1,1,67,78,0.0857038,"\int \frac{\sin ^4(x)}{i+\tan (x)} \, dx","Integrate[Sin[x]^4/(I + Tan[x]),x]","\frac{\sec (x) \left(-32 \sin (x)-27 \sin (3 x)+5 \sin (5 x)-56 i \cos (x)-9 i \cos (3 x)+i \cos (5 x)+24 \tan ^{-1}(\tan (x)) (\cos (x)-i \sin (x))\right)}{384 (\tan (x)+i)}","-\frac{i x}{16}-\frac{i}{8 (-\tan (x)+i)}-\frac{3 i}{16 (\tan (x)+i)}-\frac{1}{32 (-\tan (x)+i)^2}-\frac{5}{32 (\tan (x)+i)^2}+\frac{i}{24 (\tan (x)+i)^3}",1,"(Sec[x]*((-56*I)*Cos[x] - (9*I)*Cos[3*x] + I*Cos[5*x] + 24*ArcTan[Tan[x]]*(Cos[x] - I*Sin[x]) - 32*Sin[x] - 27*Sin[3*x] + 5*Sin[5*x]))/(384*(I + Tan[x]))","A",1
2,1,51,29,0.0205956,"\int \frac{\sin ^3(x)}{i+\tan (x)} \, dx","Integrate[Sin[x]^3/(I + Tan[x]),x]","\frac{\sin (x)}{8}-\frac{1}{16} \sin (3 x)+\frac{1}{80} \sin (5 x)+\frac{1}{8} i \cos (x)+\frac{1}{48} i \cos (3 x)-\frac{1}{80} i \cos (5 x)","\frac{\sin ^5(x)}{5}-\frac{1}{5} i \cos ^5(x)+\frac{1}{3} i \cos ^3(x)",1,"(I/8)*Cos[x] + (I/48)*Cos[3*x] - (I/80)*Cos[5*x] + Sin[x]/8 - Sin[3*x]/16 + Sin[5*x]/80","A",1
3,1,39,50,0.0959891,"\int \frac{\sin ^2(x)}{i+\tan (x)} \, dx","Integrate[Sin[x]^2/(I + Tan[x]),x]","-\frac{i \left(-3 i \sin (2 x)+\cos (2 x)+2 \tan ^{-1}(\tan (x)) (\tan (x)+i)+3\right)}{16 (\tan (x)+i)}","-\frac{i x}{8}-\frac{i}{8 (-\tan (x)+i)}-\frac{i}{4 (\tan (x)+i)}-\frac{1}{8 (\tan (x)+i)^2}",1,"((-1/16*I)*(3 + Cos[2*x] - (3*I)*Sin[2*x] + 2*ArcTan[Tan[x]]*(I + Tan[x])))/(I + Tan[x])","A",1
4,1,33,19,0.0130608,"\int \frac{\sin (x)}{i+\tan (x)} \, dx","Integrate[Sin[x]/(I + Tan[x]),x]","\frac{\sin (x)}{4}-\frac{1}{12} \sin (3 x)+\frac{1}{4} i \cos (x)+\frac{1}{12} i \cos (3 x)","\frac{\sin ^3(x)}{3}+\frac{1}{3} i \cos ^3(x)",1,"(I/4)*Cos[x] + (I/12)*Cos[3*x] + Sin[x]/4 - Sin[3*x]/12","A",1
5,1,31,16,0.0189658,"\int \frac{\csc (x)}{i+\tan (x)} \, dx","Integrate[Csc[x]/(I + Tan[x]),x]","\sin (x)-i \cos (x)-i \log \left(\sin \left(\frac{x}{2}\right)\right)+i \log \left(\cos \left(\frac{x}{2}\right)\right)","\sin (x)-i \cos (x)+i \tanh ^{-1}(\cos (x))",1,"(-I)*Cos[x] + I*Log[Cos[x/2]] - I*Log[Sin[x/2]] + Sin[x]","A",1
6,1,15,18,0.0194798,"\int \frac{\csc ^2(x)}{i+\tan (x)} \, dx","Integrate[Csc[x]^2/(I + Tan[x]),x]","i x+i \cot (x)+\log (\sin (x))","i x+i \cot (x)+\log (\tan (x))+\log (\cos (x))",1,"I*x + I*Cot[x] + Log[Sin[x]]","A",1
7,1,75,24,0.0217032,"\int \frac{\csc ^3(x)}{i+\tan (x)} \, dx","Integrate[Csc[x]^3/(I + Tan[x]),x]","-\frac{1}{2} \tan \left(\frac{x}{2}\right)-\frac{1}{2} \cot \left(\frac{x}{2}\right)+\frac{1}{8} i \csc ^2\left(\frac{x}{2}\right)-\frac{1}{8} i \sec ^2\left(\frac{x}{2}\right)+\frac{1}{2} i \log \left(\sin \left(\frac{x}{2}\right)\right)-\frac{1}{2} i \log \left(\cos \left(\frac{x}{2}\right)\right)","-\csc (x)-\frac{1}{2} i \tanh ^{-1}(\cos (x))+\frac{1}{2} i \cot (x) \csc (x)",1,"-1/2*Cot[x/2] + (I/8)*Csc[x/2]^2 - (I/2)*Log[Cos[x/2]] + (I/2)*Log[Sin[x/2]] - (I/8)*Sec[x/2]^2 - Tan[x/2]/2","B",1
8,1,29,19,0.0193189,"\int \frac{\csc ^4(x)}{i+\tan (x)} \, dx","Integrate[Csc[x]^4/(I + Tan[x]),x]","-\frac{1}{3} i \cot (x)-\frac{\csc ^2(x)}{2}+\frac{1}{3} i \cot (x) \csc ^2(x)","-\frac{\cot ^2(x)}{2}+\frac{1}{3} i \cot ^3(x)",1,"(-1/3*I)*Cot[x] - Csc[x]^2/2 + (I/3)*Cot[x]*Csc[x]^2","A",1
9,1,139,40,0.0246943,"\int \frac{\csc ^5(x)}{i+\tan (x)} \, dx","Integrate[Csc[x]^5/(I + Tan[x]),x]","-\frac{1}{12} \tan \left(\frac{x}{2}\right)-\frac{1}{12} \cot \left(\frac{x}{2}\right)+\frac{1}{64} i \csc ^4\left(\frac{x}{2}\right)-\frac{1}{32} i \csc ^2\left(\frac{x}{2}\right)-\frac{1}{64} i \sec ^4\left(\frac{x}{2}\right)+\frac{1}{32} i \sec ^2\left(\frac{x}{2}\right)+\frac{1}{8} i \log \left(\sin \left(\frac{x}{2}\right)\right)-\frac{1}{8} i \log \left(\cos \left(\frac{x}{2}\right)\right)-\frac{1}{24} \cot \left(\frac{x}{2}\right) \csc ^2\left(\frac{x}{2}\right)-\frac{1}{24} \tan \left(\frac{x}{2}\right) \sec ^2\left(\frac{x}{2}\right)","-\frac{\csc ^3(x)}{3}-\frac{1}{8} i \tanh ^{-1}(\cos (x))+\frac{1}{4} i \cot (x) \csc ^3(x)-\frac{1}{8} i \cot (x) \csc (x)",1,"-1/12*Cot[x/2] - (I/32)*Csc[x/2]^2 - (Cot[x/2]*Csc[x/2]^2)/24 + (I/64)*Csc[x/2]^4 - (I/8)*Log[Cos[x/2]] + (I/8)*Log[Sin[x/2]] + (I/32)*Sec[x/2]^2 - (I/64)*Sec[x/2]^4 - Tan[x/2]/12 - (Sec[x/2]^2*Tan[x/2])/24","B",1
10,1,41,37,0.0203449,"\int \frac{\csc ^6(x)}{i+\tan (x)} \, dx","Integrate[Csc[x]^6/(I + Tan[x]),x]","-\frac{2}{15} i \cot (x)-\frac{\csc ^4(x)}{4}+\frac{1}{5} i \cot (x) \csc ^4(x)-\frac{1}{15} i \cot (x) \csc ^2(x)","\frac{1}{5} i \cot ^5(x)-\frac{\cot ^4(x)}{4}+\frac{1}{3} i \cot ^3(x)-\frac{\cot ^2(x)}{2}",1,"((-2*I)/15)*Cot[x] - (I/15)*Cot[x]*Csc[x]^2 - Csc[x]^4/4 + (I/5)*Cot[x]*Csc[x]^4","A",1
11,1,103,101,0.0475219,"\int \sin ^5(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sin[c + d*x]^5*(a + b*Tan[c + d*x]),x]","-\frac{5 a \cos (c+d x)}{8 d}+\frac{5 a \cos (3 (c+d x))}{48 d}-\frac{a \cos (5 (c+d x))}{80 d}-\frac{b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cos ^5(c+d x)}{5 d}+\frac{2 a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (5*a*Cos[c + d*x])/(8*d) + (5*a*Cos[3*(c + d*x)])/(48*d) - (a*Cos[5*(c + d*x)])/(80*d) - (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d) - (b*Sin[c + d*x]^5)/(5*d)","A",1
12,1,82,83,0.0817862,"\int \sin ^4(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sin[c + d*x]^4*(a + b*Tan[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 \left(\frac{1}{4} \cos ^4(c+d x)-\cos ^2(c+d x)+\log (\cos (c+d x))\right)}{d}","-\frac{\sin ^3(c+d x) \cos (c+d x) (a+b \tan (c+d x))}{4 d}-\frac{\sin (c+d x) \cos (c+d x) (3 a+4 b \tan (c+d x))}{8 d}+\frac{3 a x}{8}-\frac{b \log (\cos (c+d x))}{d}",1,"(3*a*(c + d*x))/(8*d) - (b*(-Cos[c + d*x]^2 + Cos[c + d*x]^4/4 + Log[Cos[c + d*x]]))/d - (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)","A",1
13,1,71,69,0.0275788,"\int \sin ^3(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x]),x]","-\frac{3 a \cos (c+d x)}{4 d}+\frac{a \cos (3 (c+d x))}{12 d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (3*a*Cos[c + d*x])/(4*d) + (a*Cos[3*(c + d*x)])/(12*d) - (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d)","A",1
14,1,56,49,0.0495718,"\int \sin ^2(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sin[c + d*x]^2*(a + b*Tan[c + d*x]),x]","\frac{a (c+d x)}{2 d}-\frac{a \sin (2 (c+d x))}{4 d}-\frac{b \left(\log (\cos (c+d x))-\frac{1}{2} \cos ^2(c+d x)\right)}{d}","-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))}{2 d}+\frac{a x}{2}-\frac{b \log (\cos (c+d x))}{d}",1,"(a*(c + d*x))/(2*d) - (b*(-1/2*Cos[c + d*x]^2 + Log[Cos[c + d*x]]))/d - (a*Sin[2*(c + d*x)])/(4*d)","A",1
15,1,48,37,0.0269724,"\int \sin (c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sin[c + d*x]*(a + b*Tan[c + d*x]),x]","\frac{a \sin (c) \sin (d x)}{d}-\frac{a \cos (c) \cos (d x)}{d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \cos (c+d x)}{d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (a*Cos[c]*Cos[d*x])/d + (a*Sin[c]*Sin[d*x])/d - (b*Sin[c + d*x])/d","A",1
16,1,52,26,0.0177597,"\int \csc (c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Csc[c + d*x]*(a + b*Tan[c + d*x]),x]","\frac{a \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","\frac{b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d - (a*Log[Cos[c/2 + (d*x)/2]])/d + (a*Log[Sin[c/2 + (d*x)/2]])/d","A",1
17,1,36,25,0.0620539,"\int \csc ^2(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Csc[c + d*x]^2*(a + b*Tan[c + d*x]),x]","-\frac{a \cot (c+d x)}{d}-\frac{b (\log (\cos (c+d x))-\log (\sin (c+d x)))}{d}","\frac{b \log (\tan (c+d x))}{d}-\frac{a \cot (c+d x)}{d}",1,"-((a*Cot[c + d*x])/d) - (b*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]]))/d","A",1
18,1,107,60,0.0282475,"\int \csc ^3(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Csc[c + d*x]^3*(a + b*Tan[c + d*x]),x]","-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{b \csc (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sin ^2(c+d x)\right)}{d}","-\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"-1/8*(a*Csc[(c + d*x)/2]^2)/d - (b*Csc[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[c + d*x]^2])/d - (a*Log[Cos[(c + d*x)/2]])/(2*d) + (a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","C",1
19,1,72,57,0.255341,"\int \csc ^4(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Csc[c + d*x]^4*(a + b*Tan[c + d*x]),x]","-\frac{2 a \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac{b \left(\csc ^2(c+d x)-2 \log (\sin (c+d x))+2 \log (\cos (c+d x))\right)}{2 d}","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d}",1,"(-2*a*Cot[c + d*x])/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) - (b*(Csc[c + d*x]^2 + 2*Log[Cos[c + d*x]] - 2*Log[Sin[c + d*x]]))/(2*d)","A",1
20,1,151,98,0.0386349,"\int \csc ^5(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Csc[c + d*x]^5*(a + b*Tan[c + d*x]),x]","-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{3 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{b \csc ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sin ^2(c+d x)\right)}{3 d}","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-3*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (b*Csc[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Sin[c + d*x]^2])/(3*d) - (3*a*Log[Cos[(c + d*x)/2]])/(8*d) + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) + (3*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","C",1
21,1,104,87,0.5560412,"\int \csc ^6(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Csc[c + d*x]^6*(a + b*Tan[c + d*x]),x]","-\frac{8 a \cot (c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac{b \left(\csc ^4(c+d x)+2 \csc ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right)}{4 d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{b \cot ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d}",1,"(-8*a*Cot[c + d*x])/(15*d) - (4*a*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d) - (b*(2*Csc[c + d*x]^2 + Csc[c + d*x]^4 + 4*Log[Cos[c + d*x]] - 4*Log[Sin[c + d*x]]))/(4*d)","A",1
22,1,240,113,3.5066499,"\int \sin ^4(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sin[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]","\frac{b \left(\frac{2 \left(3 b^2-2 a^2\right) \sin (2 (c+d x))}{b}+\frac{4 \left(3 b^2-2 a^2\right) \tan ^{-1}(\tan (c+d x))}{b}+4 \left(\frac{a^2-3 b^2}{\sqrt{-b^2}}+2 a\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+4 \left(\frac{3 b^2-a^2}{\sqrt{-b^2}}+2 a\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+\frac{3 \left(a^2-b^2\right) \left(\sin (2 (c+d x))+2 \tan ^{-1}(\tan (c+d x))\right)}{2 b}+\frac{2 \left(a^2-b^2\right) \sin (c+d x) \cos ^3(c+d x)}{b}-4 a \cos ^4(c+d x)+16 a \cos ^2(c+d x)+8 b \tan (c+d x)\right)}{8 d}","\frac{3}{8} x \left(a^2-5 b^2\right)+\frac{\cos ^2(c+d x) (7 b-5 a \tan (c+d x)) (a+b \tan (c+d x))}{8 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{\sin (c+d x) \cos ^3(c+d x) (a+b \tan (c+d x))^2}{4 d}+\frac{b^2 \tan (c+d x)}{d}",1,"(b*((4*(-2*a^2 + 3*b^2)*ArcTan[Tan[c + d*x]])/b + 16*a*Cos[c + d*x]^2 - 4*a*Cos[c + d*x]^4 + 4*(2*a + (a^2 - 3*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 4*(2*a + (-a^2 + 3*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + (2*(a^2 - b^2)*Cos[c + d*x]^3*Sin[c + d*x])/b + (2*(-2*a^2 + 3*b^2)*Sin[2*(c + d*x)])/b + (3*(a^2 - b^2)*(2*ArcTan[Tan[c + d*x]] + Sin[2*(c + d*x)]))/(2*b) + 8*b*Tan[c + d*x]))/(8*d)","B",1
23,1,152,122,0.9894742,"\int \sin ^3(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]","\frac{\sec (c+d x) \left(\left(20 b^2-8 a^2\right) \cos (2 (c+d x))+\left(a^2-b^2\right) \cos (4 (c+d x))-9 a^2-28 a b \sin (2 (c+d x))+2 a b \sin (4 (c+d x))-48 a b \cos (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+48 a b \cos (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 b^2\right)}{24 d}","\frac{a^2 \cos ^3(c+d x)}{3 d}-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 \cos ^3(c+d x)}{3 d}+\frac{2 b^2 \cos (c+d x)}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"(Sec[c + d*x]*(-9*a^2 + 45*b^2 + (-8*a^2 + 20*b^2)*Cos[2*(c + d*x)] + (a^2 - b^2)*Cos[4*(c + d*x)] - 48*a*b*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*a*b*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 28*a*b*Sin[2*(c + d*x)] + 2*a*b*Sin[4*(c + d*x)]))/(24*d)","A",1
24,1,162,76,2.4966551,"\int \sin ^2(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^2,x]","\frac{b \left(\frac{\left(b^2-a^2\right) \sin (2 (c+d x))}{2 b}+\frac{\left(b^2-a^2\right) \tan ^{-1}(\tan (c+d x))}{b}+\left(\frac{a^2-2 b^2}{\sqrt{-b^2}}+2 a\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\left(\frac{2 b^2-a^2}{\sqrt{-b^2}}+2 a\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+2 a \cos ^2(c+d x)+2 b \tan (c+d x)\right)}{2 d}","\frac{1}{2} x \left(a^2-3 b^2\right)-\frac{2 a b \log (\cos (c+d x))}{d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^2}{2 d}+\frac{3 b^2 \tan (c+d x)}{2 d}",1,"(b*(((-a^2 + b^2)*ArcTan[Tan[c + d*x]])/b + 2*a*Cos[c + d*x]^2 + (2*a + (a^2 - 2*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + (2*a + (-a^2 + 2*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + ((-a^2 + b^2)*Sin[2*(c + d*x)])/(2*b) + 2*b*Tan[c + d*x]))/(2*d)","B",1
25,1,111,68,0.4685096,"\int \sin (c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sin[c + d*x]*(a + b*Tan[c + d*x])^2,x]","-\frac{\sec (c+d x) \left(\left(a^2-b^2\right) \cos (2 (c+d x))+a^2+2 a b \sin (2 (c+d x))+4 a b \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^2\right)}{2 d}","-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \cos (c+d x)}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"-1/2*(Sec[c + d*x]*(a^2 - 3*b^2 + (a^2 - b^2)*Cos[2*(c + d*x)] + 4*a*b*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]]) + 2*a*b*Sin[2*(c + d*x)]))/d","A",1
26,1,97,43,0.2500286,"\int \csc (c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Csc[c + d*x]*(a + b*Tan[c + d*x])^2,x]","\frac{a \left(a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b^2 \sec (c+d x)}{d}","-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"(a*(-(a*Log[Cos[(c + d*x)/2]]) - 2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + a*Log[Sin[(c + d*x)/2]] + 2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + b^2*Sec[c + d*x])/d","B",1
27,1,91,42,0.5785076,"\int \csc ^2(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^2,x]","-\frac{\cos (c+d x) (a+b \tan (c+d x))^2 \left(a \cos (c+d x) (a \cot (c+d x)+2 b (\log (\cos (c+d x))-\log (\sin (c+d x))))-b^2 \sin (c+d x)\right)}{d (a \cos (c+d x)+b \sin (c+d x))^2}","-\frac{a^2 \cot (c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"-((Cos[c + d*x]*(a*Cos[c + d*x]*(a*Cot[c + d*x] + 2*b*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]])) - b^2*Sin[c + d*x])*(a + b*Tan[c + d*x])^2)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2))","B",1
28,1,250,95,1.917989,"\int \csc ^3(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^2,x]","\frac{4 \left(a^2+2 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \left(a^2+2 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-a^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)+a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-8 a b \tan \left(\frac{1}{2} (c+d x)\right)-8 a b \cot \left(\frac{1}{2} (c+d x)\right)-16 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+16 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 b^2 \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{8 b^2 \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)}+8 b^2}{8 d}","-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{d}",1,"(8*b^2 - 8*a*b*Cot[(c + d*x)/2] - a^2*Csc[(c + d*x)/2]^2 - 4*(a^2 + 2*b^2)*Log[Cos[(c + d*x)/2]] - 16*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*(a^2 + 2*b^2)*Log[Sin[(c + d*x)/2]] + 16*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x)/2]^2 + (8*b^2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (8*b^2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 8*a*b*Tan[(c + d*x)/2])/(8*d)","B",1
