1,1,34,36,0.0061805,"\int \sin ^3(x) (a \cos (x)+b \sin (x)) \, dx","Integrate[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 (2 x)+\frac{1}{32} b \sin (4 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 + (a*Sin[x]^4)/4 - (b*Sin[2*x])/4 + (b*Sin[4*x])/32","A",1
2,1,26,24,0.0040181,"\int \sin ^2(x) (a \cos (x)+b \sin (x)) \, dx","Integrate[Sin[x]^2*(a*Cos[x] + b*Sin[x]),x]","\frac{1}{3} a \sin ^3(x)-\frac{3}{4} b \cos (x)+\frac{1}{12} b \cos (3 x)","\frac{1}{3} a \sin ^3(x)+\frac{1}{3} b \cos ^3(x)-b \cos (x)",1,"(-3*b*Cos[x])/4 + (b*Cos[3*x])/12 + (a*Sin[x]^3)/3","A",1
3,1,25,25,0.0041005,"\int \sin (x) (a \cos (x)+b \sin (x)) \, dx","Integrate[Sin[x]*(a*Cos[x] + b*Sin[x]),x]","-\frac{1}{2} a \cos ^2(x)+\frac{b x}{2}-\frac{1}{4} b \sin (2 x)","\frac{1}{2} a \sin ^2(x)+\frac{b x}{2}-\frac{1}{2} b \sin (x) \cos (x)",1,"(b*x)/2 - (a*Cos[x]^2)/2 - (b*Sin[2*x])/4","A",1
4,1,10,10,0.0019688,"\int (a \cos (x)+b \sin (x)) \, dx","Integrate[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",1
5,1,9,9,0.0057617,"\int \csc (x) (a \cos (x)+b \sin (x)) \, dx","Integrate[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",1
6,1,25,12,0.0074554,"\int \csc ^2(x) (a \cos (x)+b \sin (x)) \, dx","Integrate[Csc[x]^2*(a*Cos[x] + b*Sin[x]),x]","-a \csc (x)+b \log \left(\sin \left(\frac{x}{2}\right)\right)-b \log \left(\cos \left(\frac{x}{2}\right)\right)","-a \csc (x)-b \tanh ^{-1}(\cos (x))",1,"-(a*Csc[x]) - b*Log[Cos[x/2]] + b*Log[Sin[x/2]]","B",1
7,1,15,15,0.008114,"\int \csc ^3(x) (a \cos (x)+b \sin (x)) \, dx","Integrate[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",1
8,1,94,91,0.1976719,"\int \frac{\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[Sin[x]^3/(a*Cos[x] + b*Sin[x]),x]","\frac{-2 a^3 \log \left((a \cos (x)+b \sin (x))^2\right)-4 i a^3 x+4 i a^3 \tan ^{-1}(\tan (x))+a \left(a^2+b^2\right) \cos (2 x)+6 a^2 b x-a^2 b \sin (2 x)+2 b^3 x-b^3 \sin (2 x)}{4 \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,"((-4*I)*a^3*x + 6*a^2*b*x + 2*b^3*x + (4*I)*a^3*ArcTan[Tan[x]] + a*(a^2 + b^2)*Cos[2*x] - 2*a^3*Log[(a*Cos[x] + b*Sin[x])^2] - a^2*b*Sin[2*x] - b^3*Sin[2*x])/(4*(a^2 + b^2)^2)","C",1
9,1,62,68,0.1632518,"\int \frac{\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[Sin[x]^2/(a*Cos[x] + b*Sin[x]),x]","\frac{2 a^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}-\frac{a \sin (x)+b \cos (x)}{a^2+b^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,"(2*a^2*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (b*Cos[x] + a*Sin[x])/(a^2 + b^2)","A",1
10,1,47,35,0.0585216,"\int \frac{\sin (x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[Sin[x]/(a*Cos[x] + b*Sin[x]),x]","\frac{2 x (b-i a)-a \log \left((a \cos (x)+b \sin (x))^2\right)+2 i a \tan ^{-1}(\tan (x))}{2 \left(a^2+b^2\right)}","\frac{b x}{a^2+b^2}-\frac{a \log (a \cos (x)+b \sin (x))}{a^2+b^2}",1,"(2*((-I)*a + b)*x + (2*I)*a*ArcTan[Tan[x]] - a*Log[(a*Cos[x] + b*Sin[x])^2])/(2*(a^2 + b^2))","C",1
11,1,38,36,0.0259253,"\int \frac{1}{a \cos (x)+b \sin (x)} \, dx","Integrate[(a*Cos[x] + b*Sin[x])^(-1),x]","\frac{2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\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,"(2*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2]","A",1
12,1,20,23,0.048139,"\int \frac{\csc (x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[Csc[x]/(a*Cos[x] + b*Sin[x]),x]","\frac{\log (\sin (x))-\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]] - Log[a*Cos[x] + b*Sin[x]])/a","A",1
13,1,67,55,0.1295869,"\int \frac{\csc ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[Csc[x]^2/(a*Cos[x] + b*Sin[x]),x]","\frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\sqrt{a^2+b^2}}\right)-a \csc (x)+b \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)}{a^2}","-\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,"(2*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]] - a*Csc[x] + b*(Log[Cos[x/2]] - Log[Sin[x/2]]))/a^2","A",1
14,1,48,55,0.1611066,"\int \frac{\csc ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[Csc[x]^3/(a*Cos[x] + b*Sin[x]),x]","\frac{2 \left(a^2+b^2\right) (\log (\sin (x))-\log (a \cos (x)+b \sin (x)))-a^2 \csc ^2(x)+2 a b \cot (x)}{2 a^3}","\frac{b \cot (x)}{a^2}+\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{\csc ^2(x)}{2 a}",1,"(2*a*b*Cot[x] - a^2*Csc[x]^2 + 2*(a^2 + b^2)*(Log[Sin[x]] - Log[a*Cos[x] + b*Sin[x]]))/(2*a^3)","A",1
15,1,107,107,0.4381529,"\int \frac{\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[Sin[x]^3/(a*Cos[x] + b*Sin[x])^2,x]","\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}}","\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,"(6*a^2*b*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (3*a*(a^2 - b^2) + a*(a^2 + b^2)*Cos[2*x] - b*(a^2 + b^2)*Sin[2*x])/(2*(a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))","A",1
16,1,121,64,0.2643295,"\int \frac{\sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[Sin[x]^2/(a*Cos[x] + b*Sin[x])^2,x]","\frac{\sin (x) \left(a^3-a^2 b x+a b^2 (1-2 i x)-a b^2 \log \left((a \cos (x)+b \sin (x))^2\right)+b^3 x\right)-a \cos (x) \left(a b \log \left((a \cos (x)+b \sin (x))^2\right)+x (a+i b)^2\right)+2 i a b \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (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}",1,"(-(a*Cos[x]*((a + I*b)^2*x + a*b*Log[(a*Cos[x] + b*Sin[x])^2])) + (a^3 + a*b^2*(1 - (2*I)*x) - a^2*b*x + b^3*x - a*b^2*Log[(a*Cos[x] + b*Sin[x])^2])*Sin[x] + (2*I)*a*b*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x]))/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))","C",1
17,1,62,60,0.1647496,"\int \frac{\sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[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{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)^{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,"(2*b*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) + a/((a^2 + b^2)*(a*Cos[x] + b*Sin[x]))","A",1
18,1,17,17,0.0211322,"\int \frac{1}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(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
19,1,72,63,0.3305351,"\int \frac{\csc (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[Csc[x]/(a*Cos[x] + b*Sin[x])^2,x]","\frac{-\frac{2 b \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}+\frac{a \csc (x)}{a \cot (x)+b}+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)}{a^2}","\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,"((-2*b*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + (a*Csc[x])/(b + a*Cot[x]) - Log[Cos[x/2]] + Log[Sin[x/2]])/a^2","A",1
20,1,76,49,0.2006442,"\int \frac{\csc ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[Csc[x]^2/(a*Cos[x] + b*Sin[x])^2,x]","\frac{a^2 \left(-\cot ^2(x)\right)+a^2+2 b^2 \log (a \cos (x)+b \sin (x))-a b \cot (x) (-2 \log (a \cos (x)+b \sin (x))+2 \log (\sin (x))+1)-2 b^2 \log (\sin (x))+b^2}{a^3 (a \cot (x)+b)}","-\frac{2 b \log (\tan (x))}{a^3}+\frac{2 b \log (a+b \tan (x))}{a^3}-\frac{\frac{b}{a^2}+\frac{1}{b}}{a+b \tan (x)}-\frac{\cot (x)}{a^2}",1,"(a^2 + b^2 - a^2*Cot[x]^2 - 2*b^2*Log[Sin[x]] - a*b*Cot[x]*(1 + 2*Log[Sin[x]] - 2*Log[a*Cos[x] + b*Sin[x]]) + 2*b^2*Log[a*Cos[x] + b*Sin[x]])/(a^3*(b + a*Cot[x]))","A",1
21,1,270,118,1.9085657,"\int \frac{\csc ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[Csc[x]^3/(a*Cos[x] + b*Sin[x])^2,x]","\frac{8 a^3 \csc (x)+a^3 \cot (x) \sec ^2\left(\frac{x}{2}\right)-12 a^3 \cot (x) \log \left(\cos \left(\frac{x}{2}\right)\right)+12 a^3 \cot (x) \log \left(\sin \left(\frac{x}{2}\right)\right)-48 b \sqrt{a^2+b^2} (a \cot (x)+b) \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\sqrt{a^2+b^2}}\right)+a^2 b \sec ^2\left(\frac{x}{2}\right)+12 a^2 b \log \left(\sin \left(\frac{x}{2}\right)\right)-12 a^2 b \log \left(\cos \left(\frac{x}{2}\right)\right)+8 a^2 b \tan \left(\frac{x}{2}\right) \cot (x)-a \csc ^2\left(\frac{x}{2}\right) \left(a^2 \cot (x)+b (a-4 b \sin (x))-4 a b \cos (x)\right)+8 a b^2 \tan \left(\frac{x}{2}\right)+8 a b^2 \csc (x)-24 a b^2 \cot (x) \log \left(\cos \left(\frac{x}{2}\right)\right)+24 a b^2 \cot (x) \log \left(\sin \left(\frac{x}{2}\right)\right)+24 b^3 \log \left(\sin \left(\frac{x}{2}\right)\right)-24 b^3 \log \left(\cos \left(\frac{x}{2}\right)\right)}{8 a^4 (a \cot (x)+b)}","-\frac{2 b^2 \tanh ^{-1}(\cos (x))}{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{\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{a^2+b^2}{a^3 (a \cos (x)+b \sin (x))}",1,"(-48*b*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]]*(b + a*Cot[x]) + 8*a^3*Csc[x] + 8*a*b^2*Csc[x] - 12*a^2*b*Log[Cos[x/2]] - 24*b^3*Log[Cos[x/2]] - 12*a^3*Cot[x]*Log[Cos[x/2]] - 24*a*b^2*Cot[x]*Log[Cos[x/2]] + 12*a^2*b*Log[Sin[x/2]] + 24*b^3*Log[Sin[x/2]] + 12*a^3*Cot[x]*Log[Sin[x/2]] + 24*a*b^2*Cot[x]*Log[Sin[x/2]] + a^2*b*Sec[x/2]^2 + a^3*Cot[x]*Sec[x/2]^2 - a*Csc[x/2]^2*(-4*a*b*Cos[x] + a^2*Cot[x] + b*(a - 4*b*Sin[x])) + 8*a*b^2*Tan[x/2] + 8*a^2*b*Cot[x]*Tan[x/2])/(8*a^4*(b + a*Cot[x]))","B",1
22,1,114,98,0.860284,"\int \frac{\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx","Integrate[Sin[x]^3/(a*Cos[x] + b*Sin[x])^3,x]","\frac{a^3}{2 (a-i b)^2 (a+i b)^2 (a \cos (x)+b \sin (x))^2}+\frac{b x \left(b^2-3 a^2\right)}{\left(a^2+b^2\right)^3}+\frac{3 a b \sin (x)}{\left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}+\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*(a^2 - 3*b^2)*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + a^3/(2*(a - I*b)^2*(a + I*b)^2*(a*Cos[x] + b*Sin[x])^2) + (3*a*b*Sin[x])/((a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))","C",1
23,1,92,92,0.4294176,"\int \frac{\sin ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx","Integrate[Sin[x]^2/(a*Cos[x] + b*Sin[x])^3,x]","\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}}","\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,"-(((a^2 - 2*b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (a*(3*a*b*Cos[x] + (a^2 + 4*b^2)*Sin[x]))/(2*(a^2 + b^2)^2*(a*Cos[x] + b*Sin[x])^2)","A",1
24,1,47,15,0.0958242,"\int \frac{\sin (x)}{(a \cos (x)+b \sin (x))^3} \, dx","Integrate[Sin[x]/(a*Cos[x] + b*Sin[x])^3,x]","\frac{a (a+b \sin (2 x))+2 b^2 \sin ^2(x)}{2 a \left(a^2+b^2\right) (a \cos (x)+b \sin (x))^2}","\frac{1}{2 a (a \cot (x)+b)^2}",1,"(2*b^2*Sin[x]^2 + a*(a + b*Sin[2*x]))/(2*a*(a^2 + b^2)*(a*Cos[x] + b*Sin[x])^2)","B",1
25,1,101,73,0.1711435,"\int \frac{1}{(a \cos (x)+b \sin (x))^3} \, dx","Integrate[(a*Cos[x] + b*Sin[x])^(-3),x]","\frac{\left(a^2+b^2\right) (a \sin (x)-b \cos (x))+2 \sqrt{a^2+b^2} (a \cos (x)+b \sin (x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\sqrt{a^2+b^2}}\right)}{2 (a-i b)^2 (a+i b)^2 (a \cos (x)+b \sin (x))^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,"((a^2 + b^2)*(-(b*Cos[x]) + a*Sin[x]) + 2*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]]*(a*Cos[x] + b*Sin[x])^2)/(2*(a - I*b)^2*(a + I*b)^2*(a*Cos[x] + b*Sin[x])^2)","C",1
26,1,96,59,0.2248439,"\int \frac{\csc (x)}{(a \cos (x)+b \sin (x))^3} \, dx","Integrate[Csc[x]/(a*Cos[x] + b*Sin[x])^3,x]","\frac{2 a^2 \cot ^2(x) (\log (\sin (x))-\log (a \cos (x)+b \sin (x)))+a^2 \csc ^2(x)+2 b^2 (-\log (a \cos (x)+b \sin (x))+\log (\sin (x))-1)+2 a b \cot (x) (-2 \log (a \cos (x)+b \sin (x))+2 \log (\sin (x))-1)}{2 a^3 (a \cot (x)+b)^2}","-\frac{\log (a+b \tan (x))}{a^3}+\frac{\log (\tan (x))}{a^3}+\frac{\frac{1}{a^2}-\frac{1}{b^2}}{a+b \tan (x)}+\frac{\frac{a}{b^2}+\frac{1}{a}}{2 (a+b \tan (x))^2}",1,"(a^2*Csc[x]^2 + 2*a*b*Cot[x]*(-1 + 2*Log[Sin[x]] - 2*Log[a*Cos[x] + b*Sin[x]]) + 2*b^2*(-1 + Log[Sin[x]] - Log[a*Cos[x] + b*Sin[x]]) + 2*a^2*Cot[x]^2*(Log[Sin[x]] - Log[a*Cos[x] + b*Sin[x]]))/(2*a^3*(b + a*Cot[x])^2)","A",1
27,1,193,184,0.7951891,"\int \frac{\csc ^2(x)}{(a \cos (x)+b \sin (x))^3} \, dx","Integrate[Csc[x]^2/(a*Cos[x] + b*Sin[x])^3,x]","\frac{\csc ^3(x) (a \cos (x)+b \sin (x)) \left(a \left(a^2+b^2\right) \sin (x)+\frac{6 \left(a^2+2 b^2\right) (a \cos (x)+b \sin (x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}-5 a b (a \cos (x)+b \sin (x))+6 b \log \left(\cos \left(\frac{x}{2}\right)\right) (a \cos (x)+b \sin (x))^2-6 b \log \left(\sin \left(\frac{x}{2}\right)\right) (a \cos (x)+b \sin (x))^2-a \tan \left(\frac{x}{2}\right) (a \cos (x)+b \sin (x))^2-a \cot \left(\frac{x}{2}\right) (a \cos (x)+b \sin (x))^2\right)}{2 a^4 (a \cot (x)+b)^3}","\frac{3 b \tanh ^{-1}(\cos (x))}{a^4}-\frac{2 b}{a^3 (a \cos (x)+b \sin (x))}-\frac{\csc (x)}{a^3}-\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{b \cos (x)-a \sin (x)}{2 a^2 (a \cos (x)+b \sin (x))^2}-\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}",1,"(Csc[x]^3*(a*Cos[x] + b*Sin[x])*(a*(a^2 + b^2)*Sin[x] - 5*a*b*(a*Cos[x] + b*Sin[x]) + (6*(a^2 + 2*b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]]*(a*Cos[x] + b*Sin[x])^2)/Sqrt[a^2 + b^2] - a*Cot[x/2]*(a*Cos[x] + b*Sin[x])^2 + 6*b*Log[Cos[x/2]]*(a*Cos[x] + b*Sin[x])^2 - 6*b*Log[Sin[x/2]]*(a*Cos[x] + b*Sin[x])^2 - a*(a*Cos[x] + b*Sin[x])^2*Tan[x/2]))/(2*a^4*(b + a*Cot[x])^3)","A",1
28,1,208,117,0.8348928,"\int \frac{\csc ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx","Integrate[Csc[x]^3/(a*Cos[x] + b*Sin[x])^3,x]","\frac{a^4 \csc ^2(x)+6 a^3 b \cot ^3(x)+2 b^2 \left(2 \left(a^2+3 b^2\right) \log (\sin (x))-2 \left(a^2+3 b^2\right) \log (a \cos (x)+b \sin (x))-3 \left(a^2+b^2\right)\right)-2 a b \cot (x) \left(-4 \left(a^2+3 b^2\right) \log (\sin (x))+4 a^2 \log (a \cos (x)+b \sin (x))+a^2 \csc ^2(x)+3 a^2+12 b^2 \log (a \cos (x)+b \sin (x))\right)+\cot ^2(x) \left(4 a^2 \left(\left(a^2+3 b^2\right) \log (\sin (x))-\left(a^2+3 b^2\right) \log (a \cos (x)+b \sin (x))+3 b^2\right)-a^4 \csc ^2(x)\right)}{2 a^5 (a \cot (x)+b)^2}","\frac{3 b \cot (x)}{a^4}-\frac{\cot ^2(x)}{2 a^3}+\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{\left(a^2-3 b^2\right) \left(a^2+b^2\right)}{a^4 b^2 (a+b \tan (x))}+\frac{\left(a^2+b^2\right)^2}{2 a^3 b^2 (a+b \tan (x))^2}",1,"(6*a^3*b*Cot[x]^3 + a^4*Csc[x]^2 - 2*a*b*Cot[x]*(3*a^2 + a^2*Csc[x]^2 - 4*(a^2 + 3*b^2)*Log[Sin[x]] + 4*a^2*Log[a*Cos[x] + b*Sin[x]] + 12*b^2*Log[a*Cos[x] + b*Sin[x]]) + 2*b^2*(-3*(a^2 + b^2) + 2*(a^2 + 3*b^2)*Log[Sin[x]] - 2*(a^2 + 3*b^2)*Log[a*Cos[x] + b*Sin[x]]) + Cot[x]^2*(-(a^4*Csc[x]^2) + 4*a^2*(3*b^2 + (a^2 + 3*b^2)*Log[Sin[x]] - (a^2 + 3*b^2)*Log[a*Cos[x] + b*Sin[x]])))/(2*a^5*(b + a*Cot[x])^2)","A",1
29,1,367,66,3.7473005,"\int \sin ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n/Sin[c + d*x]^n,x]","-\frac{4 \sin \left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sin ^{-n}(c+d x) (a (\cos (c+d x)+i \sin (c+d x)))^n \left(F_1\left(1-n;-2 n,1;2-n;-i \tan \left(\frac{1}{2} (c+d x)\right),i \tan \left(\frac{1}{2} (c+d x)\right)\right)+\, _2F_1\left(1-2 n,1-n;2-n;-i \tan \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d (n-1) \left(2 F_1\left(1-n;-2 n,1;2-n;-i \tan \left(\frac{1}{2} (c+d x)\right),i \tan \left(\frac{1}{2} (c+d x)\right)\right)+\frac{\left(1-i \tan \left(\frac{1}{2} (c+d x)\right)\right) \left(-2 n (i \sin (c+d x)+\cos (c+d x)-1) F_1\left(2-n;1-2 n,1;3-n;-i \tan \left(\frac{1}{2} (c+d x)\right),i \tan \left(\frac{1}{2} (c+d x)\right)\right)-(i \sin (c+d x)+\cos (c+d x)-1) F_1\left(2-n;-2 n,2;3-n;-i \tan \left(\frac{1}{2} (c+d x)\right),i \tan \left(\frac{1}{2} (c+d x)\right)\right)+(n-2) (\cos (c+d x)+1) \left(1+i \tan \left(\frac{1}{2} (c+d x)\right)\right)^{2 n}\right)}{n-2}\right)}","-\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,"(-4*Cos[(c + d*x)/2]*(AppellF1[1 - n, -2*n, 1, 2 - n, (-I)*Tan[(c + d*x)/2], I*Tan[(c + d*x)/2]] + Hypergeometric2F1[1 - 2*n, 1 - n, 2 - n, (-I)*Tan[(c + d*x)/2]])*Sin[(c + d*x)/2]*(a*(Cos[c + d*x] + I*Sin[c + d*x]))^n)/(d*(-1 + n)*Sin[c + d*x]^n*(2*AppellF1[1 - n, -2*n, 1, 2 - n, (-I)*Tan[(c + d*x)/2], I*Tan[(c + d*x)/2]] + ((-2*n*AppellF1[2 - n, 1 - 2*n, 1, 3 - n, (-I)*Tan[(c + d*x)/2], I*Tan[(c + d*x)/2]]*(-1 + Cos[c + d*x] + I*Sin[c + d*x]) - AppellF1[2 - n, -2*n, 2, 3 - n, (-I)*Tan[(c + d*x)/2], I*Tan[(c + d*x)/2]]*(-1 + Cos[c + d*x] + I*Sin[c + d*x]) + (-2 + n)*(1 + Cos[c + d*x])*(1 + I*Tan[(c + d*x)/2])^(2*n))*(1 - I*Tan[(c + d*x)/2]))/(-2 + n)))","C",0
30,1,57,87,0.1165837,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a (45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+60 c+60 d x)-32 b \cos ^6(c+d x)}{192 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,"(-32*b*Cos[c + d*x]^6 + a*(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)]))/(192*d)","A",1
31,1,60,60,0.0155471,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[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,"-1/5*(b*Cos[c + d*x]^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",1
32,1,62,65,0.0901902,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{3 a (c+d x)}{8 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \sin (4 (c+d x))}{32 d}-\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*(c + d*x))/(8*d) - (b*Cos[c + d*x]^4)/(4*d) + (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)","A",1
33,1,44,44,0.0111708,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[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,"-1/3*(b*Cos[c + d*x]^3)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)","A",1
34,1,46,43,0.0487116,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{b \cos ^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*(c + d*x))/(2*d) - (b*Cos[c + d*x]^2)/(2*d) + (a*Sin[2*(c + d*x)])/(4*d)","A",1
35,1,46,24,0.0120647,"\int (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[a*Cos[c + d*x] + b*Sin[c + d*x],x]","\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d}","\frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d}",1,"-((b*Cos[c]*Cos[d*x])/d) + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d + (b*Sin[c]*Sin[d*x])/d","A",1
36,1,17,17,0.0169449,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[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",1
37,1,24,24,0.0139448,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[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",1
38,1,28,28,0.0137825,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[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",1
39,1,52,52,0.0166243,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[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",1
40,1,41,44,0.0892812,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{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] + Tan[c + d*x]^3/3))/d","A",1
41,1,68,74,0.2101732,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{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,"(b*Sec[c + d*x]^5)/(5*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*a*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d)","A",1
42,1,53,60,0.1596732,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{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] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
43,1,154,137,0.3759753,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","-\frac{-3675 a^2 \sin (c+d x)-735 a^2 \sin (3 (c+d x))-147 a^2 \sin (5 (c+d x))-15 a^2 \sin (7 (c+d x))+1050 a b \cos (c+d x)+630 a b \cos (3 (c+d x))+210 a b \cos (5 (c+d x))+30 a b \cos (7 (c+d x))-525 b^2 \sin (c+d x)+35 b^2 \sin (3 (c+d x))+63 b^2 \sin (5 (c+d x))+15 b^2 \sin (7 (c+d x))}{6720 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,"-1/6720*(1050*a*b*Cos[c + d*x] + 630*a*b*Cos[3*(c + d*x)] + 210*a*b*Cos[5*(c + d*x)] + 30*a*b*Cos[7*(c + d*x)] - 3675*a^2*Sin[c + d*x] - 525*b^2*Sin[c + d*x] - 735*a^2*Sin[3*(c + d*x)] + 35*b^2*Sin[3*(c + d*x)] - 147*a^2*Sin[5*(c + d*x)] + 63*b^2*Sin[5*(c + d*x)] - 15*a^2*Sin[7*(c + d*x)] + 15*b^2*Sin[7*(c + d*x)])/d","A",1
44,1,147,174,0.2480117,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\left(5 a^2+b^2\right) (c+d x)}{16 d}+\frac{\left(15 a^2+b^2\right) \sin (2 (c+d x))}{64 d}+\frac{\left(3 a^2-b^2\right) \sin (4 (c+d x))}{64 d}+\frac{\left(a^2-b^2\right) \sin (6 (c+d x))}{192 d}-\frac{5 a b \cos (2 (c+d x))}{32 d}-\frac{a b \cos (4 (c+d x))}{16 d}-\frac{a b \cos (6 (c+d x))}{96 d}","\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 + b^2)*(c + d*x))/(16*d) - (5*a*b*Cos[2*(c + d*x)])/(32*d) - (a*b*Cos[4*(c + d*x)])/(16*d) - (a*b*Cos[6*(c + d*x)])/(96*d) + ((15*a^2 + b^2)*Sin[2*(c + d*x)])/(64*d) + ((3*a^2 - b^2)*Sin[4*(c + d*x)])/(64*d) + ((a^2 - b^2)*Sin[6*(c + d*x)])/(192*d)","A",1
45,1,116,103,0.1714976,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{150 a^2 \sin (c+d x)+25 a^2 \sin (3 (c+d x))+3 a^2 \sin (5 (c+d x))-60 a b \cos (c+d x)-30 a b \cos (3 (c+d x))-6 a b \cos (5 (c+d x))+30 b^2 \sin (c+d x)-5 b^2 \sin (3 (c+d x))-3 b^2 \sin (5 (c+d x))}{240 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,"(-60*a*b*Cos[c + d*x] - 30*a*b*Cos[3*(c + d*x)] - 6*a*b*Cos[5*(c + d*x)] + 150*a^2*Sin[c + d*x] + 30*b^2*Sin[c + d*x] + 25*a^2*Sin[3*(c + d*x)] - 5*b^2*Sin[3*(c + d*x)] + 3*a^2*Sin[5*(c + d*x)] - 3*b^2*Sin[5*(c + d*x)])/(240*d)","A",1
46,1,98,126,0.2272763,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\left(3 a^2+b^2\right) (c+d x)}{8 d}+\frac{\left(a^2-b^2\right) \sin (4 (c+d x))}{32 d}+\frac{a^2 \sin (2 (c+d x))}{4 d}-\frac{a b \cos (2 (c+d x))}{4 d}-\frac{a b \cos (4 (c+d x))}{16 d}","\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 + b^2)*(c + d*x))/(8*d) - (a*b*Cos[2*(c + d*x)])/(4*d) - (a*b*Cos[4*(c + d*x)])/(16*d) + (a^2*Sin[2*(c + d*x)])/(4*d) + ((a^2 - b^2)*Sin[4*(c + d*x)])/(32*d)","A",1
47,1,64,67,0.3936379,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\sin (c+d x) \left(\left(a^2-b^2\right) \cos (2 (c+d x))+5 a^2+b^2\right)-3 a b \cos (c+d x)-a b \cos (3 (c+d x))}{6 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,"(-3*a*b*Cos[c + d*x] - a*b*Cos[3*(c + d*x)] + (5*a^2 + b^2 + (a^2 - b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(6*d)","A",1
48,1,52,55,0.0981648,"\int (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{2 \left(a^2+b^2\right) (c+d x)+\left(a^2-b^2\right) \sin (2 (c+d x))-2 a b \cos (2 (c+d x))}{4 d}","\frac{1}{2} x \left(a^2+b^2\right)-\frac{(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{2 d}",1,"(2*(a^2 + b^2)*(c + d*x) - 2*a*b*Cos[2*(c + d*x)] + (a^2 - b^2)*Sin[2*(c + d*x)])/(4*d)","A",1
49,1,84,55,0.1557549,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-b^2\right) \sin (c+d x)-2 a b \cos (c+d x)+b^2 \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{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,"(-2*a*b*Cos[c + d*x] + b^2*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (a^2 - b^2)*Sin[c + d*x])/d","A",1
50,1,69,39,0.1354702,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{2 b^2 \tan (c+d x)-i \left((a+i b)^2 \log (-\tan (c+d x)+i)-(a-i b)^2 \log (\tan (c+d x)+i)\right)}{2 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,"((-I)*((a + I*b)^2*Log[I - Tan[c + d*x]] - (a - I*b)^2*Log[I + Tan[c + d*x]]) + 2*b^2*Tan[c + d*x])/(2*d)","C",1
51,1,67,67,0.0443959,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[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",1
52,1,46,30,0.0416764,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^3(c+d x)}{3 d}","\frac{\tan ^3(c+d x) (a \cot (c+d x)+b)^3}{3 b d}",1,"(a^2*Tan[c + d*x])/d + (a*b*Tan[c + d*x]^2)/d + (b^2*Tan[c + d*x]^3)/(3*d)","A",1
53,1,120,120,0.076983,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[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",1
54,1,54,85,0.1909506,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{(a+b \tan (c+d x))^3 \left(a^2-3 a b \tan (c+d x)+6 b^2 \tan ^2(c+d x)+10 b^2\right)}{30 b^3 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 + b*Tan[c + d*x])^3*(a^2 + 10*b^2 - 3*a*b*Tan[c + d*x] + 6*b^2*Tan[c + d*x]^2))/(30*b^3*d)","A",1
55,1,104,168,0.5941271,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{15 \left(6 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))+10 \left(6 a^2-b^2\right) \tan (c+d x) \sec ^3(c+d x)+15 \left(6 a^2-b^2\right) \tan (c+d x) \sec (c+d x)+8 b \sec ^5(c+d x) (12 a+5 b \tan (c+d x))}{240 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,"(15*(6*a^2 - b^2)*ArcTanh[Sin[c + d*x]] + 15*(6*a^2 - b^2)*Sec[c + d*x]*Tan[c + d*x] + 10*(6*a^2 - b^2)*Sec[c + d*x]^3*Tan[c + d*x] + 8*b*Sec[c + d*x]^5*(12*a + 5*b*Tan[c + d*x]))/(240*d)","A",1
56,1,104,125,0.6887269,"\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\tan (c+d x) \left(21 \left(a^2+2 b^2\right) \tan ^4(c+d x)+35 \left(2 a^2+b^2\right) \tan ^2(c+d x)+105 a^2+35 a b \tan ^5(c+d x)+105 a b \tan ^3(c+d x)+105 a b \tan (c+d x)+15 b^2 \tan ^6(c+d x)\right)}{105 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,"(Tan[c + d*x]*(105*a^2 + 105*a*b*Tan[c + d*x] + 35*(2*a^2 + b^2)*Tan[c + d*x]^2 + 105*a*b*Tan[c + d*x]^3 + 21*(a^2 + 2*b^2)*Tan[c + d*x]^4 + 35*a*b*Tan[c + d*x]^5 + 15*b^2*Tan[c + d*x]^6))/(105*d)","A",1
57,1,235,265,0.4723383,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{5 a \left(7 a^2+3 b^2\right) (c+d x)}{128 d}+\frac{a \left(14 a^2+3 b^2\right) \sin (2 (c+d x))}{64 d}+\frac{a \left(7 a^2-3 b^2\right) \sin (4 (c+d x))}{128 d}+\frac{a \left(2 a^2-3 b^2\right) \sin (6 (c+d x))}{192 d}+\frac{a \left(a^2-3 b^2\right) \sin (8 (c+d x))}{1024 d}-\frac{3 b \left(7 a^2+b^2\right) \cos (2 (c+d x))}{128 d}-\frac{b \left(21 a^2+b^2\right) \cos (4 (c+d x))}{256 d}-\frac{b \left(9 a^2-b^2\right) \cos (6 (c+d x))}{384 d}-\frac{b \left(3 a^2-b^2\right) \cos (8 (c+d x))}{1024 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^2 b \cos ^8(c+d x)}{8 d}-\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,"(5*a*(7*a^2 + 3*b^2)*(c + d*x))/(128*d) - (3*b*(7*a^2 + b^2)*Cos[2*(c + d*x)])/(128*d) - (b*(21*a^2 + b^2)*Cos[4*(c + d*x)])/(256*d) - (b*(9*a^2 - b^2)*Cos[6*(c + d*x)])/(384*d) - (b*(3*a^2 - b^2)*Cos[8*(c + d*x)])/(1024*d) + (a*(14*a^2 + 3*b^2)*Sin[2*(c + d*x)])/(64*d) + (a*(7*a^2 - 3*b^2)*Sin[4*(c + d*x)])/(128*d) + (a*(2*a^2 - 3*b^2)*Sin[6*(c + d*x)])/(192*d) + (a*(a^2 - 3*b^2)*Sin[8*(c + d*x)])/(1024*d)","A",1
58,1,204,175,0.4161036,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{1225 a^3 \sin (c+d x)+245 a^3 \sin (3 (c+d x))+49 a^3 \sin (5 (c+d x))+5 a^3 \sin (7 (c+d x))-35 \left(9 a^2 b+b^3\right) \cos (3 (c+d x))-105 b \left(5 a^2+b^2\right) \cos (c+d x)-105 a^2 b \cos (5 (c+d x))-15 a^2 b \cos (7 (c+d x))+525 a b^2 \sin (c+d x)-35 a b^2 \sin (3 (c+d x))-63 a b^2 \sin (5 (c+d x))-15 a b^2 \sin (7 (c+d x))+7 b^3 \cos (5 (c+d x))+5 b^3 \cos (7 (c+d x))}{2240 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^2 b \cos ^7(c+d x)}{7 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,"(-105*b*(5*a^2 + b^2)*Cos[c + d*x] - 35*(9*a^2*b + b^3)*Cos[3*(c + d*x)] - 105*a^2*b*Cos[5*(c + d*x)] + 7*b^3*Cos[5*(c + d*x)] - 15*a^2*b*Cos[7*(c + d*x)] + 5*b^3*Cos[7*(c + d*x)] + 1225*a^3*Sin[c + d*x] + 525*a*b^2*Sin[c + d*x] + 245*a^3*Sin[3*(c + d*x)] - 35*a*b^2*Sin[3*(c + d*x)] + 49*a^3*Sin[5*(c + d*x)] - 63*a*b^2*Sin[5*(c + d*x)] + 5*a^3*Sin[7*(c + d*x)] - 15*a*b^2*Sin[7*(c + d*x)])/(2240*d)","A",1
59,1,171,216,0.3010228,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{a \left(5 a^2+3 b^2\right) (c+d x)}{16 d}+\frac{3 a \left(5 a^2+b^2\right) \sin (2 (c+d x))}{64 d}+\frac{3 a \left(a^2-b^2\right) \sin (4 (c+d x))}{64 d}+\frac{a \left(a^2-3 b^2\right) \sin (6 (c+d x))}{192 d}-\frac{3 b \left(5 a^2+b^2\right) \cos (2 (c+d x))}{64 d}-\frac{b \left(3 a^2-b^2\right) \cos (6 (c+d x))}{192 d}-\frac{3 a^2 b \cos (4 (c+d x))}{32 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^2 b \cos ^6(c+d x)}{2 d}-\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,"(a*(5*a^2 + 3*b^2)*(c + d*x))/(16*d) - (3*b*(5*a^2 + b^2)*Cos[2*(c + d*x)])/(64*d) - (3*a^2*b*Cos[4*(c + d*x)])/(32*d) - (b*(3*a^2 - b^2)*Cos[6*(c + d*x)])/(192*d) + (3*a*(5*a^2 + b^2)*Sin[2*(c + d*x)])/(64*d) + (3*a*(a^2 - b^2)*Sin[4*(c + d*x)])/(64*d) + (a*(a^2 - 3*b^2)*Sin[6*(c + d*x)])/(192*d)","A",1
