sin[x]^2)/(a*Cos[x] + b*Sin[x]),x]","\frac{a x}{8 \left(a^2+b^2\right)}-\frac{a b^2 x}{2 \left(a^2+b^2\right)^2}+\frac{a^3 b^2 x}{\left(a^2+b^2\right)^3}-\frac{a^2 b \sin ^2(x)}{2 \left(a^2+b^2\right)^2}-\frac{b \cos ^4(x)}{4 \left(a^2+b^2\right)}-\frac{a \sin (x) \cos ^3(x)}{4 \left(a^2+b^2\right)}+\frac{a \sin (x) \cos (x)}{8 \left(a^2+b^2\right)}-\frac{a b^2 \sin (x) \cos (x)}{2 \left(a^2+b^2\right)^2}+\frac{a^2 b^3 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}","\frac{a x}{8 \left(a^2+b^2\right)}-\frac{a b^2 x}{2 \left(a^2+b^2\right)^2}+\frac{a^3 b^2 x}{\left(a^2+b^2\right)^3}-\frac{a^2 b \sin ^2(x)}{2 \left(a^2+b^2\right)^2}-\frac{b \cos ^4(x)}{4 \left(a^2+b^2\right)}-\frac{a \sin (x) \cos ^3(x)}{4 \left(a^2+b^2\right)}+\frac{a \sin (x) \cos (x)}{8 \left(a^2+b^2\right)}-\frac{a b^2 \sin (x) \cos (x)}{2 \left(a^2+b^2\right)^2}+\frac{a^2 b^3 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}",1,"(a^3*b^2*x)/(a^2 + b^2)^3 - (a*b^2*x)/(2*(a^2 + b^2)^2) + (a*x)/(8*(a^2 + b^2)) - (b*Cos[x]^4)/(4*(a^2 + b^2)) + (a^2*b^3*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (a*b^2*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (a*Cos[x]*Sin[x])/(8*(a^2 + b^2)) - (a*Cos[x]^3*Sin[x])/(4*(a^2 + b^2)) - (a^2*b*Sin[x]^2)/(2*(a^2 + b^2)^2)","A",13,9,20,0.4500,1,"{3109, 2565, 30, 2568, 2635, 8, 2564, 3098, 3133}"
283,1,193,0,0.3586711,"\int \frac{\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Int[(Cos[x]^3*Sin[x]^3)/(a*Cos[x] + b*Sin[x]),x]","-\frac{b \sin ^5(x)}{5 \left(a^2+b^2\right)}+\frac{b \sin ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a^2 b \sin ^3(x)}{3 \left(a^2+b^2\right)^2}+\frac{a^2 b^3 \sin (x)}{\left(a^2+b^2\right)^3}+\frac{a \cos ^5(x)}{5 \left(a^2+b^2\right)}-\frac{a \cos ^3(x)}{3 \left(a^2+b^2\right)}+\frac{a b^2 \cos ^3(x)}{3 \left(a^2+b^2\right)^2}-\frac{a^3 b^2 \cos (x)}{\left(a^2+b^2\right)^3}+\frac{a^3 b^3 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}","-\frac{b \sin ^5(x)}{5 \left(a^2+b^2\right)}+\frac{b \sin ^3(x)}{3 \left(a^2+b^2\right)}-\frac{a^2 b \sin ^3(x)}{3 \left(a^2+b^2\right)^2}+\frac{a^2 b^3 \sin (x)}{\left(a^2+b^2\right)^3}+\frac{a \cos ^5(x)}{5 \left(a^2+b^2\right)}-\frac{a \cos ^3(x)}{3 \left(a^2+b^2\right)}+\frac{a b^2 \cos ^3(x)}{3 \left(a^2+b^2\right)^2}-\frac{a^3 b^2 \cos (x)}{\left(a^2+b^2\right)^3}+\frac{a^3 b^3 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}",1,"(a^3*b^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (a^3*b^2*Cos[x])/(a^2 + b^2)^3 + (a*b^2*Cos[x]^3)/(3*(a^2 + b^2)^2) - (a*Cos[x]^3)/(3*(a^2 + b^2)) + (a*Cos[x]^5)/(5*(a^2 + b^2)) + (a^2*b^3*Sin[x])/(a^2 + b^2)^3 - (a^2*b*Sin[x]^3)/(3*(a^2 + b^2)^2) + (b*Sin[x]^3)/(3*(a^2 + b^2)) - (b*Sin[x]^5)/(5*(a^2 + b^2))","A",17,9,20,0.4500,1,"{3109, 2564, 14, 2565, 30, 2637, 2638, 3074, 206}"
284,1,87,0,0.1664607,"\int \frac{\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(Cos[x]*Sin[x])/(a*Cos[x] + b*Sin[x])^2,x]","\frac{2 a b x}{\left(a^2+b^2\right)^2}-\frac{b \sin (x)}{\left(a^2+b^2\right) (a \cos (x)+b \sin (x))}-\frac{a^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}+\frac{b^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}","\frac{2 a b x}{\left(a^2+b^2\right)^2}-\frac{b \sin (x)}{\left(a^2+b^2\right) (a \cos (x)+b \sin (x))}-\frac{\left(a^2-b^2\right) \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}",1,"(2*a*b*x)/(a^2 + b^2)^2 - (a^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 + (b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 - (b*Sin[x])/((a^2 + b^2)*(a*Cos[x] + b*Sin[x]))","A",6,5,16,0.3125,1,"{3111, 3098, 3133, 3097, 3075}"