29,1,127,79,1.4338491,"\int \csc ^4(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^2,x]","-\frac{(a+b \tan (c+d x))^2 \left(\cos ^2(c+d x) \left(\left(2 a^2+3 b^2\right) \cot (c+d x)+6 a b (\log (\cos (c+d x))-\log (\sin (c+d x)))\right)+a^2 \cot ^3(c+d x)+3 a b \cot ^2(c+d x)-\frac{3}{2} b^2 \sin (2 (c+d x))\right)}{3 d (a \cos (c+d x)+b \sin (c+d x))^2}","-\frac{\left(a^2+b^2\right) \cot (c+d x)}{d}-\frac{a^2 \cot ^3(c+d x)}{3 d}-\frac{a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"-1/3*((3*a*b*Cot[c + d*x]^2 + a^2*Cot[c + d*x]^3 + Cos[c + d*x]^2*((2*a^2 + 3*b^2)*Cot[c + d*x] + 6*a*b*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]])) - (3*b^2*Sin[2*(c + d*x)])/2)*(a + b*Tan[c + d*x])^2)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)","A",1
30,1,994,165,6.2188215,"\int \csc ^5(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^2,x]","-\frac{a^2 \cos ^2(c+d x) (a+b \tan (c+d x))^2 \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\left(-3 a^2-4 b^2\right) \cos ^2(c+d x) (a+b \tan (c+d x))^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{a b \cos ^2(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^2 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{12 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{7 a b \cos ^2(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^2}{6 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{a b \cos ^2(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^2}{12 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{a^2 \cos ^2(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^2}{64 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{b^2 \cos ^2(c+d x) (a+b \tan (c+d x))^2}{d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{\left(3 a^2+4 b^2\right) \cos ^2(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^2}{32 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{7 a b \cos ^2(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^2}{6 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{3 \left(a^2+4 b^2\right) \cos ^2(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^2}{8 d (a \cos (c+d x)+b \sin (c+d x))^2}-\frac{2 a b \cos ^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+b \tan (c+d x))^2}{d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{3 \left(a^2+4 b^2\right) \cos ^2(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^2}{8 d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{2 a b \cos ^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+b \tan (c+d x))^2}{d (a \cos (c+d x)+b \sin (c+d x))^2}+\frac{b^2 \cos ^2(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^2}{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))^2}-\frac{b^2 \cos ^2(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^2}{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))^2}","-\frac{3 a^2 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 b^2 \sec (c+d x)}{2 d}-\frac{3 b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}",1,"(b^2*Cos[c + d*x]^2*(a + b*Tan[c + d*x])^2)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (7*a*b*Cos[c + d*x]^2*Cot[(c + d*x)/2]*(a + b*Tan[c + d*x])^2)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + ((-3*a^2 - 4*b^2)*Cos[c + d*x]^2*Csc[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^2)/(32*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (a*b*Cos[c + d*x]^2*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^2)/(12*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (a^2*Cos[c + d*x]^2*Csc[(c + d*x)/2]^4*(a + b*Tan[c + d*x])^2)/(64*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (3*(a^2 + 4*b^2)*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2]]*(a + b*Tan[c + d*x])^2)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (2*a*b*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^2)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (3*(a^2 + 4*b^2)*Cos[c + d*x]^2*Log[Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^2)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (2*a*b*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^2)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + ((3*a^2 + 4*b^2)*Cos[c + d*x]^2*Sec[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^2)/(32*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (a^2*Cos[c + d*x]^2*Sec[(c + d*x)/2]^4*(a + b*Tan[c + d*x])^2)/(64*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) + (b^2*Cos[c + d*x]^2*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^2)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (b^2*Cos[c + d*x]^2*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^2)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (7*a*b*Cos[c + d*x]^2*Tan[(c + d*x)/2]*(a + b*Tan[c + d*x])^2)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2) - (a*b*Cos[c + d*x]^2*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(a + b*Tan[c + d*x])^2)/(12*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)","B",0
31,1,114,122,1.5085527,"\int \csc ^6(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^6*(a + b*Tan[c + d*x])^2,x]","-\frac{2 \cot (c+d x) \left(\left(4 a^2+5 b^2\right) \csc ^2(c+d x)+3 a^2 \csc ^4(c+d x)+8 a^2+25 b^2\right)+15 b \left(a \csc ^4(c+d x)+2 a \csc ^2(c+d x)-4 a \log (\sin (c+d x))+4 a \log (\cos (c+d x))-2 b \tan (c+d x)\right)}{30 d}","-\frac{\left(2 a^2+b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2+2 b^2\right) \cot (c+d x)}{d}-\frac{a^2 \cot ^5(c+d x)}{5 d}-\frac{a b \cot ^4(c+d x)}{2 d}-\frac{2 a b \cot ^2(c+d x)}{d}+\frac{2 a b \log (\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"-1/30*(2*Cot[c + d*x]*(8*a^2 + 25*b^2 + (4*a^2 + 5*b^2)*Csc[c + d*x]^2 + 3*a^2*Csc[c + d*x]^4) + 15*b*(2*a*Csc[c + d*x]^2 + a*Csc[c + d*x]^4 + 4*a*Log[Cos[c + d*x]] - 4*a*Log[Sin[c + d*x]] - 2*b*Tan[c + d*x]))/d","A",1
32,1,771,205,6.304952,"\int \sin ^3(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^3,x]","\frac{\left(5 b^3-6 a^2 b\right) \cos ^3(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)}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left(6 a^2 b-5 b^3\right) \cos ^3(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 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 a \left(a^2-7 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^3}{4 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{b \left(3 a^2-b^2\right) \sin (3 (c+d x)) \cos ^3(c+d x) (a+b \tan (c+d x))^3}{12 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{a \left(a^2-3 b^2\right) \cos (3 (c+d x)) \cos ^3(c+d x) (a+b \tan (c+d x))^3}{12 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 b \left(5 a^2-3 b^2\right) \sin (c+d x) \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 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))^3}-\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 a b^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3(c+d x) (a+b \tan (c+d x))^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))^3}+\frac{3 a b^2 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 a b^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3(c+d x) (a+b \tan (c+d x))^3}{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))^3}","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{a^3 \cos (c+d x)}{d}-\frac{a^2 b \sin ^3(c+d x)}{d}-\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b^2 \cos ^3(c+d x)}{d}+\frac{6 a b^2 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{5 b^3 \sin ^3(c+d x)}{6 d}+\frac{5 b^3 \sin (c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d}-\frac{5 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(3*a*b^2*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*a*(a^2 - 7*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^3)/(4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (a*(a^2 - 3*b^2)*Cos[c + d*x]^3*Cos[3*(c + d*x)]*(a + b*Tan[c + d*x])^3)/(12*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + ((-6*a^2*b + 5*b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + ((6*a^2*b - 5*b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (b^3*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*a*b^2*Cos[c + d*x]^3*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (b^3*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*a*b^2*Cos[c + d*x]^3*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*b*(5*a^2 - 3*b^2)*Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x])^3)/(4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (b*(3*a^2 - b^2)*Cos[c + d*x]^3*Sin[3*(c + d*x)]*(a + b*Tan[c + d*x])^3)/(12*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)","B",1
33,1,203,103,4.523658,"\int \sin ^2(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]","\frac{b \left(-\frac{a \left(a^2-3 b^2\right) \sin (2 (c+d x))}{2 b}+\left(3 a^2-b^2\right) \cos ^2(c+d x)-\frac{a \left(a^2-3 b^2\right) \tan ^{-1}(\tan (c+d x))}{b}+\left(\frac{a^3-6 a b^2}{\sqrt{-b^2}}+3 a^2-2 b^2\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\left(\frac{6 a b^2-a^3}{\sqrt{-b^2}}+3 a^2-2 b^2\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+6 a b \tan (c+d x)+b^2 \tan ^2(c+d x)\right)}{2 d}","-\frac{b \left(3 a^2-2 b^2\right) \log (\cos (c+d x))}{d}+\frac{1}{2} a x \left(a^2-9 b^2\right)+\frac{9 a b^2 \tan (c+d x)}{2 d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^3}{2 d}+\frac{b^3 \tan ^2(c+d x)}{d}",1,"(b*(-((a*(a^2 - 3*b^2)*ArcTan[Tan[c + d*x]])/b) + (3*a^2 - b^2)*Cos[c + d*x]^2 + (3*a^2 - 2*b^2 + (a^3 - 6*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + (3*a^2 - 2*b^2 + (-a^3 + 6*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] - (a*(a^2 - 3*b^2)*Sin[2*(c + d*x)])/(2*b) + 6*a*b*Tan[c + d*x] + b^2*Tan[c + d*x]^2))/(2*d)","A",1
34,1,637,133,6.1716806,"\int \sin (c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sin[c + d*x]*(a + b*Tan[c + d*x])^3,x]","-\frac{3 \left(2 a^2 b-b^3\right) \cos ^3(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)}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 \left(2 a^2 b-b^3\right) \cos ^3(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 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a \left(a^2-3 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^3}{d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{b \left(3 a^2-b^2\right) \sin (c+d x) \cos ^3(c+d x) (a+b \tan (c+d x))^3}{d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 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))^3}-\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 a b^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3(c+d x) (a+b \tan (c+d x))^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))^3}+\frac{3 a b^2 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 a b^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3(c+d x) (a+b \tan (c+d x))^3}{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))^3}","-\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a b^2 \cos (c+d x)}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{3 b^3 \sin (c+d x)}{2 d}+\frac{b^3 \sin (c+d x) \tan ^2(c+d x)}{2 d}-\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(3*a*b^2*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (a*(a^2 - 3*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^3)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*(2*a^2*b - b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*(2*a^2*b - b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (b^3*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*a*b^2*Cos[c + d*x]^3*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (b^3*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*a*b^2*Cos[c + d*x]^3*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (b*(3*a^2 - b^2)*Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x])^3)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)","B",1
35,1,241,86,2.2887785,"\int \csc (c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Csc[c + d*x]*(a + b*Tan[c + d*x])^3,x]","\frac{4 a^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 a^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-12 a^2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 a b^2 \sin ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)+12 a b^2+\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}+2 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 b^3 \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^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(12*a*b^2 - 4*a^3*Log[Cos[(c + d*x)/2]] - 12*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*a^3*Log[Sin[(c + d*x)/2]] + 12*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b^3/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + 24*a*b^2*Sec[c + d*x]*Sin[(c + d*x)/2]^2 - b^3/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(4*d)","B",1
36,1,126,64,1.0088906,"\int \csc ^2(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^3,x]","-\frac{\csc (c+d x) \sec ^2(c+d x) \left(\left(a^3+3 a b^2\right) \cos (3 (c+d x))+3 a \left(a^2-b^2\right) \cos (c+d x)-2 b \sin (c+d x) \left(3 a^2 \log (\sin (c+d x))-3 a^2 \log (\cos (c+d x))-3 a^2 \cos (2 (c+d x)) (\log (\cos (c+d x))-\log (\sin (c+d x)))+b^2\right)\right)}{4 d}","-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a^2 b \log (\tan (c+d x))}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d}",1,"-1/4*(Csc[c + d*x]*Sec[c + d*x]^2*(3*a*(a^2 - b^2)*Cos[c + d*x] + (a^3 + 3*a*b^2)*Cos[3*(c + d*x)] - 2*b*(b^2 - 3*a^2*Log[Cos[c + d*x]] - 3*a^2*Cos[2*(c + d*x)]*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]]) + 3*a^2*Log[Sin[c + d*x]])*Sin[c + d*x]))/d","A",1
37,1,897,141,6.1938167,"\int \csc ^3(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^3,x]","-\frac{3 a^2 b \cos ^3(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 a b^2 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a^3 \cos ^3(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{a^3 \cos ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 a^2 b \cos ^3(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left(-a^3-6 b^2 a\right) \cos ^3(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left(-b^3-6 a^2 b\right) \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+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left(a^3+6 b^2 a\right) \cos ^3(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left(b^3+6 a^2 b\right) \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+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 a b^2 \cos ^3(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^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))^3}-\frac{3 a b^2 \cos ^3(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^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))^3}+\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 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))^3}-\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 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))^3}","-\frac{a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^3 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{3 a b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(3*a*b^2*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*a^2*b*Cos[c + d*x]^3*Cot[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (a^3*Cos[c + d*x]^3*Csc[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^3)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + ((-a^3 - 6*a*b^2)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + ((-6*a^2*b - b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + ((a^3 + 6*a*b^2)*Cos[c + d*x]^3*Log[Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + ((6*a^2*b + b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (a^3*Cos[c + d*x]^3*Sec[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^3)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (b^3*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*a*b^2*Cos[c + d*x]^3*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (b^3*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*a*b^2*Cos[c + d*x]^3*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*a^2*b*Cos[c + d*x]^3*Tan[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)","B",1