60,1,150,140,0.289748,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{150 a^3 \sin (c+d x)+25 a^3 \sin (3 (c+d x))+3 a^3 \sin (5 (c+d x))-5 \left(9 a^2 b+b^3\right) \cos (3 (c+d x))-30 b \left(3 a^2+b^2\right) \cos (c+d x)-9 a^2 b \cos (5 (c+d x))+90 a b^2 \sin (c+d x)-15 a b^2 \sin (3 (c+d x))-9 a b^2 \sin (5 (c+d x))+3 b^3 \cos (5 (c+d x))}{240 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^2 b \cos ^5(c+d x)}{5 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,"(-30*b*(3*a^2 + b^2)*Cos[c + d*x] - 5*(9*a^2*b + b^3)*Cos[3*(c + d*x)] - 9*a^2*b*Cos[5*(c + d*x)] + 3*b^3*Cos[5*(c + d*x)] + 150*a^3*Sin[c + d*x] + 90*a*b^2*Sin[c + d*x] + 25*a^3*Sin[3*(c + d*x)] - 15*a*b^2*Sin[3*(c + d*x)] + 3*a^3*Sin[5*(c + d*x)] - 9*a*b^2*Sin[5*(c + d*x)])/(240*d)","A",1
61,1,94,78,0.390576,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{8 a^3 \sin (2 (c+d x))-4 \left(3 a^2 b+b^3\right) \cos (2 (c+d x))+\left(b^3-3 a^2 b\right) \cos (4 (c+d x))+12 a \left(a^2+b^2\right) (c+d x)+a \left(a^2-3 b^2\right) \sin (4 (c+d x))}{32 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,"(12*a*(a^2 + b^2)*(c + d*x) - 4*(3*a^2*b + b^3)*Cos[2*(c + d*x)] + (-3*a^2*b + b^3)*Cos[4*(c + d*x)] + 8*a^3*Sin[2*(c + d*x)] + a*(a^2 - 3*b^2)*Sin[4*(c + d*x)])/(32*d)","A",1
62,1,81,58,0.3403452,"\int (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\left(b^3-3 a^2 b\right) \cos (3 (c+d x))-9 b \left(a^2+b^2\right) \cos (c+d x)+2 a \sin (c+d x) \left(\left(a^2-3 b^2\right) \cos (2 (c+d x))+5 a^2+3 b^2\right)}{12 d}","\frac{(b \cos (c+d x)-a \sin (c+d x))^3}{3 d}-\frac{\left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x))}{d}",1,"(-9*b*(a^2 + b^2)*Cos[c + d*x] + (-3*a^2*b + b^3)*Cos[3*(c + d*x)] + 2*a*(5*a^2 + 3*b^2 + (a^2 - 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(12*d)","A",1
63,1,401,91,0.8083256,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{-a^5 \sqrt{-b^2} \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+a^5 \sqrt{-b^2} \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+5 a^4 b^2+4 a^3 \left(-b^2\right)^{3/2} \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)-4 a^3 \left(-b^2\right)^{3/2} \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+2 a^2 b^4+2 a^2 b^4 \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+2 a^2 b^4 \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+a b \left(a^4-2 a^2 b^2-3 b^4\right) \sin (2 (c+d x))+\left(-3 a^4 b^2-2 a^2 b^4+b^6\right) \cos (2 (c+d x))-3 a \left(-b^2\right)^{5/2} \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+3 a \sqrt{-b^2} b^4 \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)-b^6+2 b^6 \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+2 b^6 \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)}{4 b d \left(a^2+b^2\right)}","\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,"(5*a^4*b^2 + 2*a^2*b^4 - b^6 + (-3*a^4*b^2 - 2*a^2*b^4 + b^6)*Cos[2*(c + d*x)] + 2*a^2*b^4*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 2*b^6*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - a^5*Sqrt[-b^2]*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 4*a^3*(-b^2)^(3/2)*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 3*a*(-b^2)^(5/2)*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 2*a^2*b^4*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + 2*b^6*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + a^5*Sqrt[-b^2]*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + 3*a*b^4*Sqrt[-b^2]*Log[Sqrt[-b^2] + b*Tan[c + d*x]] - 4*a^3*(-b^2)^(3/2)*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + a*b*(a^4 - 2*a^2*b^2 - 3*b^4)*Sin[2*(c + d*x)])/(4*b*(a^2 + b^2)*d)","B",1
64,1,131,86,1.0714295,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\sec (c+d x) \left(a^3 \sin (2 (c+d x))+\left(b^3-3 a^2 b\right) \cos (2 (c+d x))-3 a^2 b-3 a b^2 \sin (2 (c+d x))-6 a b^2 \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 b^3\right)}{2 d}","\frac{a^3 \sin (c+d x)}{d}-\frac{3 a^2 b \cos (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,"(Sec[c + d*x]*(-3*a^2*b + 3*b^3 + (-3*a^2*b + b^3)*Cos[2*(c + d*x)] - 6*a*b^2*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + a^3*Sin[2*(c + d*x)] - 3*a*b^2*Sin[2*(c + d*x)]))/(2*d)","A",1
65,1,79,72,0.2670992,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{6 a b^2 \tan (c+d x)+(-b+i a)^3 \log (-\tan (c+d x)+i)-(b+i a)^3 \log (\tan (c+d x)+i)+b^3 \tan ^2(c+d x)}{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,"((I*a - b)^3*Log[I - Tan[c + d*x]] - (I*a + b)^3*Log[I + Tan[c + d*x]] + 6*a*b^2*Tan[c + d*x] + b^3*Tan[c + d*x]^2)/(2*d)","C",1
66,1,293,103,1.6311195,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{12 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-6 a \left(2 a^2-3 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b \sin ^2\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(\left(18 a^2-5 b^2\right) \cos (2 (c+d x))+18 a^2+2 b^2 \cos (c+d x)-b^2\right)+36 a^2 b+\frac{9 a b^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{9 a b^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-18 a b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{b^3}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{b^3}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-10 b^3}{12 d}","\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a^2 b \sec (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,"(36*a^2*b - 10*b^3 - 6*a*(2*a^2 - 3*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 18*a*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (9*a*b^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + b^3/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + 2*b*(18*a^2 - b^2 + 2*b^2*Cos[c + d*x] + (18*a^2 - 5*b^2)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2]^2 - (9*a*b^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + b^3/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(12*d)","B",1
67,1,57,30,0.1781572,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\tan (c+d x) \left(4 a^3+6 a^2 b \tan (c+d x)+4 a b^2 \tan ^2(c+d x)+b^3 \tan ^3(c+d x)\right)}{4 d}","\frac{\tan ^4(c+d x) (a \cot (c+d x)+b)^4}{4 b d}",1,"(Tan[c + d*x]*(4*a^3 + 6*a^2*b*Tan[c + d*x] + 4*a*b^2*Tan[c + d*x]^2 + b^3*Tan[c + d*x]^3))/(4*d)","A",1
68,1,464,158,1.3248064,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\sec ^5(c+d x) \left(240 a^3 \sin (2 (c+d x))+120 a^3 \sin (4 (c+d x))-300 a^3 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-60 a^3 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+300 a^3 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+60 a^3 \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+320 \left(3 a^2 b-b^3\right) \cos (2 (c+d x))-150 a \left(4 a^2-3 b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+960 a^2 b+540 a b^2 \sin (2 (c+d x))-90 a b^2 \sin (4 (c+d x))+225 a b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+45 a b^2 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-225 a b^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-45 a b^2 \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+64 b^3\right)}{1920 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{a^2 b \sec ^3(c+d x)}{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,"(Sec[c + d*x]^5*(960*a^2*b + 64*b^3 + 320*(3*a^2*b - b^3)*Cos[2*(c + d*x)] - 300*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 225*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 60*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 45*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 150*a*(4*a^2 - 3*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 300*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 225*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 45*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 240*a^3*Sin[2*(c + d*x)] + 540*a*b^2*Sin[2*(c + d*x)] + 120*a^3*Sin[4*(c + d*x)] - 90*a*b^2*Sin[4*(c + d*x)]))/(1920*d)","B",1
69,1,54,120,0.3729966,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{(a+b \tan (c+d x))^4 \left(a^2-4 a b \tan (c+d x)+10 b^2 \tan ^2(c+d x)+15 b^2\right)}{60 b^3 d}","\frac{a^3 \tan (c+d x)}{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{3 a b^2 \tan ^5(c+d x)}{5 d}+\frac{b^3 \tan ^6(c+d x)}{6 d}",1,"((a + b*Tan[c + d*x])^4*(a^2 + 15*b^2 - 4*a*b*Tan[c + d*x] + 10*b^2*Tan[c + d*x]^2))/(60*b^3*d)","A",1
70,1,637,210,2.131319,"\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\sec ^7(c+d x) \left(4340 a^3 \sin (2 (c+d x))+2800 a^3 \sin (4 (c+d x))+420 a^3 \sin (6 (c+d x))-4410 a^3 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-1470 a^3 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-210 a^3 \cos (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4410 a^3 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+1470 a^3 \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+210 a^3 \cos (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+3584 \left(3 a^2 b-b^3\right) \cos (2 (c+d x))-3675 a \left(2 a^2-b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+10752 a^2 b+6790 a b^2 \sin (2 (c+d x))-1400 a b^2 \sin (4 (c+d x))-210 a b^2 \sin (6 (c+d x))+2205 a b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+735 a b^2 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 a b^2 \cos (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2205 a b^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-735 a b^2 \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 a b^2 \cos (7 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+1536 b^3\right)}{35840 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^2 b \sec ^5(c+d x)}{5 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,"(Sec[c + d*x]^7*(10752*a^2*b + 1536*b^3 + 3584*(3*a^2*b - b^3)*Cos[2*(c + d*x)] - 4410*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2205*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 1470*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 735*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 210*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*a*b^2*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3675*a*(2*a^2 - b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4410*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2205*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 1470*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 735*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 210*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 105*a*b^2*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4340*a^3*Sin[2*(c + d*x)] + 6790*a*b^2*Sin[2*(c + d*x)] + 2800*a^3*Sin[4*(c + d*x)] - 1400*a*b^2*Sin[4*(c + d*x)] + 420*a^3*Sin[6*(c + d*x)] - 210*a*b^2*Sin[6*(c + d*x)]))/(35840*d)","B",1
71,1,115,174,0.6349457,"\int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\frac{1}{3} \left(3 a^2+b^2\right) (a+b \tan (c+d x))^6-\frac{4}{5} a \left(a^2+b^2\right) (a+b \tan (c+d x))^5+\frac{1}{4} \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^4+\frac{1}{8} (a+b \tan (c+d x))^8-\frac{4}{7} a (a+b \tan (c+d x))^7}{b^5 d}","\frac{a^3 \tan (c+d x)}{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{3 a b^2 \tan ^7(c+d x)}{7 d}+\frac{b^3 \tan ^8(c+d x)}{8 d}",1,"(((a^2 + b^2)^2*(a + b*Tan[c + d*x])^4)/4 - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^5)/5 + ((3*a^2 + b^2)*(a + b*Tan[c + d*x])^6)/3 - (4*a*(a + b*Tan[c + d*x])^7)/7 + (a + b*Tan[c + d*x])^8/8)/(b^5*d)","A",1
72,1,810,259,4.0247324,"\int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\sec ^9(c+d x) \left(-211680 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) a^3-90720 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) a^3-22680 \cos (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) a^3-2520 \cos (9 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) a^3+211680 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) a^3+90720 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) a^3+22680 \cos (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) a^3+2520 \cos (9 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) a^3+223776 \sin (2 (c+d x)) a^3+167328 \sin (4 (c+d x)) a^3+43680 \sin (6 (c+d x)) a^3+5040 \sin (8 (c+d x)) a^3+442368 b a^2+79380 b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) a+34020 b^2 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) a+8505 b^2 \cos (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) a+945 b^2 \cos (9 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) a-39690 \left(8 a^2-3 b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)\right) a-79380 b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) a-34020 b^2 \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) a-8505 b^2 \cos (7 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) a-945 b^2 \cos (9 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) a+303156 b^2 \sin (2 (c+d x)) a-62748 b^2 \sin (4 (c+d x)) a-16380 b^2 \sin (6 (c+d x)) a-1890 b^2 \sin (8 (c+d x)) a+81920 b^3+147456 \left(3 a^2 b-b^3\right) \cos (2 (c+d x))\right)}{2064384 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{3 a^2 b \sec ^7(c+d x)}{7 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,"(Sec[c + d*x]^9*(442368*a^2*b + 81920*b^3 + 147456*(3*a^2*b - b^3)*Cos[2*(c + d*x)] - 211680*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 79380*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 90720*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 34020*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 22680*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8505*a*b^2*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2520*a^3*Cos[9*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 945*a*b^2*Cos[9*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 39690*a*(8*a^2 - 3*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 211680*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 79380*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 90720*a^3*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 34020*a*b^2*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 22680*a^3*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 8505*a*b^2*Cos[7*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2520*a^3*Cos[9*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 945*a*b^2*Cos[9*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 223776*a^3*Sin[2*(c + d*x)] + 303156*a*b^2*Sin[2*(c + d*x)] + 167328*a^3*Sin[4*(c + d*x)] - 62748*a*b^2*Sin[4*(c + d*x)] + 43680*a^3*Sin[6*(c + d*x)] - 16380*a*b^2*Sin[6*(c + d*x)] + 5040*a^3*Sin[8*(c + d*x)] - 1890*a*b^2*Sin[8*(c + d*x)]))/(2064384*d)","B",1
73,1,177,213,2.0734631,"\int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\frac{3}{8} \left(5 a^2+b^2\right) (a+b \tan (c+d x))^8-\frac{4}{7} a \left(5 a^2+3 b^2\right) (a+b \tan (c+d x))^7+\frac{1}{2} \left(a^2+b^2\right) \left(5 a^2+b^2\right) (a+b \tan (c+d x))^6-\frac{6}{5} a \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^5+\frac{1}{4} \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^4+\frac{1}{10} (a+b \tan (c+d x))^{10}-\frac{2}{3} a (a+b \tan (c+d x))^9}{b^7 d}","\frac{a^3 \tan (c+d x)}{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 b^2 \tan ^9(c+d x)}{3 d}+\frac{b^3 \tan ^{10}(c+d x)}{10 d}",1,"(((a^2 + b^2)^3*(a + b*Tan[c + d*x])^4)/4 - (6*a*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^5)/5 + ((a^2 + b^2)*(5*a^2 + b^2)*(a + b*Tan[c + d*x])^6)/2 - (4*a*(5*a^2 + 3*b^2)*(a + b*Tan[c + d*x])^7)/7 + (3*(5*a^2 + b^2)*(a + b*Tan[c + d*x])^8)/8 - (2*a*(a + b*Tan[c + d*x])^9)/3 + (a + b*Tan[c + d*x])^10/10)/(b^7*d)","A",1
74,1,237,279,0.7094971,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{420 \left(21 a^4-b^4\right) \sin (3 (c+d x))-5040 a^3 b \cos (5 (c+d x))-2520 a b \left(7 a^2+3 b^2\right) \cos (c+d x)-1680 a b \left(7 a^2+2 b^2\right) \cos (3 (c+d x))-180 a b \left(7 a^2-3 b^2\right) \cos (7 (c+d x))-140 a b \left(a^2-b^2\right) \cos (9 (c+d x))+1890 \left(21 a^4+14 a^2 b^2+b^4\right) \sin (c+d x)+252 \left(9 a^4-12 a^2 b^2-b^4\right) \sin (5 (c+d x))+45 \left(9 a^4-30 a^2 b^2+b^4\right) \sin (7 (c+d x))+35 \left(a^4-6 a^2 b^2+b^4\right) \sin (9 (c+d x))}{80640 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^3 b \cos ^9(c+d x)}{9 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 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,"(-2520*a*b*(7*a^2 + 3*b^2)*Cos[c + d*x] - 1680*a*b*(7*a^2 + 2*b^2)*Cos[3*(c + d*x)] - 5040*a^3*b*Cos[5*(c + d*x)] - 180*a*b*(7*a^2 - 3*b^2)*Cos[7*(c + d*x)] - 140*a*b*(a^2 - b^2)*Cos[9*(c + d*x)] + 1890*(21*a^4 + 14*a^2*b^2 + b^4)*Sin[c + d*x] + 420*(21*a^4 - b^4)*Sin[3*(c + d*x)] + 252*(9*a^4 - 12*a^2*b^2 - b^4)*Sin[5*(c + d*x)] + 45*(9*a^4 - 30*a^2*b^2 + b^4)*Sin[7*(c + d*x)] + 35*(a^4 - 6*a^2*b^2 + b^4)*Sin[9*(c + d*x)])/(80640*d)","A",1
75,1,222,381,0.6100571,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{96 a^2 \left(7 a^2+3 b^2\right) \sin (2 (c+d x))+32 a^2 \left(a^2-3 b^2\right) \sin (6 (c+d x))-96 a b \left(7 a^2+3 b^2\right) \cos (2 (c+d x))-48 a b \left(7 a^2+b^2\right) \cos (4 (c+d x))-32 a b \left(3 a^2-b^2\right) \cos (6 (c+d x))-12 a b \left(a^2-b^2\right) \cos (8 (c+d x))+24 \left(35 a^4+30 a^2 b^2+3 b^4\right) (c+d x)+24 \left(7 a^4-6 a^2 b^2-b^4\right) \sin (4 (c+d x))+3 \left(a^4-6 a^2 b^2+b^4\right) \sin (8 (c+d x))}{3072 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^3 b \cos ^8(c+d x)}{2 d}-\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 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,"(24*(35*a^4 + 30*a^2*b^2 + 3*b^4)*(c + d*x) - 96*a*b*(7*a^2 + 3*b^2)*Cos[2*(c + d*x)] - 48*a*b*(7*a^2 + b^2)*Cos[4*(c + d*x)] - 32*a*b*(3*a^2 - b^2)*Cos[6*(c + d*x)] - 12*a*b*(a^2 - b^2)*Cos[8*(c + d*x)] + 96*a^2*(7*a^2 + 3*b^2)*Sin[2*(c + d*x)] + 24*(7*a^4 - 6*a^2*b^2 - b^4)*Sin[4*(c + d*x)] + 32*a^2*(a^2 - 3*b^2)*Sin[6*(c + d*x)] + 3*(a^4 - 6*a^2*b^2 + b^4)*Sin[8*(c + d*x)])/(3072*d)","A",1
76,1,204,220,0.5374754,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{-140 a b \left(5 a^2+3 b^2\right) \cos (c+d x)-140 a b \left(3 a^2+b^2\right) \cos (3 (c+d x))-28 a b \left(5 a^2-b^2\right) \cos (5 (c+d x))-20 a b \left(a^2-b^2\right) \cos (7 (c+d x))+35 \left(35 a^4+30 a^2 b^2+3 b^4\right) \sin (c+d x)+35 \left(7 a^4-2 a^2 b^2-b^4\right) \sin (3 (c+d x))+7 \left(7 a^4-18 a^2 b^2-b^4\right) \sin (5 (c+d x))+5 \left(a^4-6 a^2 b^2+b^4\right) \sin (7 (c+d x))}{2240 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^3 b \cos ^7(c+d x)}{7 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 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,"(-140*a*b*(5*a^2 + 3*b^2)*Cos[c + d*x] - 140*a*b*(3*a^2 + b^2)*Cos[3*(c + d*x)] - 28*a*b*(5*a^2 - b^2)*Cos[5*(c + d*x)] - 20*a*b*(a^2 - b^2)*Cos[7*(c + d*x)] + 35*(35*a^4 + 30*a^2*b^2 + 3*b^4)*Sin[c + d*x] + 35*(7*a^4 - 2*a^2*b^2 - b^4)*Sin[3*(c + d*x)] + 7*(7*a^4 - 18*a^2*b^2 - b^4)*Sin[5*(c + d*x)] + 5*(a^4 - 6*a^2*b^2 + b^4)*Sin[7*(c + d*x)])/(2240*d)","A",1
77,1,178,301,0.4255744,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{-24 a^3 b \cos (4 (c+d x))+12 (a-i b) (a+i b) \left(5 a^2+b^2\right) (c+d x)-12 a b \left(5 a^2+3 b^2\right) \cos (2 (c+d x))-4 a b \left(a^2-b^2\right) \cos (6 (c+d x))+3 \left(15 a^4+6 a^2 b^2-b^4\right) \sin (2 (c+d x))+3 \left(3 a^4-6 a^2 b^2-b^4\right) \sin (4 (c+d x))+\left(a^4-6 a^2 b^2+b^4\right) \sin (6 (c+d x))}{192 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^3 b \cos ^6(c+d x)}{3 d}-\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 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,"(12*(a - I*b)*(a + I*b)*(5*a^2 + b^2)*(c + d*x) - 12*a*b*(5*a^2 + 3*b^2)*Cos[2*(c + d*x)] - 24*a^3*b*Cos[4*(c + d*x)] - 4*a*b*(a^2 - b^2)*Cos[6*(c + d*x)] + 3*(15*a^4 + 6*a^2*b^2 - b^4)*Sin[2*(c + d*x)] + 3*(3*a^4 - 6*a^2*b^2 - b^4)*Sin[4*(c + d*x)] + (a^4 - 6*a^2*b^2 + b^4)*Sin[6*(c + d*x)])/(192*d)","C",1
78,1,146,165,0.4276424,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{-120 a b \left(a^2+b^2\right) \cos (c+d x)-20 a b \left(3 a^2+b^2\right) \cos (3 (c+d x))-12 a b \left(a^2-b^2\right) \cos (5 (c+d x))+30 \left(5 a^4+6 a^2 b^2+b^4\right) \sin (c+d x)+5 \left(5 a^4-6 a^2 b^2-3 b^4\right) \sin (3 (c+d x))+3 \left(a^4-6 a^2 b^2+b^4\right) \sin (5 (c+d x))}{240 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^3 b \cos ^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 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,"(-120*a*b*(a^2 + b^2)*Cos[c + d*x] - 20*a*b*(3*a^2 + b^2)*Cos[3*(c + d*x)] - 12*a*b*(a^2 - b^2)*Cos[5*(c + d*x)] + 30*(5*a^4 + 6*a^2*b^2 + b^4)*Sin[c + d*x] + 5*(5*a^4 - 6*a^2*b^2 - 3*b^4)*Sin[3*(c + d*x)] + 3*(a^4 - 6*a^2*b^2 + b^4)*Sin[5*(c + d*x)])/(240*d)","A",1
79,1,107,108,0.4323839,"\int (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{8 \left(a^4-b^4\right) \sin (2 (c+d x))+12 \left(a^2+b^2\right)^2 (c+d x)-16 a b \left(a^2+b^2\right) \cos (2 (c+d x))-4 a b \left(a^2-b^2\right) \cos (4 (c+d x))+\left(a^4-6 a^2 b^2+b^4\right) \sin (4 (c+d x))}{32 d}","-\frac{3 \left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{8 d}+\frac{3}{8} x \left(a^2+b^2\right)^2-\frac{(b \cos (c+d x)-a \sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{4 d}",1,"(12*(a^2 + b^2)^2*(c + d*x) - 16*a*b*(a^2 + b^2)*Cos[2*(c + d*x)] - 4*a*b*(a^2 - b^2)*Cos[4*(c + d*x)] + 8*(a^4 - b^4)*Sin[2*(c + d*x)] + (a^4 - 6*a^2*b^2 + b^4)*Sin[4*(c + d*x)])/(32*d)","A",1
80,1,181,150,0.9529847,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{9 a^4 \sin (c+d x)+a^4 \sin (3 (c+d x))+\left(4 a b^3-4 a^3 b\right) \cos (3 (c+d x))+18 a^2 b^2 \sin (c+d x)-6 a^2 b^2 \sin (3 (c+d x))-12 a b \left(a^2+3 b^2\right) \cos (c+d x)-15 b^4 \sin (c+d x)+b^4 \sin (3 (c+d x))-12 b^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{12 d}","-\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin (c+d x)}{d}-\frac{4 a^3 b \cos ^3(c+d x)}{3 d}+\frac{2 a^2 b^2 \sin ^3(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,"(-12*a*b*(a^2 + 3*b^2)*Cos[c + d*x] + (-4*a^3*b + 4*a*b^3)*Cos[3*(c + d*x)] - 12*b^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*a^4*Sin[c + d*x] + 18*a^2*b^2*Sin[c + d*x] - 15*b^4*Sin[c + d*x] + a^4*Sin[3*(c + d*x)] - 6*a^2*b^2*Sin[3*(c + d*x)] + b^4*Sin[3*(c + d*x)])/(12*d)","A",1
81,1,477,119,6.2611309,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{b^3 \left(\frac{\cos ^2(c+d x) (a+b \tan (c+d x))^5 \left(a b \tan (c+d x)+b^2\right)}{2 b^4 \left(a^2+b^2\right)}-\frac{\left(3 b^2-5 a^2\right) \left(b \left(6 a^2-b^2\right) \tan (c+d x)+\frac{1}{2} \left(\frac{a^4-6 a^2 b^2+b^4}{\sqrt{-b^2}}+4 a (a-b) (a+b)\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\frac{1}{2} \left(4 a (a-b) (a+b)-\frac{a^4-6 a^2 b^2+b^4}{\sqrt{-b^2}}\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+2 a b^2 \tan ^2(c+d x)+\frac{1}{3} b^3 \tan ^3(c+d x)\right)+4 a \left(\frac{1}{2} b^2 \left(10 a^2-b^2\right) \tan ^2(c+d x)+5 a b \left(2 a^2-b^2\right) \tan (c+d x)+\frac{1}{2} \left(5 a^4-10 a^2 b^2+\frac{a^5-10 a^3 b^2+5 a b^4}{\sqrt{-b^2}}+b^4\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\frac{1}{2} \left(5 a^4-10 a^2 b^2-\frac{a^5-10 a^3 b^2+5 a b^4}{\sqrt{-b^2}}+b^4\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+\frac{5}{3} a b^3 \tan ^3(c+d x)+\frac{1}{4} b^4 \tan ^4(c+d x)\right)}{2 b^2 \left(a^2+b^2\right)}\right)}{d}","\frac{\sin ^2(c+d x) \left(4 a b \left(a^2-b^2\right)+\left(a^4-6 a^2 b^2+b^4\right) \cot (c+d x)\right)}{2 d}+\frac{1}{2} x \left(a^4+6 a^2 b^2-3 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,"(b^3*((Cos[c + d*x]^2*(a + b*Tan[c + d*x])^5*(b^2 + a*b*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)) - ((-5*a^2 + 3*b^2)*(((4*a*(a - b)*(a + b) + (a^4 - 6*a^2*b^2 + b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/2 + ((4*a*(a - b)*(a + b) - (a^4 - 6*a^2*b^2 + b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/2 + b*(6*a^2 - b^2)*Tan[c + d*x] + 2*a*b^2*Tan[c + d*x]^2 + (b^3*Tan[c + d*x]^3)/3) + 4*a*(((5*a^4 - 10*a^2*b^2 + b^4 + (a^5 - 10*a^3*b^2 + 5*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/2 + ((5*a^4 - 10*a^2*b^2 + b^4 - (a^5 - 10*a^3*b^2 + 5*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/2 + 5*a*b*(2*a^2 - b^2)*Tan[c + d*x] + (b^2*(10*a^2 - b^2)*Tan[c + d*x]^2)/2 + (5*a*b^3*Tan[c + d*x]^3)/3 + (b^4*Tan[c + d*x]^4)/4))/(2*b^2*(a^2 + b^2))))/d","B",1
82,1,268,151,2.5558613,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{4 a^4 \sin (c+d x)-24 a^2 b^2 \sin (c+d x)-16 a b \left(a^2-b^2\right) \cos (c+d x)-24 a^2 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+24 a^2 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+32 a b^3 \sin ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)+16 a b^3+4 b^4 \sin (c+d x)+\frac{b^4}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^4}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+6 b^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 b^4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{a^4 \sin (c+d x)}{d}-\frac{4 a^3 b \cos (c+d x)}{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 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,"(16*a*b^3 - 16*a*b*(a^2 - b^2)*Cos[c + d*x] - 24*a^2*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*b^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 24*a^2*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 6*b^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b^4/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + 32*a*b^3*Sec[c + d*x]*Sin[(c + d*x)/2]^2 - b^4/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 4*a^4*Sin[c + d*x] - 24*a^2*b^2*Sin[c + d*x] + 4*b^4*Sin[c + d*x])/(4*d)","A",1
83,1,105,103,0.40363,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{-6 b^2 \left(b^2-6 a^2\right) \tan (c+d x)+12 a b^3 \tan ^2(c+d x)+3 i (a-i b)^4 \log (\tan (c+d x)+i)-3 i (a+i b)^4 \log (-\tan (c+d x)+i)+2 b^4 \tan ^3(c+d x)}{6 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(a^4-6 a^2 b^2+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,"((-3*I)*(a + I*b)^4*Log[I - Tan[c + d*x]] + (3*I)*(a - I*b)^4*Log[I + Tan[c + d*x]] - 6*b^2*(-6*a^2 + b^2)*Tan[c + d*x] + 12*a*b^3*Tan[c + d*x]^2 + 2*b^4*Tan[c + d*x]^3)/(6*d)","C",1
84,1,936,168,6.2349389,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{2 a b \left(6 a^2-5 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{3 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-8 a^4+24 b^2 a^2-3 b^4\right) \cos ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(8 a^4-24 b^2 a^2+3 b^4\right) \cos ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{2 a b^3 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{2 \cos ^4(c+d x) \left(6 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)-5 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{2 \cos ^4(c+d x) \left(6 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)-5 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-15 b^4+16 a b^3+72 a^2 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(15 b^4+16 a b^3-72 a^2 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{2 a b^3 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^4 \cos ^4(c+d x) (a+b \tan (c+d x))^4}{16 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{b^4 \cos ^4(c+d x) (a+b \tan (c+d x))^4}{16 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^4}","\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a^3 b \sec (c+d x)}{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 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,"(2*a*b*(6*a^2 - 5*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-8*a^4 + 24*a^2*b^2 - 3*b^4)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((8*a^4 - 24*a^2*b^2 + 3*b^4)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (b^4*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(16*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((72*a^2*b^2 + 16*a*b^3 - 15*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(48*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (2*a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (b^4*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(16*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (2*a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-72*a^2*b^2 + 16*a*b^3 + 15*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(48*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (2*Cos[c + d*x]^4*(6*a^3*b*Sin[(c + d*x)/2] - 5*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (2*Cos[c + d*x]^4*(6*a^3*b*Sin[(c + d*x)/2] - 5*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)","B",1