285,1,152,0,0.2393388,"\int \frac{\cos (x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(Cos[x]*Sin[x]^2)/(a*Cos[x] + b*Sin[x])^2,x]","-\frac{a^2 \sin (x)}{\left(a^2+b^2\right)^2}+\frac{b^2 \sin (x)}{\left(a^2+b^2\right)^2}-\frac{2 a b \cos (x)}{\left(a^2+b^2\right)^2}-\frac{a^2 b}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}-\frac{a^3 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}+\frac{2 a b^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}","-\frac{\left(a^2-b^2\right) \sin (x)}{\left(a^2+b^2\right)^2}-\frac{2 a b \cos (x)}{\left(a^2+b^2\right)^2}-\frac{a^2 b}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}-\frac{a \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}",1,"-((a^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (2*a*b^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (2*a*b*Cos[x])/(a^2 + b^2)^2 - (a^2*Sin[x])/(a^2 + b^2)^2 + (b^2*Sin[x])/(a^2 + b^2)^2 - (a^2*b)/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))","A",13,8,18,0.4444,1,"{3111, 3109, 2637, 2638, 3074, 206, 3099, 3154}"
286,1,198,0,0.5064728,"\int \frac{\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(Cos[x]*Sin[x]^3)/(a*Cos[x] + b*Sin[x])^2,x]","\frac{a^3 b x}{\left(a^2+b^2\right)^3}+\frac{a b x}{\left(a^2+b^2\right)^2}-\frac{a b^3 x}{\left(a^2+b^2\right)^3}+\frac{a b x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^3}-\frac{a^2 \sin ^2(x)}{2 \left(a^2+b^2\right)^2}+\frac{b^2 \sin ^2(x)}{2 \left(a^2+b^2\right)^2}-\frac{a^2 b}{\left(a^2+b^2\right)^2 (a \cot (x)+b)}-\frac{a b \sin (x) \cos (x)}{\left(a^2+b^2\right)^2}-\frac{a^4 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}+\frac{3 a^2 b^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}","\frac{b x \left(3 a^3-a b^2\right)}{\left(a^2+b^2\right)^3}-\frac{\left(a^2-b^2\right) \sin ^2(x)}{2 \left(a^2+b^2\right)^2}-\frac{a^2 b \sin (x)}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}-\frac{a b \sin (x) \cos (x)}{\left(a^2+b^2\right)^2}-\frac{a^2 \left(a^2-3 b^2\right) \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}",1,"(a^3*b*x)/(a^2 + b^2)^3 - (a*b^3*x)/(a^2 + b^2)^3 + (a*b*(a^2 - b^2)*x)/(a^2 + b^2)^3 + (a*b*x)/(a^2 + b^2)^2 - (a^2*b)/((a^2 + b^2)^2*(b + a*Cot[x])) - (a^4*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (3*a^2*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (a*b*Cos[x]*Sin[x])/(a^2 + b^2)^2 - (a^2*Sin[x]^2)/(2*(a^2 + b^2)^2) + (b^2*Sin[x]^2)/(2*(a^2 + b^2)^2)","A",17,13,18,0.7222,1,"{3111, 3109, 2564, 30, 2635, 8, 3097, 3133, 3099, 3085, 3483, 3531, 3530}"
287,1,151,0,0.2592929,"\int \frac{\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(Cos[x]^2*Sin[x])/(a*Cos[x] + b*Sin[x])^2,x]","\frac{2 a b \sin (x)}{\left(a^2+b^2\right)^2}+\frac{b^2 \cos (x)}{\left(a^2+b^2\right)^2}-\frac{a^2 \cos (x)}{\left(a^2+b^2\right)^2}+\frac{a b^2}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}-\frac{b^3 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}+\frac{2 a^2 b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}","\frac{2 a b \sin (x)}{\left(a^2+b^2\right)^2}-\frac{\left(a^2-b^2\right) \cos (x)}{\left(a^2+b^2\right)^2}+\frac{a