38,1,212,113,2.2465477,"\int \csc ^4(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (a \cot (c+d x)+b)^3 \left(-16 a^3 \cos (c+d x)-2 \sin (c+d x) \left(2 a^3 \sin (4 (c+d x))+6 \left(3 a^2 b+b^3\right) \cos (2 (c+d x))-3 b \left(3 a^2+b^2\right) \cos (4 (c+d x)) (\log (\cos (c+d x))-\log (\sin (c+d x)))-9 a^2 b \log (\sin (c+d x))+9 a^2 b \log (\cos (c+d x))+18 a^2 b+18 a b^2 \sin (4 (c+d x))-3 b^3 \log (\sin (c+d x))+3 b^3 \log (\cos (c+d x))-6 b^3\right)\right)}{48 d (a \cos (c+d x)+b \sin (c+d x))^3}","-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a \left(a^2+3 b^2\right) \cot (c+d x)}{d}+\frac{b \left(3 a^2+b^2\right) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^2(c+d x)}{2 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d}",1,"((b + a*Cot[c + d*x])^3*Sec[c + d*x]^2*(-16*a^3*Cos[c + d*x] - 2*Sin[c + d*x]*(18*a^2*b - 6*b^3 + 6*(3*a^2*b + b^3)*Cos[2*(c + d*x)] + 9*a^2*b*Log[Cos[c + d*x]] + 3*b^3*Log[Cos[c + d*x]] - 3*b*(3*a^2 + b^2)*Cos[4*(c + d*x)]*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]]) - 9*a^2*b*Log[Sin[c + d*x]] - 3*b^3*Log[Sin[c + d*x]] + 2*a^3*Sin[4*(c + d*x)] + 18*a*b^2*Sin[4*(c + d*x)])))/(48*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)","A",1
39,1,1229,229,6.2168813,"\int \csc ^5(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^3,x]","-\frac{a^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3 \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 \left(a^3+4 b^2 a\right) \cos ^3(c+d x) (a+b \tan (c+d x))^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a^2 b \cos ^3(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{\left(-2 \cos \left(\frac{1}{2} (c+d x)\right) b^3-7 a^2 \cos \left(\frac{1}{2} (c+d x)\right) b\right) \cos ^3(c+d x) (a+b \tan (c+d x))^3 \csc \left(\frac{1}{2} (c+d x)\right)}{4 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{a^2 b \cos ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{a^3 \cos ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^3}{64 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 a b^2 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 \left(a^3+4 b^2 a\right) \cos ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^3}{32 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 \left(a^3+12 b^2 a\right) \cos ^3(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 \left(b^3+2 a^2 b\right) \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+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 \left(a^3+12 b^2 a\right) \cos ^3(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^3}{8 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 \left(b^3+2 a^2 b\right) \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+b \tan (c+d x))^3}{2 d (a \cos (c+d x)+b \sin (c+d x))^3}+\frac{3 a b^2 \cos ^3(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^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))^3}+\frac{\cos ^3(c+d x) \sec \left(\frac{1}{2} (c+d x)\right) \left(-2 \sin \left(\frac{1}{2} (c+d x)\right) b^3-7 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b\right) (a+b \tan (c+d x))^3}{4 d (a \cos (c+d x)+b \sin (c+d x))^3}-\frac{3 a b^2 \cos ^3(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^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))^3}+\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 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))^3}-\frac{b^3 \cos ^3(c+d x) (a+b \tan (c+d x))^3}{4 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))^3}","-\frac{3 a^3 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{9 a b^2 \sec (c+d x)}{2 d}-\frac{9 a b^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{3 a b^2 \csc ^2(c+d x) \sec (c+d x)}{2 d}-\frac{3 b^3 \csc (c+d x)}{2 d}+\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}",1,"(3*a*b^2*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + ((-7*a^2*b*Cos[(c + d*x)/2] - 2*b^3*Cos[(c + d*x)/2])*Cos[c + d*x]^3*Csc[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*(a^3 + 4*a*b^2)*Cos[c + d*x]^3*Csc[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^3)/(32*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (a^2*b*Cos[c + d*x]^3*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^3)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (a^3*Cos[c + d*x]^3*Csc[(c + d*x)/2]^4*(a + b*Tan[c + d*x])^3)/(64*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*(a^3 + 12*a*b^2)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*(2*a^2*b + b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*(a^3 + 12*a*b^2)*Cos[c + d*x]^3*Log[Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*(2*a^2*b + b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^3)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*(a^3 + 4*a*b^2)*Cos[c + d*x]^3*Sec[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^3)/(32*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (a^3*Cos[c + d*x]^3*Sec[(c + d*x)/2]^4*(a + b*Tan[c + d*x])^3)/(64*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (b^3*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (3*a*b^2*Cos[c + d*x]^3*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (b^3*Cos[c + d*x]^3*(a + b*Tan[c + d*x])^3)/(4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (3*a*b^2*Cos[c + d*x]^3*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) + (Cos[c + d*x]^3*Sec[(c + d*x)/2]*(-7*a^2*b*Sin[(c + d*x)/2] - 2*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^3)/(4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3) - (a^2*b*Cos[c + d*x]^3*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(a + b*Tan[c + d*x])^3)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)","B",0
40,1,515,167,1.8375072,"\int \csc ^6(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^6*(a + b*Tan[c + d*x])^3,x]","-\frac{\csc ^5(c+d x) \sec ^2(c+d x) \left(8 \left(a^3+15 a b^2\right) \cos (3 (c+d x))-24 a^3 \cos (5 (c+d x))+8 a^3 \cos (7 (c+d x))+40 a \left(5 a^2+3 b^2\right) \cos (c+d x)+360 a^2 b \sin (c+d x)+270 a^2 b \sin (3 (c+d x))-90 a^2 b \sin (5 (c+d x))-225 a^2 b \sin (c+d x) \log (\sin (c+d x))-45 a^2 b \sin (3 (c+d x)) \log (\sin (c+d x))+135 a^2 b \sin (5 (c+d x)) \log (\sin (c+d x))-45 a^2 b \sin (7 (c+d x)) \log (\sin (c+d x))+225 a^2 b \sin (c+d x) \log (\cos (c+d x))+45 a^2 b \sin (3 (c+d x)) \log (\cos (c+d x))-135 a^2 b \sin (5 (c+d x)) \log (\cos (c+d x))+45 a^2 b \sin (7 (c+d x)) \log (\cos (c+d x))-360 a b^2 \cos (5 (c+d x))+120 a b^2 \cos (7 (c+d x))-240 b^3 \sin (c+d x)+180 b^3 \sin (3 (c+d x))-60 b^3 \sin (5 (c+d x))-150 b^3 \sin (c+d x) \log (\sin (c+d x))-30 b^3 \sin (3 (c+d x)) \log (\sin (c+d x))+90 b^3 \sin (5 (c+d x)) \log (\sin (c+d x))-30 b^3 \sin (7 (c+d x)) \log (\sin (c+d x))+150 b^3 \sin (c+d x) \log (\cos (c+d x))+30 b^3 \sin (3 (c+d x)) \log (\cos (c+d x))-90 b^3 \sin (5 (c+d x)) \log (\cos (c+d x))+30 b^3 \sin (7 (c+d x)) \log (\cos (c+d x))\right)}{960 d}","-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{a \left(2 a^2+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{b \left(6 a^2+b^2\right) \cot ^2(c+d x)}{2 d}-\frac{a \left(a^2+6 b^2\right) \cot (c+d x)}{d}+\frac{b \left(3 a^2+2 b^2\right) \log (\tan (c+d x))}{d}-\frac{3 a^2 b \cot ^4(c+d x)}{4 d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^2(c+d x)}{2 d}",1,"-1/960*(Csc[c + d*x]^5*Sec[c + d*x]^2*(40*a*(5*a^2 + 3*b^2)*Cos[c + d*x] + 8*(a^3 + 15*a*b^2)*Cos[3*(c + d*x)] - 24*a^3*Cos[5*(c + d*x)] - 360*a*b^2*Cos[5*(c + d*x)] + 8*a^3*Cos[7*(c + d*x)] + 120*a*b^2*Cos[7*(c + d*x)] + 360*a^2*b*Sin[c + d*x] - 240*b^3*Sin[c + d*x] + 225*a^2*b*Log[Cos[c + d*x]]*Sin[c + d*x] + 150*b^3*Log[Cos[c + d*x]]*Sin[c + d*x] - 225*a^2*b*Log[Sin[c + d*x]]*Sin[c + d*x] - 150*b^3*Log[Sin[c + d*x]]*Sin[c + d*x] + 270*a^2*b*Sin[3*(c + d*x)] + 180*b^3*Sin[3*(c + d*x)] + 45*a^2*b*Log[Cos[c + d*x]]*Sin[3*(c + d*x)] + 30*b^3*Log[Cos[c + d*x]]*Sin[3*(c + d*x)] - 45*a^2*b*Log[Sin[c + d*x]]*Sin[3*(c + d*x)] - 30*b^3*Log[Sin[c + d*x]]*Sin[3*(c + d*x)] - 90*a^2*b*Sin[5*(c + d*x)] - 60*b^3*Sin[5*(c + d*x)] - 135*a^2*b*Log[Cos[c + d*x]]*Sin[5*(c + d*x)] - 90*b^3*Log[Cos[c + d*x]]*Sin[5*(c + d*x)] + 135*a^2*b*Log[Sin[c + d*x]]*Sin[5*(c + d*x)] + 90*b^3*Log[Sin[c + d*x]]*Sin[5*(c + d*x)] + 45*a^2*b*Log[Cos[c + d*x]]*Sin[7*(c + d*x)] + 30*b^3*Log[Cos[c + d*x]]*Sin[7*(c + d*x)] - 45*a^2*b*Log[Sin[c + d*x]]*Sin[7*(c + d*x)] - 30*b^3*Log[Sin[c + d*x]]*Sin[7*(c + d*x)]))/d","B",1
41,1,1017,275,6.270449,"\int \sin ^3(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]","-\frac{\left(3 a^4-42 b^2 a^2+11 b^4\right) (a+b \tan (c+d x))^4 \cos ^5(c+d x)}{4 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{a b \left(a^2-b^2\right) \sin (3 (c+d x)) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{3 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(a^4-6 b^2 a^2+b^4\right) \cos (3 (c+d x)) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{12 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{2 \left(2 a^3 b-5 a b^3\right) \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 \cos ^4(c+d x)}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{2 \left(2 a^3 b-5 a b^3\right) \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 \cos ^4(c+d x)}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^4 \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 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(36 a^2 b^2 \sin \left(\frac{1}{2} (c+d x)\right)-17 b^4 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 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(17 b^4 \sin \left(\frac{1}{2} (c+d x)\right)-36 a^2 b^2 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 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{a b \left(5 a^2-9 b^2\right) \sin (c+d x) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{b^2 \left(17 b^2-36 a^2\right) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(b^4+12 a b^3\right) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{12 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(b^4-12 a b^3\right) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{12 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{b^4 \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4 \cos ^4(c+d x)}{6 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{a^4 \cos ^3(c+d x)}{3 d}-\frac{a^4 \cos (c+d x)}{d}-\frac{4 a^3 b \sin ^3(c+d x)}{3 d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a^2 b^2 \cos ^3(c+d x)}{d}+\frac{12 a^2 b^2 \cos (c+d x)}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}+\frac{10 a b^3 \sin ^3(c+d x)}{3 d}+\frac{10 a b^3 \sin (c+d x)}{d}+\frac{2 a b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{d}-\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^4 \cos ^3(c+d x)}{3 d}-\frac{3 b^4 \cos (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{3 b^4 \sec (c+d x)}{d}",1,"-1/6*(b^2*(-36*a^2 + 17*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - ((3*a^4 - 42*a^2*b^2 + 11*b^4)*Cos[c + d*x]^5*(a + b*Tan[c + d*x])^4)/(4*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((a^4 - 6*a^2*b^2 + b^4)*Cos[c + d*x]^4*Cos[3*(c + d*x)]*(a + b*Tan[c + d*x])^4)/(12*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (2*(2*a^3*b - 5*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (2*(2*a^3*b - 5*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((12*a*b^3 + b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (b^4*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*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*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-12*a*b^3 + b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(12*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*(36*a^2*b^2*Sin[(c + d*x)/2] - 17*b^4*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*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*(-36*a^2*b^2*Sin[(c + d*x)/2] + 17*b^4*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a*b*(5*a^2 - 9*b^2)*Cos[c + d*x]^4*Sin[c + d*x]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (a*b*(a^2 - b^2)*Cos[c + d*x]^4*Sin[3*(c + d*x)]*(a + b*Tan[c + d*x])^4)/(3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)","B",1
42,1,263,139,6.2860534,"\int \sin ^2(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]","\frac{b \left(2 b \left(3 a^2-b^2\right) \tan (c+d x)-\frac{\left(a^4-6 a^2 b^2+b^4\right) \tan ^{-1}(\tan (c+d x))}{2 b}-\frac{\left(a^4-6 a^2 b^2+b^4\right) \sin (c+d x) \cos (c+d x)}{2 b}+\frac{1}{2} \left(4 a^3+\frac{a^4-12 a^2 b^2+3 b^4}{\sqrt{-b^2}}-8 a b^2\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\frac{1}{2} \left(4 a^3-\frac{a^4-12 a^2 b^2+3 b^4}{\sqrt{-b^2}}-8 a b^2\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)+2 a b^2 \tan ^2(c+d x)+2 a (a-b) (a+b) \cos ^2(c+d x)+\frac{1}{3} b^3 \tan ^3(c+d x)\right)}{d}","\frac{b^2 \left(18 a^2-5 b^2\right) \tan (c+d x)}{2 d}-\frac{4 a b \left(a^2-2 b^2\right) \log (\cos (c+d x))}{d}+\frac{1}{2} x \left(a^4-18 a^2 b^2+5 b^4\right)+\frac{4 a b^3 \tan ^2(c+d x)}{d}-\frac{\sin (c+d x) \cos (c+d x) (a+b \tan (c+d x))^4}{2 d}+\frac{5 b^4 \tan ^3(c+d x)}{6 d}",1,"(b*(-1/2*((a^4 - 6*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]])/b + 2*a*(a - b)*(a + b)*Cos[c + d*x]^2 + ((4*a^3 - 8*a*b^2 + (a^4 - 12*a^2*b^2 + 3*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/2 + ((4*a^3 - 8*a*b^2 - (a^4 - 12*a^2*b^2 + 3*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/2 - ((a^4 - 6*a^2*b^2 + b^4)*Cos[c + d*x]*Sin[c + d*x])/(2*b) + 2*b*(3*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))/d","A",1
43,1,383,180,5.2601736,"\int \sin (c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Sin[c + d*x]*(a + b*Tan[c + d*x])^4,x]","\frac{-48 a b \left(a^2-b^2\right) \sin (c+d x)+\frac{2 b^2 \left(36 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^2 \left(11 b^2-36 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)}-24 a b \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)+24 a b \left(2 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)+72 a^2 b^2-12 \left(a^4-6 a^2 b^2+b^4\right) \cos (c+d x)+\frac{b^3 (12 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^3 (b-12 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^4 \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^4 \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^4}{12 d}","-\frac{a^4 \cos (c+d x)}{d}-\frac{4 a^3 b \sin (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{6 a^2 b^2 \cos (c+d x)}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}+\frac{6 a b^3 \sin (c+d x)}{d}+\frac{2 a b^3 \sin (c+d x) \tan ^2(c+d x)}{d}-\frac{6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^4 \cos (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{2 b^4 \sec (c+d x)}{d}",1,"(72*a^2*b^2 - 22*b^4 - 12*(a^4 - 6*a^2*b^2 + b^4)*Cos[c + d*x] - 24*a*b*(2*a^2 - 3*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 24*a*b*(2*a^2 - 3*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^3*(12*a + b))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*b^4*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (2*b^2*(36*a^2 - 11*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (2*b^4*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (b^3*(-12*a + b))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (2*b^2*(-36*a^2 + 11*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 48*a*b*(a^2 - b^2)*Sin[c + d*x])/(12*d)","B",1