85,1,73,30,0.3159263,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\tan (c+d x) \left(5 a^4+10 a^3 b \tan (c+d x)+10 a^2 b^2 \tan ^2(c+d x)+5 a b^3 \tan ^3(c+d x)+b^4 \tan ^4(c+d x)\right)}{5 d}","\frac{\tan ^5(c+d x) (a \cot (c+d x)+b)^5}{5 b d}",1,"(Tan[c + d*x]*(5*a^4 + 10*a^3*b*Tan[c + d*x] + 10*a^2*b^2*Tan[c + d*x]^2 + 5*a*b^3*Tan[c + d*x]^3 + b^4*Tan[c + d*x]^4))/(5*d)","B",1
86,1,1342,258,6.2512636,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{a b \left(20 a^2-11 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{30 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-8 a^4+12 b^2 a^2-b^4\right) \cos ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(8 a^4-12 b^2 a^2+b^4\right) \cos ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{16 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{a b^3 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{5 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(20 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)-11 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{30 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(20 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)-11 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{30 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(11 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)-20 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{30 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(11 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)-20 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{30 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(120 a^4+160 b a^3-180 b^2 a^2-88 b^3 a+15 b^4\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{480 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-120 a^4+160 b a^3+180 b^2 a^2-88 b^3 a-15 b^4\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{480 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-5 b^4+8 a b^3+30 a^2 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{80 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(5 b^4+8 a b^3-30 a^2 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{80 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{a b^3 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{5 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^4 \cos ^4(c+d x) (a+b \tan (c+d x))^4}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^6 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{b^4 \cos ^4(c+d x) (a+b \tan (c+d x))^4}{48 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^6 (a \cos (c+d x)+b \sin (c+d x))^4}","\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^3 b \sec ^3(c+d x)}{3 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 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*b*(20*a^2 - 11*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(30*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-8*a^4 + 12*a^2*b^2 - b^4)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(16*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((8*a^4 - 12*a^2*b^2 + b^4)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(16*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (b^4*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(48*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((30*a^2*b^2 + 8*a*b^3 - 5*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(80*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((120*a^4 + 160*a^3*b - 180*a^2*b^2 - 88*a*b^3 + 15*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(480*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(5*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (b^4*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(48*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-30*a^2*b^2 + 8*a*b^3 + 5*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(80*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-120*a^4 + 160*a^3*b + 180*a^2*b^2 - 88*a*b^3 - 15*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(480*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(20*a^3*b*Sin[(c + d*x)/2] - 11*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(30*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(20*a^3*b*Sin[(c + d*x)/2] - 11*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(30*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(-20*a^3*b*Sin[(c + d*x)/2] + 11*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(30*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(-20*a^3*b*Sin[(c + d*x)/2] + 11*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(30*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)","B",1
87,1,54,143,0.5523904,"\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{(a+b \tan (c+d x))^5 \left(a^2-5 a b \tan (c+d x)+15 b^2 \tan ^2(c+d x)+21 b^2\right)}{105 b^3 d}","\frac{a^4 \tan (c+d x)}{d}+\frac{2 a^3 b \tan ^2(c+d x)}{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 b^3 \tan ^6(c+d x)}{3 d}+\frac{b^4 \tan ^7(c+d x)}{7 d}",1,"((a + b*Tan[c + d*x])^5*(a^2 + 21*b^2 - 5*a*b*Tan[c + d*x] + 15*b^2*Tan[c + d*x]^2))/(105*b^3*d)","A",1
88,1,1732,330,6.3914417,"\int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{a b \left(42 a^2-17 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{140 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{3 \left(16 a^4-16 b^2 a^2+b^4\right) \cos ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{128 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{3 \left(16 a^4-16 b^2 a^2+b^4\right) \cos ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{128 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{a b^3 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{14 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^7 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(42 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)-17 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{140 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(42 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)-17 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{140 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(7 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)-2 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{35 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(2 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)-7 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{35 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(17 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)-42 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{140 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(17 a b^3 \sin \left(\frac{1}{2} (c+d x)\right)-42 a^3 b \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{140 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(1680 a^4+1344 b a^3-1680 b^2 a^2-544 b^3 a+105 b^4\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{8960 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-1680 a^4+1344 b a^3+1680 b^2 a^2-544 b^3 a-105 b^4\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{8960 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(560 a^4+896 b a^3-256 b^3 a-35 b^4\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{8960 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-560 a^4+896 b a^3-256 b^3 a+35 b^4\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{8960 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-7 b^4+16 a b^3+56 a^2 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{448 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^6 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(7 b^4+16 a b^3-56 a^2 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{448 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^6 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{a b^3 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{14 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^7 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^4 \cos ^4(c+d x) (a+b \tan (c+d x))^4}{128 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^8 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{b^4 \cos ^4(c+d x) (a+b \tan (c+d x))^4}{128 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^8 (a \cos (c+d x)+b \sin (c+d x))^4}","\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^3 b \sec ^5(c+d x)}{5 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 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,"(a*b*(42*a^2 - 17*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(140*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (3*(16*a^4 - 16*a^2*b^2 + b^4)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(128*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (3*(16*a^4 - 16*a^2*b^2 + b^4)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(128*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (b^4*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(128*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((56*a^2*b^2 + 16*a*b^3 - 7*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(448*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((560*a^4 + 896*a^3*b - 256*a*b^3 - 35*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(8960*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((1680*a^4 + 1344*a^3*b - 1680*a^2*b^2 - 544*a*b^3 + 105*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(8960*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(14*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (b^4*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(128*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a*b^3*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(14*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-56*a^2*b^2 + 16*a*b^3 + 7*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(448*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-560*a^4 + 896*a^3*b - 256*a*b^3 + 35*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(8960*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-1680*a^4 + 1344*a^3*b + 1680*a^2*b^2 - 544*a*b^3 - 105*b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(8960*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(42*a^3*b*Sin[(c + d*x)/2] - 17*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(140*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(42*a^3*b*Sin[(c + d*x)/2] - 17*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(140*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(7*a^3*b*Sin[(c + d*x)/2] - 2*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(35*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(-7*a^3*b*Sin[(c + d*x)/2] + 2*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(35*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(-42*a^3*b*Sin[(c + d*x)/2] + 17*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(140*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(-42*a^3*b*Sin[(c + d*x)/2] + 17*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(140*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)","B",1
89,1,115,201,0.9045358,"\int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\frac{2}{7} \left(3 a^2+b^2\right) (a+b \tan (c+d x))^7-\frac{2}{3} a \left(a^2+b^2\right) (a+b \tan (c+d x))^6+\frac{1}{5} \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^5+\frac{1}{9} (a+b \tan (c+d x))^9-\frac{1}{2} a (a+b \tan (c+d x))^8}{b^5 d}","\frac{a^4 \tan (c+d x)}{d}+\frac{2 a^3 b \tan ^2(c+d x)}{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{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{\left(a^4+12 a^2 b^2+b^4\right) \tan ^5(c+d x)}{5 d}+\frac{a b^3 \tan ^8(c+d x)}{2 d}+\frac{b^4 \tan ^9(c+d x)}{9 d}",1,"(((a^2 + b^2)^2*(a + b*Tan[c + d*x])^5)/5 - (2*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^6)/3 + (2*(3*a^2 + b^2)*(a + b*Tan[c + d*x])^7)/7 - (a*(a + b*Tan[c + d*x])^8)/2 + (a + b*Tan[c + d*x])^9/9)/(b^5*d)","A",1
90,1,242,408,1.3124167,"\int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{10 \sec ^9(c+d x) \left(32768 a b \left(27 a^2+b^2\right)+189 \left(592 a^4+1604 a^2 b^2+739 b^4\right) \tan (c+d x)\right)-80640 \left(80 a^4-60 a^2 b^2+3 b^4\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 \sec ^{10}(c+d x) \left(983040 a b \left(a^2-b^2\right) \cos (3 (c+d x))+420 \left(1552 a^4+1908 a^2 b^2-505 b^4\right) \sin (3 (c+d x))+7 \left(80 a^4-60 a^2 b^2+3 b^4\right) (628 \sin (5 (c+d x))+145 \sin (7 (c+d x))+15 \sin (9 (c+d x)))\right)}{20643840 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^3 b \sec ^7(c+d x)}{7 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 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,"(-80640*(80*a^4 - 60*a^2*b^2 + 3*b^4)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*Sec[c + d*x]^10*(983040*a*b*(a^2 - b^2)*Cos[3*(c + d*x)] + 420*(1552*a^4 + 1908*a^2*b^2 - 505*b^4)*Sin[3*(c + d*x)] + 7*(80*a^4 - 60*a^2*b^2 + 3*b^4)*(628*Sin[5*(c + d*x)] + 145*Sin[7*(c + d*x)] + 15*Sin[9*(c + d*x)])) + 10*Sec[c + d*x]^9*(32768*a*b*(27*a^2 + b^2) + 189*(592*a^4 + 1604*a^2*b^2 + 739*b^4)*Tan[c + d*x]))/(20643840*d)","A",1
91,1,175,254,1.8043104,"\int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^12*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\frac{1}{3} \left(5 a^2+b^2\right) (a+b \tan (c+d x))^9-\frac{1}{2} a \left(5 a^2+3 b^2\right) (a+b \tan (c+d x))^8+\frac{3}{7} \left(a^2+b^2\right) \left(5 a^2+b^2\right) (a+b \tan (c+d x))^7-a \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^6+\frac{1}{5} \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^5+\frac{1}{11} (a+b \tan (c+d x))^{11}-\frac{3}{5} a (a+b \tan (c+d x))^{10}}{b^7 d}","\frac{a^4 \tan (c+d x)}{d}+\frac{2 a^3 b \tan ^2(c+d x)}{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{2 a b \left(a^2+b^2\right) \tan ^6(c+d x)}{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{\left(a^4+18 a^2 b^2+3 b^4\right) \tan ^7(c+d x)}{7 d}+\frac{\left(3 a^4+18 a^2 b^2+b^4\right) \tan ^5(c+d x)}{5 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^2 + b^2)^3*(a + b*Tan[c + d*x])^5)/5 - a*(a^2 + b^2)^2*(a + b*Tan[c + d*x])^6 + (3*(a^2 + b^2)*(5*a^2 + b^2)*(a + b*Tan[c + d*x])^7)/7 - (a*(5*a^2 + 3*b^2)*(a + b*Tan[c + d*x])^8)/2 + ((5*a^2 + b^2)*(a + b*Tan[c + d*x])^9)/3 - (3*a*(a + b*Tan[c + d*x])^10)/5 + (a + b*Tan[c + d*x])^11/11)/(b^7*d)","A",1
92,1,307,515,1.2274183,"\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{-1200 a^2 b \left(3 a^2+b^2\right) \cos (4 (c+d x))-300 a^2 b \left(a^2-b^2\right) \cos (8 (c+d x))+120 a \left(63 a^4+70 a^2 b^2+15 b^4\right) (c+d x)+300 a \left(21 a^4+14 a^2 b^2+b^4\right) \sin (2 (c+d x))+600 a \left(3 a^4-2 a^2 b^2-b^4\right) \sin (4 (c+d x))+50 a \left(9 a^4-26 a^2 b^2-3 b^4\right) \sin (6 (c+d x))+75 a \left(a^4-6 a^2 b^2+b^4\right) \sin (8 (c+d x))+6 a \left(a^4-10 a^2 b^2+5 b^4\right) \sin (10 (c+d x))-300 b \left(21 a^4+14 a^2 b^2+b^4\right) \cos (2 (c+d x))+50 b \left(-27 a^4+6 a^2 b^2+b^4\right) \cos (6 (c+d x))-6 b \left(5 a^4-10 a^2 b^2+b^4\right) \cos (10 (c+d x))}{30720 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^4 b \cos ^{10}(c+d x)}{2 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^2 b^3 \cos ^{10}(c+d x)}{d}-\frac{5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\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,"(120*a*(63*a^4 + 70*a^2*b^2 + 15*b^4)*(c + d*x) - 300*b*(21*a^4 + 14*a^2*b^2 + b^4)*Cos[2*(c + d*x)] - 1200*a^2*b*(3*a^2 + b^2)*Cos[4*(c + d*x)] + 50*b*(-27*a^4 + 6*a^2*b^2 + b^4)*Cos[6*(c + d*x)] - 300*a^2*b*(a^2 - b^2)*Cos[8*(c + d*x)] - 6*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[10*(c + d*x)] + 300*a*(21*a^4 + 14*a^2*b^2 + b^4)*Sin[2*(c + d*x)] + 600*a*(3*a^4 - 2*a^2*b^2 - b^4)*Sin[4*(c + d*x)] + 50*a*(9*a^4 - 26*a^2*b^2 - 3*b^4)*Sin[6*(c + d*x)] + 75*a*(a^4 - 6*a^2*b^2 + b^4)*Sin[8*(c + d*x)] + 6*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[10*(c + d*x)])/(30720*d)","A",1
93,1,278,337,1.0170293,"\int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{420 a \left(21 a^4-5 b^4\right) \sin (3 (c+d x))+252 b \left(b^4-25 a^4\right) \cos (5 (c+d x))+630 a \left(63 a^4+70 a^2 b^2+15 b^4\right) \sin (c+d x)+252 a \left(9 a^4-20 a^2 b^2-5 b^4\right) \sin (5 (c+d x))+45 a \left(9 a^4-50 a^2 b^2+5 b^4\right) \sin (7 (c+d x))+35 a \left(a^4-10 a^2 b^2+5 b^4\right) \sin (9 (c+d x))-630 b \left(35 a^4+30 a^2 b^2+3 b^4\right) \cos (c+d x)-420 b \left(35 a^4+20 a^2 b^2+b^4\right) \cos (3 (c+d x))+45 b \left(-35 a^4+30 a^2 b^2+b^4\right) \cos (7 (c+d x))-35 b \left(5 a^4-10 a^2 b^2+b^4\right) \cos (9 (c+d x))}{80640 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^4 b \cos ^9(c+d x)}{9 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 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,"(-630*b*(35*a^4 + 30*a^2*b^2 + 3*b^4)*Cos[c + d*x] - 420*b*(35*a^4 + 20*a^2*b^2 + b^4)*Cos[3*(c + d*x)] + 252*b*(-25*a^4 + b^4)*Cos[5*(c + d*x)] + 45*b*(-35*a^4 + 30*a^2*b^2 + b^4)*Cos[7*(c + d*x)] - 35*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[9*(c + d*x)] + 630*a*(63*a^4 + 70*a^2*b^2 + 15*b^4)*Sin[c + d*x] + 420*a*(21*a^4 - 5*b^4)*Sin[3*(c + d*x)] + 252*a*(9*a^4 - 20*a^2*b^2 - 5*b^4)*Sin[5*(c + d*x)] + 45*a*(9*a^4 - 50*a^2*b^2 + 5*b^4)*Sin[7*(c + d*x)] + 35*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[9*(c + d*x)])/(80640*d)","A",1
94,1,259,426,0.8658542,"\int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{120 a (a-i b) (a+i b) \left(7 a^2+3 b^2\right) (c+d x)+24 a \left(7 a^4-10 a^2 b^2-5 b^4\right) \sin (4 (c+d x))+3 a \left(a^4-10 a^2 b^2+5 b^4\right) \sin (8 (c+d x))-24 b \left(35 a^4+30 a^2 b^2+3 b^4\right) \cos (2 (c+d x))+12 b \left(-35 a^4-10 a^2 b^2+b^4\right) \cos (4 (c+d x))+8 b \left(-15 a^4+10 a^2 b^2+b^4\right) \cos (6 (c+d x))-3 b \left(5 a^4-10 a^2 b^2+b^4\right) \cos (8 (c+d x))+96 a^3 \left(7 a^2+5 b^2\right) \sin (2 (c+d x))+32 a^3 \left(a^2-5 b^2\right) \sin (6 (c+d x))}{3072 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^4 b \cos ^8(c+d x)}{8 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^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 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,"(120*a*(a - I*b)*(a + I*b)*(7*a^2 + 3*b^2)*(c + d*x) - 24*b*(35*a^4 + 30*a^2*b^2 + 3*b^4)*Cos[2*(c + d*x)] + 12*b*(-35*a^4 - 10*a^2*b^2 + b^4)*Cos[4*(c + d*x)] + 8*b*(-15*a^4 + 10*a^2*b^2 + b^4)*Cos[6*(c + d*x)] - 3*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[8*(c + d*x)] + 96*a^3*(7*a^2 + 5*b^2)*Sin[2*(c + d*x)] + 24*a*(7*a^4 - 10*a^2*b^2 - 5*b^4)*Sin[4*(c + d*x)] + 32*a^3*(a^2 - 5*b^2)*Sin[6*(c + d*x)] + 3*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[8*(c + d*x)])/(3072*d)","C",1
95,1,236,275,0.7462357,"\int \cos ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Cos[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{525 a \left(7 a^4+10 a^2 b^2+3 b^4\right) \sin (c+d x)+35 a \left(21 a^4-10 a^2 b^2-15 b^4\right) \sin (3 (c+d x))+21 a \left(7 a^4-30 a^2 b^2-5 b^4\right) \sin (5 (c+d x))+15 a \left(a^4-10 a^2 b^2+5 b^4\right) \sin (7 (c+d x))-525 b \left(5 a^4+6 a^2 b^2+b^4\right) \cos (c+d x)-35 b \left(45 a^4+30 a^2 b^2+b^4\right) \cos (3 (c+d x))+21 b \left(-25 a^4+10 a^2 b^2+3 b^4\right) \cos (5 (c+d x))-15 b \left(5 a^4-10 a^2 b^2+b^4\right) \cos (7 (c+d x))}{6720 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^4 b \cos ^7(c+d x)}{7 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 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,"(-525*b*(5*a^4 + 6*a^2*b^2 + b^4)*Cos[c + d*x] - 35*b*(45*a^4 + 30*a^2*b^2 + b^4)*Cos[3*(c + d*x)] + 21*b*(-25*a^4 + 10*a^2*b^2 + 3*b^4)*Cos[5*(c + d*x)] - 15*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[7*(c + d*x)] + 525*a*(7*a^4 + 10*a^2*b^2 + 3*b^4)*Sin[c + d*x] + 35*a*(21*a^4 - 10*a^2*b^2 - 15*b^4)*Sin[3*(c + d*x)] + 21*a*(7*a^4 - 30*a^2*b^2 - 5*b^4)*Sin[5*(c + d*x)] + 15*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[7*(c + d*x)])/(6720*d)","A",1
96,1,188,126,0.6136679,"\int \cos (c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Cos[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{6 b \left(b^4-5 a^4\right) \cos (4 (c+d x))+60 a \left(a^2+b^2\right)^2 (c+d x)+15 a \left(3 a^4+2 a^2 b^2-b^4\right) \sin (2 (c+d x))+3 a \left(3 a^4-10 a^2 b^2-5 b^4\right) \sin (4 (c+d x))+a \left(a^4-10 a^2 b^2+5 b^4\right) \sin (6 (c+d x))-15 b \left(5 a^4+6 a^2 b^2+b^4\right) \cos (2 (c+d x))-b \left(5 a^4-10 a^2 b^2+b^4\right) \cos (6 (c+d x))}{192 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,"(60*a*(a^2 + b^2)^2*(c + d*x) - 15*b*(5*a^4 + 6*a^2*b^2 + b^4)*Cos[2*(c + d*x)] + 6*b*(-5*a^4 + b^4)*Cos[4*(c + d*x)] - b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[6*(c + d*x)] + 15*a*(3*a^4 + 2*a^2*b^2 - b^4)*Sin[2*(c + d*x)] + 3*a*(3*a^4 - 10*a^2*b^2 - 5*b^4)*Sin[4*(c + d*x)] + a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[6*(c + d*x)])/(192*d)","A",1
97,1,156,94,0.4638557,"\int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{150 a \left(a^2+b^2\right)^2 \sin (c+d x)-150 b \left(a^2+b^2\right)^2 \cos (c+d x)+25 a \left(a^4-2 a^2 b^2-3 b^4\right) \sin (3 (c+d x))+3 a \left(a^4-10 a^2 b^2+5 b^4\right) \sin (5 (c+d x))+25 b \left(-3 a^4-2 a^2 b^2+b^4\right) \cos (3 (c+d x))-3 b \left(5 a^4-10 a^2 b^2+b^4\right) \cos (5 (c+d x))}{240 d}","\frac{2 \left(a^2+b^2\right) (b \cos (c+d x)-a \sin (c+d x))^3}{3 d}-\frac{\left(a^2+b^2\right)^2 (b \cos (c+d x)-a \sin (c+d x))}{d}-\frac{(b \cos (c+d x)-a \sin (c+d x))^5}{5 d}",1,"(-150*b*(a^2 + b^2)^2*Cos[c + d*x] + 25*b*(-3*a^4 - 2*a^2*b^2 + b^4)*Cos[3*(c + d*x)] - 3*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[5*(c + d*x)] + 150*a*(a^2 + b^2)^2*Sin[c + d*x] + 25*a*(a^4 - 2*a^2*b^2 - 3*b^4)*Sin[3*(c + d*x)] + 3*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[5*(c + d*x)])/(240*d)","A",1
98,1,711,170,6.4446765,"\int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{b^5 \left(\frac{\cos ^4(c+d x) (a+b \tan (c+d x))^6 \left(a b \tan (c+d x)+b^2\right)}{4 b^6 \left(a^2+b^2\right)}-\frac{\frac{\cos ^2(c+d x) (a+b \tan (c+d x))^6 \left(b \left(a \left(2 b^2-3 a^2\right)+3 a b^2\right) \tan (c+d x)-3 a^2 b^2+b^2 \left(2 b^2-3 a^2\right)\right)}{2 b^4 \left(a^2+b^2\right)}-\frac{\left(3 a^4-29 a^2 b^2+5 a^2 \left(3 a^2-5 b^2\right)+8 b^4\right) \left(\frac{1}{2} b^2 \left(10 a^2-b^2\right) \tan ^2(c+d x)+5 a b \left(2 a^2-b^2\right) \tan (c+d x)+\frac{1}{2} \left(5 a^4-10 a^2 b^2+\frac{a^5-10 a^3 b^2+5 a b^4}{\sqrt{-b^2}}+b^4\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\frac{1}{2} \left(5 a^4-10 a^2 b^2-\frac{a^5-10 a^3 b^2+5 a b^4}{\sqrt{-b^2}}+b^4\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+\frac{5}{3} a b^3 \tan ^3(c+d x)+\frac{1}{4} b^4 \tan ^4(c+d x)\right)-5 a \left(3 a^2-5 b^2\right) \left(a b^2 \left(10 a^2-3 b^2\right) \tan ^2(c+d x)+\frac{1}{3} b^3 \left(15 a^2-b^2\right) \tan ^3(c+d x)+b \left(15 a^4-15 a^2 b^2+b^4\right) \tan (c+d x)+\frac{1}{2} \left(6 a^5-20 a^3 b^2+\frac{a^6-15 a^4 b^2+15 a^2 b^4-b^6}{\sqrt{-b^2}}+6 a b^4\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\frac{1}{2} \left(6 a^5-20 a^3 b^2-\frac{a^6-15 a^4 b^2+15 a^2 b^4-b^6}{\sqrt{-b^2}}+6 a b^4\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+\frac{3}{2} a b^4 \tan ^4(c+d x)+\frac{1}{5} b^5 \tan ^5(c+d x)\right)}{2 b^2 \left(a^2+b^2\right)}}{4 b^2 \left(a^2+b^2\right)}\right)}{d}","-\frac{\sin ^4(c+d x) \left(a \left(a^4-10 a^2 b^2+5 b^4\right) \cot (c+d x)+b \left(5 a^4-10 a^2 b^2+b^4\right)\right)}{4 d}+\frac{\sin ^2(c+d x) \left(4 b \left(5 a^4-b^4\right)+5 a \left(a^2-3 b^2\right) \left(a^2+b^2\right) \cot (c+d x)\right)}{8 d}+\frac{1}{8} a x \left(3 a^4+10 a^2 b^2+15 b^4\right)-\frac{b^5 \log (\sin (c+d x))}{d}+\frac{b^5 \log (\tan (c+d x))}{d}",1,"(b^5*((Cos[c + d*x]^4*(a + b*Tan[c + d*x])^6*(b^2 + a*b*Tan[c + d*x]))/(4*b^6*(a^2 + b^2)) - ((Cos[c + d*x]^2*(a + b*Tan[c + d*x])^6*(-3*a^2*b^2 + b^2*(-3*a^2 + 2*b^2) + b*(3*a*b^2 + a*(-3*a^2 + 2*b^2))*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)) - ((3*a^4 - 29*a^2*b^2 + 8*b^4 + 5*a^2*(3*a^2 - 5*b^2))*(((5*a^4 - 10*a^2*b^2 + b^4 + (a^5 - 10*a^3*b^2 + 5*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/2 + ((5*a^4 - 10*a^2*b^2 + b^4 - (a^5 - 10*a^3*b^2 + 5*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/2 + 5*a*b*(2*a^2 - b^2)*Tan[c + d*x] + (b^2*(10*a^2 - b^2)*Tan[c + d*x]^2)/2 + (5*a*b^3*Tan[c + d*x]^3)/3 + (b^4*Tan[c + d*x]^4)/4) - 5*a*(3*a^2 - 5*b^2)*(((6*a^5 - 20*a^3*b^2 + 6*a*b^4 + (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/2 + ((6*a^5 - 20*a^3*b^2 + 6*a*b^4 - (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/2 + b*(15*a^4 - 15*a^2*b^2 + b^4)*Tan[c + d*x] + a*b^2*(10*a^2 - 3*b^2)*Tan[c + d*x]^2 + (b^3*(15*a^2 - b^2)*Tan[c + d*x]^3)/3 + (3*a*b^4*Tan[c + d*x]^4)/2 + (b^5*Tan[c + d*x]^5)/5))/(2*b^2*(a^2 + b^2)))/(4*b^2*(a^2 + b^2))))/d","B",1
99,1,632,205,6.2958279,"\int \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","-\frac{b \left(5 a^4+30 a^2 b^2-7 b^4\right) \cos ^6(c+d x) (a+b \tan (c+d x))^5}{4 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{a \left(a^4-10 a^2 b^2+5 b^4\right) \sin (3 (c+d x)) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{12 d (a \cos (c+d x)+b \sin (c+d x))^5}-\frac{b \left(5 a^4-10 a^2 b^2+b^4\right) \cos (3 (c+d x)) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{12 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{a \left(3 a^4+10 a^2 b^2-25 b^4\right) \sin (c+d x) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{4 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{b^5 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{b^5 \cos ^5(c+d x) (a+b \tan (c+d x))^5}{d (a \cos (c+d x)+b \sin (c+d x))^5}-\frac{b^5 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^5}-\frac{5 a b^4 \cos ^5(c+d x) (a+b \tan (c+d x))^5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{5 a b^4 \cos ^5(c+d x) (a+b \tan (c+d x))^5 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d (a \cos (c+d x)+b \sin (c+d x))^5}","-\frac{a^5 \sin ^3(c+d x)}{3 d}+\frac{a^5 \sin (c+d x)}{d}-\frac{5 a^4 b \cos ^3(c+d x)}{3 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 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,"(b^5*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) - (b*(5*a^4 + 30*a^2*b^2 - 7*b^4)*Cos[c + d*x]^6*(a + b*Tan[c + d*x])^5)/(4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) - (b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x]^5*Cos[3*(c + d*x)]*(a + b*Tan[c + d*x])^5)/(12*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) - (5*a*b^4*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^5)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (5*a*b^4*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^5)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (b^5*Cos[c + d*x]^5*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^5)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) - (b^5*Cos[c + d*x]^5*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^5)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (a*(3*a^4 + 10*a^2*b^2 - 25*b^4)*Cos[c + d*x]^5*Sin[c + d*x]*(a + b*Tan[c + d*x])^5)/(4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (a*(a^4 - 10*a^2*b^2 + 5*b^4)*Cos[c + d*x]^5*Sin[3*(c + d*x)]*(a + b*Tan[c + d*x])^5)/(12*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5)","B",1