b^2}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}-\frac{b \left(b^2-2 a^2\right) \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}",1,"(2*a^2*b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (b^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a^2*Cos[x])/(a^2 + b^2)^2 + (b^2*Cos[x])/(a^2 + b^2)^2 + (2*a*b*Sin[x])/(a^2 + b^2)^2 + (a*b^2)/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))","A",13,8,18,0.4444,1,"{3111, 3100, 2637, 3074, 206, 3109, 2638, 3155}"
288,1,186,0,0.544877,"\int \frac{\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(Cos[x]^2*Sin[x]^2)/(a*Cos[x] + b*Sin[x])^2,x]","\frac{a^2 x}{2 \left(a^2+b^2\right)^2}-\frac{4 a^2 b^2 x}{\left(a^2+b^2\right)^3}+\frac{b^2 x}{2 \left(a^2+b^2\right)^2}+\frac{a b \sin ^2(x)}{\left(a^2+b^2\right)^2}-\frac{a^2 \sin (x) \cos (x)}{2 \left(a^2+b^2\right)^2}+\frac{a b^2 \sin (x)}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}+\frac{b^2 \sin (x) \cos (x)}{2 \left(a^2+b^2\right)^2}+\frac{2 a^3 b \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}-\frac{2 a b^3 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}","\frac{x \left(-6 a^2 b^2+a^4+b^4\right)}{2 \left(a^2+b^2\right)^3}+\frac{a b \sin ^2(x)}{\left(a^2+b^2\right)^2}+\frac{a b^2 \sin (x)}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}+\frac{\left(b^2-a^2\right) \sin (x) \cos (x)}{2 \left(a^2+b^2\right)^2}+\frac{2 a b \left(a^2-b^2\right) \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}",1,"(-4*a^2*b^2*x)/(a^2 + b^2)^3 + (a^2*x)/(2*(a^2 + b^2)^2) + (b^2*x)/(2*(a^2 + b^2)^2) + (2*a^3*b*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (2*a*b^3*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (a^2*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (b^2*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (a*b*Sin[x]^2)/(a^2 + b^2)^2 + (a*b^2*Sin[x])/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))","A",21,10,20,0.5000,1,"{3111, 3109, 2635, 8, 2564, 30, 3098, 3133, 3097, 3075}"
289,1,238,0,0.6773155,"\int \frac{\cos ^2(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(Cos[x]^2*Sin[x]^3)/(a*Cos[x] + b*Sin[x])^2,x]","\frac{2 a b \sin ^3(x)}{3 \left(a^2+b^2\right)^2}+\frac{2 a^3 b \sin (x)}{\left(a^2+b^2\right)^3}-\frac{2 a b^3 \sin (x)}{\left(a^2+b^2\right)^3}+\frac{a^2 \cos ^3(x)}{3 \left(a^2+b^2\right)^2}-\frac{b^2 \cos ^3(x)}{3 \left(a^2+b^2\right)^2}-\frac{a^2 \cos (x)}{\left(a^2+b^2\right)^2}+\frac{4 a^2 b^2 \cos (x)}{\left(a^2+b^2\right)^3}+\frac{a^3 b^2}{\left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}+\frac{2 a^4 b \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}-\frac{3 a^2 b^3 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}","\frac{2 a b \sin ^3(x)}{3 \left(a^2+b^2\right)^2}+\frac{2 a b \left(a^2-b^2\right) \sin (x)}{\left(a^2+b^2\right)^3}+\frac{\left(a^2-b^2\right) \cos ^3(x)}{3 \left(a^2+b^2\right)^2}-\frac{a^2 \left(a^2-3 b^2\right) \cos (x)}{\left(a^2+b^2\right)^3}+\frac{a^3 b^2}{\left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}+\frac{a^2 b \left(2 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}",1,"(2*a^4*b*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (3*a^2*b^3*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (4*a^2*b^2*Cos[x])/(a^2 + b^2)^3 - (a^2*Cos[x])/(a^2 + b^2)^2 + (a^2*Cos[x]^3)/(3*(a^2 + b^2)^2) - (b^2*Cos[x]^3)/(3*(a^2 + b^2)^2) + (2*a^3*b*Sin[x])/(a^2 + b^2)^3 - (2*a*b^3*Sin[x])/(a^2 + b^2)^3 + (2*a*b*Sin[x]^3)/(3*(a^2 + b^2)^2) + (a^3*b^2)/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))","A",33,12,20,0.6000,1,"{3111, 3109, 2565, 30, 2564, 2637, 2638, 3074, 206, 2633, 3099, 3154}"