44,1,352,118,5.0000212,"\int \csc (c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Csc[c + d*x]*(a + b*Tan[c + d*x])^4,x]","\frac{12 a^4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-12 a^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-48 a^3 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+48 a^3 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 b^2 \sin ^2\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(\left(36 a^2-5 b^2\right) \cos (2 (c+d x))+36 a^2+2 b^2 \cos (c+d x)-b^2\right)+72 a^2 b^2+\frac{12 a b^3}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{12 a b^3}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+24 a b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-24 a b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\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}-10 b^4}{12 d}","-\frac{a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}-\frac{b^4 \sec (c+d x)}{d}",1,"(72*a^2*b^2 - 10*b^4 - 12*a^4*Log[Cos[(c + d*x)/2]] - 48*a^3*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 24*a*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^4*Log[Sin[(c + d*x)/2]] + 48*a^3*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 24*a*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (12*a*b^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + b^4/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + 2*b^2*(36*a^2 - b^2 + 2*b^2*Cos[c + d*x] + (36*a^2 - 5*b^2)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2]^2 - (12*a*b^3)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + b^4/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(12*d)","B",1
45,1,162,83,1.2068277,"\int \csc ^2(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^4,x]","-\frac{\csc (c+d x) \sec ^3(c+d x) \left(4 \left(3 a^4+b^4\right) \cos (2 (c+d x))+\left(3 a^4+18 a^2 b^2-b^4\right) \cos (4 (c+d x))+3 \left(3 a^4+4 a^3 b \sin (4 (c+d x)) (\log (\cos (c+d x))-\log (\sin (c+d x)))+8 a b \sin (2 (c+d x)) \left(-a^2 \log (\sin (c+d x))+a^2 \log (\cos (c+d x))-b^2\right)-6 a^2 b^2-b^4\right)\right)}{24 d}","-\frac{a^4 \cot (c+d x)}{d}+\frac{4 a^3 b \log (\tan (c+d x))}{d}+\frac{6 a^2 b^2 \tan (c+d x)}{d}+\frac{2 a b^3 \tan ^2(c+d x)}{d}+\frac{b^4 \tan ^3(c+d x)}{3 d}",1,"-1/24*(Csc[c + d*x]*Sec[c + d*x]^3*(4*(3*a^4 + b^4)*Cos[2*(c + d*x)] + (3*a^4 + 18*a^2*b^2 - b^4)*Cos[4*(c + d*x)] + 3*(3*a^4 - 6*a^2*b^2 - b^4 + 8*a*b*(-b^2 + a^2*Log[Cos[c + d*x]] - a^2*Log[Sin[c + d*x]])*Sin[2*(c + d*x)] + 4*a^3*b*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]])*Sin[4*(c + d*x)])))/d","A",1
46,1,1128,161,6.2070585,"\int \csc ^3(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^4,x]","-\frac{2 a^3 b \cos ^4(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^2 \left(36 a^2+b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{a^4 \cos ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{a^4 \cos ^4(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{2 a^3 b \cos ^4(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-a^4-12 b^2 a^2\right) \cos ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{2 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{2 \left(2 b a^3+b^3 a\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}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(a^4+12 b^2 a^2\right) \cos ^4(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \tan (c+d x))^4}{2 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{2 \left(2 b a^3+b^3 a\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}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^4 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{6 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(-\sin \left(\frac{1}{2} (c+d x)\right) b^4-36 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^2\right) (a+b \tan (c+d x))^4}{6 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(\sin \left(\frac{1}{2} (c+d x)\right) b^4+36 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^2\right) (a+b \tan (c+d x))^4}{6 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(b^4+12 a b^3\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 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(b^4-12 a b^3\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 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{b^4 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{6 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{a^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a^4 \cot (c+d x) \csc (c+d x)}{2 d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{6 a^2 b^2 \sec (c+d x)}{d}-\frac{6 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}+\frac{2 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \tan (c+d x) \sec (c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}",1,"(b^2*(36*a^2 + b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (2*a^3*b*Cos[c + d*x]^4*Cot[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a^4*Cos[c + d*x]^4*Csc[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^4)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-a^4 - 12*a^2*b^2)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (2*(2*a^3*b + a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((a^4 + 12*a^2*b^2)*Cos[c + d*x]^4*Log[Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (2*(2*a^3*b + a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (a^4*Cos[c + d*x]^4*Sec[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^4)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((12*a*b^3 + b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (b^4*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*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*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-12*a*b^3 + b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(12*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*(-36*a^2*b^2*Sin[(c + d*x)/2] - b^4*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*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*(36*a^2*b^2*Sin[(c + d*x)/2] + b^4*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (2*a^3*b*Cos[c + d*x]^4*Tan[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)","B",1
47,1,188,137,3.8435934,"\int \csc ^4(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^4,x]","-\frac{\sin (c+d x) \tan ^3(c+d x) (a \cot (c+d x)+b)^4 \left(-2 b^2 \left(9 a^2+b^2\right) \sin (c+d x) \cos ^2(c+d x)+2 a \cos ^3(c+d x) \left(a \left(a^2+9 b^2\right) \cot (c+d x)+6 b \left(a^2+b^2\right) (\log (\cos (c+d x))-\log (\sin (c+d x)))\right)+\cos (c+d x) \left(a^4 \cot ^3(c+d x)+6 a^3 b \cot ^2(c+d x)-6 a b^3\right)+b^4 (-\sin (c+d x))\right)}{3 d (a \cos (c+d x)+b \sin (c+d x))^4}","-\frac{a^4 \cot ^3(c+d x)}{3 d}-\frac{2 a^3 b \cot ^2(c+d x)}{d}+\frac{b^2 \left(6 a^2+b^2\right) \tan (c+d x)}{d}-\frac{a^2 \left(a^2+6 b^2\right) \cot (c+d x)}{d}+\frac{4 a b \left(a^2+b^2\right) \log (\tan (c+d x))}{d}+\frac{2 a b^3 \tan ^2(c+d x)}{d}+\frac{b^4 \tan ^3(c+d x)}{3 d}",1,"-1/3*((b + a*Cot[c + d*x])^4*Sin[c + d*x]*(Cos[c + d*x]*(-6*a*b^3 + 6*a^3*b*Cot[c + d*x]^2 + a^4*Cot[c + d*x]^3) + 2*a*Cos[c + d*x]^3*(a*(a^2 + 9*b^2)*Cot[c + d*x] + 6*b*(a^2 + b^2)*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]])) - b^4*Sin[c + d*x] - 2*b^2*(9*a^2 + b^2)*Cos[c + d*x]^2*Sin[c + d*x])*Tan[c + d*x]^3)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)","A",1
48,1,1491,274,6.3024212,"\int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^4,x]","-\frac{a^3 b \cos ^4(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^2 \left(36 a^2+7 b^2\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{a^4 \cos ^4(c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{64 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{a^4 \cos ^4(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{64 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{3 \left(a^4+8 b^2 a^2\right) \cos ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{32 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac{a^3 b \cos ^4(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{3 \left(a^4+8 b^2 a^2\right) \cos ^4(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{32 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-7 b \cos \left(\frac{1}{2} (c+d x)\right) a^3-6 b^3 \cos \left(\frac{1}{2} (c+d x)\right) a\right) \cos ^4(c+d x) \csc \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{3 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(-3 a^4-72 b^2 a^2-8 b^4\right) \cos ^4(c+d x) \log \left(\cos \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 \left(2 b a^3+3 b^3 a\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}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\left(3 a^4+72 b^2 a^2+8 b^4\right) \cos ^4(c+d x) \log \left(\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 \left(2 b a^3+3 b^3 a\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}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{b^4 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{6 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) \sec \left(\frac{1}{2} (c+d x)\right) \left(-7 b \sin \left(\frac{1}{2} (c+d x)\right) a^3-6 b^3 \sin \left(\frac{1}{2} (c+d x)\right) a\right) (a+b \tan (c+d x))^4}{3 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac{\cos ^4(c+d x) \left(-7 \sin \left(\frac{1}{2} (c+d x)\right) b^4-36 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^2\right) (a+b \tan (c+d x))^4}{6 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(7 \sin \left(\frac{1}{2} (c+d x)\right) b^4+36 a^2 \sin \left(\frac{1}{2} (c+d x)\right) b^2\right) (a+b \tan (c+d x))^4}{6 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(b^4+12 a b^3\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 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(b^4-12 a b^3\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 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{b^4 \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \tan (c+d x))^4}{6 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{3 a^4 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{9 a^2 b^2 \sec (c+d x)}{d}-\frac{9 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac{3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}-\frac{6 a b^3 \csc (c+d x)}{d}+\frac{6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac{b^4 \sec ^3(c+d x)}{3 d}+\frac{b^4 \sec (c+d x)}{d}-\frac{b^4 \tanh ^{-1}(\cos (c+d x))}{d}",1,"(b^2*(36*a^2 + 7*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-7*a^3*b*Cos[(c + d*x)/2] - 6*a*b^3*Cos[(c + d*x)/2])*Cos[c + d*x]^4*Csc[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (3*(a^4 + 8*a^2*b^2)*Cos[c + d*x]^4*Csc[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^4)/(32*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a^3*b*Cos[c + d*x]^4*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a^4*Cos[c + d*x]^4*Csc[(c + d*x)/2]^4*(a + b*Tan[c + d*x])^4)/(64*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-3*a^4 - 72*a^2*b^2 - 8*b^4)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (2*(2*a^3*b + 3*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((3*a^4 + 72*a^2*b^2 + 8*b^4)*Cos[c + d*x]^4*Log[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) + (2*(2*a^3*b + 3*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (3*(a^4 + 8*a^2*b^2)*Cos[c + d*x]^4*Sec[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^4)/(32*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (a^4*Cos[c + d*x]^4*Sec[(c + d*x)/2]^4*(a + b*Tan[c + d*x])^4)/(64*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((12*a*b^3 + b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (b^4*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*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*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-12*a*b^3 + b^4)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(12*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*Sec[(c + d*x)/2]*(-7*a^3*b*Sin[(c + d*x)/2] - 6*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(-36*a^2*b^2*Sin[(c + d*x)/2] - 7*b^4*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*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*(36*a^2*b^2*Sin[(c + d*x)/2] + 7*b^4*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a^3*b*Cos[c + d*x]^4*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)","B",0
49,1,233,194,4.0121532,"\int \csc ^6(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Csc[c + d*x]^6*(a + b*Tan[c + d*x])^4,x]","-\frac{(a+b \tan (c+d x))^4 \left(3 a^4 \cot ^5(c+d x)+15 a^3 b \cot ^4(c+d x)-5 b^2 \left(18 a^2+5 b^2\right) \sin (c+d x) \cos ^3(c+d x)+2 a \cos ^2(c+d x) \left(a \left(2 a^2+15 b^2\right) \cot ^3(c+d x)+15 b \left(a^2+b^2\right) \cot ^2(c+d x)-15 b^3\right)+\cos ^4(c+d x) \left(60 a b \left(a^2+2 b^2\right) (\log (\cos (c+d x))-\log (\sin (c+d x)))+\left(8 a^4+150 a^2 b^2+15 b^4\right) \cot (c+d x)\right)-\frac{5}{2} b^4 \sin (2 (c+d x))\right)}{15 d (a \cos (c+d x)+b \sin (c+d x))^4}","-\frac{a^4 \cot ^5(c+d x)}{5 d}-\frac{a^3 b \cot ^4(c+d x)}{d}+\frac{2 b^2 \left(3 a^2+b^2\right) \tan (c+d x)}{d}-\frac{2 a^2 \left(a^2+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{2 a b \left(2 a^2+b^2\right) \cot ^2(c+d x)}{d}+\frac{4 a b \left(a^2+2 b^2\right) \log (\tan (c+d x))}{d}-\frac{\left(a^4+12 a^2 b^2+b^4\right) \cot (c+d x)}{d}+\frac{2 a b^3 \tan ^2(c+d x)}{d}+\frac{b^4 \tan ^3(c+d x)}{3 d}",1,"-1/15*((15*a^3*b*Cot[c + d*x]^4 + 3*a^4*Cot[c + d*x]^5 + 2*a*Cos[c + d*x]^2*(-15*b^3 + 15*b*(a^2 + b^2)*Cot[c + d*x]^2 + a*(2*a^2 + 15*b^2)*Cot[c + d*x]^3) + Cos[c + d*x]^4*((8*a^4 + 150*a^2*b^2 + 15*b^4)*Cot[c + d*x] + 60*a*b*(a^2 + 2*b^2)*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]])) - 5*b^2*(18*a^2 + 5*b^2)*Cos[c + d*x]^3*Sin[c + d*x] - (5*b^4*Sin[2*(c + d*x)])/2)*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)","A",1