100,1,571,169,6.3642374,"\int \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{b^3 \left(\frac{\cos ^2(c+d x) (a+b \tan (c+d x))^6 \left(a b \tan (c+d x)+b^2\right)}{2 b^4 \left(a^2+b^2\right)}-\frac{\left(4 b^2-6 a^2\right) \left(\frac{1}{2} b^2 \left(10 a^2-b^2\right) \tan ^2(c+d x)+5 a b \left(2 a^2-b^2\right) \tan (c+d x)+\frac{1}{2} \left(5 a^4-10 a^2 b^2+\frac{a^5-10 a^3 b^2+5 a b^4}{\sqrt{-b^2}}+b^4\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\frac{1}{2} \left(5 a^4-10 a^2 b^2-\frac{a^5-10 a^3 b^2+5 a b^4}{\sqrt{-b^2}}+b^4\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+\frac{5}{3} a b^3 \tan ^3(c+d x)+\frac{1}{4} b^4 \tan ^4(c+d x)\right)+5 a \left(a b^2 \left(10 a^2-3 b^2\right) \tan ^2(c+d x)+\frac{1}{3} b^3 \left(15 a^2-b^2\right) \tan ^3(c+d x)+b \left(15 a^4-15 a^2 b^2+b^4\right) \tan (c+d x)+\frac{1}{2} \left(6 a^5-20 a^3 b^2+\frac{a^6-15 a^4 b^2+15 a^2 b^4-b^6}{\sqrt{-b^2}}+6 a b^4\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\frac{1}{2} \left(6 a^5-20 a^3 b^2-\frac{a^6-15 a^4 b^2+15 a^2 b^4-b^6}{\sqrt{-b^2}}+6 a b^4\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+\frac{3}{2} a b^4 \tan ^4(c+d x)+\frac{1}{5} b^5 \tan ^5(c+d x)\right)}{2 b^2 \left(a^2+b^2\right)}\right)}{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(a^4-10 a^2 b^2+5 b^4\right) \cot (c+d x)+b \left(5 a^4-10 a^2 b^2+b^4\right)\right)}{2 d}+\frac{1}{2} a x \left(a^4+10 a^2 b^2-15 b^4\right)+\frac{5 a b^4 \tan (c+d x)}{d}+\frac{b^5 \tan ^2(c+d x)}{2 d}",1,"(b^3*((Cos[c + d*x]^2*(a + b*Tan[c + d*x])^6*(b^2 + a*b*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)) - ((-6*a^2 + 4*b^2)*(((5*a^4 - 10*a^2*b^2 + b^4 + (a^5 - 10*a^3*b^2 + 5*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/2 + ((5*a^4 - 10*a^2*b^2 + b^4 - (a^5 - 10*a^3*b^2 + 5*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/2 + 5*a*b*(2*a^2 - b^2)*Tan[c + d*x] + (b^2*(10*a^2 - b^2)*Tan[c + d*x]^2)/2 + (5*a*b^3*Tan[c + d*x]^3)/3 + (b^4*Tan[c + d*x]^4)/4) + 5*a*(((6*a^5 - 20*a^3*b^2 + 6*a*b^4 + (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/2 + ((6*a^5 - 20*a^3*b^2 + 6*a*b^4 - (a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/2 + b*(15*a^4 - 15*a^2*b^2 + b^4)*Tan[c + d*x] + a*b^2*(10*a^2 - 3*b^2)*Tan[c + d*x]^2 + (b^3*(15*a^2 - b^2)*Tan[c + d*x]^3)/3 + (3*a*b^4*Tan[c + d*x]^4)/2 + (b^5*Tan[c + d*x]^5)/5))/(2*b^2*(a^2 + b^2))))/d","B",0
101,1,397,204,5.8990251,"\int \sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{120 a^2 b^3-30 a b^2 \left(4 a^2-3 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+30 a b^2 \left(4 a^2-3 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 b^3 \left(60 a^2-11 b^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{2 b^3 \left(11 b^2-60 a^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+12 a \left(a^4-10 a^2 b^2+5 b^4\right) \sin (c+d x)-12 b \left(5 a^4-10 a^2 b^2+b^4\right) \cos (c+d x)+\frac{b^4 (15 a+b)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{b^4 (b-15 a)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 b^5 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{2 b^5 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}-22 b^5}{12 d}","\frac{a^5 \sin (c+d x)}{d}-\frac{5 a^4 b \cos (c+d x)}{d}-\frac{10 a^3 b^2 \sin (c+d x)}{d}+\frac{10 a^3 b^2 \tanh ^{-1}(\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{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,"(120*a^2*b^3 - 22*b^5 - 12*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[c + d*x] - 30*a*b^2*(4*a^2 - 3*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 30*a*b^2*(4*a^2 - 3*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^4*(15*a + b))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*b^5*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (2*b^3*(60*a^2 - 11*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (2*b^5*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (b^4*(-15*a + b))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (2*b^3*(-60*a^2 + 11*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[c + d*x])/(12*d)","A",1
102,1,126,147,0.7372703,"\int \sec ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{60 a b^2 \left(2 a^2-b^2\right) \tan (c+d x)-6 b^3 \left(b^2-10 a^2\right) \tan ^2(c+d x)+20 a b^4 \tan ^3(c+d x)+6 (b-i a)^5 \log (-\tan (c+d x)+i)+6 (b+i a)^5 \log (\tan (c+d x)+i)+3 b^5 \tan ^4(c+d x)}{12 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(5 a^4-10 a^2 b^2+b^4\right) \log (\cos (c+d x))}{d}+a x \left(a^4-10 a^2 b^2+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,"(6*((-I)*a + b)^5*Log[I - Tan[c + d*x]] + 6*(I*a + b)^5*Log[I + Tan[c + d*x]] + 60*a*b^2*(2*a^2 - b^2)*Tan[c + d*x] - 6*b^3*(-10*a^2 + b^2)*Tan[c + d*x]^2 + 20*a*b^4*Tan[c + d*x]^3 + 3*b^5*Tan[c + d*x]^4)/(12*d)","C",1
103,1,1219,224,6.2936675,"\int \sec ^6(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{b \left(600 a^4-1000 b^2 a^2+89 b^4\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{120 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(-8 a^5+40 b^2 a^3-15 b^4 a\right) \cos ^5(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^5}{8 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(8 a^5-40 b^2 a^3+15 b^4 a\right) \cos ^5(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^5}{8 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{b^5 \cos ^5(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^5}{20 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(-89 \sin \left(\frac{1}{2} (c+d x)\right) b^5+1000 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^3-600 a^4 \sin \left(\frac{1}{2} (c+d x)\right) b\right) (a+b \tan (c+d x))^5}{120 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(200 a^2 b^3 \sin \left(\frac{1}{2} (c+d x)\right)-31 b^5 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^5}{120 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(31 b^5 \sin \left(\frac{1}{2} (c+d x)\right)-200 a^2 b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^5}{120 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(89 \sin \left(\frac{1}{2} (c+d x)\right) b^5-1000 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^3+600 a^4 \sin \left(\frac{1}{2} (c+d x)\right) b\right) (a+b \tan (c+d x))^5}{120 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(-31 b^5-375 a b^4+200 a^2 b^3+600 a^3 b^2\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{240 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(-31 b^5+375 a b^4+200 a^2 b^3-600 a^3 b^2\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{240 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(2 b^5+25 a b^4\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{80 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(2 b^5-25 a b^4\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{80 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^5}-\frac{b^5 \cos ^5(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^5}{20 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (a \cos (c+d x)+b \sin (c+d x))^5}","\frac{a^5 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a^4 b \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{10 a^2 b^3 \sec ^3(c+d x)}{3 d}-\frac{10 a^2 b^3 \sec (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,"(b*(600*a^4 - 1000*a^2*b^2 + 89*b^4)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(120*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-8*a^5 + 40*a^3*b^2 - 15*a*b^4)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^5)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((8*a^5 - 40*a^3*b^2 + 15*a*b^4)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^5)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((25*a*b^4 + 2*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(80*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((600*a^3*b^2 + 200*a^2*b^3 - 375*a*b^4 - 31*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(240*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (b^5*Cos[c + d*x]^5*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^5)/(20*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) - (b^5*Cos[c + d*x]^5*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^5)/(20*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-25*a*b^4 + 2*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(80*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-600*a^3*b^2 + 200*a^2*b^3 + 375*a*b^4 - 31*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(240*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(-600*a^4*b*Sin[(c + d*x)/2] + 1000*a^2*b^3*Sin[(c + d*x)/2] - 89*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(120*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(200*a^2*b^3*Sin[(c + d*x)/2] - 31*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(120*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(-200*a^2*b^3*Sin[(c + d*x)/2] + 31*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(120*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(600*a^4*b*Sin[(c + d*x)/2] - 1000*a^2*b^3*Sin[(c + d*x)/2] + 89*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(120*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^5)","B",1
104,1,89,30,0.4948459,"\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{\tan (c+d x) \left(6 a^5+15 a^4 b \tan (c+d x)+20 a^3 b^2 \tan ^2(c+d x)+15 a^2 b^3 \tan ^3(c+d x)+6 a b^4 \tan ^4(c+d x)+b^5 \tan ^5(c+d x)\right)}{6 d}","\frac{\tan ^6(c+d x) (a \cot (c+d x)+b)^6}{6 b d}",1,"(Tan[c + d*x]*(6*a^5 + 15*a^4*b*Tan[c + d*x] + 20*a^3*b^2*Tan[c + d*x]^2 + 15*a^2*b^3*Tan[c + d*x]^3 + 6*a*b^4*Tan[c + d*x]^4 + b^5*Tan[c + d*x]^5))/(6*d)","B",1
105,1,1677,318,6.3351043,"\int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^8*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{b \left(1400 a^4-1540 b^2 a^2+103 b^4\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{1680 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(-8 a^5+20 b^2 a^3-5 b^4 a\right) \cos ^5(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^5}{16 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(8 a^5-20 b^2 a^3+5 b^4 a\right) \cos ^5(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^5}{16 d (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{b^5 \cos ^5(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^5}{56 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^7 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(-103 \sin \left(\frac{1}{2} (c+d x)\right) b^5+1540 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^3-1400 a^4 \sin \left(\frac{1}{2} (c+d x)\right) b\right) (a+b \tan (c+d x))^5}{1680 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(-103 \sin \left(\frac{1}{2} (c+d x)\right) b^5+1540 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^3-1400 a^4 \sin \left(\frac{1}{2} (c+d x)\right) b\right) (a+b \tan (c+d x))^5}{1680 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(70 a^2 b^3 \sin \left(\frac{1}{2} (c+d x)\right)-9 b^5 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^5}{140 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(9 b^5 \sin \left(\frac{1}{2} (c+d x)\right)-70 a^2 b^3 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^5}{140 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(103 \sin \left(\frac{1}{2} (c+d x)\right) b^5-1540 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^3+1400 a^4 \sin \left(\frac{1}{2} (c+d x)\right) b\right) (a+b \tan (c+d x))^5}{1680 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\cos ^5(c+d x) \left(103 \sin \left(\frac{1}{2} (c+d x)\right) b^5-1540 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^3+1400 a^4 \sin \left(\frac{1}{2} (c+d x)\right) b\right) (a+b \tan (c+d x))^5}{1680 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(840 a^5+1400 b a^4-2100 b^2 a^3-1540 b^3 a^2+525 b^4 a+103 b^5\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{3360 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(-840 a^5+1400 b a^4+2100 b^2 a^3-1540 b^3 a^2-525 b^4 a+103 b^5\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{3360 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(-18 b^5-175 a b^4+140 a^2 b^3+350 a^3 b^2\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{560 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(-18 b^5+175 a b^4+140 a^2 b^3-350 a^3 b^2\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{560 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(3 b^5+35 a b^4\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{336 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^6 (a \cos (c+d x)+b \sin (c+d x))^5}+\frac{\left(3 b^5-35 a b^4\right) \cos ^5(c+d x) (a+b \tan (c+d x))^5}{336 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^6 (a \cos (c+d x)+b \sin (c+d x))^5}-\frac{b^5 \cos ^5(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^5}{56 d \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^7 (a \cos (c+d x)+b \sin (c+d x))^5}","\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^4 b \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{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 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,"(b*(1400*a^4 - 1540*a^2*b^2 + 103*b^4)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(1680*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-8*a^5 + 20*a^3*b^2 - 5*a*b^4)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^5)/(16*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((8*a^5 - 20*a^3*b^2 + 5*a*b^4)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^5)/(16*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((35*a*b^4 + 3*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(336*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((350*a^3*b^2 + 140*a^2*b^3 - 175*a*b^4 - 18*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(560*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((840*a^5 + 1400*a^4*b - 2100*a^3*b^2 - 1540*a^2*b^3 + 525*a*b^4 + 103*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(3360*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (b^5*Cos[c + d*x]^5*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^5)/(56*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) - (b^5*Cos[c + d*x]^5*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^5)/(56*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-35*a*b^4 + 3*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(336*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-350*a^3*b^2 + 140*a^2*b^3 + 175*a*b^4 - 18*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(560*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + ((-840*a^5 + 1400*a^4*b + 2100*a^3*b^2 - 1540*a^2*b^3 - 525*a*b^4 + 103*b^5)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^5)/(3360*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(-1400*a^4*b*Sin[(c + d*x)/2] + 1540*a^2*b^3*Sin[(c + d*x)/2] - 103*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(1680*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(-1400*a^4*b*Sin[(c + d*x)/2] + 1540*a^2*b^3*Sin[(c + d*x)/2] - 103*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(1680*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(70*a^2*b^3*Sin[(c + d*x)/2] - 9*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(140*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(-70*a^2*b^3*Sin[(c + d*x)/2] + 9*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(140*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(1400*a^4*b*Sin[(c + d*x)/2] - 1540*a^2*b^3*Sin[(c + d*x)/2] + 103*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(1680*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^5) + (Cos[c + d*x]^5*(1400*a^4*b*Sin[(c + d*x)/2] - 1540*a^2*b^3*Sin[(c + d*x)/2] + 103*b^5*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^5)/(1680*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^5)","B",1
106,1,54,177,0.4481444,"\int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{(a+b \tan (c+d x))^6 \left(a^2-6 a b \tan (c+d x)+21 b^2 \tan ^2(c+d x)+28 b^2\right)}{168 b^3 d}","\frac{a^5 \tan (c+d x)}{d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 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{b^3 \left(10 a^2+b^2\right) \tan ^6(c+d x)}{6 d}+\frac{a^3 \left(a^2+10 b^2\right) \tan ^3(c+d x)}{3 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 + b*Tan[c + d*x])^6*(a^2 + 28*b^2 - 6*a*b*Tan[c + d*x] + 21*b^2*Tan[c + d*x]^2))/(168*b^3*d)","A",1
107,1,331,391,2.1378819,"\int \sec ^{10}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^10*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{1260 a \left(656 a^4+2320 a^2 b^2+845 b^4\right) \tan (c+d x) \sec ^7(c+d x)-40320 a \left(48 a^4-80 a^2 b^2+15 b^4\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec ^9(c+d x) \left(372960 a^5 \sin (4 (c+d x))+131040 a^5 \sin (6 (c+d x))+15120 a^5 \sin (8 (c+d x))+1935360 a^4 b+453600 a^3 b^2 \sin (4 (c+d x))-218400 a^3 b^2 \sin (6 (c+d x))-25200 a^3 b^2 \sin (8 (c+d x))-184320 a^2 b^3+73728 \left(35 a^4 b-20 a^2 b^3-3 b^5\right) \cos (2 (c+d x))+129024 \left(5 a^4 b-10 a^2 b^3+b^5\right) \cos (4 (c+d x))-488250 a b^4 \sin (4 (c+d x))+40950 a b^4 \sin (6 (c+d x))+4725 a b^4 \sin (8 (c+d x))+223232 b^5\right)}{5160960 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{a^4 b \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{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{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,"(-40320*a*(48*a^4 - 80*a^2*b^2 + 15*b^4)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c + d*x]^9*(1935360*a^4*b - 184320*a^2*b^3 + 223232*b^5 + 73728*(35*a^4*b - 20*a^2*b^3 - 3*b^5)*Cos[2*(c + d*x)] + 129024*(5*a^4*b - 10*a^2*b^3 + b^5)*Cos[4*(c + d*x)] + 372960*a^5*Sin[4*(c + d*x)] + 453600*a^3*b^2*Sin[4*(c + d*x)] - 488250*a*b^4*Sin[4*(c + d*x)] + 131040*a^5*Sin[6*(c + d*x)] - 218400*a^3*b^2*Sin[6*(c + d*x)] + 40950*a*b^4*Sin[6*(c + d*x)] + 15120*a^5*Sin[8*(c + d*x)] - 25200*a^3*b^2*Sin[8*(c + d*x)] + 4725*a*b^4*Sin[8*(c + d*x)]) + 1260*a*(656*a^4 + 2320*a^2*b^2 + 845*b^4)*Sec[c + d*x]^7*Tan[c + d*x])/(5160960*d)","A",1
108,1,115,242,1.2223343,"\int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{\frac{1}{4} \left(3 a^2+b^2\right) (a+b \tan (c+d x))^8-\frac{4}{7} a \left(a^2+b^2\right) (a+b \tan (c+d x))^7+\frac{1}{6} \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^6+\frac{1}{10} (a+b \tan (c+d x))^{10}-\frac{4}{9} a (a+b \tan (c+d x))^9}{b^5 d}","\frac{a^5 \tan (c+d x)}{d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{10 a b^2 \left(a^2+b^2\right) \tan ^7(c+d x)}{7 d}+\frac{5 a^2 b \left(a^2+b^2\right) \tan ^4(c+d x)}{2 d}+\frac{b^3 \left(5 a^2+b^2\right) \tan ^8(c+d x)}{4 d}+\frac{b \left(5 a^4+20 a^2 b^2+b^4\right) \tan ^6(c+d x)}{6 d}+\frac{a \left(a^4+20 a^2 b^2+5 b^4\right) \tan ^5(c+d x)}{5 d}+\frac{2 a^3 \left(a^2+5 b^2\right) \tan ^3(c+d x)}{3 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^2 + b^2)^2*(a + b*Tan[c + d*x])^6)/6 - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^7)/7 + ((3*a^2 + b^2)*(a + b*Tan[c + d*x])^8)/4 - (4*a*(a + b*Tan[c + d*x])^9)/9 + (a + b*Tan[c + d*x])^10/10)/(b^5*d)","A",1
109,1,374,472,1.7949984,"\int \sec ^{12}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^12*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]","\frac{13860 a \left(976 a^4+2876 a^2 b^2+1207 b^4\right) \tan (c+d x) \sec ^9(c+d x)-1774080 a \left(16 a^4-20 a^2 b^2+3 b^4\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec ^{11}(c+d x) \left(6623232 a^5 \sin (4 (c+d x))+2857008 a^5 \sin (6 (c+d x))+591360 a^5 \sin (8 (c+d x))+55440 a^5 \sin (10 (c+d x))+24330240 a^4 b+5913600 a^3 b^2 \sin (4 (c+d x))-3571260 a^3 b^2 \sin (6 (c+d x))-739200 a^3 b^2 \sin (8 (c+d x))-69300 a^3 b^2 \sin (10 (c+d x))+1802240 a^2 b^3+3604480 \left(9 a^4 b-4 a^2 b^3-b^5\right) \cos (2 (c+d x))+1622016 \left(5 a^4 b-10 a^2 b^3+b^5\right) \cos (4 (c+d x))-6564096 a b^4 \sin (4 (c+d x))+535689 a b^4 \sin (6 (c+d x))+110880 a b^4 \sin (8 (c+d x))+10395 a b^4 \sin (10 (c+d x))+3031040 b^5\right)}{90832896 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{5 a^4 b \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{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{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,"(-1774080*a*(16*a^4 - 20*a^2*b^2 + 3*b^4)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c + d*x]^11*(24330240*a^4*b + 1802240*a^2*b^3 + 3031040*b^5 + 3604480*(9*a^4*b - 4*a^2*b^3 - b^5)*Cos[2*(c + d*x)] + 1622016*(5*a^4*b - 10*a^2*b^3 + b^5)*Cos[4*(c + d*x)] + 6623232*a^5*Sin[4*(c + d*x)] + 5913600*a^3*b^2*Sin[4*(c + d*x)] - 6564096*a*b^4*Sin[4*(c + d*x)] + 2857008*a^5*Sin[6*(c + d*x)] - 3571260*a^3*b^2*Sin[6*(c + d*x)] + 535689*a*b^4*Sin[6*(c + d*x)] + 591360*a^5*Sin[8*(c + d*x)] - 739200*a^3*b^2*Sin[8*(c + d*x)] + 110880*a*b^4*Sin[8*(c + d*x)] + 55440*a^5*Sin[10*(c + d*x)] - 69300*a^3*b^2*Sin[10*(c + d*x)] + 10395*a*b^4*Sin[10*(c + d*x)]) + 13860*a*(976*a^4 + 2876*a^2*b^2 + 1207*b^4)*Sec[c + d*x]^9*Tan[c + d*x])/(90832896*d)","A",1
110,1,218,227,0.4180491,"\int \frac{\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{8 a^5 \sin (2 (c+d x))+a^5 \sin (4 (c+d x))+12 a^5 c+12 a^5 d x+24 a^3 b^2 \sin (2 (c+d x))+2 a^3 b^2 \sin (4 (c+d x))+40 a^3 b^2 c+40 a^3 b^2 d x+b \left(a^2+b^2\right)^2 \cos (4 (c+d x))+4 b \left(a^4+4 a^2 b^2+3 b^4\right) \cos (2 (c+d x))+32 b^5 \log (a \cos (c+d x)+b \sin (c+d x))+16 a b^4 \sin (2 (c+d x))+a b^4 \sin (4 (c+d x))+60 a b^4 c+60 a b^4 d x}{32 d \left(a^2+b^2\right)^3}","\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{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^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{b^3 \cos ^2(c+d x)}{2 d \left(a^2+b^2\right)^2}",1,"(12*a^5*c + 40*a^3*b^2*c + 60*a*b^4*c + 12*a^5*d*x + 40*a^3*b^2*d*x + 60*a*b^4*d*x + 4*b*(a^4 + 4*a^2*b^2 + 3*b^4)*Cos[2*(c + d*x)] + b*(a^2 + b^2)^2*Cos[4*(c + d*x)] + 32*b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]] + 8*a^5*Sin[2*(c + d*x)] + 24*a^3*b^2*Sin[2*(c + d*x)] + 16*a*b^4*Sin[2*(c + d*x)] + a^5*Sin[4*(c + d*x)] + 2*a^3*b^2*Sin[4*(c + d*x)] + a*b^4*Sin[4*(c + d*x)])/(32*(a^2 + b^2)^3*d)","A",1
111,1,137,166,0.9861559,"\int \frac{\cos ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{\sqrt{a^2+b^2} \left(3 b \left(a^2+5 b^2\right) \cos (c+d x)+b \left(a^2+b^2\right) \cos (3 (c+d x))+2 a \sin (c+d x) \left(\left(a^2+b^2\right) \cos (2 (c+d x))+5 a^2+11 b^2\right)\right)+24 b^4 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{12 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^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{b^3 \cos (c+d x)}{d \left(a^2+b^2\right)^2}",1,"(24*b^4*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sqrt[a^2 + b^2]*(3*b*(a^2 + 5*b^2)*Cos[c + d*x] + b*(a^2 + b^2)*Cos[3*(c + d*x)] + 2*a*(5*a^2 + 11*b^2 + (a^2 + b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(12*(a^2 + b^2)^(5/2)*d)","A",1
112,1,143,119,0.2312157,"\int \frac{\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{a^3 \sin (2 (c+d x))+2 a^3 c+2 a^3 d x+b \left(a^2+b^2\right) \cos (2 (c+d x))+2 b^3 \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+a b^2 \sin (2 (c+d x))+6 a b^2 c+6 a b^2 d x-4 i b^3 \tan ^{-1}(\tan (c+d x))+4 i b^3 c+4 i b^3 d x}{4 d \left(a^2+b^2\right)^2}","\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{a b^2 x}{\left(a^2+b^2\right)^2}+\frac{a x}{2 \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}",1,"(2*a^3*c + 6*a*b^2*c + (4*I)*b^3*c + 2*a^3*d*x + 6*a*b^2*d*x + (4*I)*b^3*d*x - (4*I)*b^3*ArcTan[Tan[c + d*x]] + b*(a^2 + b^2)*Cos[2*(c + d*x)] + 2*b^3*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2] + a^3*Sin[2*(c + d*x)] + a*b^2*Sin[2*(c + d*x)])/(4*(a^2 + b^2)^2*d)","C",1
113,1,79,91,0.1799463,"\int \frac{\cos ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{\sqrt{a^2+b^2} (a \sin (c+d x)+b \cos (c+d x))+2 b^2 \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)^{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,"(2*b^2*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sqrt[a^2 + b^2]*(b*Cos[c + d*x] + a*Sin[c + d*x]))/((a^2 + b^2)^(3/2)*d)","A",1
114,1,41,45,0.0616624,"\int \frac{\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[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))+a (c+d x)}{d \left(a^2+b^2\right)}","\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*(c + d*x) + b*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)","A",1
115,1,45,47,0.0323633,"\int \frac{1}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-1),x]","\frac{2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{d \sqrt{a^2+b^2}}","-\frac{\tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \sqrt{a^2+b^2}}",1,"(2*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d)","A",1
116,1,18,41,0.025739,"\int \frac{\sec (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{\log (a+b \tan (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[a + b*Tan[c + d*x]]/(b*d)","A",1
117,1,109,80,0.1388072,"\int \frac{\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{2 \sqrt{a^2+b^2} \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)+a \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b \sec (c+d x)}{b^2 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,"(2*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + a*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + b*Sec[c + d*x])/(b^2*d)","A",1
118,1,52,88,0.1458181,"\int \frac{\sec ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{\left(a^2+b^2\right) \log (a+b \tan (c+d x))-a b \tan (c+d x)+\frac{1}{2} b^2 \tan ^2(c+d x)}{b^3 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[a + b*Tan[c + d*x]] - a*b*Tan[c + d*x] + (b^2*Tan[c + d*x]^2)/2)/(b^3*d)","A",1
119,1,321,153,1.9998143,"\int \frac{\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{48 \left(a^2+b^2\right)^{3/2} \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)+\sec ^3(c+d x) \left(6 a^3 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a^3 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 b \left(a^2+b^2\right) \cos (2 (c+d x))+9 a \left(2 a^2+3 b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+12 a^2 b-6 a b^2 \sin (2 (c+d x))+9 a b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-9 a b^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+20 b^3\right)}{24 b^4 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{\left(a^2+b^2\right) \sec (c+d x)}{b^3 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,"(48*(a^2 + b^2)^(3/2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sec[c + d*x]^3*(12*a^2*b + 20*b^3 + 12*b*(a^2 + b^2)*Cos[2*(c + d*x)] + 6*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*a*(2*a^2 + 3*b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 6*a^3*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 9*a*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 6*a*b^2*Sin[2*(c + d*x)]))/(24*b^4*d)","B",1
120,1,99,158,1.1734177,"\int \frac{\sec ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{6 b^2 \left(a^2+b^2\right) \tan ^2(c+d x)-12 a b \left(a^2+2 b^2\right) \tan (c+d x)+12 \left(a^2+b^2\right)^2 \log (a+b \tan (c+d x))-4 a b^3 \tan ^3(c+d x)+3 b^4 \sec ^4(c+d x)}{12 b^5 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 \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{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,"(12*(a^2 + b^2)^2*Log[a + b*Tan[c + d*x]] + 3*b^4*Sec[c + d*x]^4 - 12*a*b*(a^2 + 2*b^2)*Tan[c + d*x] + 6*b^2*(a^2 + b^2)*Tan[c + d*x]^2 - 4*a*b^3*Tan[c + d*x]^3)/(12*b^5*d)","A",1