290,1,196,0,0.4076335,"\int \frac{\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(Cos[x]^3*Sin[x])/(a*Cos[x] + b*Sin[x])^2,x]","\frac{a b^3 x}{\left(a^2+b^2\right)^3}+\frac{a b x}{\left(a^2+b^2\right)^2}-\frac{a^3 b x}{\left(a^2+b^2\right)^3}-\frac{a b x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^3}+\frac{a^2 \sin ^2(x)}{2 \left(a^2+b^2\right)^2}+\frac{b^2 \cos ^2(x)}{2 \left(a^2+b^2\right)^2}+\frac{a b^2}{\left(a^2+b^2\right)^2 (a+b \tan (x))}+\frac{a b \sin (x) \cos (x)}{\left(a^2+b^2\right)^2}+\frac{b^4 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}-\frac{3 a^2 b^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}","-\frac{a b x \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^3}+\frac{\left(a^2-b^2\right) \sin ^2(x)}{2 \left(a^2+b^2\right)^2}+\frac{a b^2 \cos (x)}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}+\frac{a b \sin (x) \cos (x)}{\left(a^2+b^2\right)^2}-\frac{b^2 \left(3 a^2-b^2\right) \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}",1,"-((a^3*b*x)/(a^2 + b^2)^3) + (a*b^3*x)/(a^2 + b^2)^3 - (a*b*(a^2 - b^2)*x)/(a^2 + b^2)^3 + (a*b*x)/(a^2 + b^2)^2 + (b^2*Cos[x]^2)/(2*(a^2 + b^2)^2) - (3*a^2*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (b^4*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (a*b*Cos[x]*Sin[x])/(a^2 + b^2)^2 + (a^2*Sin[x]^2)/(2*(a^2 + b^2)^2) + (a*b^2)/((a^2 + b^2)^2*(a + b*Tan[x]))","A",17,13,18,0.7222,1,"{3111, 3100, 2635, 8, 3098, 3133, 3109, 2564, 30, 3086, 3483, 3531, 3530}"
291,1,238,0,0.6965051,"\int \frac{\cos ^3(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(Cos[x]^3*Sin[x]^2)/(a*Cos[x] + b*Sin[x])^2,x]","-\frac{b^2 \sin ^3(x)}{3 \left(a^2+b^2\right)^2}+\frac{a^2 \sin ^3(x)}{3 \left(a^2+b^2\right)^2}+\frac{b^2 \sin (x)}{\left(a^2+b^2\right)^2}-\frac{4 a^2 b^2 \sin (x)}{\left(a^2+b^2\right)^3}-\frac{2 a b \cos ^3(x)}{3 \left(a^2+b^2\right)^2}-\frac{2 a b^3 \cos (x)}{\left(a^2+b^2\right)^3}+\frac{2 a^3 b \cos (x)}{\left(a^2+b^2\right)^3}-\frac{a^2 b^3}{\left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}+\frac{2 a b^4 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}-\frac{3 a^3 b^2 \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}","\frac{\left(a^2-b^2\right) \sin ^3(x)}{3 \left(a^2+b^2\right)^2}-\frac{b^2 \left(3 a^2-b^2\right) \sin (x)}{\left(a^2+b^2\right)^3}-\frac{2 a b \cos ^3(x)}{3 \left(a^2+b^2\right)^2}+\frac{2 a b \left(a^2-b^2\right) \cos (x)}{\left(a^2+b^2\right)^3}-\frac{a^2 b^3}{\left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}-\frac{a b^2 \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{b \cos (x)-a \sin (x)}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}",1,"(-3*a^3*b^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (2*a*b^4*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (2*a^3*b*Cos[x])/(a^2 + b^2)^3 - (2*a*b^3*Cos[x])/(a^2 + b^2)^3 - (2*a*b*Cos[x]^3)/(3*(a^2 + b^2)^2) - (4*a^2*b^2*Sin[x])/(a^2 + b^2)^3 + (b^2*Sin[x])/(a^2 + b^2)^2 + (a^2*Sin[x]^3)/(3*(a^2 + b^2)^2) - (b^2*Sin[x]^3)/(3*(a^2 + b^2)^2) - (a^2*b^3)/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))","A",33,12,20,0.6000,1,"{3111, 3109, 2633, 2565, 30, 3100, 2637, 3074, 206, 2564, 2638, 3155}"