50,1,660,402,6.2662978,"\int \csc ^7(c+d x) (a+b \tan (c+d x))^4 \, dx","Integrate[Csc[c + d*x]^7*(a + b*Tan[c + d*x])^4,x]","-\frac{2 \left(2 a^3 b+5 a b^3\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4 \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))^4}+\frac{2 \left(2 a^3 b+5 a b^3\right) \cos ^4(c+d x) (a+b \tan (c+d x))^4 \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))^4}-\frac{5 \left(a^4+36 a^2 b^2+8 b^4\right) \cos ^4(c+d x) \log \left(\cos \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{5 \left(a^4+36 a^2 b^2+8 b^4\right) \cos ^4(c+d x) \log \left(\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{\cot (c+d x) \csc ^5(c+d x) (a+b \tan (c+d x))^4 \left(-2760 a^4 \cos (2 (c+d x))+60 a^4 \cos (4 (c+d x))+200 a^4 \cos (6 (c+d x))-75 a^4 \cos (8 (c+d x))-2545 a^4-15744 a^3 b \sin (2 (c+d x))-1152 a^3 b \sin (4 (c+d x))+3200 a^3 b \sin (6 (c+d x))-960 a^3 b \sin (8 (c+d x))-7200 a^2 b^2 \cos (2 (c+d x))+2160 a^2 b^2 \cos (4 (c+d x))+7200 a^2 b^2 \cos (6 (c+d x))-2700 a^2 b^2 \cos (8 (c+d x))+540 a^2 b^2-8640 a b^3 \sin (2 (c+d x))-2880 a b^3 \sin (4 (c+d x))+8000 a b^3 \sin (6 (c+d x))-2400 a b^3 \sin (8 (c+d x))-6720 b^4 \cos (2 (c+d x))+480 b^4 \cos (4 (c+d x))+1600 b^4 \cos (6 (c+d x))-600 b^4 \cos (8 (c+d x))+5240 b^4\right)}{30720 d (a \cos (c+d x)+b \sin (c+d x))^4}","-\frac{5 a^4 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a^4 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a^4 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a^4 \cot (c+d x) \csc (c+d x)}{16 d}-\frac{4 a^3 b \csc ^5(c+d x)}{5 d}-\frac{4 a^3 b \csc ^3(c+d x)}{3 d}-\frac{4 a^3 b \csc (c+d x)}{d}+\frac{4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{45 a^2 b^2 \sec (c+d x)}{4 d}-\frac{45 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac{3 a^2 b^2 \csc ^4(c+d x) \sec (c+d x)}{2 d}-\frac{15 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{4 d}-\frac{10 a b^3 \csc ^3(c+d x)}{3 d}-\frac{10 a b^3 \csc (c+d x)}{d}+\frac{10 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{d}+\frac{5 b^4 \sec ^3(c+d x)}{6 d}+\frac{5 b^4 \sec (c+d x)}{2 d}-\frac{5 b^4 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{b^4 \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}",1,"(-5*(a^4 + 36*a^2*b^2 + 8*b^4)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(16*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (2*(2*a^3*b + 5*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (5*(a^4 + 36*a^2*b^2 + 8*b^4)*Cos[c + d*x]^4*Log[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) + (2*(2*a^3*b + 5*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cot[c + d*x]*Csc[c + d*x]^5*(-2545*a^4 + 540*a^2*b^2 + 5240*b^4 - 2760*a^4*Cos[2*(c + d*x)] - 7200*a^2*b^2*Cos[2*(c + d*x)] - 6720*b^4*Cos[2*(c + d*x)] + 60*a^4*Cos[4*(c + d*x)] + 2160*a^2*b^2*Cos[4*(c + d*x)] + 480*b^4*Cos[4*(c + d*x)] + 200*a^4*Cos[6*(c + d*x)] + 7200*a^2*b^2*Cos[6*(c + d*x)] + 1600*b^4*Cos[6*(c + d*x)] - 75*a^4*Cos[8*(c + d*x)] - 2700*a^2*b^2*Cos[8*(c + d*x)] - 600*b^4*Cos[8*(c + d*x)] - 15744*a^3*b*Sin[2*(c + d*x)] - 8640*a*b^3*Sin[2*(c + d*x)] - 1152*a^3*b*Sin[4*(c + d*x)] - 2880*a*b^3*Sin[4*(c + d*x)] + 3200*a^3*b*Sin[6*(c + d*x)] + 8000*a*b^3*Sin[6*(c + d*x)] - 960*a^3*b*Sin[8*(c + d*x)] - 2400*a*b^3*Sin[8*(c + d*x)])*(a + b*Tan[c + d*x])^4)/(30720*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)","A",1
51,1,289,274,3.1863412,"\int \frac{\sin ^5(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sin[c + d*x]^5/(a + b*Tan[c + d*x]),x]","\frac{\sqrt{a^2+b^2} \left(-3 a^5 \cos (5 (c+d x))+330 a^4 b \sin (c+d x)-35 a^4 b \sin (3 (c+d x))+3 a^4 b \sin (5 (c+d x))-6 a^3 b^2 \cos (5 (c+d x))+120 a^2 b^3 \sin (c+d x)-50 a^2 b^3 \sin (3 (c+d x))+6 a^2 b^3 \sin (5 (c+d x))-30 a \left(5 a^4-4 a^2 b^2-b^4\right) \cos (c+d x)+5 a \left(5 a^4+6 a^2 b^2+b^4\right) \cos (3 (c+d x))-3 a b^4 \cos (5 (c+d x))+30 b^5 \sin (c+d x)-15 b^5 \sin (3 (c+d x))+3 b^5 \sin (5 (c+d x))\right)-480 a^5 b \tanh ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)-b}{\sqrt{a^2+b^2}}\right)}{240 d \left(a^2+b^2\right)^{7/2}}","\frac{b \sin ^5(c+d x)}{5 d \left(a^2+b^2\right)}+\frac{a^2 b \sin ^3(c+d x)}{3 d \left(a^2+b^2\right)^2}-\frac{a \cos ^5(c+d x)}{5 d \left(a^2+b^2\right)}+\frac{2 a \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)}-\frac{a b^2 \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)^2}-\frac{a \cos (c+d x)}{d \left(a^2+b^2\right)}+\frac{a b^2 \cos (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a^5 b \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)^{7/2}}+\frac{a^4 b \sin (c+d x)}{d \left(a^2+b^2\right)^3}+\frac{a^3 b^2 \cos (c+d x)}{d \left(a^2+b^2\right)^3}",1,"(-480*a^5*b*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sqrt[a^2 + b^2]*(-30*a*(5*a^4 - 4*a^2*b^2 - b^4)*Cos[c + d*x] + 5*a*(5*a^4 + 6*a^2*b^2 + b^4)*Cos[3*(c + d*x)] - 3*a^5*Cos[5*(c + d*x)] - 6*a^3*b^2*Cos[5*(c + d*x)] - 3*a*b^4*Cos[5*(c + d*x)] + 330*a^4*b*Sin[c + d*x] + 120*a^2*b^3*Sin[c + d*x] + 30*b^5*Sin[c + d*x] - 35*a^4*b*Sin[3*(c + d*x)] - 50*a^2*b^3*Sin[3*(c + d*x)] - 15*b^5*Sin[3*(c + d*x)] + 3*a^4*b*Sin[5*(c + d*x)] + 6*a^2*b^3*Sin[5*(c + d*x)] + 3*b^5*Sin[5*(c + d*x)]))/(240*(a^2 + b^2)^(7/2)*d)","A",1
52,1,249,158,2.8114097,"\int \frac{\sin ^4(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Tan[c + d*x]),x]","-\frac{8 a^4 \left(-2 b^2 \log (a+b \tan (c+d x))+\left(a \sqrt{-b^2}+b^2\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\left(b^2-a \sqrt{-b^2}\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)\right)-4 b^2 \left(a^2+b^2\right)^2 \cos ^4(c+d x)-4 a b \left(a^2+b^2\right)^2 \sin (c+d x) \cos ^3(c+d x)+a \left(5 a^4 b+6 a^2 b^3+b^5\right) \sin (2 (c+d x))+8 b^2 \left(2 a^4+3 a^2 b^2+b^4\right) \cos ^2(c+d x)+2 a b \left(5 a^4+6 a^2 b^2+b^4\right) \tan ^{-1}(\tan (c+d x))}{16 b d \left(a^2+b^2\right)^3}","\frac{\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left(a^2+b^2\right)}-\frac{\cos ^2(c+d x) \left(a \left(5 a^2+b^2\right) \tan (c+d x)+4 b \left(2 a^2+b^2\right)\right)}{8 d \left(a^2+b^2\right)^2}+\frac{a^4 b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^3}+\frac{a x \left(3 a^4-6 a^2 b^2-b^4\right)}{8 \left(a^2+b^2\right)^3}",1,"-1/16*(2*a*b*(5*a^4 + 6*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]] + 8*b^2*(2*a^4 + 3*a^2*b^2 + b^4)*Cos[c + d*x]^2 - 4*b^2*(a^2 + b^2)^2*Cos[c + d*x]^4 + 8*a^4*((b^2 + a*Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 2*b^2*Log[a + b*Tan[c + d*x]] + (b^2 - a*Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]]) - 4*a*b*(a^2 + b^2)^2*Cos[c + d*x]^3*Sin[c + d*x] + a*(5*a^4*b + 6*a^2*b^3 + b^5)*Sin[2*(c + d*x)])/(b*(a^2 + b^2)^3*d)","A",1
53,1,139,168,0.8910061,"\int \frac{\sin ^3(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sin[c + d*x]^3/(a + b*Tan[c + d*x]),x]","\frac{\sqrt{a^2+b^2} \left(\left(3 a b^2-9 a^3\right) \cos (c+d x)+a \left(a^2+b^2\right) \cos (3 (c+d x))-2 b \sin (c+d x) \left(\left(a^2+b^2\right) \cos (2 (c+d x))-7 a^2-b^2\right)\right)-24 a^3 b \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{b \sin ^3(c+d x)}{3 d \left(a^2+b^2\right)}+\frac{a^2 b \sin (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a \cos ^3(c+d x)}{3 d \left(a^2+b^2\right)}-\frac{a \cos (c+d x)}{d \left(a^2+b^2\right)}+\frac{a b^2 \cos (c+d x)}{d \left(a^2+b^2\right)^2}+\frac{a^3 b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{d \left(a^2+b^2\right)^{5/2}}",1,"(-24*a^3*b*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sqrt[a^2 + b^2]*((-9*a^3 + 3*a*b^2)*Cos[c + d*x] + a*(a^2 + b^2)*Cos[3*(c + d*x)] - 2*b*(-7*a^2 - 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
54,1,170,94,0.7711142,"\int \frac{\sin ^2(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Tan[c + d*x]),x]","-\frac{2 b^2 \left(a^2+b^2\right) \cos ^2(c+d x)+2 a b \left(a^2+b^2\right) \tan ^{-1}(\tan (c+d x))+a \left(b \left(a^2+b^2\right) \sin (2 (c+d x))+2 a \left(-2 b^2 \log (a+b \tan (c+d x))+\left(a \sqrt{-b^2}+b^2\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\left(b^2-a \sqrt{-b^2}\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)\right)\right)}{4 b d \left(a^2+b^2\right)^2}","-\frac{\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left(a^2+b^2\right)}+\frac{a^2 b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^2}+\frac{a x \left(a^2-b^2\right)}{2 \left(a^2+b^2\right)^2}",1,"-1/4*(2*a*b*(a^2 + b^2)*ArcTan[Tan[c + d*x]] + 2*b^2*(a^2 + b^2)*Cos[c + d*x]^2 + a*(2*a*((b^2 + a*Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 2*b^2*Log[a + b*Tan[c + d*x]] + (b^2 - a*Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]]) + b*(a^2 + b^2)*Sin[2*(c + d*x)]))/(b*(a^2 + b^2)^2*d)","A",1
55,1,79,90,0.3659606,"\int \frac{\sin (c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sin[c + d*x]/(a + b*Tan[c + d*x]),x]","\frac{\sqrt{a^2+b^2} (b \sin (c+d x)-a \cos (c+d x))-2 a b \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{b \sin (c+d x)}{d \left(a^2+b^2\right)}-\frac{a \cos (c+d x)}{d \left(a^2+b^2\right)}+\frac{a b \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*b*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + Sqrt[a^2 + b^2]*(-(a*Cos[c + d*x]) + b*Sin[c + d*x]))/((a^2 + b^2)^(3/2)*d)","A",1
56,1,75,66,0.1084989,"\int \frac{\csc (c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Csc[c + d*x]/(a + b*Tan[c + d*x]),x]","\frac{-\frac{2 b \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}}+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a d}","\frac{b \tanh ^{-1}\left(\frac{b \cos (c+d x)-a \sin (c+d x)}{\sqrt{a^2+b^2}}\right)}{a d \sqrt{a^2+b^2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a d}",1,"((-2*b*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] - Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]])/(a*d)","A",1
57,1,47,50,0.1335418,"\int \frac{\csc ^2(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Tan[c + d*x]),x]","\frac{b (\log (a \cos (c+d x)+b \sin (c+d x))-\log (\sin (c+d x)))-a \cot (c+d x)}{a^2 d}","-\frac{b \log (\tan (c+d x))}{a^2 d}+\frac{b \log (a+b \tan (c+d x))}{a^2 d}-\frac{\cot (c+d x)}{a d}",1,"(-(a*Cot[c + d*x]) + b*(-Log[Sin[c + d*x]] + Log[a*Cos[c + d*x] + b*Sin[c + d*x]]))/(a^2*d)","A",1
58,1,179,122,0.8053552,"\int \frac{\csc ^3(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Csc[c + d*x]^3/(a + b*Tan[c + d*x]),x]","\frac{-16 b \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^2 \left(-\csc ^2\left(\frac{1}{2} (c+d x)\right)\right)+a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)+4 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 a b \tan \left(\frac{1}{2} (c+d x)\right)+4 a b \cot \left(\frac{1}{2} (c+d x)\right)+8 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 a^3 d}","-\frac{b^2 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{b \csc (c+d x)}{a^2 d}+\frac{b \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)}{a^3 d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"(-16*b*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]] + 4*a*b*Cot[(c + d*x)/2] - a^2*Csc[(c + d*x)/2]^2 - 4*a^2*Log[Cos[(c + d*x)/2]] - 8*b^2*Log[Cos[(c + d*x)/2]] + 4*a^2*Log[Sin[(c + d*x)/2]] + 8*b^2*Log[Sin[(c + d*x)/2]] + a^2*Sec[(c + d*x)/2]^2 + 4*a*b*Tan[(c + d*x)/2])/(8*a^3*d)","A",1
59,1,95,108,0.493696,"\int \frac{\csc ^4(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Tan[c + d*x]),x]","\frac{-2 \cot (c+d x) \left(a^3 \csc ^2(c+d x)+2 a^3+3 a b^2\right)-6 b \left(a^2+b^2\right) (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))+3 a^2 b \csc ^2(c+d x)}{6 a^4 d}","\frac{b \cot ^2(c+d x)}{2 a^2 d}-\frac{b \left(a^2+b^2\right) \log (\tan (c+d x))}{a^4 d}+\frac{b \left(a^2+b^2\right) \log (a+b \tan (c+d x))}{a^4 d}-\frac{\left(a^2+b^2\right) \cot (c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a d}",1,"(3*a^2*b*Csc[c + d*x]^2 - 2*Cot[c + d*x]*(2*a^3 + 3*a*b^2 + a^3*Csc[c + d*x]^2) - 6*b*(a^2 + b^2)*(Log[Sin[c + d*x]] - Log[a*Cos[c + d*x] + b*Sin[c + d*x]]))/(6*a^4*d)","A",1
60,1,150,169,2.1018531,"\int \frac{\csc ^6(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Csc[c + d*x]^6/(a + b*Tan[c + d*x]),x]","\frac{15 b \left(a^4 \csc ^4(c+d x)+2 a^2 \left(a^2+b^2\right) \csc ^2(c+d x)-4 \left(a^2+b^2\right)^2 (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))\right)-4 \cot (c+d x) \left(3 a^5 \csc ^4(c+d x)+8 a^5+25 a^3 b^2+a^3 \left(4 a^2+5 b^2\right) \csc ^2(c+d x)+15 a b^4\right)}{60 a^6 d}","\frac{b \cot ^4(c+d x)}{4 a^2 d}-\frac{b \left(a^2+b^2\right)^2 \log (\tan (c+d x))}{a^6 d}+\frac{b \left(a^2+b^2\right)^2 \log (a+b \tan (c+d x))}{a^6 d}-\frac{\left(a^2+b^2\right)^2 \cot (c+d x)}{a^5 d}+\frac{b \left(2 a^2+b^2\right) \cot ^2(c+d x)}{2 a^4 d}-\frac{\left(2 a^2+b^2\right) \cot ^3(c+d x)}{3 a^3 d}-\frac{\cot ^5(c+d x)}{5 a d}",1,"(-4*Cot[c + d*x]*(8*a^5 + 25*a^3*b^2 + 15*a*b^4 + a^3*(4*a^2 + 5*b^2)*Csc[c + d*x]^2 + 3*a^5*Csc[c + d*x]^4) + 15*b*(2*a^2*(a^2 + b^2)*Csc[c + d*x]^2 + a^4*Csc[c + d*x]^4 - 4*(a^2 + b^2)^2*(Log[Sin[c + d*x]] - Log[a*Cos[c + d*x] + b*Sin[c + d*x]])))/(60*a^6*d)","A",1
61,1,603,297,6.5530426,"\int \frac{\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]","\frac{b \left(-\frac{a \cos ^6(c+d x)}{3 \left(a^2+b^2\right)^2}+\frac{a \left(3 a^2+b^2\right) \cos ^4(c+d x)}{2 \left(a^2+b^2\right)^3}-\frac{\left(a^2-b^2\right) \sin (c+d x) \cos ^5(c+d x)}{6 b \left(a^2+b^2\right)^2}-\frac{5 \left(a^2-b^2\right) \left(3 b^2 \left(\frac{\tan ^{-1}(\tan (c+d x))}{b^3}+\frac{\sin (c+d x) \cos (c+d x)}{b^3}\right)+\frac{2 \sin (c+d x) \cos ^3(c+d x)}{b}\right)}{48 \left(a^2+b^2\right)^2}-\frac{a^6}{\left(a^2+b^2\right)^4 (a+b \tan (c+d x))}-\frac{3 a^5 \cos ^2(c+d x)}{\left(a^2+b^2\right)^4}+\frac{2 a^5 \left(a^2-3 b^2\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^5}+\frac{\left(3 a^4-3 a^2 b^2-2 b^4\right) \sin (c+d x) \cos ^3(c+d x)}{4 b \left(a^2+b^2\right)^3}+\frac{3 \left(3 a^4-3 a^2 b^2-2 b^4\right) \left(\frac{\tan ^{-1}(\tan (c+d x))}{b}+\frac{\sin (c+d x) \cos (c+d x)}{b}\right)}{8 \left(a^2+b^2\right)^3}-\frac{\left(3 a^6-6 a^4 b^2-4 a^2 b^4-b^6\right) \tan ^{-1}(\tan (c+d x))}{2 b \left(a^2+b^2\right)^4}-\frac{\left(3 a^6-6 a^4 b^2-4 a^2 b^4-b^6\right) \sin (c+d x) \cos (c+d x)}{2 b \left(a^2+b^2\right)^4}-\frac{a^5 \left(-\frac{a^3-7 a b^2}{\sqrt{-b^2}}+2 a^2-6 b^2\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^5}-\frac{a^5 \left(\frac{a^3-7 a b^2}{\sqrt{-b^2}}+2 a^2-6 b^2\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^5}\right)}{d}","-\frac{\cos ^6(c+d x) \left(\left(a^2-b^2\right) \tan (c+d x)+2 a b\right)}{6 d \left(a^2+b^2\right)^2}-\frac{a^6 b}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}+\frac{2 a^5 b \left(a^2-3 b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}+\frac{\cos ^4(c+d x) \left(12 a b \left(3 a^2+b^2\right)+\left(13 a^4-18 a^2 b^2-7 b^4\right) \tan (c+d x)\right)}{24 d \left(a^2+b^2\right)^3}+\frac{x \left(5 a^8-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+b^8\right)}{16 \left(a^2+b^2\right)^5}-\frac{\cos ^2(c+d x) \left(48 a^5 b+\left(11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right) \tan (c+d x)\right)}{16 d \left(a^2+b^2\right)^4}",1,"(b*(-1/2*((3*a^6 - 6*a^4*b^2 - 4*a^2*b^4 - b^6)*ArcTan[Tan[c + d*x]])/(b*(a^2 + b^2)^4) - (3*a^5*Cos[c + d*x]^2)/(a^2 + b^2)^4 + (a*(3*a^2 + b^2)*Cos[c + d*x]^4)/(2*(a^2 + b^2)^3) - (a*Cos[c + d*x]^6)/(3*(a^2 + b^2)^2) - (a^5*(2*a^2 - 6*b^2 - (a^3 - 7*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(2*(a^2 + b^2)^5) + (2*a^5*(a^2 - 3*b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^5 - (a^5*(2*a^2 - 6*b^2 + (a^3 - 7*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^5) - ((3*a^6 - 6*a^4*b^2 - 4*a^2*b^4 - b^6)*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 + b^2)^4) + ((3*a^4 - 3*a^2*b^2 - 2*b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*(a^2 + b^2)^3) - ((a^2 - b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(6*b*(a^2 + b^2)^2) + (3*(3*a^4 - 3*a^2*b^2 - 2*b^4)*(ArcTan[Tan[c + d*x]]/b + (Cos[c + d*x]*Sin[c + d*x])/b))/(8*(a^2 + b^2)^3) - (5*(a^2 - b^2)*((2*Cos[c + d*x]^3*Sin[c + d*x])/b + 3*b^2*(ArcTan[Tan[c + d*x]]/b^3 + (Cos[c + d*x]*Sin[c + d*x])/b^3)))/(48*(a^2 + b^2)^2) - a^6/((a^2 + b^2)^4*(a + b*Tan[c + d*x]))))/d","B",1
62,1,373,217,3.9854376,"\int \frac{\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]","\frac{b \left(4 a \left(a^2+b^2\right)^2 \cos ^4(c+d x)+\frac{3 \left(a^2-b^2\right) \left(a^2+b^2\right)^2 \left(\sin (2 (c+d x))+2 \tan ^{-1}(\tan (c+d x))\right)}{2 b}+\frac{2 \left(a^2-b^2\right) \left(a^2+b^2\right)^2 \sin (c+d x) \cos ^3(c+d x)}{b}-\frac{8 a^4 \left(a^2+b^2\right)}{a+b \tan (c+d x)}+\frac{2 \left(a^2+b^2\right) \left(-2 a^4+3 a^2 b^2+b^4\right) \sin (2 (c+d x))}{b}+\frac{4 \left(a^2+b^2\right) \left(-2 a^4+3 a^2 b^2+b^4\right) \tan ^{-1}(\tan (c+d x))}{b}-16 a^3 \left(a^2+b^2\right) \cos ^2(c+d x)-4 a^3 \left(\frac{5 a b^2-a^3}{\sqrt{-b^2}}+2 a^2-4 b^2\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+16 a^3 \left(a^2-2 b^2\right) \log (a+b \tan (c+d x))-4 a^3 \left(\frac{a^3-5 a b^2}{\sqrt{-b^2}}+2 a^2-4 b^2\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)\right)}{8 d \left(a^2+b^2\right)^4}","\frac{\cos ^4(c+d x) \left(\left(a^2-b^2\right) \tan (c+d x)+2 a b\right)}{4 d \left(a^2+b^2\right)^2}-\frac{a^4 b}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}+\frac{2 a^3 b \left(a^2-2 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(3 a^6-33 a^4 b^2+13 a^2 b^4+b^6\right)}{8 \left(a^2+b^2\right)^4}-\frac{\cos ^2(c+d x) \left(16 a^3 b+\left(5 a^4-12 a^2 b^2-b^4\right) \tan (c+d x)\right)}{8 d \left(a^2+b^2\right)^3}",1,"(b*((4*(a^2 + b^2)*(-2*a^4 + 3*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]])/b - 16*a^3*(a^2 + b^2)*Cos[c + d*x]^2 + 4*a*(a^2 + b^2)^2*Cos[c + d*x]^4 - 4*a^3*(2*a^2 - 4*b^2 + (-a^3 + 5*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] + 16*a^3*(a^2 - 2*b^2)*Log[a + b*Tan[c + d*x]] - 4*a^3*(2*a^2 - 4*b^2 + (a^3 - 5*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + (2*(a^2 - b^2)*(a^2 + b^2)^2*Cos[c + d*x]^3*Sin[c + d*x])/b + (2*(a^2 + b^2)*(-2*a^4 + 3*a^2*b^2 + b^4)*Sin[2*(c + d*x)])/b + (3*(a^2 - b^2)*(a^2 + b^2)^2*(2*ArcTan[Tan[c + d*x]] + Sin[2*(c + d*x)]))/(2*b) - (8*a^4*(a^2 + b^2))/(a + b*Tan[c + d*x])))/(8*(a^2 + b^2)^4*d)","A",1