121,1,661,262,5.0290679,"\int \frac{\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^6/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]","\frac{\sec (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(240 a^4 b+520 a^2 b^3+480 \left(a^2+b^2\right)^{5/2} \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)+\frac{2 b^3 \left(20 a^2+29 b^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{2 b^3 \left(20 a^2+29 b^2\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 b \left(120 a^4+260 a^2 b^2+149 b^4\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{2 b \left(120 a^4+260 a^2 b^2+149 b^4\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+30 a \left(8 a^4+20 a^2 b^2+15 b^4\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-30 a \left(8 a^4+20 a^2 b^2+15 b^4\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{b^2 \left(-60 a^3+20 a^2 b-105 a b^2+29 b^3\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{b^2 \left(60 a^3+20 a^2 b+105 a b^2+29 b^3\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{3 b^4 (2 b-5 a)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{3 b^4 (5 a+2 b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{12 b^5 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}-\frac{12 b^5 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}+298 b^5\right)}{240 b^6 d (a+b \tan (c+d x))}","-\frac{a \left(a^2+b^2\right)^2 \tanh ^{-1}(\sin (c+d x))}{b^6 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{\left(a^2+b^2\right)^2 \sec (c+d x)}{b^5 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) \sec ^3(c+d x)}{3 b^3 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,"(Sec[c + d*x]*(240*a^4*b + 520*a^2*b^3 + 298*b^5 + 480*(a^2 + b^2)^(5/2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + 30*a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 30*a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*b^4*(-5*a + 2*b))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4 + (b^2*(-60*a^3 + 20*a^2*b - 105*a*b^2 + 29*b^3))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*b^5*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 + (2*b^3*(20*a^2 + 29*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (2*b*(120*a^4 + 260*a^2*b^2 + 149*b^4)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (12*b^5*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 + (3*b^4*(5*a + 2*b))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - (2*b^3*(20*a^2 + 29*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (b^2*(60*a^3 + 20*a^2*b + 105*a*b^2 + 29*b^3))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (2*b*(120*a^4 + 260*a^2*b^2 + 149*b^4)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(240*b^6*d*(a + b*Tan[c + d*x]))","B",1
122,1,149,145,0.9750325,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-b^2\right) \left(a^2+b^2\right) \sin (2 (c+d x))+2 a b \left(a^2+b^2\right) \cos (2 (c+d x))+\frac{4 b^4 \left(a^2+b^2\right) \sin (c+d x)}{a (a \cos (c+d x)+b \sin (c+d x))}+2 \left(a^4+6 a^2 b^2-3 b^4\right) (c+d x)+16 a b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{4 d \left(a^2+b^2\right)^3}","-\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{b^4}{a d \left(a^2+b^2\right)^2 (a \cot (c+d x)+b)}+\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(a^4+6 a^2 b^2-3 b^4\right)}{2 \left(a^2+b^2\right)^3}",1,"(2*(a^4 + 6*a^2*b^2 - 3*b^4)*(c + d*x) + 2*a*b*(a^2 + b^2)*Cos[2*(c + d*x)] + 16*a*b^3*Log[a*Cos[c + d*x] + b*Sin[c + d*x]] + (4*b^4*(a^2 + b^2)*Sin[c + d*x])/(a*(a*Cos[c + d*x] + b*Sin[c + d*x])) + (a^2 - b^2)*(a^2 + b^2)*Sin[2*(c + d*x)])/(4*(a^2 + b^2)^3*d)","A",1
123,1,130,138,0.7556728,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\frac{a \left(a^2+b^2\right) \sin (2 (c+d x))+b \left(a^2+b^2\right) \cos (2 (c+d x))+3 b \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))}+\frac{12 a b^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}}{2 d}","\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{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}}-\frac{b^3}{d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))}",1,"((12*a*b^2*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (3*b*(a^2 - b^2) + b*(a^2 + b^2)*Cos[2*(c + d*x)] + a*(a^2 + b^2)*Sin[2*(c + d*x)])/((a^2 + b^2)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])))/(2*d)","A",1
124,1,192,82,0.4025459,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{b \sin (c+d x) \left(a^2 b \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+(a+i b) \left(a^2 (c+d x)+a b (i c+i d x+1)-i b^2\right)\right)+a^2 \cos (c+d x) \left(a b \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)+(a+i b)^2 (c+d x)\right)-2 i a^2 b \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))}{a d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d 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}",1,"(a^2*Cos[c + d*x]*((a + I*b)^2*(c + d*x) + a*b*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2]) + b*((a + I*b)*((-I)*b^2 + a*b*(1 + I*c + I*d*x) + a^2*(c + d*x)) + a^2*b*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])*Sin[c + d*x] - (2*I)*a^2*b*ArcTan[Tan[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(a*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","C",1
125,1,79,83,0.2170506,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{\frac{2 a \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}-\frac{b}{\left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))}}{d}","-\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,"((2*a*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - b/((a^2 + b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x])))/d","A",1
126,1,32,32,0.0317226,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-2),x]","\frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))}","\frac{\sin (c+d x)}{a d (a \cos (c+d x)+b \sin (c+d x))}",1,"Sin[c + d*x]/(a*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","A",1
127,1,120,92,0.78693,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","-\frac{\frac{2 a \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}+\frac{b \sec (c+d x)}{a+b \tan (c+d x)}+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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,"-(((2*a*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b*Sec[c + d*x])/(a + b*Tan[c + d*x]))/(b^2*d))","A",1
128,1,51,75,0.26521,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{-\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)}{b^3 d}","-\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{\frac{a}{b^2}+\frac{1}{a}}{d (a \cot (c+d x)+b)}+\frac{\tan (c+d x)}{b^2 d}",1,"(-2*a*Log[a + b*Tan[c + d*x]] + b*Tan[c + d*x] - (a^2 + b^2)/(a + b*Tan[c + d*x]))/(b^3*d)","A",1
129,1,709,179,6.1152198,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","-\frac{3 \left(2 a^2+b^2\right) \sec ^2(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{2 b^4 d (a+b \tan (c+d x))^2}+\frac{3 \left(2 a^2+b^2\right) \sec ^2(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{2 b^4 d (a+b \tan (c+d x))^2}-\frac{6 a \sqrt{a^2+b^2} \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left(\frac{\sqrt{a^2+b^2} \left(a \sin \left(\frac{1}{2} (c+d x)\right)-b \cos \left(\frac{1}{2} (c+d x)\right)\right)}{a^2 \cos \left(\frac{1}{2} (c+d x)\right)+b^2 \cos \left(\frac{1}{2} (c+d x)\right)}\right)}{b^4 d (a+b \tan (c+d x))^2}-\frac{2 a \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^2}-\frac{2 a \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d (a+b \tan (c+d x))^2}+\frac{2 a \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^2}-\frac{(a-i b) (a+i b) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{b^3 d (a+b \tan (c+d x))^2}+\frac{\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{4 b^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a+b \tan (c+d x))^2}-\frac{\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{4 b^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 (a+b \tan (c+d x))^2}","\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{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{a^2+b^2}{b^3 d (a \cos (c+d x)+b \sin (c+d x))}-\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,"-(((a - I*b)*(a + I*b)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(b^3*d*(a + b*Tan[c + d*x])^2)) - (2*a*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^3*d*(a + b*Tan[c + d*x])^2) - (6*a*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a^2 + b^2]*(-(b*Cos[(c + d*x)/2]) + a*Sin[(c + d*x)/2]))/(a^2*Cos[(c + d*x)/2] + b^2*Cos[(c + d*x)/2])]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^4*d*(a + b*Tan[c + d*x])^2) - (3*(2*a^2 + b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*b^4*d*(a + b*Tan[c + d*x])^2) + (3*(2*a^2 + b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*b^4*d*(a + b*Tan[c + d*x])^2) + (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(4*b^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a + b*Tan[c + d*x])^2) - (2*a*Sec[c + d*x]^2*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^2) - (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(4*b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a + b*Tan[c + d*x])^2) + (2*a*Sec[c + d*x]^2*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^2)","C",1
130,1,122,141,2.7786269,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^2,x]","\frac{4 b \left(2 a^2+b^2\right) \tan (c+d x)+\frac{b^4 \sec ^4(c+d x)-4 \left(a^2+b^2\right) \left(3 a^2 \log (a+b \tan (c+d x))+a^2+3 a b \tan (c+d x) \log (a+b \tan (c+d x))+b^2\right)}{a+b \tan (c+d x)}-2 a b^2 \tan ^2(c+d x)}{3 b^5 d}","-\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{\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{a \tan ^2(c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}",1,"(4*b*(2*a^2 + b^2)*Tan[c + d*x] - 2*a*b^2*Tan[c + d*x]^2 + (b^4*Sec[c + d*x]^4 - 4*(a^2 + b^2)*(a^2 + b^2 + 3*a^2*Log[a + b*Tan[c + d*x]] + 3*a*b*Log[a + b*Tan[c + d*x]]*Tan[c + d*x]))/(a + b*Tan[c + d*x]))/(3*b^5*d)","A",1
131,1,211,216,1.1178861,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\frac{2 a \left(a^2-3 b^2\right) \sin (c+d x)}{\left(a^2+b^2\right)^3}-\frac{2 b \left(b^2-3 a^2\right) \cos (c+d x)}{\left(a^2+b^2\right)^3}-\frac{6 b^2 \left(b^2-4 a^2\right) \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{7/2}}-\frac{b^3 \left(8 a^2+b^2\right)}{a \left(a^2+b^2\right)^3 (a \cos (c+d x)+b \sin (c+d x))}+\frac{b^4 \sin (c+d x)}{a (a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x))^2}}{2 d}","\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{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}}+\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))}",1,"((-6*b^2*(-4*a^2 + b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (2*b*(-3*a^2 + b^2)*Cos[c + d*x])/(a^2 + b^2)^3 + (2*a*(a^2 - 3*b^2)*Sin[c + d*x])/(a^2 + b^2)^3 + (b^4*Sin[c + d*x])/(a*(a - I*b)^2*(a + I*b)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (b^3*(8*a^2 + b^2))/(a*(a^2 + b^2)^3*(a*Cos[c + d*x] + b*Sin[c + d*x])))/(2*d)","C",1
132,1,154,122,1.2601208,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\frac{2 a \left(a^2-3 b^2\right) (c+d x)}{\left(a^2+b^2\right)^3}+\frac{6 b^2 \sin (c+d x)}{\left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))}-\frac{2 b \left(b^2-3 a^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{\left(a^2+b^2\right)^3}-\frac{b^3}{(a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x))^2}}{2 d}","-\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,"((2*a*(a^2 - 3*b^2)*(c + d*x))/(a^2 + b^2)^3 - (2*b*(-3*a^2 + b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a^2 + b^2)^3 - b^3/((a - I*b)^2*(a + I*b)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (6*b^2*Sin[c + d*x])/((a^2 + b^2)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])))/(2*d)","C",1
133,1,119,119,0.7050626,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\frac{2 \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)}{\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)}{\left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^2}}{2 d}","\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*(2*a^2 - b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (b*((4*a^2 + b^2)*Cos[c + d*x] + 3*a*b*Sin[c + d*x]))/((a^2 + b^2)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2))/(2*d)","A",1
134,1,57,22,0.1221245,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{a \sin (2 (c+d x))-b \cos (2 (c+d x))}{2 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2}","-\frac{1}{2 b d (a+b \tan (c+d x))^2}",1,"(-(b*Cos[2*(c + d*x)]) + a*Sin[2*(c + d*x)])/(2*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)","B",1
135,1,132,103,0.2688675,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-3),x]","\frac{\left(a^2+b^2\right) (a \sin (c+d x)-b \cos (c+d x))+2 \sqrt{a^2+b^2} (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{2 d (a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x))^2}","-\frac{b \cos (c+d x)-a \sin (c+d x)}{2 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{\tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{2 d \left(a^2+b^2\right)^{3/2}}",1,"((a^2 + b^2)*(-(b*Cos[c + d*x]) + a*Sin[c + d*x]) + 2*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*(a - I*b)^2*(a + I*b)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)","C",1
136,1,57,86,0.5207223,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{-\frac{a^2+b^2}{2 (a+b \tan (c+d x))^2}+\frac{2 a}{a+b \tan (c+d x)}+\log (a+b \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,"(Log[a + b*Tan[c + d*x]] - (a^2 + b^2)/(2*(a + b*Tan[c + d*x])^2) + (2*a)/(a + b*Tan[c + d*x]))/(b^3*d)","A",1
137,1,396,260,2.4398999,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(\frac{b^2 \left(a^2+b^2\right) \sin (c+d x)}{a}+\frac{6 \left(2 a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}+\frac{2 b \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{2 b \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+2 b (a \cos (c+d x)+b \sin (c+d x))^2+\frac{b (2 a-b) (2 a+b) (a \cos (c+d x)+b \sin (c+d x))}{a}+6 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2-6 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2\right)}{2 b^4 d (a+b \tan (c+d x))^3}","-\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{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{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,"(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*((b^2*(a^2 + b^2)*Sin[c + d*x])/a + ((2*a - b)*b*(2*a + b)*(a*Cos[c + d*x] + b*Sin[c + d*x]))/a + 2*b*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (6*(2*a^2 + b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/Sqrt[a^2 + b^2] + 6*a*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - 6*a*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (2*b*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (2*b*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(2*b^4*d*(a + b*Tan[c + d*x])^3)","A",1
138,1,140,161,2.9982667,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{-2 a \left(-\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)\right)+2 \left(a^2+b^2\right) \left(\frac{3 a^2+4 a b \tan (c+d x)-b^2}{2 (a+b \tan (c+d x))^2}+\log (a+b \tan (c+d x))\right)+\frac{b^4 \sec ^4(c+d x)}{2 (a+b \tan (c+d x))^2}}{b^5 d}","\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{\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{\left(a^2+b^2\right)^2}{2 a^2 b^3 d (a \cot (c+d x)+b)^2}-\frac{3 a \tan (c+d x)}{b^4 d}+\frac{\tan ^2(c+d x)}{2 b^3 d}",1,"((b^4*Sec[c + d*x]^4)/(2*(a + b*Tan[c + d*x])^2) - 2*a*(-2*a*Log[a + b*Tan[c + d*x]] + b*Tan[c + d*x] - (a^2 + b^2)/(a + b*Tan[c + d*x])) + 2*(a^2 + b^2)*(Log[a + b*Tan[c + d*x]] + (3*a^2 - b^2 + 4*a*b*Tan[c + d*x])/(2*(a + b*Tan[c + d*x])^2)))/(b^5*d)","A",1
139,1,688,383,2.4636231,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(\frac{6 b^2 \left(a^2+b^2\right)^2 \sin (c+d x)}{a}+2 b \left(36 a^2+13 b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2+\frac{2 b \left(36 a^2+13 b^2\right) \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{2 b \left(36 a^2+13 b^2\right) \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{6 b (a-i b) (a+i b) \left(8 a^2-b^2\right) (a \cos (c+d x)+b \sin (c+d x))}{a}+30 a \left(4 a^2+3 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2-30 a \left(4 a^2+3 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^2+60 \sqrt{a^2+b^2} \left(4 a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)+\frac{2 b^3 \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{2 b^3 \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{b^2 (b-9 a) (a \cos (c+d x)+b \sin (c+d x))^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{b^2 (9 a+b) (a \cos (c+d x)+b \sin (c+d x))^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{12 b^6 d (a+b \tan (c+d x))^3}","-\frac{4 a^3 \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\frac{4 a^2 \sec (c+d x)}{b^5 d}-\frac{6 a \left(a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{b^6 d}-\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{2 \left(a^2+b^2\right) \sec (c+d x)}{b^5 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{\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,"(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*((6*b^2*(a^2 + b^2)^2*Sin[c + d*x])/a + (6*(a - I*b)*(a + I*b)*b*(8*a^2 - b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x]))/a + 2*b*(36*a^2 + 13*b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + 60*Sqrt[a^2 + b^2]*(4*a^2 + b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + 30*a*(4*a^2 + 3*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 - 30*a*(4*a^2 + 3*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + (b^2*(-9*a + b)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*b^3*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (2*b*(36*a^2 + 13*b^2)*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (2*b^3*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (b^2*(9*a + b)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (2*b*(36*a^2 + 13*b^2)*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(12*b^6*d*(a + b*Tan[c + d*x])^3)","C",1
140,1,272,232,1.3356023,"\int \frac{\sec ^5(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^5/(a*Cos[c + d*x] + b*Sin[c + d*x])^3,x]","\frac{4 a^2 b^4 \tan ^4(c+d x)+b^4 \sec ^4(c+d x) \left(a^2-2 a b \tan (c+d x)+3 b^2\right)-20 a b^3 \left(a^2+b^2\right) \tan ^3(c+d x)+4 b^2 \tan ^2(c+d x) \left(-13 a^4-10 a^2 b^2+3 \left(5 a^4+6 a^2 b^2+b^4\right) \log (a+b \tan (c+d x))\right)+2 \left(a^2+b^2\right) \left(19 a^4+6 a^2 \left(5 a^2+b^2\right) \log (a+b \tan (c+d x))+16 a^2 b^2-3 b^4\right)+4 a b \tan (c+d x) \left(4 a^4+17 a^2 b^2+6 \left(5 a^4+6 a^2 b^2+b^4\right) \log (a+b \tan (c+d x))+11 b^4\right)+b^6 \sec ^6(c+d x)}{4 b^7 d (a+b \tan (c+d x))^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 \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{3 \left(2 a^2+b^2\right) \tan ^2(c+d x)}{2 b^5 d}-\frac{\left(a^2+b^2\right)^3}{2 a^2 b^5 d (a \cot (c+d x)+b)^2}-\frac{a \tan ^3(c+d x)}{b^4 d}+\frac{\tan ^4(c+d x)}{4 b^3 d}",1,"(2*(a^2 + b^2)*(19*a^4 + 16*a^2*b^2 - 3*b^4 + 6*a^2*(5*a^2 + b^2)*Log[a + b*Tan[c + d*x]]) + b^6*Sec[c + d*x]^6 + 4*a*b*(4*a^4 + 17*a^2*b^2 + 11*b^4 + 6*(5*a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x] + 4*b^2*(-13*a^4 - 10*a^2*b^2 + 3*(5*a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x]^2 - 20*a*b^3*(a^2 + b^2)*Tan[c + d*x]^3 + 4*a^2*b^4*Tan[c + d*x]^4 + b^4*Sec[c + d*x]^4*(a^2 + 3*b^2 - 2*a*b*Tan[c + d*x]))/(4*b^7*d*(a + b*Tan[c + d*x])^2)","A",1
141,1,419,165,6.2383713,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\left(a^2-2 a b-b^2\right) \left(a^2+2 a b-b^2\right) (c+d x)}{d (a-i b)^4 (a+i b)^4}+\frac{2 \left(9 a^2 b^2 \sin (c+d x)-2 b^4 \sin (c+d x)\right)}{3 a d (a-i b)^3 (a+i b)^3 (a \cos (c+d x)+b \sin (c+d x))}-\frac{b^3 \left(6 a^2+b^2\right)}{3 a d (a-i b)^3 (a+i b)^3 (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{4 i \left(a^3 b-a b^3\right) \tan ^{-1}(\tan (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{2 \left(a^3 b-a b^3\right) \log \left((a \cos (c+d x)+b \sin (c+d x))^2\right)}{d \left(a^2+b^2\right)^4}+\frac{4 \left(i a^{10} b+a^9 b^2+2 i a^8 b^3+2 a^7 b^4-2 i a^4 b^7-2 a^3 b^8-i a^2 b^9-a b^{10}\right) (c+d x)}{d (a-i b)^8 (a+i b)^7}+\frac{b^4 \sin (c+d x)}{3 a d (a-i b)^2 (a+i b)^2 (a \cos (c+d x)+b \sin (c+d x))^3}","-\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(a^4-6 a^2 b^2+b^4\right)}{\left(a^2+b^2\right)^4}",1,"((a^2 - 2*a*b - b^2)*(a^2 + 2*a*b - b^2)*(c + d*x))/((a - I*b)^4*(a + I*b)^4*d) + (4*(I*a^10*b + a^9*b^2 + (2*I)*a^8*b^3 + 2*a^7*b^4 - (2*I)*a^4*b^7 - 2*a^3*b^8 - I*a^2*b^9 - a*b^10)*(c + d*x))/((a - I*b)^8*(a + I*b)^7*d) - ((4*I)*(a^3*b - a*b^3)*ArcTan[Tan[c + d*x]])/((a^2 + b^2)^4*d) + (2*(a^3*b - a*b^3)*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2])/((a^2 + b^2)^4*d) + (b^4*Sin[c + d*x])/(3*a*(a - I*b)^2*(a + I*b)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (b^3*(6*a^2 + b^2))/(3*a*(a - I*b)^3*(a + I*b)^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (2*(9*a^2*b^2*Sin[c + d*x] - 2*b^4*Sin[c + d*x]))/(3*a*(a - I*b)^3*(a + I*b)^3*d*(a*Cos[c + d*x] + b*Sin[c + d*x]))","C",1
142,1,165,157,1.1184257,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\frac{6 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)}{\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(3 a^4 b-a^2 b^3+b^5\right) \cos (2 (c+d x))}{(a-i b)^3 (a+i b)^3 (a \cos (c+d x)+b \sin (c+d x))^3}}{6 d}","\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}}+\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(3 a^4 b-a^2 b^3+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}",1,"((6*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) + (-3*(3*a^4*b - a^2*b^3 + b^5)*Cos[2*(c + d*x)] + (b*(-9*a^2 + b^2)*(2*(a^2 + b^2) + 3*a*b*Sin[2*(c + d*x)]))/2)/((a - I*b)^3*(a + I*b)^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3))/(6*d)","C",1
143,1,124,30,0.6550137,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\left(2 a b^3-6 a^3 b\right) \cos (3 (c+d x))-6 a b \left(a^2+b^2\right) \cos (c+d x)+2 \left(a^2-b^2\right) \sin (c+d x) \left(\left(3 a^2-b^2\right) \cos (2 (c+d x))+3 a^2+b^2\right)}{12 a d \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^3}","-\frac{\cot ^3(c+d x)}{3 b d (a \cot (c+d x)+b)^3}",1,"(-6*a*b*(a^2 + b^2)*Cos[c + d*x] + (-6*a^3*b + 2*a*b^3)*Cos[3*(c + d*x)] + 2*(a^2 - b^2)*(3*a^2 + b^2 + (3*a^2 - b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(12*a*(a^2 + b^2)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)","B",1
144,1,128,141,0.7280488,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[Cos[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{\frac{6 a \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{5/2}}+\frac{3 \left(a^3-a b^2\right) \sin (2 (c+d x))-4 b \left(a^2+b^2\right)-6 a^2 b \cos (2 (c+d x))}{2 \left(a^2+b^2\right)^2 (a \cos (c+d x)+b \sin (c+d x))^3}}{6 d}","-\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,"((6*a*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (-4*b*(a^2 + b^2) - 6*a^2*b*Cos[2*(c + d*x)] + 3*(a^3 - a*b^2)*Sin[2*(c + d*x)])/(2*(a^2 + b^2)^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3))/(6*d)","A",1
145,1,85,98,0.2920887,"\int \frac{1}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^(-4),x]","\frac{\sin (c+d x) \left(\left(a^2-b^2\right) \cos (2 (c+d x))+2 a^2+b^2\right)-a b \cos (3 (c+d x))}{3 a d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^3}","\frac{2 \sin (c+d x)}{3 a d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))}-\frac{b \cos (c+d x)-a \sin (c+d x)}{3 d \left(a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^3}",1,"(-(a*b*Cos[3*(c + d*x)]) + (2*a^2 + b^2 + (a^2 - b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(3*a*(a^2 + b^2)*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)","A",1
146,1,290,231,3.2758832,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[Sec[c + d*x]/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(\frac{3 b \left(2 a^2+b^2\right) \cos (c+d x) (a+b \tan (c+d x))^2}{a^2+b^2}+\frac{6 a \left(2 a^2+3 b^2\right) \cos ^2(c+d x) (a+b \tan (c+d x))^3 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\left(a^2+b^2\right)^{3/2}}+3 b^2 \tan (c+d x) (a \cos (c+d x)+b \sin (c+d x))+6 \cos ^2(c+d x) (a+b \tan (c+d x))^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \cos ^2(c+d x) (a+b \tan (c+d x))^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 b^3 \sec (c+d x)\right)}{6 b^4 d (a+b \tan (c+d x))^4}","\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)}{2 b^2 d \left(a^2+b^2\right)^{3/2}}+\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{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,"-1/6*(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(2*b^3*Sec[c + d*x] + 3*b^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*Tan[c + d*x] + (3*b*(2*a^2 + b^2)*Cos[c + d*x]*(a + b*Tan[c + d*x])^2)/(a^2 + b^2) + (6*a*(2*a^2 + 3*b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*Cos[c + d*x]^2*(a + b*Tan[c + d*x])^3)/(a^2 + b^2)^(3/2) + 6*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3 - 6*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3))/(b^4*d*(a + b*Tan[c + d*x])^4)","A",1
147,1,133,138,2.1673738,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^2/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{-4 \left(a^2+b^2\right) \left(a^2+3 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)+b^2\right)+6 a (a+b \tan (c+d x)) \left(a^2-4 a (a+b \tan (c+d x))-2 (a+b \tan (c+d x))^2 \log (a+b \tan (c+d x))+b^2\right)+3 b^4 \sec ^4(c+d x)}{3 b^5 d (a+b \tan (c+d x))^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{\left(a^2+b^2\right)^2}{3 a^3 b^2 d (a \cot (c+d x)+b)^3}-\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,"(3*b^4*Sec[c + d*x]^4 - 4*(a^2 + b^2)*(a^2 + b^2 + 3*a*b*Tan[c + d*x] + 3*b^2*Tan[c + d*x]^2) + 6*a*(a + b*Tan[c + d*x])*(a^2 + b^2 - 4*a*(a + b*Tan[c + d*x]) - 2*Log[a + b*Tan[c + d*x]]*(a + b*Tan[c + d*x])^2))/(3*b^5*d*(a + b*Tan[c + d*x])^3)","A",1
148,1,538,400,3.4654176,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","-\frac{\sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(18 b^2 \left(a^2+b^2\right) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))+6 b \left(12 a^2+b^2\right) (a \cos (c+d x)+b \sin (c+d x))^2+30 \left(4 a^2+b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^3-30 \left(4 a^2+b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b \sin (c+d x))^3+\frac{60 a \left(4 a^2+3 b^2\right) (a \cos (c+d x)+b \sin (c+d x))^3 \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}+4 b^3 \left(a^2+b^2\right)-\frac{3 b^2 (a \cos (c+d x)+b \sin (c+d x))^3}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{3 b^2 (a \cos (c+d x)+b \sin (c+d x))^3}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+48 a b (a \cos (c+d x)+b \sin (c+d x))^3+\frac{48 a b \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^3}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{48 a b \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^3}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{12 b^6 d (a+b \tan (c+d x))^4}","\frac{8 a^2 \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) \tanh ^{-1}(\sin (c+d x))}{b^6 d}+\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{2 \left(a^2+b^2\right)}{b^5 d (a \cos (c+d x)+b \sin (c+d x))}+\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{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{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,"-1/12*(Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])*(4*b^3*(a^2 + b^2) + 18*b^2*(a^2 + b^2)*Sin[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x]) + 6*b*(12*a^2 + b^2)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 + 48*a*b*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 + (60*a*(4*a^2 + 3*b^2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/Sqrt[a^2 + b^2] + 30*(4*a^2 + b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 - 30*(4*a^2 + b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 - (3*b^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (48*a*b*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (3*b^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (48*a*b*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(b^6*d*(a + b*Tan[c + d*x])^4)","A",1