292,1,289,0,1.2507276,"\int \frac{\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Int[(Cos[x]^3*Sin[x]^3)/(a*Cos[x] + b*Sin[x])^2,x]","-\frac{a^3 b x}{\left(a^2+b^2\right)^3}+\frac{6 a^3 b^3 x}{\left(a^2+b^2\right)^4}+\frac{a b x}{4 \left(a^2+b^2\right)^2}-\frac{a b^3 x}{\left(a^2+b^2\right)^3}+\frac{a^2 \sin ^4(x)}{4 \left(a^2+b^2\right)^2}-\frac{2 a^2 b^2 \sin ^2(x)}{\left(a^2+b^2\right)^3}-\frac{b^2 \cos ^4(x)}{4 \left(a^2+b^2\right)^2}-\frac{a b \sin (x) \cos ^3(x)}{2 \left(a^2+b^2\right)^2}+\frac{a^3 b \sin (x) \cos (x)}{\left(a^2+b^2\right)^3}-\frac{a^2 b^3 \sin (x)}{\left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}+\frac{a b \sin (x) \cos (x)}{4 \left(a^2+b^2\right)^2}-\frac{a b^3 \sin (x) \cos (x)}{\left(a^2+b^2\right)^3}-\frac{3 a^4 b^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^4}+\frac{3 a^2 b^4 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^4}","-\frac{3 a b x \left(-6 a^2 b^2+a^4+b^4\right)}{4 \left(a^2+b^2\right)^4}+\frac{a^2 \sin ^4(x)}{4 \left(a^2+b^2\right)^2}-\frac{2 a^2 b^2 \sin ^2(x)}{\left(a^2+b^2\right)^3}-\frac{b^2 \cos ^4(x)}{4 \left(a^2+b^2\right)^2}-\frac{a b \sin (x) \cos ^3(x)}{2 \left(a^2+b^2\right)^2}+\frac{a b \left(5 a^2-3 b^2\right) \sin (x) \cos (x)}{4 \left(a^2+b^2\right)^3}-\frac{a^2 b^3 \sin (x)}{\left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}-\frac{3 a^2 b^2 \left(a^2-b^2\right) \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^4}",1,"(6*a^3*b^3*x)/(a^2 + b^2)^4 - (a^3*b*x)/(a^2 + b^2)^3 - (a*b^3*x)/(a^2 + b^2)^3 + (a*b*x)/(4*(a^2 + b^2)^2) - (b^2*Cos[x]^4)/(4*(a^2 + b^2)^2) - (3*a^4*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^4 + (3*a^2*b^4*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^4 + (a^3*b*Cos[x]*Sin[x])/(a^2 + b^2)^3 - (a*b^3*Cos[x]*Sin[x])/(a^2 + b^2)^3 + (a*b*Cos[x]*Sin[x])/(4*(a^2 + b^2)^2) - (a*b*Cos[x]^3*Sin[x])/(2*(a^2 + b^2)^2) - (2*a^2*b^2*Sin[x]^2)/(a^2 + b^2)^3 + (a^2*Sin[x]^4)/(4*(a^2 + b^2)^2) - (a^2*b^3*Sin[x])/((a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))","A",48,12,20,0.6000,1,"{3111, 3109, 2565, 30, 2568, 2635, 8, 2564, 3098, 3133, 3097, 3075}"
293,1,47,0,0.0789635,"\int \frac{\tan (x)}{b \cos (x)+a \sin (x)} \, dx","Int[Tan[x]/(b*Cos[x] + a*Sin[x]),x]","\frac{b \tanh ^{-1}\left(\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right)}{a \sqrt{a^2+b^2}}+\frac{\tanh ^{-1}(\sin (x))}{a}","\frac{b \tanh ^{-1}\left(\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right)}{a \sqrt{a^2+b^2}}+\frac{\tanh ^{-1}(\sin (x))}{a}",1,"ArcTanh[Sin[x]]/a + (b*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2])","A",5,4,14,0.2857,1,"{3110, 3770, 3074, 206}"
294,1,48,0,0.0783539,"\int \frac{\cot (x)}{b \cos (x)+a \sin (x)} \, dx","Int[Cot[x]/(b*Cos[x] + a*Sin[x]),x]","\frac{a \tanh ^{-1}\left(\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right)}{b \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cos (x))}{b}","\frac{a \tanh ^{-1}\left(\frac{a \cos (x)-b \sin (x)}{\sqrt{a^2+b^2}}\right)}{b \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cos (x))}{b}",1,"-(ArcTanh[Cos[x]]/b) + (a*ArcTanh[(a*Cos[x] - b*Sin[x])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])","A",5,4,14,0.2857,1,"{3110, 3770, 3074, 206}"