63,1,246,148,3.3873228,"\int \frac{\sin ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Tan[c + d*x])^2,x]","-\frac{b \left(\frac{(a-b) (a+b) \left(a^2+b^2\right) \sin (2 (c+d x))}{2 b}+2 a \left(a^2+b^2\right) \cos ^2(c+d x)+\frac{\left(a^2-b^2\right) \left(a^2+b^2\right) \tan ^{-1}(\tan (c+d x))}{b}+\frac{2 a^2 \left(a^2+b^2\right)}{a+b \tan (c+d x)}+a \left(\frac{3 a b^2-a^3}{\sqrt{-b^2}}+2 a^2-2 b^2\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+a \left(\frac{a^3-3 a b^2}{\sqrt{-b^2}}+2 a^2-2 b^2\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)-4 a (a-b) (a+b) \log (a+b \tan (c+d x))\right)}{2 d \left(a^2+b^2\right)^3}","-\frac{a^2 b}{d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left(\left(a^2-b^2\right) \tan (c+d x)+2 a b\right)}{2 d \left(a^2+b^2\right)^2}+\frac{2 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)^3}+\frac{x \left(a^4-6 a^2 b^2+b^4\right)}{2 \left(a^2+b^2\right)^3}",1,"-1/2*(b*(((a^2 - b^2)*(a^2 + b^2)*ArcTan[Tan[c + d*x]])/b + 2*a*(a^2 + b^2)*Cos[c + d*x]^2 + a*(2*a^2 - 2*b^2 + (-a^3 + 3*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 4*a*(a - b)*(a + b)*Log[a + b*Tan[c + d*x]] + a*(2*a^2 - 2*b^2 + (a^3 - 3*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + ((a - b)*(a + b)*(a^2 + b^2)*Sin[2*(c + d*x)])/(2*b) + (2*a^2*(a^2 + b^2))/(a + b*Tan[c + d*x])))/((a^2 + b^2)^3*d)","A",1
64,1,109,72,0.4009174,"\int \frac{\csc ^2(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Tan[c + d*x])^2,x]","\frac{-a^2 \cot ^2(c+d x)+b^2 (2 \log (a \cos (c+d x)+b \sin (c+d x))-2 \log (\sin (c+d x))+1)-a b \cot (c+d x) (-2 \log (a \cos (c+d x)+b \sin (c+d x))+2 \log (\sin (c+d x))+1)}{a^3 d (a \cot (c+d x)+b)}","-\frac{2 b \log (\tan (c+d x))}{a^3 d}+\frac{2 b \log (a+b \tan (c+d x))}{a^3 d}-\frac{b}{a^2 d (a+b \tan (c+d x))}-\frac{\cot (c+d x)}{a^2 d}",1,"(-(a^2*Cot[c + d*x]^2) - a*b*Cot[c + d*x]*(1 + 2*Log[Sin[c + d*x]] - 2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]) + b^2*(1 - 2*Log[Sin[c + d*x]] + 2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]))/(a^3*d*(b + a*Cot[c + d*x]))","A",1
65,1,244,140,2.6619373,"\int \frac{\csc ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]","\frac{3 b^2 \left(-2 \left(a^2+2 b^2\right) \log (\sin (c+d x))+2 a^2 \log (a \cos (c+d x)+b \sin (c+d x))+a^2 \csc ^2(c+d x)+a^2+4 b^2 \log (a \cos (c+d x)+b \sin (c+d x))+b^2\right)+a b \cot (c+d x) \left(-6 \left(a^2+2 b^2\right) \log (\sin (c+d x))+6 a^2 \log (a \cos (c+d x)+b \sin (c+d x))+2 a^2 \csc ^2(c+d x)-2 a^2+12 b^2 \log (a \cos (c+d x)+b \sin (c+d x))-9 b^2\right)-\cot ^2(c+d x) \left(a^4 \csc ^2(c+d x)+2 a^4+9 a^2 b^2\right)}{3 a^5 d (a \cot (c+d x)+b)}","\frac{b \cot ^2(c+d x)}{a^3 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}-\frac{2 b \left(a^2+2 b^2\right) \log (\tan (c+d x))}{a^5 d}+\frac{2 b \left(a^2+2 b^2\right) \log (a+b \tan (c+d x))}{a^5 d}-\frac{b \left(a^2+b^2\right)}{a^4 d (a+b \tan (c+d x))}-\frac{\left(a^2+3 b^2\right) \cot (c+d x)}{a^4 d}",1,"(-(Cot[c + d*x]^2*(2*a^4 + 9*a^2*b^2 + a^4*Csc[c + d*x]^2)) + 3*b^2*(a^2 + b^2 + a^2*Csc[c + d*x]^2 - 2*(a^2 + 2*b^2)*Log[Sin[c + d*x]] + 2*a^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]] + 4*b^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]) + a*b*Cot[c + d*x]*(-2*a^2 - 9*b^2 + 2*a^2*Csc[c + d*x]^2 - 6*(a^2 + 2*b^2)*Log[Sin[c + d*x]] + 6*a^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]] + 12*b^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]))/(3*a^5*d*(b + a*Cot[c + d*x]))","A",1
66,1,589,219,6.2486494,"\int \frac{\csc ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^2,x]","\frac{b \csc ^4(c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{2 a^3 d (a+b \tan (c+d x))^2}-\frac{\csc ^5(c+d x) \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{5 a^2 d (a+b \tan (c+d x))^2}+\frac{b \left(a^2+2 b^2\right) \csc ^2(c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{a^5 d (a+b \tan (c+d x))^2}+\frac{\csc ^3(c+d x) \sec ^2(c+d x) \left(-4 a^2 \cos (c+d x)-15 b^2 \cos (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{15 a^4 d (a+b \tan (c+d x))^2}-\frac{2 \left(a^4 b+4 a^2 b^3+3 b^5\right) \sec ^2(c+d x) \log (\sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2}{a^7 d (a+b \tan (c+d x))^2}+\frac{2 \left(a^4 b+4 a^2 b^3+3 b^5\right) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \log (a \cos (c+d x)+b \sin (c+d x))}{a^7 d (a+b \tan (c+d x))^2}+\frac{\sec ^2(c+d x) \left(a^4 b^2 \sin (c+d x)+2 a^2 b^4 \sin (c+d x)+b^6 \sin (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))}{a^7 d (a+b \tan (c+d x))^2}+\frac{\csc (c+d x) \sec ^2(c+d x) \left(-8 a^4 \cos (c+d x)-75 a^2 b^2 \cos (c+d x)-75 b^4 \cos (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{15 a^6 d (a+b \tan (c+d x))^2}","\frac{b \cot ^4(c+d x)}{2 a^3 d}-\frac{\cot ^5(c+d x)}{5 a^2 d}-\frac{2 b \left(a^2+b^2\right) \left(a^2+3 b^2\right) \log (\tan (c+d x))}{a^7 d}+\frac{2 b \left(a^2+b^2\right) \left(a^2+3 b^2\right) \log (a+b \tan (c+d x))}{a^7 d}-\frac{b \left(a^2+b^2\right)^2}{a^6 d (a+b \tan (c+d x))}-\frac{\left(a^2+b^2\right) \left(a^2+5 b^2\right) \cot (c+d x)}{a^6 d}+\frac{2 b \left(a^2+b^2\right) \cot ^2(c+d x)}{a^5 d}-\frac{\left(2 a^2+3 b^2\right) \cot ^3(c+d x)}{3 a^4 d}",1,"-1/5*(Csc[c + d*x]^5*Sec[c + d*x]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(a^2*d*(a + b*Tan[c + d*x])^2) + ((-8*a^4*Cos[c + d*x] - 75*a^2*b^2*Cos[c + d*x] - 75*b^4*Cos[c + d*x])*Csc[c + d*x]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(15*a^6*d*(a + b*Tan[c + d*x])^2) + (b*(a^2 + 2*b^2)*Csc[c + d*x]^2*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(a^5*d*(a + b*Tan[c + d*x])^2) + ((-4*a^2*Cos[c + d*x] - 15*b^2*Cos[c + d*x])*Csc[c + d*x]^3*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(15*a^4*d*(a + b*Tan[c + d*x])^2) + (b*Csc[c + d*x]^4*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*a^3*d*(a + b*Tan[c + d*x])^2) - (2*(a^4*b + 4*a^2*b^3 + 3*b^5)*Log[Sin[c + d*x]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(a^7*d*(a + b*Tan[c + d*x])^2) + (2*(a^4*b + 4*a^2*b^3 + 3*b^5)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(a^7*d*(a + b*Tan[c + d*x])^2) + (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])*(a^4*b^2*Sin[c + d*x] + 2*a^2*b^4*Sin[c + d*x] + b^6*Sin[c + d*x]))/(a^7*d*(a + b*Tan[c + d*x])^2)","B",1
67,1,683,382,6.6884173,"\int \frac{\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^6/(a + b*Tan[c + d*x])^3,x]","\frac{b \left(-\frac{\left(3 a^2-b^2\right) \cos ^6(c+d x)}{6 \left(a^2+b^2\right)^3}-\frac{a \left(a^2-3 b^2\right) \sin (c+d x) \cos ^5(c+d x)}{6 b \left(a^2+b^2\right)^3}-\frac{5 a \left(a^2-3 b^2\right) \left(3 b^2 \left(\frac{\tan ^{-1}(\tan (c+d x))}{b^3}+\frac{\sin (c+d x) \cos (c+d x)}{b^3}\right)+\frac{2 \sin (c+d x) \cos ^3(c+d x)}{b}\right)}{48 \left(a^2+b^2\right)^3}-\frac{a^6}{2 \left(a^2+b^2\right)^4 (a+b \tan (c+d x))^2}-\frac{3 a^5 \left(a^2-7 b^2\right) \tan ^{-1}(\tan (c+d x))}{2 b \left(a^2+b^2\right)^5}-\frac{2 a^5 \left(a^2-3 b^2\right)}{\left(a^2+b^2\right)^5 (a+b \tan (c+d x))}-\frac{3 a^5 \left(a^2-7 b^2\right) \sin (c+d x) \cos (c+d x)}{2 b \left(a^2+b^2\right)^5}-\frac{3 a^4 \left(3 a^2-5 b^2\right) \cos ^2(c+d x)}{2 \left(a^2+b^2\right)^5}+\frac{\left(9 a^4-4 a^2 b^2-b^4\right) \cos ^4(c+d x)}{4 \left(a^2+b^2\right)^4}+\frac{a^4 \left(3 a^4-22 a^2 b^2+15 b^4\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^6}+\frac{3 a \left(a^4-4 a^2 b^2-b^4\right) \sin (c+d x) \cos ^3(c+d x)}{4 b \left(a^2+b^2\right)^4}+\frac{9 a \left(a^4-4 a^2 b^2-b^4\right) \left(\frac{\tan ^{-1}(\tan (c+d x))}{b}+\frac{\sin (c+d x) \cos (c+d x)}{b}\right)}{8 \left(a^2+b^2\right)^4}-\frac{a^4 \left(3 a^4-22 a^2 b^2-\frac{a^5-18 a^3 b^2+21 a b^4}{\sqrt{-b^2}}+15 b^4\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^6}-\frac{a^4 \left(3 a^4-22 a^2 b^2+\frac{a^5-18 a^3 b^2+21 a b^4}{\sqrt{-b^2}}+15 b^4\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^6}\right)}{d}","-\frac{\cos ^6(c+d x) \left(a \left(a^2-3 b^2\right) \tan (c+d x)+b \left(3 a^2-b^2\right)\right)}{6 d \left(a^2+b^2\right)^3}-\frac{a^6 b}{2 d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))^2}-\frac{2 a^5 b \left(a^2-3 b^2\right)}{d \left(a^2+b^2\right)^5 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left(a \left(13 a^4-62 a^2 b^2-3 b^4\right) \tan (c+d x)+6 b \left(9 a^4-4 a^2 b^2-b^4\right)\right)}{24 d \left(a^2+b^2\right)^4}+\frac{a^4 b \left(3 a^4-22 a^2 b^2+15 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^6}+\frac{a x \left(5 a^8-180 a^6 b^2+390 a^4 b^4-68 a^2 b^6-3 b^8\right)}{16 \left(a^2+b^2\right)^6}-\frac{a \cos ^2(c+d x) \left(24 a^3 b \left(3 a^2-5 b^2\right)+\left(11 a^6-119 a^4 b^2+65 a^2 b^4+3 b^6\right) \tan (c+d x)\right)}{16 d \left(a^2+b^2\right)^5}",1,"(b*((-3*a^5*(a^2 - 7*b^2)*ArcTan[Tan[c + d*x]])/(2*b*(a^2 + b^2)^5) - (3*a^4*(3*a^2 - 5*b^2)*Cos[c + d*x]^2)/(2*(a^2 + b^2)^5) + ((9*a^4 - 4*a^2*b^2 - b^4)*Cos[c + d*x]^4)/(4*(a^2 + b^2)^4) - ((3*a^2 - b^2)*Cos[c + d*x]^6)/(6*(a^2 + b^2)^3) - (a^4*(3*a^4 - 22*a^2*b^2 + 15*b^4 - (a^5 - 18*a^3*b^2 + 21*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(2*(a^2 + b^2)^6) + (a^4*(3*a^4 - 22*a^2*b^2 + 15*b^4)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^6 - (a^4*(3*a^4 - 22*a^2*b^2 + 15*b^4 + (a^5 - 18*a^3*b^2 + 21*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^6) - (3*a^5*(a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2*b*(a^2 + b^2)^5) + (3*a*(a^4 - 4*a^2*b^2 - b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*(a^2 + b^2)^4) - (a*(a^2 - 3*b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(6*b*(a^2 + b^2)^3) + (9*a*(a^4 - 4*a^2*b^2 - b^4)*(ArcTan[Tan[c + d*x]]/b + (Cos[c + d*x]*Sin[c + d*x])/b))/(8*(a^2 + b^2)^4) - (5*a*(a^2 - 3*b^2)*((2*Cos[c + d*x]^3*Sin[c + d*x])/b + 3*b^2*(ArcTan[Tan[c + d*x]]/b^3 + (Cos[c + d*x]*Sin[c + d*x])/b^3)))/(48*(a^2 + b^2)^3) - a^6/(2*(a^2 + b^2)^4*(a + b*Tan[c + d*x])^2) - (2*a^5*(a^2 - 3*b^2))/((a^2 + b^2)^5*(a + b*Tan[c + d*x]))))/d","A",1
68,1,501,285,6.4929136,"\int \frac{\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^3,x]","\frac{b \left(\frac{\left(3 a^2-b^2\right) \cos ^4(c+d x)}{4 \left(a^2+b^2\right)^3}-\frac{3 a^2 (a-b) (a+b) \cos ^2(c+d x)}{\left(a^2+b^2\right)^4}+\frac{a \left(a^2-3 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{4 b \left(a^2+b^2\right)^3}+\frac{3 a \left(a^2-3 b^2\right) \left(\frac{\tan ^{-1}(\tan (c+d x))}{b}+\frac{\sin (c+d x) \cos (c+d x)}{b}\right)}{8 \left(a^2+b^2\right)^3}-\frac{a^4}{2 \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}+\frac{3 a^2 \left(a^4-5 a^2 b^2+2 b^4\right) \log (a+b \tan (c+d x))}{\left(a^2+b^2\right)^5}-\frac{a^3 \left(a^2-5 b^2\right) \tan ^{-1}(\tan (c+d x))}{b \left(a^2+b^2\right)^4}-\frac{2 a^3 \left(a^2-2 b^2\right)}{\left(a^2+b^2\right)^4 (a+b \tan (c+d x))}-\frac{a^3 \left(a^2-5 b^2\right) \sin (c+d x) \cos (c+d x)}{b \left(a^2+b^2\right)^4}-\frac{a^2 \left(3 a^4-15 a^2 b^2-\frac{a^5-13 a^3 b^2+10 a b^4}{\sqrt{-b^2}}+6 b^4\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^5}-\frac{a^2 \left(3 a^4-15 a^2 b^2+\frac{a^5-13 a^3 b^2+10 a b^4}{\sqrt{-b^2}}+6 b^4\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)}{2 \left(a^2+b^2\right)^5}\right)}{d}","\frac{\cos ^4(c+d x) \left(a \left(a^2-3 b^2\right) \tan (c+d x)+b \left(3 a^2-b^2\right)\right)}{4 d \left(a^2+b^2\right)^3}-\frac{a^4 b}{2 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}-\frac{a \cos ^2(c+d x) \left(24 a b \left(a^2-b^2\right)+\left(5 a^4-34 a^2 b^2+9 b^4\right) \tan (c+d x)\right)}{8 d \left(a^2+b^2\right)^4}+\frac{3 a^2 b \left(a^4-5 a^2 b^2+2 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}-\frac{2 a^3 b \left(a^2-2 b^2\right)}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}+\frac{3 a x \left(a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right)}{8 \left(a^2+b^2\right)^5}",1,"(b*(-((a^3*(a^2 - 5*b^2)*ArcTan[Tan[c + d*x]])/(b*(a^2 + b^2)^4)) - (3*a^2*(a - b)*(a + b)*Cos[c + d*x]^2)/(a^2 + b^2)^4 + ((3*a^2 - b^2)*Cos[c + d*x]^4)/(4*(a^2 + b^2)^3) - (a^2*(3*a^4 - 15*a^2*b^2 + 6*b^4 - (a^5 - 13*a^3*b^2 + 10*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(2*(a^2 + b^2)^5) + (3*a^2*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^5 - (a^2*(3*a^4 - 15*a^2*b^2 + 6*b^4 + (a^5 - 13*a^3*b^2 + 10*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^5) - (a^3*(a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x])/(b*(a^2 + b^2)^4) + (a*(a^2 - 3*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*(a^2 + b^2)^3) + (3*a*(a^2 - 3*b^2)*(ArcTan[Tan[c + d*x]]/b + (Cos[c + d*x]*Sin[c + d*x])/b))/(8*(a^2 + b^2)^3) - a^4/(2*(a^2 + b^2)^3*(a + b*Tan[c + d*x])^2) - (2*a^3*(a^2 - 2*b^2))/((a^2 + b^2)^4*(a + b*Tan[c + d*x]))))/d","A",1
69,1,316,206,4.0046086,"\int \frac{\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]","-\frac{b \left(\frac{4 \left(a^5-a b^4\right)}{a+b \tan (c+d x)}+\frac{a \left(a^2-3 b^2\right) \left(a^2+b^2\right) \sin (2 (c+d x))}{2 b}+\left(3 a^2-b^2\right) \left(a^2+b^2\right) \cos ^2(c+d x)+\frac{a \left(a^2-3 b^2\right) \left(a^2+b^2\right) \tan ^{-1}(\tan (c+d x))}{b}+\frac{a^2 \left(a^2+b^2\right)^2}{(a+b \tan (c+d x))^2}-2 \left(3 a^4-8 a^2 b^2+b^4\right) \log (a+b \tan (c+d x))+\left(3 a^4-8 a^2 b^2-\frac{a^5-8 a^3 b^2+3 a b^4}{\sqrt{-b^2}}+b^4\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+\left(3 a^4-8 a^2 b^2+\frac{a^5-8 a^3 b^2+3 a b^4}{\sqrt{-b^2}}+b^4\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)\right)}{2 d \left(a^2+b^2\right)^4}","-\frac{a^2 b}{2 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^2}-\frac{2 a b \left(a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left(a \left(a^2-3 b^2\right) \tan (c+d x)+b \left(3 a^2-b^2\right)\right)}{2 d \left(a^2+b^2\right)^3}+\frac{b \left(3 a^4-8 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^4}+\frac{a x \left(a^4-14 a^2 b^2+9 b^4\right)}{2 \left(a^2+b^2\right)^4}",1,"-1/2*(b*((a*(a^2 - 3*b^2)*(a^2 + b^2)*ArcTan[Tan[c + d*x]])/b + (3*a^2 - b^2)*(a^2 + b^2)*Cos[c + d*x]^2 + (3*a^4 - 8*a^2*b^2 + b^4 - (a^5 - 8*a^3*b^2 + 3*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 2*(3*a^4 - 8*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]] + (3*a^4 - 8*a^2*b^2 + b^4 + (a^5 - 8*a^3*b^2 + 3*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + (a*(a^2 - 3*b^2)*(a^2 + b^2)*Sin[2*(c + d*x)])/(2*b) + (a^2*(a^2 + b^2)^2)/(a + b*Tan[c + d*x])^2 + (4*(a^5 - a*b^4))/(a + b*Tan[c + d*x])))/((a^2 + b^2)^4*d)","A",1