149,1,295,232,2.0140527,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^4/(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]","\frac{3 b^4 \sec ^4(c+d x) \left(a^2-a b \tan (c+d x)+2 b^2\right)-2 \left(37 a^6+36 a^4 b^2-6 a^2 b^4 \tan ^4(c+d x)+3 a^2 b^4+6 a b^3 \tan ^3(c+d x) \left(\left(5 a^2+3 b^2\right) \log (a+b \tan (c+d x))-3 a^2\right)+6 a^4 \left(5 a^2+3 b^2\right) \log (a+b \tan (c+d x))+6 b^2 \tan ^2(c+d x) \left(6 a^4+3 a^2 \left(5 a^2+3 b^2\right) \log (a+b \tan (c+d x))+11 a^2 b^2+2 b^4\right)+3 a b \tan (c+d x) \left(27 a^4+6 a^2 \left(5 a^2+3 b^2\right) \log (a+b \tan (c+d x))+30 a^2 b^2+b^4\right)+4 b^6\right)+b^6 \sec ^6(c+d x)}{3 b^7 d (a+b \tan (c+d x))^3}","-\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{\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{10 a^6+9 a^4 b^2+b^6}{a^3 b^6 d (a \cot (c+d x)+b)}+\frac{2 a^6+3 a^4 b^2-b^6}{a^3 b^5 d (a \cot (c+d x)+b)^2}-\frac{2 a \tan ^2(c+d x)}{b^5 d}+\frac{\tan ^3(c+d x)}{3 b^4 d}",1,"(b^6*Sec[c + d*x]^6 + 3*b^4*Sec[c + d*x]^4*(a^2 + 2*b^2 - a*b*Tan[c + d*x]) - 2*(37*a^6 + 36*a^4*b^2 + 3*a^2*b^4 + 4*b^6 + 6*a^4*(5*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]] + 3*a*b*(27*a^4 + 30*a^2*b^2 + b^4 + 6*a^2*(5*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x] + 6*b^2*(6*a^4 + 11*a^2*b^2 + 2*b^4 + 3*a^2*(5*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x]^2 + 6*a*b^3*(-3*a^2 + (5*a^2 + 3*b^2)*Log[a + b*Tan[c + d*x]])*Tan[c + d*x]^3 - 6*a^2*b^4*Tan[c + d*x]^4))/(3*b^7*d*(a + b*Tan[c + d*x])^3)","A",1
150,1,82,99,0.1371488,"\int \frac{\cos ^5(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+15 i \cos (2 (c+d x))+6 i \cos (4 (c+d x))+i \cos (6 (c+d x))+60 c+60 d x}{192 a d}","\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,"(60*c + 60*d*x + (15*I)*Cos[2*(c + d*x)] + (6*I)*Cos[4*(c + d*x)] + I*Cos[6*(c + d*x)] + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)])/(192*a*d)","A",1
151,1,111,70,0.0669092,"\int \frac{\cos ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{5 \sin (c+d x)}{8 a d}+\frac{5 \sin (3 (c+d x))}{48 a d}+\frac{\sin (5 (c+d x))}{80 a d}+\frac{i \cos (c+d x)}{8 a d}+\frac{i \cos (3 (c+d x))}{16 a d}+\frac{i \cos (5 (c+d x))}{80 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/8)*Cos[c + d*x])/(a*d) + ((I/16)*Cos[3*(c + d*x)])/(a*d) + ((I/80)*Cos[5*(c + d*x)])/(a*d) + (5*Sin[c + d*x])/(8*a*d) + (5*Sin[3*(c + d*x)])/(48*a*d) + Sin[5*(c + d*x)]/(80*a*d)","A",1
152,1,60,75,0.09943,"\int \frac{\cos ^3(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{8 \sin (2 (c+d x))+\sin (4 (c+d x))+4 i \cos (2 (c+d x))+i \cos (4 (c+d x))+12 c+12 d x}{32 a d}","\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,"(12*c + 12*d*x + (4*I)*Cos[2*(c + d*x)] + I*Cos[4*(c + d*x)] + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])/(32*a*d)","A",1
153,1,73,52,0.0694037,"\int \frac{\cos ^2(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{3 \sin (c+d x)}{4 a d}+\frac{\sin (3 (c+d x))}{12 a d}+\frac{i \cos (c+d x)}{4 a d}+\frac{i \cos (3 (c+d x))}{12 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/4)*Cos[c + d*x])/(a*d) + ((I/12)*Cos[3*(c + d*x)])/(a*d) + (3*Sin[c + d*x])/(4*a*d) + Sin[3*(c + d*x)]/(12*a*d)","A",1
154,1,38,46,0.0661575,"\int \frac{\cos (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{2 (c+d x)+\sin (2 (c+d x))+i \cos (2 (c+d x))}{4 a d}","\frac{x}{2 a}+\frac{i \cos (c+d x)}{2 d (a \cos (c+d x)+i a \sin (c+d x))}",1,"(2*(c + d*x) + I*Cos[2*(c + d*x)] + Sin[2*(c + d*x)])/(4*a*d)","A",1
155,1,29,29,0.0328476,"\int \frac{1}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-1),x]","\frac{i}{d (a \cos (c+d x)+i a \sin (c+d x))}","\frac{i}{d (a \cos (c+d x)+i a \sin (c+d x))}",1,"I/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x]))","A",1
156,1,23,23,0.0657941,"\int \frac{\sec (c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","\frac{i \log (\cos (c+d x))+c+d x}{a d}","\frac{x}{a}+\frac{i \log (\cos (c+d x))}{a d}",1,"(c + d*x + I*Log[Cos[c + d*x]])/(a*d)","A",1
157,1,35,31,0.2189948,"\int \frac{\sec ^2(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","-\frac{i \left(\sec (c+d x)+2 i \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)\right)}{a d}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{i \sec (c+d x)}{a d}",1,"((-I)*((2*I)*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + Sec[c + d*x]))/(a*d)","A",1
158,1,35,34,0.1917056,"\int \frac{\sec ^3(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","-\frac{i \sec (c+d x) (\sec (c+d x)+2 i \sec (c) \sin (d x))}{2 a d}","\frac{\tan (c+d x)}{a d}-\frac{i \sec ^2(c+d x)}{2 a d}",1,"((-1/2*I)*Sec[c + d*x]*(Sec[c + d*x] + (2*I)*Sec[c]*Sin[d*x]))/(a*d)","A",1
159,1,54,60,0.2461338,"\int \frac{\sec ^4(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","-\frac{i \left((4+3 i \sin (2 (c+d x))) \sec ^3(c+d x)+12 i \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)\right)}{12 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,"((-1/12*I)*((12*I)*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + Sec[c + d*x]^3*(4 + (3*I)*Sin[2*(c + d*x)])))/(a*d)","A",1
160,1,53,52,0.279292,"\int \frac{\sec ^5(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","-\frac{i \sec ^4(c+d x) (i \sec (c) (4 \sin (c+2 d x)+\sin (3 c+4 d x))-3 i \tan (c)+3)}{12 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,"((-1/12*I)*Sec[c + d*x]^4*(3 + I*Sec[c]*(4*Sin[c + 2*d*x] + Sin[3*c + 4*d*x]) - (3*I)*Tan[c]))/(a*d)","A",1
161,1,66,84,0.4649589,"\int \frac{\sec ^6(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","-\frac{i \left((70 i \sin (2 (c+d x))+15 i \sin (4 (c+d x))+64) \sec ^5(c+d x)+240 i \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)\right)}{320 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,"((-1/320*I)*((240*I)*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + Sec[c + d*x]^5*(64 + (70*I)*Sin[2*(c + d*x)] + (15*I)*Sin[4*(c + d*x)])))/(a*d)","A",1
162,1,67,70,0.3715524,"\int \frac{\sec ^7(c+d x)}{a \cos (c+d x)+i a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^7/(a*Cos[c + d*x] + I*a*Sin[c + d*x]),x]","-\frac{i \sec (c) \sec ^6(c+d x) (10 \cos (c)-i (-15 \sin (c+2 d x)-6 \sin (3 c+4 d x)-\sin (5 c+6 d x)+10 \sin (c)))}{60 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,"((-1/60*I)*Sec[c]*Sec[c + d*x]^6*(10*Cos[c] - I*(10*Sin[c] - 15*Sin[c + 2*d*x] - 6*Sin[3*c + 4*d*x] - Sin[5*c + 6*d*x])))/(a*d)","A",1
163,1,149,85,0.1052205,"\int \frac{\cos ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{15 \sin (c+d x)}{32 a^2 d}+\frac{11 \sin (3 (c+d x))}{96 a^2 d}+\frac{\sin (5 (c+d x))}{32 a^2 d}+\frac{\sin (7 (c+d x))}{224 a^2 d}+\frac{5 i \cos (c+d x)}{32 a^2 d}+\frac{3 i \cos (3 (c+d x))}{32 a^2 d}+\frac{i \cos (5 (c+d x))}{32 a^2 d}+\frac{i \cos (7 (c+d x))}{224 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,"(((5*I)/32)*Cos[c + d*x])/(a^2*d) + (((3*I)/32)*Cos[3*(c + d*x)])/(a^2*d) + ((I/32)*Cos[5*(c + d*x)])/(a^2*d) + ((I/224)*Cos[7*(c + d*x)])/(a^2*d) + (15*Sin[c + d*x])/(32*a^2*d) + (11*Sin[3*(c + d*x)])/(96*a^2*d) + Sin[5*(c + d*x)]/(32*a^2*d) + Sin[7*(c + d*x)]/(224*a^2*d)","A",1
164,1,82,101,0.1227371,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{21 \sin (2 (c+d x))+6 \sin (4 (c+d x))+\sin (6 (c+d x))+15 i \cos (2 (c+d x))+6 i \cos (4 (c+d x))+i \cos (6 (c+d x))+24 c+24 d x}{96 a^2 d}","-\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,"(24*c + 24*d*x + (15*I)*Cos[2*(c + d*x)] + (6*I)*Cos[4*(c + d*x)] + I*Cos[6*(c + d*x)] + 21*Sin[2*(c + d*x)] + 6*Sin[4*(c + d*x)] + Sin[6*(c + d*x)])/(96*a^2*d)","A",1
165,1,111,68,0.0818844,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{\sin (c+d x)}{2 a^2 d}+\frac{\sin (3 (c+d x))}{8 a^2 d}+\frac{\sin (5 (c+d x))}{40 a^2 d}+\frac{i \cos (c+d x)}{4 a^2 d}+\frac{i \cos (3 (c+d x))}{8 a^2 d}+\frac{i \cos (5 (c+d x))}{40 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,"((I/4)*Cos[c + d*x])/(a^2*d) + ((I/8)*Cos[3*(c + d*x)])/(a^2*d) + ((I/40)*Cos[5*(c + d*x)])/(a^2*d) + Sin[c + d*x]/(2*a^2*d) + Sin[3*(c + d*x)]/(8*a^2*d) + Sin[5*(c + d*x)]/(40*a^2*d)","A",1
166,1,60,89,0.099785,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{4 \sin (2 (c+d x))+\sin (4 (c+d x))+4 i \cos (2 (c+d x))+i \cos (4 (c+d x))+4 c+4 d x}{16 a^2 d}","\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,"(4*c + 4*d*x + (4*I)*Cos[2*(c + d*x)] + I*Cos[4*(c + d*x)] + 4*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])/(16*a^2*d)","A",1
167,1,73,52,0.0572548,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{\sin (c+d x)}{2 a^2 d}+\frac{\sin (3 (c+d x))}{6 a^2 d}+\frac{i \cos (c+d x)}{2 a^2 d}+\frac{i \cos (3 (c+d x))}{6 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,"((I/2)*Cos[c + d*x])/(a^2*d) + ((I/6)*Cos[3*(c + d*x)])/(a^2*d) + Sin[c + d*x]/(2*a^2*d) + Sin[3*(c + d*x)]/(6*a^2*d)","A",1
168,1,31,31,0.0409872,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-2),x]","\frac{i}{2 d (a \cos (c+d x)+i a \sin (c+d x))^2}","\frac{i}{2 d (a \cos (c+d x)+i a \sin (c+d x))^2}",1,"(I/2)/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2)","A",1
169,1,184,46,0.2421072,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) \left(\cos \left(\frac{3}{2} (c+d x)\right)+i \sin \left(\frac{3}{2} (c+d x)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 i\right)+\sin \left(\frac{1}{2} (c+d x)\right) \left(i \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-i \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2\right)\right)}{a^2 d (\tan (c+d x)-i)^2}","\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,"-((Sec[c + d*x]^2*(Cos[(c + d*x)/2]*(2*I + Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (2 + I*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - I*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[(c + d*x)/2])*(Cos[(3*(c + d*x))/2] + I*Sin[(3*(c + d*x))/2]))/(a^2*d*(-I + Tan[c + d*x])^2))","B",1
170,1,71,55,0.4233541,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{4 \tan ^{-1}(\tan (d x))+i \sec (c) \sec (c+d x) \left(\cos (d x) \log \left(\cos ^2(c+d x)\right)+\cos (2 c+d x) \log \left(\cos ^2(c+d x)\right)+2 i \sin (d x)\right)}{2 a^2 d}","-\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,"(4*ArcTan[Tan[d*x]] + I*Sec[c]*Sec[c + d*x]*(Cos[d*x]*Log[Cos[c + d*x]^2] + Cos[2*c + d*x]*Log[Cos[c + d*x]^2] + (2*I)*Sin[d*x]))/(2*a^2*d)","A",1
171,1,146,56,0.43161,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) \left(2 \sin (c+d x)+8 i \cos (c+d x)+3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \cos (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 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,"-1/4*(Sec[c + d*x]^2*((8*I)*Cos[c + d*x] + 3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*Sin[c + d*x]))/(a^2*d)","B",1
172,1,68,34,0.2603997,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{\sec (c) \sec ^3(c+d x) (3 \sin (2 c+d x)-2 \sin (2 c+3 d x)+3 i \cos (2 c+d x)-3 \sin (d x)+3 i \cos (d x))}{6 a^2 d}","-\frac{i \tan ^3(c+d x) (-\cot (c+d x)+i)^3}{3 a^2 d}",1,"-1/6*(Sec[c]*Sec[c + d*x]^3*((3*I)*Cos[d*x] + (3*I)*Cos[2*c + d*x] - 3*Sin[d*x] + 3*Sin[2*c + d*x] - 2*Sin[2*c + 3*d*x]))/(a^2*d)","A",1
173,1,215,84,0.9967611,"\int \frac{\sec ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","-\frac{\sec ^4(c+d x) \left(18 \sin (c+d x)-30 \sin (3 (c+d x))+128 i \cos (c+d x)+45 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+60 \cos (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+15 \cos (4 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-45 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 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,"-1/192*(Sec[c + d*x]^4*((128*I)*Cos[c + d*x] + 45*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 60*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 15*Cos[4*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 45*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 18*Sin[c + d*x] - 30*Sin[3*(c + d*x)]))/(a^2*d)","B",1
174,1,77,70,0.4121248,"\int \frac{\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^2,x]","\frac{\sec (c) \sec ^5(c+d x) (-5 \sin (2 c+d x)+5 \sin (2 c+3 d x)+\sin (4 c+5 d x)-5 i \cos (2 c+d x)+5 \sin (d x)-5 i \cos (d x))}{20 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,"(Sec[c]*Sec[c + d*x]^5*((-5*I)*Cos[d*x] - (5*I)*Cos[2*c + d*x] + 5*Sin[d*x] - 5*Sin[2*c + d*x] + 5*Sin[2*c + 3*d*x] + Sin[4*c + 5*d*x]))/(20*a^2*d)","A",1
175,1,106,125,0.1858209,"\int \frac{\cos ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{132 \sin (2 (c+d x))+60 \sin (4 (c+d x))+20 \sin (6 (c+d x))+3 \sin (8 (c+d x))+108 i \cos (2 (c+d x))+60 i \cos (4 (c+d x))+20 i \cos (6 (c+d x))+3 i \cos (8 (c+d x))+120 c+120 d x}{768 a^3 d}","-\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,"(120*c + 120*d*x + (108*I)*Cos[2*(c + d*x)] + (60*I)*Cos[4*(c + d*x)] + (20*I)*Cos[6*(c + d*x)] + (3*I)*Cos[8*(c + d*x)] + 132*Sin[2*(c + d*x)] + 60*Sin[4*(c + d*x)] + 20*Sin[6*(c + d*x)] + 3*Sin[8*(c + d*x)])/(768*a^3*d)","A",1
176,1,149,106,0.0792064,"\int \frac{\cos ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{5 \sin (c+d x)}{16 a^3 d}+\frac{\sin (3 (c+d x))}{8 a^3 d}+\frac{\sin (5 (c+d x))}{20 a^3 d}+\frac{\sin (7 (c+d x))}{112 a^3 d}+\frac{3 i \cos (c+d x)}{16 a^3 d}+\frac{i \cos (3 (c+d x))}{8 a^3 d}+\frac{i \cos (5 (c+d x))}{20 a^3 d}+\frac{i \cos (7 (c+d x))}{112 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,"(((3*I)/16)*Cos[c + d*x])/(a^3*d) + ((I/8)*Cos[3*(c + d*x)])/(a^3*d) + ((I/20)*Cos[5*(c + d*x)])/(a^3*d) + ((I/112)*Cos[7*(c + d*x)])/(a^3*d) + (5*Sin[c + d*x])/(16*a^3*d) + Sin[3*(c + d*x)]/(8*a^3*d) + Sin[5*(c + d*x)]/(20*a^3*d) + Sin[7*(c + d*x)]/(112*a^3*d)","A",1
177,1,84,131,0.1130575,"\int \frac{\cos ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{18 \sin (2 (c+d x))+9 \sin (4 (c+d x))+2 \sin (6 (c+d x))+18 i \cos (2 (c+d x))+9 i \cos (4 (c+d x))+2 i \cos (6 (c+d x))+12 c+12 d x}{96 a^3 d}","\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,"(12*c + 12*d*x + (18*I)*Cos[2*(c + d*x)] + (9*I)*Cos[4*(c + d*x)] + (2*I)*Cos[6*(c + d*x)] + 18*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + 2*Sin[6*(c + d*x)])/(96*a^3*d)","A",1
178,1,111,90,0.0725355,"\int \frac{\cos ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{\sin (c+d x)}{4 a^3 d}+\frac{\sin (3 (c+d x))}{6 a^3 d}+\frac{\sin (5 (c+d x))}{20 a^3 d}+\frac{i \cos (c+d x)}{4 a^3 d}+\frac{i \cos (3 (c+d x))}{6 a^3 d}+\frac{i \cos (5 (c+d x))}{20 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/4)*Cos[c + d*x])/(a^3*d) + ((I/6)*Cos[3*(c + d*x)])/(a^3*d) + ((I/20)*Cos[5*(c + d*x)])/(a^3*d) + Sin[c + d*x]/(4*a^3*d) + Sin[3*(c + d*x)]/(6*a^3*d) + Sin[5*(c + d*x)]/(20*a^3*d)","A",1
179,1,77,32,0.0577125,"\int \frac{\cos (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{\sin (2 (c+d x))}{4 a^3 d}+\frac{\sin (4 (c+d x))}{8 a^3 d}+\frac{i \cos (2 (c+d x))}{4 a^3 d}+\frac{i \cos (4 (c+d x))}{8 a^3 d}","\frac{i \cot ^2(c+d x)}{2 a^3 d (\cot (c+d x)+i)^2}",1,"((I/4)*Cos[2*(c + d*x)])/(a^3*d) + ((I/8)*Cos[4*(c + d*x)])/(a^3*d) + Sin[2*(c + d*x)]/(4*a^3*d) + Sin[4*(c + d*x)]/(8*a^3*d)","B",1
180,1,31,31,0.0376623,"\int \frac{1}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^(-3),x]","\frac{i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3}","\frac{i}{3 d (a \cos (c+d x)+i a \sin (c+d x))^3}",1,"(I/3)/(d*(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3)","A",1
181,1,91,61,0.2817283,"\int \frac{\sec (c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \sec ^2(c+d x) (\sin (2 (c+d x))-i \cos (2 (c+d x))) (\log (\cos (c+d x))+\tan (c+d x) (i \log (\cos (c+d x))+d x+i)-i d x-1)}{a^3 d (\tan (c+d x)-i)^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,"(I*Sec[c + d*x]^2*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)])*(-1 - I*d*x + Log[Cos[c + d*x]] + (I + d*x + I*Log[Cos[c + d*x]])*Tan[c + d*x]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
182,1,109,62,0.3292988,"\int \frac{\sec ^2(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","-\frac{i \sec ^3(c+d x) (\cos (d x)+i \sin (d x))^3 \left((\tan (c+d x)-5 i) (\cos (2 c-d x)+i \sin (2 c-d x))+6 (\cos (3 c)+i \sin (3 c)) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)\right)}{a^3 d (\tan (c+d x)-i)^3}","\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,"((-I)*Sec[c + d*x]^3*(Cos[d*x] + I*Sin[d*x])^3*(6*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]]*(Cos[3*c] + I*Sin[3*c]) + (Cos[2*c - d*x] + I*Sin[2*c - d*x])*(-5*I + Tan[c + d*x])))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
183,1,110,75,0.6162619,"\int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \sec (c) \sec ^2(c+d x) (\cos (c) (4 \log (\cos (c+d x))-4 i d x+1)-i (2 \cos (c+2 d x) (d x+i \log (\cos (c+d x)))+2 \cos (3 c+2 d x) (d x+i \log (\cos (c+d x)))-6 \sin (d x) \cos (c+d x)))}{2 a^3 d}","\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,"((I/2)*Sec[c]*Sec[c + d*x]^2*(Cos[c]*(1 - (4*I)*d*x + 4*Log[Cos[c + d*x]]) - I*(2*Cos[c + 2*d*x]*(d*x + I*Log[Cos[c + d*x]]) + 2*Cos[3*c + 2*d*x]*(d*x + I*Log[Cos[c + d*x]]) - 6*Cos[c + d*x]*Sin[d*x])))/(a^3*d)","A",1
184,1,64,76,0.4683644,"\int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^4/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \left(\sec ^3(c+d x) (9 i \sin (2 (c+d x))-24 \cos (2 (c+d x))-20)-60 i \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)\right)}{12 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,"((I/12)*((-60*I)*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]] + Sec[c + d*x]^3*(-20 - 24*Cos[2*(c + d*x)] + (9*I)*Sin[2*(c + d*x)])))/(a^3*d)","A",1
185,1,90,34,0.4659345,"\int \frac{\sec ^5(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^5/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","-\frac{i \sec (c) \sec ^4(c+d x) (2 i \sin (c+2 d x)-2 i \sin (3 c+2 d x)+i \sin (3 c+4 d x)+2 \cos (c+2 d x)+2 \cos (3 c+2 d x)-3 i \sin (c)+3 \cos (c))}{4 a^3 d}","\frac{i \tan ^4(c+d x) (-\cot (c+d x)+i)^4}{4 a^3 d}",1,"((-1/4*I)*Sec[c]*Sec[c + d*x]^4*(3*Cos[c] + 2*Cos[c + 2*d*x] + 2*Cos[3*c + 2*d*x] - (3*I)*Sin[c] + (2*I)*Sin[c + 2*d*x] - (2*I)*Sin[3*c + 2*d*x] + I*Sin[3*c + 4*d*x]))/(a^3*d)","B",1
186,1,115,104,0.4324275,"\int \frac{\sec ^6(c+d x)}{(a \cos (c+d x)+i a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^6/(a*Cos[c + d*x] + I*a*Sin[c + d*x])^3,x]","\frac{i \sec ^8(c+d x) (\sin (3 (c+d x))-i \cos (3 (c+d x))) \left(-150 i \sin (2 (c+d x))+105 i \sin (4 (c+d x))+640 \cos (2 (c+d x))+1680 i \cos ^5(c+d x) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{d x}{2}\right)+\sin (c)\right)+448\right)}{960 a^3 d (\tan (c+d x)-i)^3}","\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,"((I/960)*Sec[c + d*x]^8*((-I)*Cos[3*(c + d*x)] + Sin[3*(c + d*x)])*(448 + (1680*I)*ArcTanh[Sin[c] + Cos[c]*Tan[(d*x)/2]]*Cos[c + d*x]^5 + 640*Cos[2*(c + d*x)] - (150*I)*Sin[2*(c + d*x)] + (105*I)*Sin[4*(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^3)","A",1
187,1,90,66,2.1657467,"\int \cos ^{-n}(c+d x) (a \cos (c+d x)+i a \sin (c+d x))^n \, dx","Integrate[(a*Cos[c + d*x] + I*a*Sin[c + d*x])^n/Cos[c + d*x]^n,x]","\frac{\cos ^{-n}(c+d x) \left(n (\tan (c+d x)-i) \, _2F_1\left(1,n+1;n+2;\frac{1}{2} (i \tan (c+d x)+1)\right)-2 i (n+1)\right) (a (\cos (c+d x)+i \sin (c+d x)))^n}{4 d n (n+1)}","-\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,"((a*(Cos[c + d*x] + I*Sin[c + d*x]))^n*((-2*I)*(1 + n) + n*Hypergeometric2F1[1, 1 + n, 2 + n, (1 + I*Tan[c + d*x])/2]*(-I + Tan[c + d*x])))/(4*d*n*(1 + n)*Cos[c + d*x]^n)","A",1
188,1,16,5,0.0185053,"\int \frac{1}{\sec (x)+\tan (x)} \, dx","Integrate[(Sec[x] + Tan[x])^(-1),x]","2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\log (\sin (x)+1)",1,"2*Log[Cos[x/2] + Sin[x/2]]","B",1
189,1,19,10,0.0210924,"\int \frac{\sin (x)}{\sec (x)+\tan (x)} \, dx","Integrate[Sin[x]/(Sec[x] + Tan[x]),x]","\sin (x)-2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\sin (x)-\log (\sin (x)+1)",1,"-2*Log[Cos[x/2] + Sin[x/2]] + Sin[x]","A",1
190,1,4,4,0.018203,"\int \frac{\cos (x)}{\sec (x)+\tan (x)} \, dx","Integrate[Cos[x]/(Sec[x] + Tan[x]),x]","x+\cos (x)","x+\cos (x)",1,"x + Cos[x]","A",1
191,1,25,11,0.0325633,"\int \frac{\tan (x)}{\sec (x)+\tan (x)} \, dx","Integrate[Tan[x]/(Sec[x] + Tan[x]),x]","x-\frac{2 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}","x+\frac{\cos (x)}{\sin (x)+1}",1,"x - (2*Sin[x/2])/(Cos[x/2] + Sin[x/2])","B",1
192,1,20,9,0.0221015,"\int \frac{\cot (x)}{\sec (x)+\tan (x)} \, dx","Integrate[Cot[x]/(Sec[x] + Tan[x]),x]","-x+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)","-x-\tanh ^{-1}(\cos (x))",1,"-x - Log[Cos[x/2]] + Log[Sin[x/2]]","B",1
193,1,23,10,0.0169548,"\int \frac{\sec (x)}{\sec (x)+\tan (x)} \, dx","Integrate[Sec[x]/(Sec[x] + Tan[x]),x]","\frac{2 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}","-\frac{\cos (x)}{\sin (x)+1}",1,"(2*Sin[x/2])/(Cos[x/2] + Sin[x/2])","B",1
194,1,20,11,0.0218896,"\int \frac{\csc (x)}{\sec (x)+\tan (x)} \, dx","Integrate[Csc[x]/(Sec[x] + Tan[x]),x]","\log (\sin (x))-2 \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\log (\sin (x))-\log (\sin (x)+1)",1,"-2*Log[Cos[x/2] + Sin[x/2]] + Log[Sin[x]]","A",1
195,1,18,9,0.0184153,"\int \frac{1}{\sec (x)-\tan (x)} \, dx","Integrate[(Sec[x] - Tan[x])^(-1),x]","-2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)","-\log (1-\sin (x))",1,"-2*Log[Cos[x/2] - Sin[x/2]]","A",1
196,1,23,14,0.021208,"\int \frac{\sin (x)}{\sec (x)-\tan (x)} \, dx","Integrate[Sin[x]/(Sec[x] - Tan[x]),x]","-\sin (x)-2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)","-\sin (x)-\log (1-\sin (x))",1,"-2*Log[Cos[x/2] - Sin[x/2]] - Sin[x]","A",1
197,1,6,6,0.0203293,"\int \frac{\cos (x)}{\sec (x)-\tan (x)} \, dx","Integrate[Cos[x]/(Sec[x] - Tan[x]),x]","x-\cos (x)","x-\cos (x)",1,"x - Cos[x]","A",1
198,1,29,15,0.0346611,"\int \frac{\tan (x)}{\sec (x)-\tan (x)} \, dx","Integrate[Tan[x]/(Sec[x] - Tan[x]),x]","\frac{2 \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}-x","\frac{\cos (x)}{1-\sin (x)}-x",1,"-x + (2*Sin[x/2])/(Cos[x/2] - Sin[x/2])","A",1
199,1,18,7,0.0221972,"\int \frac{\cot (x)}{\sec (x)-\tan (x)} \, dx","Integrate[Cot[x]/(Sec[x] - Tan[x]),x]","x+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)","x-\tanh ^{-1}(\cos (x))",1,"x - Log[Cos[x/2]] + Log[Sin[x/2]]","B",1
200,1,25,11,0.0163801,"\int \frac{\sec (x)}{\sec (x)-\tan (x)} \, dx","Integrate[Sec[x]/(Sec[x] - Tan[x]),x]","\frac{2 \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}","\frac{\cos (x)}{1-\sin (x)}",1,"(2*Sin[x/2])/(Cos[x/2] - Sin[x/2])","B",1
201,1,22,13,0.022018,"\int \frac{\csc (x)}{\sec (x)-\tan (x)} \, dx","Integrate[Csc[x]/(Sec[x] - Tan[x]),x]","\log (\sin (x))-2 \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)","\log (\sin (x))-\log (1-\sin (x))",1,"-2*Log[Cos[x/2] - Sin[x/2]] + Log[Sin[x]]","A",1
202,1,15,23,0.0148464,"\int \csc (c+d x) (\cot (c+d x)+\csc (c+d x)) \, dx","Integrate[Csc[c + d*x]*(Cot[c + d*x] + Csc[c + d*x]),x]","-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{d}","-\frac{\cot (c+d x)}{d}-\frac{\csc (c+d x)}{d}",1,"-(Cot[(c + d*x)/2]/d)","A",1
203,1,14,6,0.0073636,"\int \frac{\sin (x)}{\cot (x)+\csc (x)} \, dx","Integrate[Sin[x]/(Cot[x] + Csc[x]),x]","2 \left(\frac{x}{2}-\frac{\sin (x)}{2}\right)","x-\sin (x)",1,"2*(x/2 - Sin[x]/2)","B",1
204,1,20,10,0.0067997,"\int \frac{\cos (x)}{\cot (x)+\csc (x)} \, dx","Integrate[Cos[x]/(Cot[x] + Csc[x]),x]","2 \log \left(\cos \left(\frac{x}{2}\right)\right)-2 \cos ^2\left(\frac{x}{2}\right)","\log (\cos (x)+1)-\cos (x)",1,"-2*Cos[x/2]^2 + 2*Log[Cos[x/2]]","A",1
205,1,36,7,0.0258011,"\int \frac{\tan (x)}{\cot (x)+\csc (x)} \, dx","Integrate[Tan[x]/(Cot[x] + Csc[x]),x]","-x-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","\tanh ^{-1}(\sin (x))-x",1,"-x - Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]]","B",1
206,1,10,12,0.0168874,"\int \frac{\cot (x)}{\cot (x)+\csc (x)} \, dx","Integrate[Cot[x]/(Cot[x] + Csc[x]),x]","x-\tan \left(\frac{x}{2}\right)","x-\frac{\sin (x)}{\cos (x)+1}",1,"x - Tan[x/2]","A",1
207,1,25,11,0.0082612,"\int \frac{\sec (x)}{\cot (x)+\csc (x)} \, dx","Integrate[Sec[x]/(Cot[x] + Csc[x]),x]","2 \log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(1-2 \cos ^2\left(\frac{x}{2}\right)\right)","\log (\cos (x)+1)-\log (\cos (x))",1,"2*Log[Cos[x/2]] - Log[1 - 2*Cos[x/2]^2]","B",1
208,1,6,9,0.0187796,"\int \frac{\csc (x)}{\cot (x)+\csc (x)} \, dx","Integrate[Csc[x]/(Cot[x] + Csc[x]),x]","\tan \left(\frac{x}{2}\right)","\frac{\sin (x)}{\cos (x)+1}",1,"Tan[x/2]","A",1
209,1,14,4,0.0065752,"\int \frac{\sin (x)}{-\cot (x)+\csc (x)} \, dx","Integrate[Sin[x]/(-Cot[x] + Csc[x]),x]","2 \left(\frac{x}{2}+\frac{\sin (x)}{2}\right)","x+\sin (x)",1,"2*(x/2 + Sin[x]/2)","B",1
210,1,20,10,0.0099941,"\int \frac{\cos (x)}{-\cot (x)+\csc (x)} \, dx","Integrate[Cos[x]/(-Cot[x] + Csc[x]),x]","2 \log \left(\sin \left(\frac{x}{2}\right)\right)-2 \sin ^2\left(\frac{x}{2}\right)","\cos (x)+\log (1-\cos (x))",1,"2*Log[Sin[x/2]] - 2*Sin[x/2]^2","A",1
211,1,46,5,0.016746,"\int \frac{\tan (x)}{-\cot (x)+\csc (x)} \, dx","Integrate[Tan[x]/(-Cot[x] + Csc[x]),x]","2 \left(\frac{x}{2}-\frac{1}{2} \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\frac{1}{2} \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)","x+\tanh ^{-1}(\sin (x))",1,"2*(x/2 - Log[Cos[x/2] - Sin[x/2]]/2 + Log[Cos[x/2] + Sin[x/2]]/2)","B",1
212,1,16,16,0.015463,"\int \frac{\cot (x)}{-\cot (x)+\csc (x)} \, dx","Integrate[Cot[x]/(-Cot[x] + Csc[x]),x]","\frac{1}{2} \left(-2 x-2 \cot \left(\frac{x}{2}\right)\right)","-x-\frac{\sin (x)}{1-\cos (x)}",1,"(-2*x - 2*Cot[x/2])/2","A",1
213,1,25,13,0.0101704,"\int \frac{\sec (x)}{-\cot (x)+\csc (x)} \, dx","Integrate[Sec[x]/(-Cot[x] + Csc[x]),x]","2 \log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(1-2 \sin ^2\left(\frac{x}{2}\right)\right)","\log (1-\cos (x))-\log (\cos (x))",1,"2*Log[Sin[x/2]] - Log[1 - 2*Sin[x/2]^2]","A",1
214,1,8,12,0.0066739,"\int \frac{\csc (x)}{-\cot (x)+\csc (x)} \, dx","Integrate[Csc[x]/(-Cot[x] + Csc[x]),x]","-\cot \left(\frac{x}{2}\right)","-\frac{\sin (x)}{1-\cos (x)}",1,"-Cot[x/2]","A",1
215,1,61,23,0.1859457,"\int \frac{1}{\csc (c+d x)+\sin (c+d x)} \, dx","Integrate[(Csc[c + d*x] + Sin[c + d*x])^(-1),x]","-\frac{\tanh ^{-1}\left(\frac{\cos (c)-(\sin (c)-i) \tan \left(\frac{d x}{2}\right)}{\sqrt{2}}\right)+\tanh ^{-1}\left(\frac{\cos (c)-(\sin (c)+i) \tan \left(\frac{d x}{2}\right)}{\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] - (-I + Sin[c])*Tan[(d*x)/2])/Sqrt[2]] + ArcTanh[(Cos[c] - (I + Sin[c])*Tan[(d*x)/2])/Sqrt[2]])/(Sqrt[2]*d))","C",1
216,1,30,51,0.0703877,"\int \frac{\sin (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Integrate[Sin[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","-\frac{\tan ^{-1}\left(\sqrt{2} \tan (c+d x)\right)}{\sqrt{2} d}+\frac{c}{d}+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,"c/d + x - ArcTan[Sqrt[2]*Tan[c + d*x]]/(Sqrt[2]*d)","A",1
217,1,20,18,0.1071778,"\int \frac{\cos (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Integrate[Cos[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","\frac{\log (3-\cos (2 (c+d x)))}{2 d}","\frac{\log \left(\sin ^2(c+d x)+1\right)}{2 d}",1,"Log[3 - Cos[2*(c + d*x)]]/(2*d)","A",1
218,1,24,29,0.0348254,"\int \frac{\tan (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Integrate[Tan[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))-\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]] + ArcTanh[Sin[c + d*x]])/(2*d)","A",1
219,1,11,11,0.038361,"\int \frac{\cot (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Integrate[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",1
220,1,30,16,0.0453716,"\int \frac{\sec (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Integrate[Sec[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","\frac{\log \left(2-\cos ^2(c+d x)\right)-2 \log (\cos (c+d x))}{4 d}","\frac{\tanh ^{-1}\left(\sin ^2(c+d x)\right)}{2 d}",1,"(-2*Log[Cos[c + d*x]] + Log[2 - Cos[c + d*x]^2])/(4*d)","A",1
221,1,22,48,0.0255514,"\int \frac{\csc (c+d x)}{\csc (c+d x)+\sin (c+d x)} \, dx","Integrate[Csc[c + d*x]/(Csc[c + d*x] + Sin[c + d*x]),x]","\frac{\tan ^{-1}\left(\sqrt{2} \tan (c+d x)\right)}{\sqrt{2} d}","\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,"ArcTan[Sqrt[2]*Tan[c + d*x]]/(Sqrt[2]*d)","A",1
222,1,10,10,0.0096266,"\int \frac{1}{\csc (c+d x)-\sin (c+d x)} \, dx","Integrate[(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",1
223,1,23,14,0.0079526,"\int \frac{\sin (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Integrate[Sin[c + d*x]/(Csc[c + d*x] - Sin[c + d*x]),x]","\frac{\tan (c+d x)}{d}-\frac{\tan ^{-1}(\tan (c+d x))}{d}","\frac{\tan (c+d x)}{d}-x",1,"-(ArcTan[Tan[c + d*x]]/d) + Tan[c + d*x]/d","A",1
224,1,12,12,0.0082715,"\int \frac{\cos (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Integrate[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",1
225,1,34,34,0.0148418,"\int \frac{\tan (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Integrate[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,"-1/2*ArcTanh[Sin[c + d*x]]/d + (Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
226,1,11,11,0.0017159,"\int \frac{\cot (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Integrate[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",1
227,1,15,15,0.0116727,"\int \frac{\sec (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Integrate[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",1
228,1,10,10,0.0038457,"\int \frac{\csc (c+d x)}{\csc (c+d x)-\sin (c+d x)} \, dx","Integrate[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",1
229,1,33,33,0.0137684,"\int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Integrate[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,"-1/3*(b*Cos[c + d*x]^3)/d - (a*Cos[c + d*x]^4)/(4*d)","A",1
230,1,38,33,0.1152874,"\int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{3 a \cos (c+d x)+a \cos (3 (c+d x))+3 b \cos (2 (c+d x))}{12 d}","\frac{b \sin ^2(c+d x)}{2 d}-\frac{a \cos ^3(c+d x)}{3 d}",1,"-1/12*(3*a*Cos[c + d*x] + 3*b*Cos[2*(c + d*x)] + a*Cos[3*(c + d*x)])/d","A",1
231,1,40,22,0.0103454,"\int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{a \cos ^2(c+d x)}{2 d}+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d}","-\frac{(a \cos (c+d x)+b)^2}{2 a d}",1,"-((b*Cos[c]*Cos[d*x])/d) - (a*Cos[c + d*x]^2)/(2*d) + (b*Sin[c]*Sin[d*x])/d","A",1
232,1,37,26,0.0177884,"\int (a \sin (c+d x)+b \tan (c+d x)) \, dx","Integrate[a*Sin[c + d*x] + b*Tan[c + d*x],x]","\frac{a \sin (c) \sin (d x)}{d}-\frac{a \cos (c) \cos (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]*Cos[d*x])/d) - (b*Log[Cos[c + d*x]])/d + (a*Sin[c]*Sin[d*x])/d","A",1
233,1,25,25,0.016658,"\int \sec (c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Integrate[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",1
234,1,28,28,0.0187719,"\int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Integrate[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",1
235,1,33,33,0.0218486,"\int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Integrate[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",1
236,1,77,106,0.3846722,"\int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{30 \left(a^2+2 b^2\right) \sin (c+d x)-5 \left(a^2+4 b^2\right) \sin (3 (c+d x))-3 a (a \sin (5 (c+d x))-20 b (c+d x)+5 b \sin (4 (c+d x)))}{240 d}","\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,"(30*(a^2 + 2*b^2)*Sin[c + d*x] - 5*(a^2 + 4*b^2)*Sin[3*(c + d*x)] - 3*a*(-20*b*(c + d*x) + 5*b*Sin[4*(c + d*x)] + a*Sin[5*(c + d*x)]))/(240*d)","A",1
237,1,82,86,0.1813565,"\int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{-3 a^2 \sin (4 (c+d x))+12 a^2 c+12 a^2 d x+48 a b \sin (c+d x)-16 a b \sin (3 (c+d x))-24 b^2 \sin (2 (c+d x))+48 b^2 c+48 b^2 d x}{96 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,"(12*a^2*c + 48*b^2*c + 12*a^2*d*x + 48*b^2*d*x + 48*a*b*Sin[c + d*x] - 24*b^2*Sin[2*(c + d*x)] - 16*a*b*Sin[3*(c + d*x)] - 3*a^2*Sin[4*(c + d*x)])/(96*d)","A",1
238,1,117,87,0.1460065,"\int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{3 \left(a^2-4 b^2\right) \sin (c+d x)+a^2 (-\sin (3 (c+d x)))-6 a b \sin (2 (c+d x))+12 a b c+12 a b d x-12 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{12 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,"(12*a*b*c + 12*a*b*d*x - 12*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(a^2 - 4*b^2)*Sin[c + d*x] - 6*a*b*Sin[2*(c + d*x)] - a^2*Sin[3*(c + d*x)])/(12*d)","A",1
239,1,116,77,0.5810675,"\int (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Integrate[(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{-2 \left(a^2-2 b^2\right) (c+d x)+\tan (c+d x) \left(a^2 \cos (2 (c+d x))+a^2-4 b^2\right)+8 a b \sin (c+d x)+8 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","-\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,"-1/4*(-2*(a^2 - 2*b^2)*(c + d*x) + 8*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 8*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*a*b*Sin[c + d*x] + (a^2 - 4*b^2 + a^2*Cos[2*(c + d*x)])*Tan[c + d*x])/d","A",1
240,1,75,90,0.1922632,"\int \sec (c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Integrate[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 a^2 \sin (c+d x)-4 a b \tan ^{-1}(\tan (c+d x))+4 a b \tan (c+d x)+b^2 \tan (c+d x) \sec (c+d x)}{2 d}","\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,"(-4*a*b*ArcTan[Tan[c + d*x]] + (2*a^2 - b^2)*ArcTanh[Sin[c + d*x]] - 2*a^2*Sin[c + d*x] + 4*a*b*Tan[c + d*x] + b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
241,1,201,99,1.1178082,"\int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{\sec ^3(c+d x) \left(2 \sin (c+d x) \left(\left(3 a^2-b^2\right) \cos (2 (c+d x))+3 a^2+6 a b \cos (c+d x)+b^2\right)-9 a \cos (c+d x) \left(a (c+d x)-b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-3 a \cos (3 (c+d x)) \left(a (c+d x)-b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{12 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,"(Sec[c + d*x]^3*(-9*a*Cos[c + d*x]*(a*(c + d*x) - b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 3*a*Cos[3*(c + d*x)]*(a*(c + d*x) - b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(3*a^2 + b^2 + 6*a*b*Cos[c + d*x] + (3*a^2 - b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(12*d)","B",1
242,1,336,125,0.6076415,"\int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{\sec ^4(c+d x) \left(12 \left(4 a^2+b^2\right) \cos (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 \left(4 a^2+b^2\right) \cos (4 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+24 a^2 \sin (c+d x)+24 a^2 \sin (3 (c+d x))+36 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-36 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+32 a b \sin (2 (c+d x))-16 a b \sin (4 (c+d x))+42 b^2 \sin (c+d x)-6 b^2 \sin (3 (c+d x))+9 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-9 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 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,"(Sec[c + d*x]^4*(36*a^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*(4*a^2 + b^2)*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*(4*a^2 + b^2)*Cos[4*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 36*a^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 9*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*a^2*Sin[c + d*x] + 42*b^2*Sin[c + d*x] + 32*a*b*Sin[2*(c + d*x)] + 24*a^2*Sin[3*(c + d*x)] - 6*b^2*Sin[3*(c + d*x)] - 16*a*b*Sin[4*(c + d*x)]))/(192*d)","B",1
243,1,114,77,0.233263,"\int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{-45 \left(a^3+8 a b^2\right) \cos (2 (c+d x))+5 a^3 \cos (6 (c+d x))-360 b \left(a^2+2 b^2\right) \cos (c+d x)-60 a^2 b \cos (3 (c+d x))+36 a^2 b \cos (5 (c+d x))+90 a b^2 \cos (4 (c+d x))+80 b^3 \cos (3 (c+d x))}{960 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}",1,"(-360*b*(a^2 + 2*b^2)*Cos[c + d*x] - 45*(a^3 + 8*a*b^2)*Cos[2*(c + d*x)] - 60*a^2*b*Cos[3*(c + d*x)] + 80*b^3*Cos[3*(c + d*x)] + 90*a*b^2*Cos[4*(c + d*x)] + 36*a^2*b*Cos[5*(c + d*x)] + 5*a^3*Cos[6*(c + d*x)])/(960*d)","A",1
244,1,106,120,0.1865886,"\int \cos ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","-\frac{-\frac{1}{5} a^3 \cos ^5(c+d x)+\frac{1}{3} a \left(a^2-3 b^2\right) \cos ^3(c+d x)+\frac{1}{2} b \left(3 a^2-b^2\right) \cos ^2(c+d x)-\frac{3}{4} a^2 b \cos ^4(c+d x)+3 a b^2 \cos (c+d x)+b^3 \log (\cos (c+d x))}{d}","\frac{a^3 \cos ^5(c+d x)}{5 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{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] + (b*(3*a^2 - b^2)*Cos[c + d*x]^2)/2 + (a*(a^2 - 3*b^2)*Cos[c + d*x]^3)/3 - (3*a^2*b*Cos[c + d*x]^4)/4 - (a^3*Cos[c + d*x]^5)/5 + b^3*Log[Cos[c + d*x]])/d)","A",1
245,1,98,112,0.2680883,"\int \cos (c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{-4 \left(a^3-6 a b^2\right) \cos (2 (c+d x))+a^3 \cos (4 (c+d x))+8 b \left(4 b^2-9 a^2\right) \cos (c+d x)+8 a^2 b \cos (3 (c+d x))-96 a b^2 \log (\cos (c+d x))+32 b^3 \sec (c+d x)}{32 d}","\frac{a^3 \cos ^4(c+d x)}{4 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{3 a b^2 \log (\cos (c+d x))}{d}+\frac{b^3 \sec (c+d x)}{d}",1,"(8*b*(-9*a^2 + 4*b^2)*Cos[c + d*x] - 4*(a^3 - 6*a*b^2)*Cos[2*(c + d*x)] + 8*a^2*b*Cos[3*(c + d*x)] + a^3*Cos[4*(c + d*x)] - 96*a*b^2*Log[Cos[c + d*x]] + 32*b^3*Sec[c + d*x])/(32*d)","A",1
246,1,102,116,0.2928011,"\int (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Integrate[(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{a^3 \cos (3 (c+d x))-9 a \left(a^2-4 b^2\right) \cos (c+d x)+9 a^2 b \cos (2 (c+d x))-36 a^2 b \log (\cos (c+d x))+36 a b^2 \sec (c+d x)+6 b^3 \sec ^2(c+d x)+12 b^3 \log (\cos (c+d x))}{12 d}","\frac{a^3 \cos ^3(c+d x)}{3 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{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"(-9*a*(a^2 - 4*b^2)*Cos[c + d*x] + 9*a^2*b*Cos[2*(c + d*x)] + a^3*Cos[3*(c + d*x)] - 36*a^2*b*Log[Cos[c + d*x]] + 12*b^3*Log[Cos[c + d*x]] + 36*a*b^2*Sec[c + d*x] + 6*b^3*Sec[c + d*x]^2)/(12*d)","A",1
247,1,100,115,0.5116313,"\int \sec (c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{3 a^3 \cos (2 (c+d x))+2 \left(-6 b \left(b^2-3 a^2\right) \sec (c+d x)-6 a \left(a^2-3 b^2\right) \log (\cos (c+d x))+9 a b^2 \sec ^2(c+d x)+2 b^3 \sec ^3(c+d x)\right)+36 a^2 b \cos (c+d x)}{12 d}","\frac{a^3 \cos ^2(c+d x)}{2 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{3 a b^2 \sec ^2(c+d x)}{2 d}+\frac{b^3 \sec ^3(c+d x)}{3 d}",1,"(36*a^2*b*Cos[c + d*x] + 3*a^3*Cos[2*(c + d*x)] + 2*(-6*a*(a^2 - 3*b^2)*Log[Cos[c + d*x]] - 6*b*(-3*a^2 + b^2)*Sec[c + d*x] + 9*a*b^2*Sec[c + d*x]^2 + 2*b^3*Sec[c + d*x]^3))/(12*d)","A",1
248,1,97,111,1.897399,"\int \sec ^2(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^2*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{4 a^3 \cos (c+d x)+\left(6 a^2 b-2 b^3\right) \sec ^2(c+d x)+4 a \left(a^2-3 b^2\right) \sec (c+d x)+12 a^2 b \log (\cos (c+d x))+4 a b^2 \sec ^3(c+d x)+b^3 \sec ^4(c+d x)}{4 d}","\frac{a^3 \cos (c+d x)}{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 b^2 \sec ^3(c+d x)}{d}+\frac{b^3 \sec ^4(c+d x)}{4 d}",1,"(4*a^3*Cos[c + d*x] + 12*a^2*b*Log[Cos[c + d*x]] + 4*a*(a^2 - 3*b^2)*Sec[c + d*x] + (6*a^2*b - 2*b^3)*Sec[c + d*x]^2 + 4*a*b^2*Sec[c + d*x]^3 + b^3*Sec[c + d*x]^4)/(4*d)","A",1
249,1,99,119,0.3171296,"\int \sec ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{60 a^3 \log (\cos (c+d x))-20 b \left(b^2-3 a^2\right) \sec ^3(c+d x)+30 a \left(a^2-3 b^2\right) \sec ^2(c+d x)-180 a^2 b \sec (c+d x)+45 a b^2 \sec ^4(c+d x)+12 b^3 \sec ^5(c+d x)}{60 d}","\frac{a^3 \log (\cos (c+d x))}{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{3 a b^2 \sec ^4(c+d x)}{4 d}+\frac{b^3 \sec ^5(c+d x)}{5 d}",1,"(60*a^3*Log[Cos[c + d*x]] - 180*a^2*b*Sec[c + d*x] + 30*a*(a^2 - 3*b^2)*Sec[c + d*x]^2 - 20*b*(-3*a^2 + b^2)*Sec[c + d*x]^3 + 45*a*b^2*Sec[c + d*x]^4 + 12*b^3*Sec[c + d*x]^5)/(60*d)","A",1
250,1,100,113,0.3684858,"\int \frac{\cos ^3(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{-\frac{4 b \cos (c+d x)}{a^2}+4 \left(\frac{b^4 \log (a \cos (c+d x)+b)}{a^3 \left(b^2-a^2\right)}+\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a+b}+\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a-b}\right)+\frac{\cos (2 (c+d x))}{a}}{4 d}","-\frac{b \cos (c+d x)}{a^2 d}-\frac{b^4 \log (a \cos (c+d x)+b)}{a^3 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 ^2(c+d x)}{2 a d}",1,"((-4*b*Cos[c + d*x])/a^2 + Cos[2*(c + d*x)]/a + 4*(Log[Cos[(c + d*x)/2]]/(a - b) + (b^4*Log[b + a*Cos[c + d*x]])/(a^3*(-a^2 + b^2)) + Log[Sin[(c + d*x)/2]]/(a + b)))/(4*d)","A",1
251,1,80,92,0.227666,"\int \frac{\cos ^2(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{\frac{b^3 \log (a \cos (c+d x)+b)}{a^4-a^2 b^2}+\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a+b}+\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b-a}+\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 + Log[Cos[(c + d*x)/2]]/(-a + b) + (b^3*Log[b + a*Cos[c + d*x]])/(a^4 - a^2*b^2) + Log[Sin[(c + d*x)/2]]/(a + b))/d","A",1
252,1,70,80,0.0976914,"\int \frac{\cos (c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Integrate[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 (a-b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+a (a+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a d (a-b) (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,"(a*(a + b)*Log[Cos[(c + d*x)/2]] - b^2*Log[b + a*Cos[c + d*x]] + a*(a - b)*Log[Sin[(c + d*x)/2]])/(a*(a - b)*(a + b)*d)","A",1
253,1,63,74,0.0931747,"\int \frac{1}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Integrate[(a*Sin[c + d*x] + b*Tan[c + d*x])^(-1),x]","\frac{(a-b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\left((a+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b \log (a \cos (c+d x)+b)}{d (a-b) (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,"(-((a + b)*Log[Cos[(c + d*x)/2]]) + b*Log[b + a*Cos[c + d*x]] + (a - b)*Log[Sin[(c + d*x)/2]])/((a - b)*(a + b)*d)","A",1
254,1,64,75,0.0625123,"\int \frac{\sec (c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{(a-b) \log (1-\cos (c+d x))+(a+b) \log (\cos (c+d x)+1)-2 a \log (a \cos (c+d x)+b)}{2 d (a-b) (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,"((a - b)*Log[1 - Cos[c + d*x]] + (a + b)*Log[1 + Cos[c + d*x]] - 2*a*Log[b + a*Cos[c + d*x]])/(2*(a - b)*(a + b)*d)","A",1
255,1,103,94,0.1507306,"\int \frac{\sec ^2(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","2 \left(-\frac{a^2 \log (a \cos (c+d x)+b)}{2 b d \left(b^2-a^2\right)}+\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d (a+b)}+\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d (b-a)}-\frac{\log (\cos (c+d x))}{2 b d}\right)","\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,"2*(Log[Cos[(c + d*x)/2]]/(2*(-a + b)*d) - Log[Cos[c + d*x]]/(2*b*d) - (a^2*Log[b + a*Cos[c + d*x]])/(2*b*(-a^2 + b^2)*d) + Log[Sin[(c + d*x)/2]]/(2*(a + b)*d))","A",1
256,1,92,108,0.2897308,"\int \frac{\sec ^3(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{\frac{a^3 \log (a \cos (c+d x)+b)}{b^4-a^2 b^2}+\frac{a \log (\cos (c+d x))}{b^2}+\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a+b}+\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{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[Cos[(c + d*x)/2]]/(a - b) + (a*Log[Cos[c + d*x]])/b^2 + (a^3*Log[b + a*Cos[c + d*x]])/(-(a^2*b^2) + b^4) + Log[Sin[(c + d*x)/2]]/(a + b) + Sec[c + d*x]/b)/d","A",1
257,1,164,243,2.9522634,"\int \frac{\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{-\frac{4 b (c+d x)}{a^3}+\frac{2 b^5 \sin (c+d x)}{a^2 (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{2 \sin (c+d x)}{a^2}+\frac{4 b^4 \left(5 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^3 \left(a^2-b^2\right)^{5/2}}+\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2}+\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2}}{2 d}","\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 x}{a^3}-\frac{b^5 \sin (c+d x)}{a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{\sin (c+d x)}{a^2 d}+\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{\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,"-1/2*((-4*b*(c + d*x))/a^3 + (4*b^4*(5*a^2 - 2*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2)) + Cot[(c + d*x)/2]/(a + b)^2 + (2*Sin[c + d*x])/a^2 + (2*b^5*Sin[c + d*x])/(a^2*(a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])) + Tan[(c + d*x)/2]/(a - b)^2)/d","A",1
258,1,151,227,2.0815995,"\int \frac{\cos ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{-\frac{4 b^3 \left(b^2-4 a^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 \left(a^2-b^2\right)^{5/2}}-\frac{2 (c+d x)}{a^2}+\frac{2 b^4 \sin (c+d x)}{a (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2}}{2 d}","-\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{b^4 \sin (c+d x)}{a d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\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,"((-2*(c + d*x))/a^2 - (4*b^3*(-4*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(5/2)) - Cot[(c + d*x)/2]/(a + b)^2 + (2*b^4*Sin[c + d*x])/(a*(a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])) + Tan[(c + d*x)/2]/(a - b)^2)/(2*d)","A",1
259,1,131,219,1.2877816,"\int \frac{\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{\frac{\csc (c+d x) \left(\left(2 a^2 b+b^3\right) \cos (2 (c+d x))-2 a \left(a^2-b^2\right) \cos (c+d x)-3 b^3\right)}{a \cos (c+d x)+b}-\frac{12 a b^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{2 d (a-b)^2 (a+b)^2}","\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{b^3 \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\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,"((-12*a*b^2*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + ((-3*b^3 - 2*a*(a^2 - b^2)*Cos[c + d*x] + (2*a^2*b + b^3)*Cos[2*(c + d*x)])*Csc[c + d*x])/(b + a*Cos[c + d*x]))/(2*(a - b)^2*(a + b)^2*d)","A",1
260,1,128,203,1.2635092,"\int \frac{1}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Integrate[(a*Sin[c + d*x] + b*Tan[c + d*x])^(-2),x]","\frac{\frac{4 b \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)}{\left(a^2-b^2\right)^{5/2}}+\frac{\frac{2 a b^2 \sin (c+d x)}{(a+b)^2 (a \cos (c+d x)+b)}+\tan \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2}}{2 d}","\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*b*(2*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - Cot[(c + d*x)/2]/(a + b)^2 + ((2*a*b^2*Sin[c + d*x])/((a + b)^2*(b + a*Cos[c + d*x])) + Tan[(c + d*x)/2])/(a - b)^2)/(2*d)","A",1
261,1,127,136,1.133148,"\int \frac{\sec (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{\frac{4 a \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)}{\left(a^2-b^2\right)^{5/2}}+\frac{\frac{2 a^2 b \sin (c+d x)}{(a+b)^2 (a \cos (c+d x)+b)}+\tan \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2}+\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2}}{2 d}","-\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,"-1/2*((4*a*(a^2 + 2*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + Cot[(c + d*x)/2]/(a + b)^2 + ((2*a^2*b*Sin[c + d*x])/((a + b)^2*(b + a*Cos[c + d*x])) + Tan[(c + d*x)/2])/(a - b)^2)/d","A",1
262,1,121,131,0.7788505,"\int \frac{\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\frac{\frac{\frac{2 a^3 \sin (c+d x)}{(a+b)^2 (a \cos (c+d x)+b)}+\tan \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2}+\frac{12 a^2 b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2}}{2 d}","\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,"((12*a^2*b*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - Cot[(c + d*x)/2]/(a + b)^2 + ((2*a^3*Sin[c + d*x])/((a + b)^2*(b + a*Cos[c + d*x])) + Tan[(c + d*x)/2])/(a - b)^2)/(2*d)","A",1
263,1,196,231,1.9686841,"\int \frac{\sec ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","-\frac{\frac{2 a^4 \sin (c+d x)}{b (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}-\frac{4 \left(a^5-4 a^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)}{b^2 \left(a^2-b^2\right)^{5/2}}+\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2}+\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2}+\frac{2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{b^2}-\frac{2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b^2}}{2 d}","\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{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{\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,"-1/2*((-4*(a^5 - 4*a^3*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*(a^2 - b^2)^(5/2)) + Cot[(c + d*x)/2]/(a + b)^2 + (2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/b^2 - (2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/b^2 + (2*a^4*Sin[c + d*x])/((a - b)^2*b*(a + b)^2*(b + a*Cos[c + d*x])) + Tan[(c + d*x)/2]/(a - b)^2)/d","A",1
264,1,713,248,6.3576722,"\int \frac{\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{b^6 \tan ^3(c+d x) (a \cos (c+d x)+b)}{2 a^3 d (b-a)^2 (a+b)^2 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{2 i \left(a^5-4 a^3 b^2-9 a b^4\right) (c+d x) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{d (a-b)^4 (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{2 b^5 \left(b^2-3 a^2\right) \tan ^3(c+d x) (a \cos (c+d x)+b)^2}{a^3 d (b-a)^3 (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{\left(-15 a^4 b^4+4 a^2 b^6-b^8\right) \tan ^3(c+d x) (a \cos (c+d x)+b)^3 \log (a \cos (c+d x)+b)}{a^3 d \left(b^2-a^2\right)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (5 b-2 a) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (-2 a-5 b) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(5 b-2 a) \tan ^3(c+d x) \log \left(\cos ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}{4 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(-2 a-5 b) \tan ^3(c+d x) \log \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}{4 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{\tan ^3(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^3}{8 d (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{\tan ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^3}{8 d (b-a)^3 (a \sin (c+d x)+b \tan (c+d x))^3}","-\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{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(15 a^4-4 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b)}{a^3 d \left(a^2-b^2\right)^4}-\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*(b + a*Cos[c + d*x])*Tan[c + d*x]^3)/(2*a^3*(-a + b)^2*(a + b)^2*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - (2*b^5*(-3*a^2 + b^2)*(b + a*Cos[c + d*x])^2*Tan[c + d*x]^3)/(a^3*(-a + b)^3*(a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((2*I)*(a^5 - 4*a^3*b^2 - 9*a*b^4)*(c + d*x)*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((a - b)^4*(a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((I/2)*(-2*a - 5*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((I/2)*(-2*a + 5*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((b + a*Cos[c + d*x])^3*Csc[(c + d*x)/2]^2*Tan[c + d*x]^3)/(8*(a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((-2*a + 5*b)*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2]^2]*Tan[c + d*x]^3)/(4*(-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((-15*a^4*b^4 + 4*a^2*b^6 - b^8)*(b + a*Cos[c + d*x])^3*Log[b + a*Cos[c + d*x]]*Tan[c + d*x]^3)/(a^3*(-a^2 + b^2)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((-2*a - 5*b)*(b + a*Cos[c + d*x])^3*Log[Sin[(c + d*x)/2]^2]*Tan[c + d*x]^3)/(4*(a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[(c + d*x)/2]^2*Tan[c + d*x]^3)/(8*(-a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3)","C",1
265,1,204,232,6.1116825,"\int \frac{\cos ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{-\frac{4 b^5}{a^2 (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac{8 b^4 \left(b^2-5 a^2\right)}{a^2 (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}+\frac{16 b^3 \left(5 a^2+b^2\right) \log (a \cos (c+d x)+b)}{\left(a^2-b^2\right)^4}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{(a+b)^3}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{(a-b)^3}-\frac{4 (a+4 b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{(a+b)^4}+\frac{4 (a-4 b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{(a-b)^4}}{8 d}","\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{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{(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,"((-4*b^5)/(a^2*(a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])^2) + (8*b^4*(-5*a^2 + b^2))/(a^2*(-a + b)^3*(a + b)^3*(b + a*Cos[c + d*x])) - Csc[(c + d*x)/2]^2/(a + b)^3 + (4*(a - 4*b)*Log[Cos[(c + d*x)/2]])/(a - b)^4 + (16*b^3*(5*a^2 + b^2)*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^4 - (4*(a + 4*b)*Log[Sin[(c + d*x)/2]])/(a + b)^4 + Sec[(c + d*x)/2]^2/(a - b)^3)/(8*d)","A",1
266,1,184,211,5.3265555,"\int \frac{\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{-\frac{48 a b^2 \left(a^2+b^2\right) \log (a \cos (c+d x)+b)}{\left(a^2-b^2\right)^4}+\frac{4 b^4}{a (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac{32 a b^3}{(b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{(a+b)^3}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{(b-a)^3}-\frac{12 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{(a+b)^4}+\frac{12 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{(a-b)^4}}{8 d}","-\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{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{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,"((4*b^4)/(a*(a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])^2) + (32*a*b^3)/((-a + b)^3*(a + b)^3*(b + a*Cos[c + d*x])) - Csc[(c + d*x)/2]^2/(a + b)^3 + (12*b*Log[Cos[(c + d*x)/2]])/(a - b)^4 - (48*a*b^2*(a^2 + b^2)*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^4 - (12*b*Log[Sin[(c + d*x)/2]])/(a + b)^4 + Sec[(c + d*x)/2]^2/(-a + b)^3)/(8*d)","A",1