70,1,241,95,2.6896236,"\int \frac{\csc ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]","\frac{b \left(a^2 \left(-b^2\right) \sec ^2(c+d x)-2 a^2 \left(a^2+b^2\right) (-3 \log (a \cos (c+d x)+b \sin (c+d x))+3 \log (\sin (c+d x))+2)-2 b^2 \tan ^2(c+d x) \left(3 \left(a^2+b^2\right) \log (\sin (c+d x))-3 \left(a^2+b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))-3 a^2-2 b^2\right)+2 a b \tan (c+d x) \left(-6 \left(a^2+b^2\right) \log (\sin (c+d x))+6 \left(a^2+b^2\right) \log (a \cos (c+d x)+b \sin (c+d x))+2 a^2+b^2\right)\right)-2 a^3 \left(a^2+b^2\right) \cot (c+d x)}{2 a^4 d \left(a^2+b^2\right) (a+b \tan (c+d x))^2}","-\frac{3 b \log (\tan (c+d x))}{a^4 d}+\frac{3 b \log (a+b \tan (c+d x))}{a^4 d}-\frac{2 b}{a^3 d (a+b \tan (c+d x))}-\frac{\cot (c+d x)}{a^3 d}-\frac{b}{2 a^2 d (a+b \tan (c+d x))^2}",1,"(-2*a^3*(a^2 + b^2)*Cot[c + d*x] + b*(-2*a^2*(a^2 + b^2)*(2 + 3*Log[Sin[c + d*x]] - 3*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]) - a^2*b^2*Sec[c + d*x]^2 + 2*a*b*(2*a^2 + b^2 - 6*(a^2 + b^2)*Log[Sin[c + d*x]] + 6*(a^2 + b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])*Tan[c + d*x] - 2*b^2*(-3*a^2 - 2*b^2 + 3*(a^2 + b^2)*Log[Sin[c + d*x]] - 3*(a^2 + b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])*Tan[c + d*x]^2))/(2*a^4*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2)","B",1
71,1,456,178,3.4060168,"\int \frac{\csc ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Tan[c + d*x])^3,x]","-\frac{b^3 \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{2 a^4 d (a+b \tan (c+d x))^3}+\frac{3 b \csc ^2(c+d x) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{2 a^4 d (a+b \tan (c+d x))^3}-\frac{\csc ^3(c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3}{3 a^3 d (a+b \tan (c+d x))^3}+\frac{\left(-3 a^2 b-10 b^3\right) \sec ^3(c+d x) \log (\sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3}{a^6 d (a+b \tan (c+d x))^3}+\frac{\left(3 a^2 b+10 b^3\right) \sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \log (a \cos (c+d x)+b \sin (c+d x))}{a^6 d (a+b \tan (c+d x))^3}+\frac{\sec ^3(c+d x) \left(3 a^2 b^2 \sin (c+d x)+4 b^4 \sin (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^2}{a^6 d (a+b \tan (c+d x))^3}-\frac{2 \csc (c+d x) \sec ^3(c+d x) \left(a^2 \cos (c+d x)+9 b^2 \cos (c+d x)\right) (a \cos (c+d x)+b \sin (c+d x))^3}{3 a^5 d (a+b \tan (c+d x))^3}","\frac{3 b \cot ^2(c+d x)}{2 a^4 d}-\frac{\cot ^3(c+d x)}{3 a^3 d}-\frac{b \left(3 a^2+10 b^2\right) \log (\tan (c+d x))}{a^6 d}+\frac{b \left(3 a^2+10 b^2\right) \log (a+b \tan (c+d x))}{a^6 d}-\frac{2 b \left(a^2+2 b^2\right)}{a^5 d (a+b \tan (c+d x))}-\frac{\left(a^2+6 b^2\right) \cot (c+d x)}{a^5 d}-\frac{b \left(a^2+b^2\right)}{2 a^4 d (a+b \tan (c+d x))^2}",1,"-1/2*(b^3*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(a^4*d*(a + b*Tan[c + d*x])^3) - (Csc[c + d*x]^3*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(3*a^3*d*(a + b*Tan[c + d*x])^3) - (2*(a^2*Cos[c + d*x] + 9*b^2*Cos[c + d*x])*Csc[c + d*x]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(3*a^5*d*(a + b*Tan[c + d*x])^3) + (3*b*Csc[c + d*x]^2*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(2*a^4*d*(a + b*Tan[c + d*x])^3) + ((-3*a^2*b - 10*b^3)*Log[Sin[c + d*x]]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(a^6*d*(a + b*Tan[c + d*x])^3) + ((3*a^2*b + 10*b^3)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]*Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^3)/(a^6*d*(a + b*Tan[c + d*x])^3) + (Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^2*(3*a^2*b^2*Sin[c + d*x] + 4*b^4*Sin[c + d*x]))/(a^6*d*(a + b*Tan[c + d*x])^3)","B",1
72,1,494,265,4.755257,"\int \frac{\csc ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^3,x]","-\frac{\csc ^5(c+d x) \left(960 b \left(3 a^4+20 a^2 b^2+21 b^4\right) \sin ^5(c+d x) (a+b \tan (c+d x))^2 (\log (\sin (c+d x))-\log (a \cos (c+d x)+b \sin (c+d x)))+5 \sec (c+d x) \left(40 a^7-27 a^5 b^2-42 a^3 b^4-3 b \left(8 a^6+89 a^4 b^2+345 a^2 b^4+210 b^6\right) \tan (c+d x)+135 a b^6\right)+\sec ^2(c+d x) \left(8 a^7 \cos (7 (c+d x))-126 a^6 b \sin (3 (c+d x))+10 a^6 b \sin (5 (c+d x))+16 a^6 b \sin (7 (c+d x))+187 a^5 b^2 \cos (7 (c+d x))+1665 a^4 b^3 \sin (3 (c+d x))-1215 a^4 b^3 \sin (5 (c+d x))+345 a^4 b^3 \sin (7 (c+d x))+210 a^3 b^4 \cos (7 (c+d x))+4635 a^2 b^5 \sin (3 (c+d x))-2565 a^2 b^5 \sin (5 (c+d x))+585 a^2 b^5 \sin (7 (c+d x))+\left(8 a^7+567 a^5 b^2+630 a^3 b^4-1215 a b^6\right) \cos (3 (c+d x))-\left(24 a^7+619 a^5 b^2+630 a^3 b^4-675 a b^6\right) \cos (5 (c+d x))-135 a b^6 \cos (7 (c+d x))+1890 b^7 \sin (3 (c+d x))-630 b^7 \sin (5 (c+d x))+90 b^7 \sin (7 (c+d x))\right)\right)}{960 a^8 d (a+b \tan (c+d x))^2}","\frac{3 b \cot ^4(c+d x)}{4 a^4 d}-\frac{\cot ^5(c+d x)}{5 a^3 d}-\frac{2 b \left(a^2+b^2\right) \left(a^2+3 b^2\right)}{a^7 d (a+b \tan (c+d x))}-\frac{b \left(a^2+b^2\right)^2}{2 a^6 d (a+b \tan (c+d x))^2}+\frac{b \left(3 a^2+5 b^2\right) \cot ^2(c+d x)}{a^6 d}-\frac{2 \left(a^2+3 b^2\right) \cot ^3(c+d x)}{3 a^5 d}-\frac{b \left(3 a^4+20 a^2 b^2+21 b^4\right) \log (\tan (c+d x))}{a^8 d}+\frac{b \left(3 a^4+20 a^2 b^2+21 b^4\right) \log (a+b \tan (c+d x))}{a^8 d}-\frac{\left(a^4+12 a^2 b^2+15 b^4\right) \cot (c+d x)}{a^7 d}",1,"-1/960*(Csc[c + d*x]^5*(Sec[c + d*x]^2*((8*a^7 + 567*a^5*b^2 + 630*a^3*b^4 - 1215*a*b^6)*Cos[3*(c + d*x)] - (24*a^7 + 619*a^5*b^2 + 630*a^3*b^4 - 675*a*b^6)*Cos[5*(c + d*x)] + 8*a^7*Cos[7*(c + d*x)] + 187*a^5*b^2*Cos[7*(c + d*x)] + 210*a^3*b^4*Cos[7*(c + d*x)] - 135*a*b^6*Cos[7*(c + d*x)] - 126*a^6*b*Sin[3*(c + d*x)] + 1665*a^4*b^3*Sin[3*(c + d*x)] + 4635*a^2*b^5*Sin[3*(c + d*x)] + 1890*b^7*Sin[3*(c + d*x)] + 10*a^6*b*Sin[5*(c + d*x)] - 1215*a^4*b^3*Sin[5*(c + d*x)] - 2565*a^2*b^5*Sin[5*(c + d*x)] - 630*b^7*Sin[5*(c + d*x)] + 16*a^6*b*Sin[7*(c + d*x)] + 345*a^4*b^3*Sin[7*(c + d*x)] + 585*a^2*b^5*Sin[7*(c + d*x)] + 90*b^7*Sin[7*(c + d*x)]) + 960*b*(3*a^4 + 20*a^2*b^2 + 21*b^4)*(Log[Sin[c + d*x]] - Log[a*Cos[c + d*x] + b*Sin[c + d*x]])*Sin[c + d*x]^5*(a + b*Tan[c + d*x])^2 + 5*Sec[c + d*x]*(40*a^7 - 27*a^5*b^2 - 42*a^3*b^4 + 135*a*b^6 - 3*b*(8*a^6 + 89*a^4*b^2 + 345*a^2*b^4 + 210*b^6)*Tan[c + d*x])))/(a^8*d*(a + b*Tan[c + d*x])^2)","A",1
73,1,564,366,5.6612239,"\int \frac{\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^4,x]","-\frac{b \left(-24 a (a-b) (a+b) \left(a^2+b^2\right)^2 \cos ^4(c+d x)+\frac{8 a^4 \left(a^2+b^2\right)^3}{(a+b \tan (c+d x))^3}+\frac{12 a^2 \left(a^2+b^2\right) \left(a^4-10 a^2 b^2+5 b^4\right) \sin (2 (c+d x))}{b}+48 a \left(a^2+b^2\right) \left(2 a^4-5 a^2 b^2+b^4\right) \cos ^2(c+d x)+\frac{24 a^2 \left(a^2+b^2\right) \left(a^4-10 a^2 b^2+5 b^4\right) \tan ^{-1}(\tan (c+d x))}{b}+\frac{72 a^2 \left(a^2+b^2\right) \left(a^4-5 a^2 b^2+2 b^4\right)}{a+b \tan (c+d x)}-96 a (a-b) (a+b) \left(a^4-8 a^2 b^2+b^4\right) \log (a+b \tan (c+d x))-\frac{9 \left(a^2+b^2\right)^2 \left(a^4-6 a^2 b^2+b^4\right) \left(\sin (2 (c+d x))+2 \tan ^{-1}(\tan (c+d x))\right)}{2 b}-\frac{6 \left(a^2+b^2\right)^2 \left(a^4-6 a^2 b^2+b^4\right) \sin (c+d x) \cos ^3(c+d x)}{b}+\frac{24 a^3 \left(a^2-2 b^2\right) \left(a^2+b^2\right)^2}{(a+b \tan (c+d x))^2}+12 a \left(4 a^6-36 a^4 b^2+36 a^2 b^4+\frac{-a^7+24 a^5 b^2-45 a^3 b^4+10 a b^6}{\sqrt{-b^2}}-4 b^6\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+12 a \left(4 a^6-36 a^4 b^2+36 a^2 b^4+\frac{a^7-24 a^5 b^2+45 a^3 b^4-10 a b^6}{\sqrt{-b^2}}-4 b^6\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)\right)}{24 d \left(a^2+b^2\right)^6}","-\frac{a^4 b}{3 d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^3}-\frac{3 a^2 b \left(a^4-5 a^2 b^2+2 b^4\right)}{d \left(a^2+b^2\right)^5 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left(4 a b \left(a^2-b^2\right)+\left(a^4-6 a^2 b^2+b^4\right) \tan (c+d x)\right)}{4 d \left(a^2+b^2\right)^4}+\frac{4 a b \left(a^2-b^2\right) \left(a^4-8 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^6}-\frac{a^3 b \left(a^2-2 b^2\right)}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))^2}-\frac{\cos ^2(c+d x) \left(16 a b \left(2 a^4-5 a^2 b^2+b^4\right)+\left(5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right) \tan (c+d x)\right)}{8 d \left(a^2+b^2\right)^5}+\frac{x \left(3 a^8-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 b^8\right)}{8 \left(a^2+b^2\right)^6}",1,"-1/24*(b*((24*a^2*(a^2 + b^2)*(a^4 - 10*a^2*b^2 + 5*b^4)*ArcTan[Tan[c + d*x]])/b + 48*a*(a^2 + b^2)*(2*a^4 - 5*a^2*b^2 + b^4)*Cos[c + d*x]^2 - 24*a*(a - b)*(a + b)*(a^2 + b^2)^2*Cos[c + d*x]^4 + 12*a*(4*a^6 - 36*a^4*b^2 + 36*a^2*b^4 - 4*b^6 + (-a^7 + 24*a^5*b^2 - 45*a^3*b^4 + 10*a*b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 96*a*(a - b)*(a + b)*(a^4 - 8*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]] + 12*a*(4*a^6 - 36*a^4*b^2 + 36*a^2*b^4 - 4*b^6 + (a^7 - 24*a^5*b^2 + 45*a^3*b^4 - 10*a*b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] - (6*(a^2 + b^2)^2*(a^4 - 6*a^2*b^2 + b^4)*Cos[c + d*x]^3*Sin[c + d*x])/b + (12*a^2*(a^2 + b^2)*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[2*(c + d*x)])/b - (9*(a^2 + b^2)^2*(a^4 - 6*a^2*b^2 + b^4)*(2*ArcTan[Tan[c + d*x]] + Sin[2*(c + d*x)]))/(2*b) + (8*a^4*(a^2 + b^2)^3)/(a + b*Tan[c + d*x])^3 + (24*a^3*(a^2 - 2*b^2)*(a^2 + b^2)^2)/(a + b*Tan[c + d*x])^2 + (72*a^2*(a^2 + b^2)*(a^4 - 5*a^2*b^2 + 2*b^4))/(a + b*Tan[c + d*x])))/((a^2 + b^2)^6*d)","A",1
74,1,395,264,3.7714129,"\int \frac{\sin ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]","-\frac{b \left(12 a (a-b) (a+b) \left(a^2+b^2\right) \cos ^2(c+d x)+\frac{2 a^2 \left(a^2+b^2\right)^3}{(a+b \tan (c+d x))^3}+\frac{6 a (a-b) (a+b) \left(a^2+b^2\right)^2}{(a+b \tan (c+d x))^2}+\frac{3 \left(a^4-6 a^2 b^2+b^4\right) \left(a^2+b^2\right) \sin (2 (c+d x))}{2 b}+\frac{3 \left(a^4-6 a^2 b^2+b^4\right) \left(a^2+b^2\right) \tan ^{-1}(\tan (c+d x))}{b}+\frac{6 \left(3 a^4-8 a^2 b^2+b^4\right) \left(a^2+b^2\right)}{a+b \tan (c+d x)}-24 a \left(a^4-5 a^2 b^2+2 b^4\right) \log (a+b \tan (c+d x))+3 \left(4 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}}+8 a b^4\right) \log \left(\sqrt{-b^2}-b \tan (c+d x)\right)+3 \left(4 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}}+8 a b^4\right) \log \left(\sqrt{-b^2}+b \tan (c+d x)\right)\right)}{6 d \left(a^2+b^2\right)^5}","-\frac{a^2 b}{3 d \left(a^2+b^2\right)^2 (a+b \tan (c+d x))^3}-\frac{a b \left(a^2-b^2\right)}{d \left(a^2+b^2\right)^3 (a+b \tan (c+d x))^2}-\frac{b \left(3 a^4-8 a^2 b^2+b^4\right)}{d \left(a^2+b^2\right)^4 (a+b \tan (c+d x))}-\frac{\cos ^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) \tan (c+d x)\right)}{2 d \left(a^2+b^2\right)^4}+\frac{4 a b \left(a^4-5 a^2 b^2+2 b^4\right) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left(a^2+b^2\right)^5}+\frac{x \left(a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right)}{2 \left(a^2+b^2\right)^5}",1,"-1/6*(b*((3*(a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]])/b + 12*a*(a - b)*(a + b)*(a^2 + b^2)*Cos[c + d*x]^2 + 3*(4*a^5 - 20*a^3*b^2 + 8*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]] - 24*a*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a + b*Tan[c + d*x]] + 3*(4*a^5 - 20*a^3*b^2 + 8*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]] + (3*(a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*Sin[2*(c + d*x)])/(2*b) + (2*a^2*(a^2 + b^2)^3)/(a + b*Tan[c + d*x])^3 + (6*a*(a - b)*(a + b)*(a^2 + b^2)^2)/(a + b*Tan[c + d*x])^2 + (6*(a^2 + b^2)*(3*a^4 - 8*a^2*b^2 + b^4))/(a + b*Tan[c + d*x])))/((a^2 + b^2)^5*d)","A",1
75,1,259,116,2.217136,"\int \frac{\csc ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Tan[c + d*x])^4,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(\frac{a^2 b^4 \tan (c+d x)}{a^2+b^2}-\frac{2 a^2 b^3 \left(3 a^2+2 b^2\right) (a+b \tan (c+d x))}{\left(a^2+b^2\right)^2}+\frac{b^2 \left(18 a^4+23 a^2 b^2+9 b^4\right) \tan (c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{\left(a^2+b^2\right)^2}-3 a \sin ^2(c+d x) (a \cot (c+d x)+b)^3-12 b \cos ^2(c+d x) \log (\sin (c+d x)) (a+b \tan (c+d x))^3+12 b \cos ^2(c+d x) (a+b \tan (c+d x))^3 \log (a \cos (c+d x)+b \sin (c+d x))\right)}{3 a^5 d (a+b \tan (c+d x))^4}","-\frac{4 b \log (\tan (c+d x))}{a^5 d}+\frac{4 b \log (a+b \tan (c+d x))}{a^5 d}-\frac{3 b}{a^4 d (a+b \tan (c+d x))}-\frac{\cot (c+d x)}{a^4 d}-\frac{b}{a^3 d (a+b \tan (c+d x))^2}-\frac{b}{3 a^2 d (a+b \tan (c+d x))^3}",1,"(Sec[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])*(-3*a*(b + a*Cot[c + d*x])^3*Sin[c + d*x]^2 + (a^2*b^4*Tan[c + d*x])/(a^2 + b^2) + (b^2*(18*a^4 + 23*a^2*b^2 + 9*b^4)*(a*Cos[c + d*x] + b*Sin[c + d*x])^2*Tan[c + d*x])/(a^2 + b^2)^2 - (2*a^2*b^3*(3*a^2 + 2*b^2)*(a + b*Tan[c + d*x]))/(a^2 + b^2)^2 - 12*b*Cos[c + d*x]^2*Log[Sin[c + d*x]]*(a + b*Tan[c + d*x])^3 + 12*b*Cos[c + d*x]^2*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]*(a + b*Tan[c + d*x])^3))/(3*a^5*d*(a + b*Tan[c + d*x])^4)","B",1
76,1,528,205,2.0785479,"\int \frac{\csc ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(-192 b \left(a^2+5 b^2\right) \log (\sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3+192 b \left(a^2+5 b^2\right) (a \cos (c+d x)+b \sin (c+d x))^3 \log (a \cos (c+d x)+b \sin (c+d x))-\frac{\csc ^3(c+d x) \left(a^8 (-\cos (6 (c+d x)))+8 a^8-3 a^7 b \sin (2 (c+d x))-6 a^7 b \sin (4 (c+d x))-3 a^7 b \sin (6 (c+d x))-22 a^6 b^2 \cos (6 (c+d x))-4 a^6 b^2+3 a^5 b^3 \sin (2 (c+d x))+84 a^5 b^3 \sin (4 (c+d x))-65 a^5 b^3 \sin (6 (c+d x))+17 a^4 b^4 \cos (6 (c+d x))-50 a^4 b^4-75 a^3 b^5 \sin (2 (c+d x))+156 a^3 b^5 \sin (4 (c+d x))-79 a^3 b^5 \sin (6 (c+d x))+55 a^2 b^6 \cos (6 (c+d x))-190 a^2 b^6+6 \left(2 a^6 b^2-17 a^4 b^4-35 a^2 b^6-15 b^8\right) \cos (4 (c+d x))+3 \left(3 a^8+10 a^6 b^2+45 a^4 b^4+115 a^2 b^6+75 b^8\right) \cos (2 (c+d x))-75 a b^7 \sin (2 (c+d x))+60 a b^7 \sin (4 (c+d x))-15 a b^7 \sin (6 (c+d x))+15 b^8 \cos (6 (c+d x))-150 b^8\right)}{a^2+b^2}\right)}{48 a^7 d (a+b \tan (c+d x))^4}","\frac{2 b \cot ^2(c+d x)}{a^5 d}-\frac{\cot ^3(c+d x)}{3 a^4 d}-\frac{4 b \left(a^2+5 b^2\right) \log (\tan (c+d x))}{a^7 d}+\frac{4 b \left(a^2+5 b^2\right) \log (a+b \tan (c+d x))}{a^7 d}-\frac{b \left(3 a^2+10 b^2\right)}{a^6 d (a+b \tan (c+d x))}-\frac{\left(a^2+10 b^2\right) \cot (c+d x)}{a^6 d}-\frac{b \left(a^2+2 b^2\right)}{a^5 d (a+b \tan (c+d x))^2}-\frac{b \left(a^2+b^2\right)}{3 a^4 d (a+b \tan (c+d x))^3}",1,"(Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])*(-192*b*(a^2 + 5*b^2)*Log[Sin[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 + 192*b*(a^2 + 5*b^2)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 - (Csc[c + d*x]^3*(8*a^8 - 4*a^6*b^2 - 50*a^4*b^4 - 190*a^2*b^6 - 150*b^8 + 3*(3*a^8 + 10*a^6*b^2 + 45*a^4*b^4 + 115*a^2*b^6 + 75*b^8)*Cos[2*(c + d*x)] + 6*(2*a^6*b^2 - 17*a^4*b^4 - 35*a^2*b^6 - 15*b^8)*Cos[4*(c + d*x)] - a^8*Cos[6*(c + d*x)] - 22*a^6*b^2*Cos[6*(c + d*x)] + 17*a^4*b^4*Cos[6*(c + d*x)] + 55*a^2*b^6*Cos[6*(c + d*x)] + 15*b^8*Cos[6*(c + d*x)] - 3*a^7*b*Sin[2*(c + d*x)] + 3*a^5*b^3*Sin[2*(c + d*x)] - 75*a^3*b^5*Sin[2*(c + d*x)] - 75*a*b^7*Sin[2*(c + d*x)] - 6*a^7*b*Sin[4*(c + d*x)] + 84*a^5*b^3*Sin[4*(c + d*x)] + 156*a^3*b^5*Sin[4*(c + d*x)] + 60*a*b^7*Sin[4*(c + d*x)] - 3*a^7*b*Sin[6*(c + d*x)] - 65*a^5*b^3*Sin[6*(c + d*x)] - 79*a^3*b^5*Sin[6*(c + d*x)] - 15*a*b^7*Sin[6*(c + d*x)]))/(a^2 + b^2)))/(48*a^7*d*(a + b*Tan[c + d*x])^4)","B",1