267,1,696,229,6.3210026,"\int \frac{1}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Integrate[(a*Sin[c + d*x] + b*Tan[c + d*x])^(-3),x]","-\frac{b^2 \left(3 a^2+b^2\right) \tan ^3(c+d x) (a \cos (c+d x)+b)^2}{d (b-a)^3 (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{2 i \left(3 a^4 b+8 a^2 b^3+b^5\right) (c+d x) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{d (a-b)^4 (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{\left(3 a^4 b+8 a^2 b^3+b^5\right) \tan ^3(c+d x) (a \cos (c+d x)+b)^3 \log (a \cos (c+d x)+b)}{d \left(b^2-a^2\right)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{b^3 \tan ^3(c+d x) (a \cos (c+d x)+b)}{2 d (b-a)^2 (a+b)^2 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (-a-2 b) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (a-2 b) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(-a-2 b) \tan ^3(c+d x) \log \left(\cos ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}{4 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(a-2 b) \tan ^3(c+d x) \log \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}{4 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{\tan ^3(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^3}{8 d (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{\tan ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^3}{8 d (b-a)^3 (a \sin (c+d x)+b \tan (c+d x))^3}","\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{\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{b^3}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b \left(3 a^4+8 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\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,"-1/2*(b^3*(b + a*Cos[c + d*x])*Tan[c + d*x]^3)/((-a + b)^2*(a + b)^2*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - (b^2*(3*a^2 + b^2)*(b + a*Cos[c + d*x])^2*Tan[c + d*x]^3)/((-a + b)^3*(a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((2*I)*(3*a^4*b + 8*a^2*b^3 + b^5)*(c + d*x)*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((a - b)^4*(a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((I/2)*(-a - 2*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((I/2)*(a - 2*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((b + a*Cos[c + d*x])^3*Csc[(c + d*x)/2]^2*Tan[c + d*x]^3)/(8*(a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((-a - 2*b)*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2]^2]*Tan[c + d*x]^3)/(4*(-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((3*a^4*b + 8*a^2*b^3 + b^5)*(b + a*Cos[c + d*x])^3*Log[b + a*Cos[c + d*x]]*Tan[c + d*x]^3)/((-a^2 + b^2)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((a - 2*b)*(b + a*Cos[c + d*x])^3*Log[Sin[(c + d*x)/2]^2]*Tan[c + d*x]^3)/(4*(a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((b + a*Cos[c + d*x])^3*Sec[(c + d*x)/2]^2*Tan[c + d*x]^3)/(8*(-a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3)","C",1
268,1,703,231,6.3799328,"\int \frac{\sec (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{2 i \left(a^5+8 a^3 b^2+3 a b^4\right) (c+d x) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{d (a-b)^4 (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{\left(-a^5-8 a^3 b^2-3 a b^4\right) \tan ^3(c+d x) (a \cos (c+d x)+b)^3 \log (a \cos (c+d x)+b)}{d \left(b^2-a^2\right)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{a b^2 \tan ^3(c+d x) (a \cos (c+d x)+b)}{2 d (b-a)^2 (a+b)^2 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{2 a b (b-i a) (b+i a) \tan ^3(c+d x) (a \cos (c+d x)+b)^2}{d (b-a)^3 (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (2 a+b) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{i (2 a-b) \tan ^{-1}(\tan (c+d x)) \tan ^3(c+d x) (a \cos (c+d x)+b)^3}{2 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(2 a+b) \tan ^3(c+d x) \log \left(\cos ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}{4 d (b-a)^4 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{(2 a-b) \tan ^3(c+d x) \log \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}{4 d (a+b)^4 (a \sin (c+d x)+b \tan (c+d x))^3}-\frac{\tan ^3(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^3}{8 d (a+b)^3 (a \sin (c+d x)+b \tan (c+d x))^3}+\frac{\tan ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^3}{8 d (b-a)^3 (a \sin (c+d x)+b \tan (c+d x))^3}","\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{\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{a \left(a^4+8 a^2 b^2+3 b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\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*(b + a*Cos[c + d*x])*Tan[c + d*x]^3)/(2*(-a + b)^2*(a + b)^2*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + (2*a*b*((-I)*a + b)*(I*a + b)*(b + a*Cos[c + d*x])^2*Tan[c + d*x]^3)/((-a + b)^3*(a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((2*I)*(a^5 + 8*a^3*b^2 + 3*a*b^4)*(c + d*x)*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((a - b)^4*(a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((I/2)*(2*a - b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((I/2)*(2*a + b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((b + a*Cos[c + d*x])^3*Csc[(c + d*x)/2]^2*Tan[c + d*x]^3)/(8*(a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((2*a + b)*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2]^2]*Tan[c + d*x]^3)/(4*(-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((-a^5 - 8*a^3*b^2 - 3*a*b^4)*(b + a*Cos[c + d*x])^3*Log[b + a*Cos[c + d*x]]*Tan[c + d*x]^3)/((-a^2 + b^2)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((2*a - b)*(b + a*Cos[c + d*x])^3*Log[Sin[(c + d*x)/2]^2]*Tan[c + d*x]^3)/(4*(a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[(c + d*x)/2]^2*Tan[c + d*x]^3)/(8*(-a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3)","C",1
269,1,217,212,6.267234,"\int \frac{\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","-\frac{a^2 \left(a^2+3 b^2\right)}{d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}-\frac{a^2 b}{2 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac{6 \left(a^4 b+a^2 b^3\right) \log (a \cos (c+d x)+b)}{d \left(b^2-a^2\right)^4}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d (a+b)^3}-\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d (b-a)^3}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d (a+b)^4}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d (b-a)^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,"-1/2*(a^2*b)/((-a + b)^2*(a + b)^2*d*(b + a*Cos[c + d*x])^2) - (a^2*(a^2 + 3*b^2))/((-a + b)^3*(a + b)^3*d*(b + a*Cos[c + d*x])) - Csc[(c + d*x)/2]^2/(8*(a + b)^3*d) - (3*a*Log[Cos[(c + d*x)/2]])/(2*(-a + b)^4*d) + (6*(a^4*b + a^2*b^3)*Log[b + a*Cos[c + d*x]])/((-a^2 + b^2)^4*d) + (3*a*Log[Sin[(c + d*x)/2]])/(2*(a + b)^4*d) - Sec[(c + d*x)/2]^2/(8*(-a + b)^3*d)","A",1
270,1,217,228,6.2325674,"\int \frac{\sec ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{4 a^3 b}{d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}+\frac{a^3}{2 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}-\frac{2 \left(a^5+5 a^3 b^2\right) \log (a \cos (c+d x)+b)}{d \left(b^2-a^2\right)^4}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d (a+b)^3}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d (b-a)^3}+\frac{(4 a+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d (a+b)^4}+\frac{(4 a-b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d (b-a)^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{\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{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{(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^3/(2*(-a + b)^2*(a + b)^2*d*(b + a*Cos[c + d*x])^2) + (4*a^3*b)/((-a + b)^3*(a + b)^3*d*(b + a*Cos[c + d*x])) - Csc[(c + d*x)/2]^2/(8*(a + b)^3*d) + ((4*a - b)*Log[Cos[(c + d*x)/2]])/(2*(-a + b)^4*d) - (2*(a^5 + 5*a^3*b^2)*Log[b + a*Cos[c + d*x]])/((-a^2 + b^2)^4*d) + ((4*a + b)*Log[Sin[(c + d*x)/2]])/(2*(a + b)^4*d) + Sec[(c + d*x)/2]^2/(8*(-a + b)^3*d)","A",1
271,1,246,155,1.4465171,"\int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^m*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]","\frac{\cos ^{m+1}(c+d x) (a+b \sec (c+d x))^3 \left(-a m \left(m^3-m^2-4 m+4\right) \left(a^2 (m+9)-12 b^2 (m+3)\right) \cos ^3(c+d x)+\left(m^3-2 m^2-m+2\right) \cos ^2(c+d x) \left(a^3 m (m+2) \cos (3 (c+d x))+2 b (m+3) \left(2 b^2 (m+2)-3 a^2 (m+4)\right)+6 a^2 b m (m+3) \cos (2 (c+d x))\right)-12 a b^2 m \left(m^4+4 m^3-m^2-16 m-12\right) \cos (c+d x)-4 b^3 m \left(m^4+5 m^3+5 m^2-5 m-6\right)\right)}{4 d (m-2) (m-1) m (m+1) (m+2) (m+3) (a \cos (c+d x)+b)^3}","\frac{a^3 \cos ^{m+3}(c+d x)}{d (m+3)}-\frac{a \left(a^2-3 b^2\right) \cos ^{m+1}(c+d x)}{d (m+1)}-\frac{b \left(3 a^2-b^2\right) \cos ^m(c+d x)}{d m}+\frac{3 a^2 b \cos ^{m+2}(c+d x)}{d (m+2)}+\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,"(Cos[c + d*x]^(1 + m)*(-4*b^3*m*(-6 - 5*m + 5*m^2 + 5*m^3 + m^4) - 12*a*b^2*m*(-12 - 16*m - m^2 + 4*m^3 + m^4)*Cos[c + d*x] - a*m*(4 - 4*m - m^2 + m^3)*(-12*b^2*(3 + m) + a^2*(9 + m))*Cos[c + d*x]^3 + (2 - m - 2*m^2 + m^3)*Cos[c + d*x]^2*(2*b*(3 + m)*(2*b^2*(2 + m) - 3*a^2*(4 + m)) + 6*a^2*b*m*(3 + m)*Cos[2*(c + d*x)] + a^3*m*(2 + m)*Cos[3*(c + d*x)]))*(a + b*Sec[c + d*x])^3)/(4*d*(-2 + m)*(-1 + m)*m*(1 + m)*(2 + m)*(3 + m)*(b + a*Cos[c + d*x])^3)","A",1
272,1,6669,264,30.5972976,"\int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^m*(a*Sin[c + d*x] + b*Tan[c + d*x])^2,x]","\text{Result too large to show}","-\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,"Result too large to show","C",0
273,1,35,39,0.085439,"\int \cos ^m(c+d x) (a \sin (c+d x)+b \tan (c+d x)) \, dx","Integrate[Cos[c + d*x]^m*(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","-\frac{\cos ^m(c+d x) (a m \cos (c+d x)+b m+b)}{d m (m+1)}","-\frac{a \cos ^{m+1}(c+d x)}{d (m+1)}-\frac{b \cos ^m(c+d x)}{d m}",1,"-((Cos[c + d*x]^m*(b + b*m + a*m*Cos[c + d*x]))/(d*m*(1 + m)))","A",1
274,1,106,144,0.3706138,"\int \frac{\cos ^m(c+d x)}{a \sin (c+d x)+b \tan (c+d x)} \, dx","Integrate[Cos[c + d*x]^m/(a*Sin[c + d*x] + b*Tan[c + d*x]),x]","\frac{\cos ^{m+2}(c+d x) \left(-2 a^2 \, _2F_1\left(1,m+2;m+3;-\frac{a \cos (c+d x)}{b}\right)+b (a+b) \, _2F_1(1,m+2;m+3;-\cos (c+d x))-b (a-b) \, _2F_1(1,m+2;m+3;\cos (c+d x))\right)}{2 b d (m+2) (a-b) (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)*(b*(a + b)*Hypergeometric2F1[1, 2 + m, 3 + m, -Cos[c + d*x]] - (a - b)*b*Hypergeometric2F1[1, 2 + m, 3 + m, Cos[c + d*x]] - 2*a^2*Hypergeometric2F1[1, 2 + m, 3 + m, -((a*Cos[c + d*x])/b)]))/(2*(a - b)*b*(a + b)*d*(2 + m))","A",1
275,1,61,65,0.1357075,"\int \frac{\cos (x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[(Cos[x]*Sin[x])/(a*Cos[x] + b*Sin[x]),x]","\frac{b \sin (x)-a \cos (x)}{a^2+b^2}-\frac{2 a 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)^{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,"(-2*a*b*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) + (-(a*Cos[x]) + b*Sin[x])/(a^2 + b^2)","A",1
276,1,153,92,0.3372569,"\int \frac{\cos (x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[(Cos[x]*Sin[x]^2)/(a*Cos[x] + b*Sin[x]),x]","-\frac{-2 a^3 x+2 a^3 \sin (2 x)+2 b \left(a^2+b^2\right) \cos (2 x)-2 i b \left(b^2-3 a^2\right) \tan ^{-1}(\tan (x))-2 \left(a^2+b^2\right) (b \log (a \cos (x)+b \sin (x))+a x)-6 i a^2 b x-3 a^2 b \log \left((a \cos (x)+b \sin (x))^2\right)+b^3 \log \left((a \cos (x)+b \sin (x))^2\right)+6 a b^2 x+2 a b^2 \sin (2 x)+2 i b^3 x}{8 \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,"-1/8*(-2*a^3*x - (6*I)*a^2*b*x + 6*a*b^2*x + (2*I)*b^3*x - (2*I)*b*(-3*a^2 + b^2)*ArcTan[Tan[x]] + 2*b*(a^2 + b^2)*Cos[2*x] - 2*(a^2 + b^2)*(a*x + b*Log[a*Cos[x] + b*Sin[x]]) - 3*a^2*b*Log[(a*Cos[x] + b*Sin[x])^2] + b^3*Log[(a*Cos[x] + b*Sin[x])^2] + 2*a^3*Sin[2*x] + 2*a*b^2*Sin[2*x])/(a^2 + b^2)^2","C",1
277,1,113,122,0.9825062,"\int \frac{\cos (x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[(Cos[x]*Sin[x]^3)/(a*Cos[x] + b*Sin[x]),x]","\frac{\left(3 a b^2-9 a^3\right) \cos (x)+a \left(a^2+b^2\right) \cos (3 x)-2 b \sin (x) \left(\left(a^2+b^2\right) \cos (2 x)-7 a^2-b^2\right)}{12 \left(a^2+b^2\right)^2}-\frac{2 a^3 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}}","\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,"(-2*a^3*b*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + ((-9*a^3 + 3*a*b^2)*Cos[x] + a*(a^2 + b^2)*Cos[3*x] - 2*b*(-7*a^2 - b^2 + (a^2 + b^2)*Cos[2*x])*Sin[x])/(12*(a^2 + b^2)^2)","A",1
278,1,82,93,0.3098469,"\int \frac{\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[(Cos[x]^2*Sin[x])/(a*Cos[x] + b*Sin[x]),x]","\frac{b \left(a^2+b^2\right) \sin (2 x)-a \left(a^2+b^2\right) \cos (2 x)+4 i a b^2 \tan ^{-1}(\tan (x))-2 b \left(a b \log \left((a \cos (x)+b \sin (x))^2\right)+x (a+i b)^2\right)}{4 \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,"((4*I)*a*b^2*ArcTan[Tan[x]] - a*(a^2 + b^2)*Cos[2*x] - 2*b*((a + I*b)^2*x + a*b*Log[(a*Cos[x] + b*Sin[x])^2]) + b*(a^2 + b^2)*Sin[2*x])/(4*(a^2 + b^2)^2)","C",1
279,1,115,112,0.6285061,"\int \frac{\cos ^2(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[(Cos[x]^2*Sin[x]^2)/(a*Cos[x] + b*Sin[x]),x]","\frac{2 a^2 b^2 \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}}-\frac{\left(3 b^3-9 a^2 b\right) \cos (x)+b \left(a^2+b^2\right) \cos (3 x)+2 a \sin (x) \left(\left(a^2+b^2\right) \cos (2 x)-a^2+5 b^2\right)}{12 \left(a^2+b^2\right)^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,"(2*a^2*b^2*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - ((-9*a^2*b + 3*b^3)*Cos[x] + b*(a^2 + b^2)*Cos[3*x] + 2*a*(-a^2 + 5*b^2 + (a^2 + b^2)*Cos[2*x])*Sin[x])/(12*(a^2 + b^2)^2)","A",1
280,1,178,176,0.5219444,"\int \frac{\cos ^2(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[(Cos[x]^2*Sin[x]^3)/(a*Cos[x] + b*Sin[x]),x]","\frac{a^5 \cos (4 x)-4 a \left(a^4-b^4\right) \cos (2 x)-12 a^4 b x+8 a^4 b \sin (2 x)-a^4 b \sin (4 x)-32 i a^3 b^2 x+2 a^3 b^2 \cos (4 x)+32 i a^3 b^2 \tan ^{-1}(\tan (x))-16 a^3 b^2 \log \left((a \cos (x)+b \sin (x))^2\right)+24 a^2 b^3 x+8 a^2 b^3 \sin (2 x)-2 a^2 b^3 \sin (4 x)+a b^4 \cos (4 x)+4 b^5 x-b^5 \sin (4 x)}{32 \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 \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^2 b^3 x}{\left(a^2+b^2\right)^3}-\frac{a^3 b^2 \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^3}",1,"(-12*a^4*b*x - (32*I)*a^3*b^2*x + 24*a^2*b^3*x + 4*b^5*x + (32*I)*a^3*b^2*ArcTan[Tan[x]] - 4*a*(a^4 - b^4)*Cos[2*x] + a^5*Cos[4*x] + 2*a^3*b^2*Cos[4*x] + a*b^4*Cos[4*x] - 16*a^3*b^2*Log[(a*Cos[x] + b*Sin[x])^2] + 8*a^4*b*Sin[2*x] + 8*a^2*b^3*Sin[2*x] - a^4*b*Sin[4*x] - 2*a^2*b^3*Sin[4*x] - b^5*Sin[4*x])/(32*(a^2 + b^2)^3)","C",1
281,1,112,123,1.0195641,"\int \frac{\cos ^3(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[(Cos[x]^3*Sin[x])/(a*Cos[x] + b*Sin[x]),x]","-\frac{3 a \left(a^2+5 b^2\right) \cos (x)+a \left(a^2+b^2\right) \cos (3 x)-2 b \sin (x) \left(\left(a^2+b^2\right) \cos (2 x)-a^2+5 b^2\right)}{12 \left(a^2+b^2\right)^2}-\frac{2 a b^3 \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}}","-\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,"(-2*a*b^3*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (3*a*(a^2 + 5*b^2)*Cos[x] + a*(a^2 + b^2)*Cos[3*x] - 2*b*(-a^2 + 5*b^2 + (a^2 + b^2)*Cos[2*x])*Sin[x])/(12*(a^2 + b^2)^2)","A",1
282,1,287,175,0.7857745,"\int \frac{\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[(Cos[x]^3*Sin[x]^2)/(a*Cos[x] + b*Sin[x]),x]","-\frac{-4 a^5 x+a^5 \sin (4 x)+4 b \left(b^4-a^4\right) \cos (2 x)+4 i a^4 b x+a^4 b \cos (4 x)-4 a^4 b \log (a \cos (x)+b \sin (x))+2 a^4 b \log \left((a \cos (x)+b \sin (x))^2\right)-24 a^3 b^2 x+8 a^3 b^2 \sin (2 x)+2 a^3 b^2 \sin (4 x)-24 i a^2 b^3 x+2 a^2 b^3 \cos (4 x)-8 a^2 b^3 \log (a \cos (x)+b \sin (x))-12 a^2 b^3 \log \left((a \cos (x)+b \sin (x))^2\right)-4 i b \left(a^4-6 a^2 b^2+b^4\right) \tan ^{-1}(\tan (x))-4 b^5 \log (a \cos (x)+b \sin (x))+2 b^5 \log \left((a \cos (x)+b \sin (x))^2\right)+12 a b^4 x+8 a b^4 \sin (2 x)+a b^4 \sin (4 x)+4 i b^5 x+b^5 \cos (4 x)}{32 \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^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^3 b^2 x}{\left(a^2+b^2\right)^3}",1,"-1/32*(-4*a^5*x + (4*I)*a^4*b*x - 24*a^3*b^2*x - (24*I)*a^2*b^3*x + 12*a*b^4*x + (4*I)*b^5*x - (4*I)*b*(a^4 - 6*a^2*b^2 + b^4)*ArcTan[Tan[x]] + 4*b*(-a^4 + b^4)*Cos[2*x] + a^4*b*Cos[4*x] + 2*a^2*b^3*Cos[4*x] + b^5*Cos[4*x] - 4*a^4*b*Log[a*Cos[x] + b*Sin[x]] - 8*a^2*b^3*Log[a*Cos[x] + b*Sin[x]] - 4*b^5*Log[a*Cos[x] + b*Sin[x]] + 2*a^4*b*Log[(a*Cos[x] + b*Sin[x])^2] - 12*a^2*b^3*Log[(a*Cos[x] + b*Sin[x])^2] + 2*b^5*Log[(a*Cos[x] + b*Sin[x])^2] + 8*a^3*b^2*Sin[2*x] + 8*a*b^4*Sin[2*x] + a^5*Sin[4*x] + 2*a^3*b^2*Sin[4*x] + a*b^4*Sin[4*x])/(a^2 + b^2)^3","C",1
283,1,223,193,1.4762417,"\int \frac{\cos ^3(x) \sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx","Integrate[(Cos[x]^3*Sin[x]^3)/(a*Cos[x] + b*Sin[x]),x]","\frac{3 a^5 \cos (5 x)-30 a^4 b \sin (x)+15 a^4 b \sin (3 x)-3 a^4 b \sin (5 x)+6 a^3 b^2 \cos (5 x)+240 a^2 b^3 \sin (x)+10 a^2 b^3 \sin (3 x)-6 a^2 b^3 \sin (5 x)-30 a \left(a^4+8 a^2 b^2-b^4\right) \cos (x)-5 a \left(a^4-2 a^2 b^2-3 b^4\right) \cos (3 x)+3 a b^4 \cos (5 x)+30 b^5 \sin (x)-5 b^5 \sin (3 x)-3 b^5 \sin (5 x)}{240 \left(a^2+b^2\right)^3}-\frac{2 a^3 b^3 \tanh ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)-b}{\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 \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^2 b^3 \sin (x)}{\left(a^2+b^2\right)^3}-\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,"(-2*a^3*b^3*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (-30*a*(a^4 + 8*a^2*b^2 - b^4)*Cos[x] - 5*a*(a^4 - 2*a^2*b^2 - 3*b^4)*Cos[3*x] + 3*a^5*Cos[5*x] + 6*a^3*b^2*Cos[5*x] + 3*a*b^4*Cos[5*x] - 30*a^4*b*Sin[x] + 240*a^2*b^3*Sin[x] + 30*b^5*Sin[x] + 15*a^4*b*Sin[3*x] + 10*a^2*b^3*Sin[3*x] - 5*b^5*Sin[3*x] - 3*a^4*b*Sin[5*x] - 6*a^2*b^3*Sin[5*x] - 3*b^5*Sin[5*x])/(240*(a^2 + b^2)^3)","A",1
284,1,144,70,0.257436,"\int \frac{\cos (x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(Cos[x]*Sin[x])/(a*Cos[x] + b*Sin[x])^2,x]","\frac{a \cos (x) \left(\left(b^2-a^2\right) \log \left((a \cos (x)+b \sin (x))^2\right)-2 i x (a+i b)^2\right)+b \sin (x) \left(\left(b^2-a^2\right) \log \left((a \cos (x)+b \sin (x))^2\right)+2 (a+i b) (a (-1-i x)+b (x+i))\right)+2 i \left(a^2-b^2\right) \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))}{2 \left(a^2+b^2\right)^2 (a \cos (x)+b \sin (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{\left(a^2-b^2\right) \log (a \cos (x)+b \sin (x))}{\left(a^2+b^2\right)^2}",1,"(a*Cos[x]*((-2*I)*(a + I*b)^2*x + (-a^2 + b^2)*Log[(a*Cos[x] + b*Sin[x])^2]) + b*(2*(a + I*b)*(a*(-1 - I*x) + b*(I + x)) + (-a^2 + b^2)*Log[(a*Cos[x] + b*Sin[x])^2])*Sin[x] + (2*I)*(a^2 - b^2)*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x]))/(2*(a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))","C",1
285,1,111,110,0.593018,"\int \frac{\cos (x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(Cos[x]*Sin[x]^2)/(a*Cos[x] + b*Sin[x])^2,x]","\frac{2 a \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}}-\frac{a \left(a^2+b^2\right) \sin (2 x)+b \left(a^2+b^2\right) \cos (2 x)+5 a^2 b-b^3}{2 \left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}","-\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,"(2*a*(a^2 - 2*b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (5*a^2*b - b^3 + b*(a^2 + b^2)*Cos[2*x] + a*(a^2 + b^2)*Sin[2*x])/(2*(a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))","A",1
286,1,226,129,1.4788331,"\int \frac{\cos (x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(Cos[x]*Sin[x]^3)/(a*Cos[x] + b*Sin[x])^2,x]","\frac{4 i a^2 \left(a^2-3 b^2\right) \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))+a \cos (x) \left(\left(a^4-b^4\right) \cos (2 x)+2 a \left(-b \left(a^2+b^2\right) \sin (2 x)-a \left(a^2-3 b^2\right) \log \left((a \cos (x)+b \sin (x))^2\right)+2 x (-b+i a)^3\right)\right)-b \sin (x) \left(\left(b^4-a^4\right) \cos (2 x)+2 a \left(b \left(a^2+b^2\right) \sin (2 x)+a \left(a^2-3 b^2\right) \log \left((a \cos (x)+b \sin (x))^2\right)+2 \left(a^3 (1+i x)-3 a^2 b x+a b^2 (1-3 i x)+b^3 x\right)\right)\right)}{4 \left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}","-\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}+\frac{b x \left(3 a^3-a b^2\right)}{\left(a^2+b^2\right)^3}",1,"((4*I)*a^2*(a^2 - 3*b^2)*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x]) + a*Cos[x]*((a^4 - b^4)*Cos[2*x] + 2*a*(2*(I*a - b)^3*x - a*(a^2 - 3*b^2)*Log[(a*Cos[x] + b*Sin[x])^2] - b*(a^2 + b^2)*Sin[2*x])) - b*Sin[x]*((-a^4 + b^4)*Cos[2*x] + 2*a*(2*(a^3*(1 + I*x) + a*b^2*(1 - (3*I)*x) - 3*a^2*b*x + b^3*x) + a*(a^2 - 3*b^2)*Log[(a*Cos[x] + b*Sin[x])^2] + b*(a^2 + b^2)*Sin[2*x])))/(4*(a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))","C",1
287,1,110,109,0.6820271,"\int \frac{\cos ^2(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(Cos[x]^2*Sin[x])/(a*Cos[x] + b*Sin[x])^2,x]","\frac{2 b \left(b^2-2 a^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}}-\frac{a^3-b \left(a^2+b^2\right) \sin (2 x)+a \left(a^2+b^2\right) \cos (2 x)-5 a b^2}{2 \left(a^2+b^2\right)^2 (a \cos (x)+b \sin (x))}","\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*b*(-2*a^2 + b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a^3 - 5*a*b^2 + a*(a^2 + b^2)*Cos[2*x] - b*(a^2 + b^2)*Sin[2*x])/(2*(a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))","A",1
288,1,145,131,1.658975,"\int \frac{\cos ^2(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(Cos[x]^2*Sin[x]^2)/(a*Cos[x] + b*Sin[x])^2,x]","\frac{\sin (x)}{8 a (a \cos (x)+b \sin (x))}-\frac{2 \left(a^4-b^4\right) \sin (2 x)+4 a b \left(a^2+b^2\right) \cos (2 x)-16 a b \left(a^2-b^2\right) \log (a \cos (x)+b \sin (x))-4 x \left(a^4-6 a^2 b^2+b^4\right)+\frac{\left(a^2+b^2\right) \left(a^4-6 a^2 b^2+b^4\right) \sin (x)}{a (a \cos (x)+b \sin (x))}}{8 \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}+\frac{x \left(a^4-6 a^2 b^2+b^4\right)}{2 \left(a^2+b^2\right)^3}",1,"Sin[x]/(8*a*(a*Cos[x] + b*Sin[x])) - (-4*(a^4 - 6*a^2*b^2 + b^4)*x + 4*a*b*(a^2 + b^2)*Cos[2*x] - 16*a*b*(a^2 - b^2)*Log[a*Cos[x] + b*Sin[x]] + ((a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*Sin[x])/(a*(a*Cos[x] + b*Sin[x])) + 2*(a^4 - b^4)*Sin[2*x])/(8*(a^2 + b^2)^3)","A",1
289,1,200,172,1.2082407,"\int \frac{\cos ^2(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(Cos[x]^2*Sin[x]^3)/(a*Cos[x] + b*Sin[x])^2,x]","\frac{-9 a^5+18 a^4 b \sin (2 x)-a^4 b \sin (4 x)+90 a^3 b^2+16 a^2 b^3 \sin (2 x)-2 a^2 b^3 \sin (4 x)+a \left(a^2+b^2\right)^2 \cos (4 x)+\left(-8 a^5+4 a^3 b^2+12 a b^4\right) \cos (2 x)-21 a b^4-2 b^5 \sin (2 x)-b^5 \sin (4 x)}{24 \left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}-\frac{2 a^2 b \left(2 a^2-3 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)^{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^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}}+\frac{a^3 b^2}{\left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}",1,"(-2*a^2*b*(2*a^2 - 3*b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) + (-9*a^5 + 90*a^3*b^2 - 21*a*b^4 + (-8*a^5 + 4*a^3*b^2 + 12*a*b^4)*Cos[2*x] + a*(a^2 + b^2)^2*Cos[4*x] + 18*a^4*b*Sin[2*x] + 16*a^2*b^3*Sin[2*x] - 2*b^5*Sin[2*x] - a^4*b*Sin[4*x] - 2*a^2*b^3*Sin[4*x] - b^5*Sin[4*x])/(24*(a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))","A",1
290,1,221,128,1.3334531,"\int \frac{\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(Cos[x]^3*Sin[x])/(a*Cos[x] + b*Sin[x])^2,x]","\frac{-4 i b^2 \left(b^2-3 a^2\right) \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))-a \cos (x) \left(\left(a^4-b^4\right) \cos (2 x)+2 b \left(-a \left(a^2+b^2\right) \sin (2 x)-b \left(b^2-3 a^2\right) \log \left((a \cos (x)+b \sin (x))^2\right)+2 x (a+i b)^3\right)\right)+b \sin (x) \left(\left(b^4-a^4\right) \cos (2 x)+2 b \left(\left(b^3-3 a^2 b\right) \log \left((a \cos (x)+b \sin (x))^2\right)-2 (a+i b) \left(a^2 x+a (b+2 i b x)-b^2 (x+i)\right)+a \left(a^2+b^2\right) \sin (2 x)\right)\right)}{4 \left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}","-\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,"((-4*I)*b^2*(-3*a^2 + b^2)*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x]) - a*Cos[x]*((a^4 - b^4)*Cos[2*x] + 2*b*(2*(a + I*b)^3*x - b*(-3*a^2 + b^2)*Log[(a*Cos[x] + b*Sin[x])^2] - a*(a^2 + b^2)*Sin[2*x])) + b*Sin[x]*((-a^4 + b^4)*Cos[2*x] + 2*b*(-2*(a + I*b)*(a^2*x - b^2*(I + x) + a*(b + (2*I)*b*x)) + (-3*a^2*b + b^3)*Log[(a*Cos[x] + b*Sin[x])^2] + a*(a^2 + b^2)*Sin[2*x])))/(4*(a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))","C",1
291,1,198,176,1.2682289,"\int \frac{\cos ^3(x) \sin ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(Cos[x]^3*Sin[x]^2)/(a*Cos[x] + b*Sin[x])^2,x]","\frac{2 a b^2 \left(3 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)^{7/2}}-\frac{-2 a^5 \sin (2 x)+a^5 \sin (4 x)-21 a^4 b+16 a^3 b^2 \sin (2 x)+2 a^3 b^2 \sin (4 x)+90 a^2 b^3+b \left(a^2+b^2\right)^2 \cos (4 x)-4 b \left(3 a^4+a^2 b^2-2 b^4\right) \cos (2 x)+18 a b^4 \sin (2 x)+a b^4 \sin (4 x)-9 b^5}{24 \left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}","\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 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}}-\frac{a^2 b^3}{\left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}",1,"(2*a*b^2*(3*a^2 - 2*b^2)*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (-21*a^4*b + 90*a^2*b^3 - 9*b^5 - 4*b*(3*a^4 + a^2*b^2 - 2*b^4)*Cos[2*x] + b*(a^2 + b^2)^2*Cos[4*x] - 2*a^5*Sin[2*x] + 16*a^3*b^2*Sin[2*x] + 18*a*b^4*Sin[2*x] + a^5*Sin[4*x] + 2*a^3*b^2*Sin[4*x] + a*b^4*Sin[4*x])/(24*(a^2 + b^2)^3*(a*Cos[x] + b*Sin[x]))","A",1
292,1,409,210,2.7570916,"\int \frac{\cos ^3(x) \sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx","Integrate[(Cos[x]^3*Sin[x]^3)/(a*Cos[x] + b*Sin[x])^2,x]","\frac{16 a b \left(a^4-b^4\right) \sin (2 x)-12 a b x \left(a^2-3 b^2\right) \left(3 a^2-b^2\right)-2 a b \left(a^2+b^2\right)^2 \sin (4 x)+\left(a^2-b^2\right) \left(a^2+b^2\right)^2 \cos (4 x)+\frac{3 \left(a^2+b^2\right)^2 \left(a \cos (x) \left(\left(b^2-a^2\right) \log \left((a \cos (x)+b \sin (x))^2\right)-2 i x (a+i b)^2\right)+b \sin (x) \left(\left(b^2-a^2\right) \log \left((a \cos (x)+b \sin (x))^2\right)+2 (a+i b) (a (-1-i x)+b (x+i))\right)+2 i \left(a^2-b^2\right) \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))\right)}{a \cos (x)+b \sin (x)}-4 \left(a^4-6 a^2 b^2+b^4\right) \left(a^2+b^2\right) \cos (2 x)+\frac{2 b \left(3 a^4-10 a^2 b^2+3 b^4\right) \left(a^2+b^2\right) \sin (x)}{a \cos (x)+b \sin (x)}+6 i x \left(a^6-15 a^4 b^2+15 a^2 b^4-b^6\right)-6 i \left(a^6-15 a^4 b^2+15 a^2 b^4-b^6\right) \tan ^{-1}(\tan (x))+3 \left(a^6-15 a^4 b^2+15 a^2 b^4-b^6\right) \log \left((a \cos (x)+b \sin (x))^2\right)}{32 \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{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}-\frac{a^2 b^3 \sin (x)}{\left(a^2+b^2\right)^3 (a \cos (x)+b \sin (x))}-\frac{3 a b x \left(a^4-6 a^2 b^2+b^4\right)}{4 \left(a^2+b^2\right)^4}",1,"(-12*a*b*(a^2 - 3*b^2)*(3*a^2 - b^2)*x + (6*I)*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*x - (6*I)*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*ArcTan[Tan[x]] - 4*(a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*Cos[2*x] + (a^2 - b^2)*(a^2 + b^2)^2*Cos[4*x] + 3*(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)*Log[(a*Cos[x] + b*Sin[x])^2] + (2*b*(a^2 + b^2)*(3*a^4 - 10*a^2*b^2 + 3*b^4)*Sin[x])/(a*Cos[x] + b*Sin[x]) + (3*(a^2 + b^2)^2*(a*Cos[x]*((-2*I)*(a + I*b)^2*x + (-a^2 + b^2)*Log[(a*Cos[x] + b*Sin[x])^2]) + b*(2*(a + I*b)*(a*(-1 - I*x) + b*(I + x)) + (-a^2 + b^2)*Log[(a*Cos[x] + b*Sin[x])^2])*Sin[x] + (2*I)*(a^2 - b^2)*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x])))/(a*Cos[x] + b*Sin[x]) + 16*a*b*(a^4 - b^4)*Sin[2*x] - 2*a*b*(a^2 + b^2)^2*Sin[4*x])/(32*(a^2 + b^2)^4)","C",1
293,1,76,47,0.1135311,"\int \frac{\tan (x)}{b \cos (x)+a \sin (x)} \, dx","Integrate[Tan[x]/(b*Cos[x] + a*Sin[x]),x]","\frac{-\frac{2 b \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-a}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{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,"((-2*b*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] - Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])/a","A",1
294,1,60,48,0.078927,"\int \frac{\cot (x)}{b \cos (x)+a \sin (x)} \, dx","Integrate[Cot[x]/(b*Cos[x] + a*Sin[x]),x]","\frac{-\frac{2 a \tanh ^{-1}\left(\frac{b \tan \left(\frac{x}{2}\right)-a}{\sqrt{a^2+b^2}}\right)}{\sqrt{a^2+b^2}}+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)}{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,"((-2*a*ArcTanh[(-a + b*Tan[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] - Log[Cos[x/2]] + Log[Sin[x/2]])/b","A",1