77,1,673,300,1.759768,"\int \frac{\csc ^6(c+d x)}{(a+b \tan (c+d x))^4} \, dx","Integrate[Csc[c + d*x]^6/(a + b*Tan[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \left(-7680 b \left(a^4+10 a^2 b^2+14 b^4\right) \log (\sin (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^3+7680 b \left(a^4+10 a^2 b^2+14 b^4\right) (a \cos (c+d x)+b \sin (c+d x))^3 \log (a \cos (c+d x)+b \sin (c+d x))+\csc ^5(c+d x) \left(16 a^8 \cos (6 (c+d x))-8 a^8 \cos (8 (c+d x))-200 a^8+264 a^7 b \sin (2 (c+d x))+144 a^7 b \sin (4 (c+d x))-24 a^7 b \sin (6 (c+d x))-24 a^7 b \sin (8 (c+d x))+776 a^6 b^2 \cos (6 (c+d x))-316 a^6 b^2 \cos (8 (c+d x))+380 a^6 b^2+372 a^5 b^3 \sin (2 (c+d x))-2476 a^5 b^3 \sin (4 (c+d x))+2756 a^5 b^3 \sin (6 (c+d x))-922 a^5 b^3 \sin (8 (c+d x))-1000 a^4 b^4 \cos (6 (c+d x))-70 a^4 b^4 \cos (8 (c+d x))+3070 a^4 b^4+4830 a^3 b^5 \sin (2 (c+d x))-9730 a^3 b^5 \sin (4 (c+d x))+7670 a^3 b^5 \sin (6 (c+d x))-2095 a^3 b^5 \sin (8 (c+d x))-8540 a^2 b^6 \cos (6 (c+d x))+1645 a^2 b^6 \cos (8 (c+d x))+11375 a^2 b^6-4 \left(52 a^8+194 a^6 b^2+1510 a^4 b^4+5705 a^2 b^6+4410 b^8\right) \cos (2 (c+d x))+4 \left(4 a^8-16 a^6 b^2+1010 a^4 b^4+4585 a^2 b^6+2205 b^8\right) \cos (4 (c+d x))+1470 a b^7 \sin (2 (c+d x))-1470 a b^7 \sin (4 (c+d x))+630 a b^7 \sin (6 (c+d x))-105 a b^7 \sin (8 (c+d x))-2520 b^8 \cos (6 (c+d x))+315 b^8 \cos (8 (c+d x))+11025 b^8\right)\right)}{1920 a^9 d (a+b \tan (c+d x))^4}","\frac{b \cot ^4(c+d x)}{a^5 d}-\frac{\cot ^5(c+d x)}{5 a^4 d}-\frac{b \left(a^2+b^2\right) \left(a^2+3 b^2\right)}{a^7 d (a+b \tan (c+d x))^2}+\frac{2 b \left(2 a^2+5 b^2\right) \cot ^2(c+d x)}{a^7 d}-\frac{b \left(a^2+b^2\right)^2}{3 a^6 d (a+b \tan (c+d x))^3}-\frac{2 \left(a^2+5 b^2\right) \cot ^3(c+d x)}{3 a^6 d}-\frac{4 b \left(a^4+10 a^2 b^2+14 b^4\right) \log (\tan (c+d x))}{a^9 d}+\frac{4 b \left(a^4+10 a^2 b^2+14 b^4\right) \log (a+b \tan (c+d x))}{a^9 d}-\frac{b \left(3 a^4+20 a^2 b^2+21 b^4\right)}{a^8 d (a+b \tan (c+d x))}-\frac{\left(a^4+20 a^2 b^2+35 b^4\right) \cot (c+d x)}{a^8 d}",1,"(Sec[c + d*x]^4*(a*Cos[c + d*x] + b*Sin[c + d*x])*(-7680*b*(a^4 + 10*a^2*b^2 + 14*b^4)*Log[Sin[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 + 7680*b*(a^4 + 10*a^2*b^2 + 14*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^3 + Csc[c + d*x]^5*(-200*a^8 + 380*a^6*b^2 + 3070*a^4*b^4 + 11375*a^2*b^6 + 11025*b^8 - 4*(52*a^8 + 194*a^6*b^2 + 1510*a^4*b^4 + 5705*a^2*b^6 + 4410*b^8)*Cos[2*(c + d*x)] + 4*(4*a^8 - 16*a^6*b^2 + 1010*a^4*b^4 + 4585*a^2*b^6 + 2205*b^8)*Cos[4*(c + d*x)] + 16*a^8*Cos[6*(c + d*x)] + 776*a^6*b^2*Cos[6*(c + d*x)] - 1000*a^4*b^4*Cos[6*(c + d*x)] - 8540*a^2*b^6*Cos[6*(c + d*x)] - 2520*b^8*Cos[6*(c + d*x)] - 8*a^8*Cos[8*(c + d*x)] - 316*a^6*b^2*Cos[8*(c + d*x)] - 70*a^4*b^4*Cos[8*(c + d*x)] + 1645*a^2*b^6*Cos[8*(c + d*x)] + 315*b^8*Cos[8*(c + d*x)] + 264*a^7*b*Sin[2*(c + d*x)] + 372*a^5*b^3*Sin[2*(c + d*x)] + 4830*a^3*b^5*Sin[2*(c + d*x)] + 1470*a*b^7*Sin[2*(c + d*x)] + 144*a^7*b*Sin[4*(c + d*x)] - 2476*a^5*b^3*Sin[4*(c + d*x)] - 9730*a^3*b^5*Sin[4*(c + d*x)] - 1470*a*b^7*Sin[4*(c + d*x)] - 24*a^7*b*Sin[6*(c + d*x)] + 2756*a^5*b^3*Sin[6*(c + d*x)] + 7670*a^3*b^5*Sin[6*(c + d*x)] + 630*a*b^7*Sin[6*(c + d*x)] - 24*a^7*b*Sin[8*(c + d*x)] - 922*a^5*b^3*Sin[8*(c + d*x)] - 2095*a^3*b^5*Sin[8*(c + d*x)] - 105*a*b^7*Sin[8*(c + d*x)])))/(1920*a^9*d*(a + b*Tan[c + d*x])^4)","B",1
78,1,41,26,0.0419769,"\int \frac{\csc (x)}{1+\tan (x)} \, dx","Integrate[Csc[x]/(1 + Tan[x]),x]","\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)+(1+i) (-1)^{3/4} \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)-1}{\sqrt{2}}\right)","\frac{\tanh ^{-1}\left(\frac{\cos (x)-\sin (x)}{\sqrt{2}}\right)}{\sqrt{2}}-\tanh ^{-1}(\cos (x))",1,"(1 + I)*(-1)^(3/4)*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] - Log[Cos[x/2]] + Log[Sin[x/2]]","C",1
79,1,205,229,2.5636979,"\int \sin ^m(c+d x) (a+b \tan (c+d x))^3 \, dx","Integrate[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^3,x]","\frac{\sin ^{m+1}(c+d x) \left(\frac{a^3 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{m+1}+b \sin (c+d x) \left(\frac{3 a^2 \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{m+2}+b \left(\frac{3 a \sqrt{\cos ^2(c+d x)} \tan (c+d x) \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(c+d x)\right)}{m+3}+\frac{b \sin ^2(c+d x) \, _2F_1\left(2,\frac{m+4}{2};\frac{m+6}{2};\sin ^2(c+d x)\right)}{m+4}\right)\right)\right)}{d}","\frac{a^3 \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^2 b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}+\frac{3 a b^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+3}(c+d x) \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(c+d x)\right)}{d (m+3)}+\frac{b^3 \sin ^{m+4}(c+d x) \, _2F_1\left(2,\frac{m+4}{2};\frac{m+6}{2};\sin ^2(c+d x)\right)}{d (m+4)}",1,"(Sin[c + d*x]^(1 + m)*((a^3*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x])/(1 + m) + b*Sin[c + d*x]*((3*a^2*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2])/(2 + m) + b*((b*Hypergeometric2F1[2, (4 + m)/2, (6 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^2)/(4 + m) + (3*a*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[c + d*x]^2]*Tan[c + d*x])/(3 + m)))))/d","A",1
80,1,166,179,1.2082358,"\int \sin ^m(c+d x) (a+b \tan (c+d x))^2 \, dx","Integrate[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^2,x]","\frac{\sin ^{m+1}(c+d x) \left(\frac{a^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{m+1}+\frac{b \sin (c+d x) \left(2 a (m+3) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)+b (m+2) \sqrt{\cos ^2(c+d x)} \tan (c+d x) \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(c+d x)\right)\right)}{(m+2) (m+3)}\right)}{d}","\frac{a^2 \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}+\frac{b^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+3}(c+d x) \, _2F_1\left(\frac{3}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(c+d x)\right)}{d (m+3)}",1,"(Sin[c + d*x]^(1 + m)*((a^2*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x])/(1 + m) + (b*Sin[c + d*x]*(2*a*(3 + m)*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2] + b*(2 + m)*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (3 + m)/2, (5 + m)/2, Sin[c + d*x]^2]*Tan[c + d*x]))/((2 + m)*(3 + m))))/d","A",1
81,1,109,109,0.2732863,"\int \sin ^m(c+d x) (a+b \tan (c+d x)) \, dx","Integrate[Sin[c + d*x]^m*(a + b*Tan[c + d*x]),x]","\frac{a \sqrt{\cos ^2(c+d x)} \sec (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1)}+\frac{b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}","\frac{a \cos (c+d x) \sin ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b \sin ^{m+2}(c+d x) \, _2F_1\left(1,\frac{m+2}{2};\frac{m+4}{2};\sin ^2(c+d x)\right)}{d (m+2)}",1,"(a*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sec[c + d*x]*Sin[c + d*x]^(1 + m))/(d*(1 + m)) + (b*Hypergeometric2F1[1, (2 + m)/2, (4 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x]^(2 + m))/(d*(2 + m))","A",1
82,0,0,765,13.2164318,"\int \frac{\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx","Integrate[Sin[c + d*x]^m/(a + b*Tan[c + d*x]),x]","\int \frac{\sin ^m(c+d x)}{a+b \tan (c+d x)} \, dx","\frac{b 2^{m+1} \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+2}{2};m+1,1;\frac{m+4}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b-\sqrt{a^2+b^2}\right)^2}\right)}{d (m+2) \sqrt{a^2+b^2} \left(b-\sqrt{a^2+b^2}\right)}-\frac{b 2^{m+1} \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+2}{2};m+1,1;\frac{m+4}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b+\sqrt{a^2+b^2}\right)^2}\right)}{d (m+2) \sqrt{a^2+b^2} \left(\sqrt{a^2+b^2}+b\right)}+\frac{a b 2^{m+1} \tan ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b-\sqrt{a^2+b^2}\right)^2}\right)}{d (m+3) \sqrt{a^2+b^2} \left(b-\sqrt{a^2+b^2}\right)^2}-\frac{a b 2^{m+1} \tan ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m F_1\left(\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{a^2 \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\left(b+\sqrt{a^2+b^2}\right)^2}\right)}{d (m+3) \sqrt{a^2+b^2} \left(\sqrt{a^2+b^2}+b\right)^2}+\frac{2^{m+1} \tan \left(\frac{1}{2} (c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}\right)^m \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^m \, _2F_1\left(\frac{m+1}{2},m+1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{a d (m+1)}",1,"Integrate[Sin[c + d*x]^m/(a + b*Tan[c + d*x]), x]","F",-1
83,0,0,24,3.4974743,"\int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n,x]","\int \sin ^m(c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\sin ^m(c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Integrate[Sin[c + d*x]^m*(a + b*Tan[c + d*x])^n, x]","A",-1
84,1,910,435,6.5977948,"\int \sin ^4(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sin[c + d*x]^4*(a + b*Tan[c + d*x])^n,x]","\frac{b \left(\frac{\, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right) (a+b \tan (c+d x))^{n+1}}{2 \sqrt{-b^2} \left(a-\sqrt{-b^2}\right) (n+1)}-\frac{\, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right) (a+b \tan (c+d x))^{n+1}}{2 \sqrt{-b^2} \left(a+\sqrt{-b^2}\right) (n+1)}+\frac{\cos ^4(c+d x) \left(b^2+a \tan (c+d x) b\right) (a+b \tan (c+d x))^{n+1}}{4 b^2 \left(a^2+b^2\right)}-\frac{\cos ^2(c+d x) \left(b^2+a \tan (c+d x) b\right) (a+b \tan (c+d x))^{n+1}}{b^2 \left(a^2+b^2\right)}+\frac{\frac{\left(\sqrt{-b^2} \left(a^2+b^2 (1-n)\right)-a b^2 n\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right) (a+b \tan (c+d x))^{n+1}}{b^2 \left(a-\sqrt{-b^2}\right) (n+1)}-\frac{\left(\sqrt{-b^2} a^2+b^2 n a-\left(-b^2\right)^{3/2} (1-n)\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right) (a+b \tan (c+d x))^{n+1}}{b^2 \left(a+\sqrt{-b^2}\right) (n+1)}}{2 \left(a^2+b^2\right)}-\frac{b^2 \left(\frac{\cos ^2(c+d x) (a+b \tan (c+d x))^{n+1} \left(\left(-3 a^2-b^2 (3-n)\right) b^2+a^2 (2-n) b^2+\left(a \left(-3 a^2-b^2 (3-n)\right)-a b^2 (2-n)\right) \tan (c+d x) b\right)}{2 b^4 \left(a^2+b^2\right)}-\frac{\frac{\left(a b^2 \left(3 a^2+b^2 (5-2 n)\right) n-\sqrt{-b^2} \left(3 a^4+b^2 \left(-n^2-2 n+6\right) a^2+b^4 \left(n^2-4 n+3\right)\right)\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right) (a+b \tan (c+d x))^{n+1}}{2 b^2 \left(a-\sqrt{-b^2}\right) (n+1)}+\frac{\left(a \left(3 a^2+b^2 (5-2 n)\right) n b^2+\sqrt{-b^2} \left(3 a^4+b^2 \left(-n^2-2 n+6\right) a^2+b^4 \left(n^2-4 n+3\right)\right)\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right) (a+b \tan (c+d x))^{n+1}}{2 b^2 \left(a+\sqrt{-b^2}\right) (n+1)}}{2 b^2 \left(a^2+b^2\right)}\right)}{4 \left(a^2+b^2\right)}\right)}{d}","\frac{\cos ^4(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{4 d \left(a^2+b^2\right)}-\frac{\cos ^2(c+d x) \left(a \left(5 a^2+b^2 (2 n+3)\right) \tan (c+d x)+b \left(a^2 (7-n)+b^2 (n+5)\right)\right) (a+b \tan (c+d x))^{n+1}}{8 d \left(a^2+b^2\right)^2}-\frac{\left(a b^2 n \left(5 a^2+b^2 (2 n+3)\right)+\sqrt{-b^2} \left(3 a^4+a^2 b^2 \left(-n^2+6 n+6\right)+b^4 \left(n^2+4 n+3\right)\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{16 b d (n+1) \left(a^2+b^2\right)^2 \left(a-\sqrt{-b^2}\right)}-\frac{\left(a b^2 n \left(5 a^2+b^2 (2 n+3)\right)-\sqrt{-b^2} \left(3 a^4+a^2 b^2 \left(-n^2+6 n+6\right)+b^4 \left(n^2+4 n+3\right)\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{16 b d (n+1) \left(a^2+b^2\right)^2 \left(a+\sqrt{-b^2}\right)}",1,"(b*((Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*Sqrt[-b^2]*(a - Sqrt[-b^2])*(1 + n)) - (Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*Sqrt[-b^2]*(a + Sqrt[-b^2])*(1 + n)) - (Cos[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n)*(b^2 + a*b*Tan[c + d*x]))/(b^2*(a^2 + b^2)) + (Cos[c + d*x]^4*(a + b*Tan[c + d*x])^(1 + n)*(b^2 + a*b*Tan[c + d*x]))/(4*b^2*(a^2 + b^2)) + (((Sqrt[-b^2]*(a^2 + b^2*(1 - n)) - a*b^2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(b^2*(a - Sqrt[-b^2])*(1 + n)) - ((a^2*Sqrt[-b^2] - (-b^2)^(3/2)*(1 - n) + a*b^2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(b^2*(a + Sqrt[-b^2])*(1 + n)))/(2*(a^2 + b^2)) - (b^2*((Cos[c + d*x]^2*(a + b*Tan[c + d*x])^(1 + n)*(b^2*(-3*a^2 - b^2*(3 - n)) + a^2*b^2*(2 - n) + b*(a*(-3*a^2 - b^2*(3 - n)) - a*b^2*(2 - n))*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)) - (((a*b^2*(3*a^2 + b^2*(5 - 2*n))*n - Sqrt[-b^2]*(3*a^4 + a^2*b^2*(6 - 2*n - n^2) + b^4*(3 - 4*n + n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*b^2*(a - Sqrt[-b^2])*(1 + n)) + ((a*b^2*(3*a^2 + b^2*(5 - 2*n))*n + Sqrt[-b^2]*(3*a^4 + a^2*b^2*(6 - 2*n - n^2) + b^4*(3 - 4*n + n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(2*b^2*(a + Sqrt[-b^2])*(1 + n)))/(2*b^2*(a^2 + b^2))))/(4*(a^2 + b^2))))/d","B",1
85,1,270,276,1.1502779,"\int \sin ^2(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sin[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]","\frac{(a+b \tan (c+d x))^{n+1} \left(2 b (n+1) \left(a^2+b^2\right) \cos (c+d x) (a \sin (c+d x)+b \cos (c+d x))+\left(a^3 \sqrt{-b^2}+a^2 b^2 (n-1)-a \left(-b^2\right)^{3/2} (2 n+1)-\left(b^4 (n+1)\right)\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)-\left(a^3 \sqrt{-b^2}-a^2 b^2 (n-1)-a \left(-b^2\right)^{3/2} (2 n+1)+b^4 (n+1)\right) \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)\right)}{4 b d (n+1) \left(a^2+b^2\right) \left(\sqrt{-b^2}-a\right) \left(a+\sqrt{-b^2}\right)}","-\frac{\left(\sqrt{-b^2} \left(a^2+b^2 (n+1)\right)+a b^2 n\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right)}{4 b d (n+1) \left(a^2+b^2\right) \left(a-\sqrt{-b^2}\right)}-\frac{\left(a b^2 n-\sqrt{-b^2} \left(a^2+b^2 (n+1)\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right)}{4 b d (n+1) \left(a^2+b^2\right) \left(a+\sqrt{-b^2}\right)}-\frac{\cos ^2(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{2 d \left(a^2+b^2\right)}",1,"(((a^3*Sqrt[-b^2] + a^2*b^2*(-1 + n) - b^4*(1 + n) - a*(-b^2)^(3/2)*(1 + 2*n))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])] - (a^3*Sqrt[-b^2] - a^2*b^2*(-1 + n) + b^4*(1 + n) - a*(-b^2)^(3/2)*(1 + 2*n))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])] + 2*b*(a^2 + b^2)*(1 + n)*Cos[c + d*x]*(b*Cos[c + d*x] + a*Sin[c + d*x]))*(a + b*Tan[c + d*x])^(1 + n))/(4*b*(a^2 + b^2)*(-a + Sqrt[-b^2])*(a + Sqrt[-b^2])*d*(1 + n))","A",1
86,1,48,48,0.9375778,"\int \csc ^2(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Csc[c + d*x]^2*(a + b*Tan[c + d*x])^n,x]","\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{a^2 d (n+1)}","\frac{b (a+b \tan (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{a^2 d (n+1)}",1,"(b*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a]*(a + b*Tan[c + d*x])^(1 + n))/(a^2*d*(1 + n))","A",1
87,1,78,140,1.3073353,"\int \csc ^4(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Csc[c + d*x]^4*(a + b*Tan[c + d*x])^n,x]","\frac{b (a+b \tan (c+d x))^{n+1} \left(a^2 \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)+b^2 \, _2F_1\left(4,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)\right)}{a^4 d (n+1)}","\frac{b (2-n) \cot ^2(c+d x) (a+b \tan (c+d x))^{n+1}}{6 a^2 d}+\frac{b \left(6 a^2+b^2 \left(n^2-3 n+2\right)\right) (a+b \tan (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \tan (c+d x)}{a}+1\right)}{6 a^4 d (n+1)}-\frac{\cot ^3(c+d x) (a+b \tan (c+d x))^{n+1}}{3 a d}",1,"(b*(a^2*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a] + b^2*Hypergeometric2F1[4, 1 + n, 2 + n, 1 + (b*Tan[c + d*x])/a])*(a + b*Tan[c + d*x])^(1 + n))/(a^4*d*(1 + n))","A",1
88,0,0,24,3.2147515,"\int \sin ^3(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]","\int \sin ^3(c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\sin ^3(c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Integrate[Sin[c + d*x]^3*(a + b*Tan[c + d*x])^n, x]","A",-1
89,0,0,22,2.2473814,"\int \sin (c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Sin[c + d*x]*(a + b*Tan[c + d*x])^n,x]","\int \sin (c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\sin (c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Integrate[Sin[c + d*x]*(a + b*Tan[c + d*x])^n, x]","A",-1
90,0,0,22,1.5904789,"\int \csc (c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Csc[c + d*x]*(a + b*Tan[c + d*x])^n,x]","\int \csc (c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\csc (c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Integrate[Csc[c + d*x]*(a + b*Tan[c + d*x])^n, x]","A",-1
91,0,0,24,15.4298617,"\int \csc ^3(c+d x) (a+b \tan (c+d x))^n \, dx","Integrate[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^n,x]","\int \csc ^3(c+d x) (a+b \tan (c+d x))^n \, dx","\text{Int}\left(\csc ^3(c+d x) (a+b \tan (c+d x))^n,x\right)",0,"Integrate[Csc[c + d*x]^3*(a + b*Tan[c + d*x])^n, x]","A",-1