1,1,218,227,0.115394,"\int (c+d x)^3 (a+a \sec (e+f x)) \, dx","Integrate[(c + d*x)^3*(a + a*Sec[e + f*x]),x]","a \left(\frac{3 i d \left(f^2 (c+d x)^2 \text{Li}_2\left(-i e^{i (e+f x)}\right)+2 i d f (c+d x) \text{Li}_3\left(-i e^{i (e+f x)}\right)-2 d^2 \text{Li}_4\left(-i e^{i (e+f x)}\right)\right)}{f^4}+\frac{3 d \left(2 d \left(f (c+d x) \text{Li}_3\left(i e^{i (e+f x)}\right)+i d \text{Li}_4\left(i e^{i (e+f x)}\right)\right)-i f^2 (c+d x)^2 \text{Li}_2\left(i e^{i (e+f x)}\right)\right)}{f^4}-\frac{2 i (c+d x)^3 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{(c+d x)^4}{4 d}\right)","-\frac{6 a d^2 (c+d x) \text{Li}_3\left(-i e^{i (e+f x)}\right)}{f^3}+\frac{6 a d^2 (c+d x) \text{Li}_3\left(i e^{i (e+f x)}\right)}{f^3}+\frac{3 i a d (c+d x)^2 \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{3 i a d (c+d x)^2 \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}-\frac{2 i a (c+d x)^3 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{a (c+d x)^4}{4 d}-\frac{6 i a d^3 \text{Li}_4\left(-i e^{i (e+f x)}\right)}{f^4}+\frac{6 i a d^3 \text{Li}_4\left(i e^{i (e+f x)}\right)}{f^4}",1,"a*((c + d*x)^4/(4*d) - ((2*I)*(c + d*x)^3*ArcTan[E^(I*(e + f*x))])/f + ((3*I)*d*(f^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))] + (2*I)*d*f*(c + d*x)*PolyLog[3, (-I)*E^(I*(e + f*x))] - 2*d^2*PolyLog[4, (-I)*E^(I*(e + f*x))]))/f^4 + (3*d*((-I)*f^2*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))] + 2*d*(f*(c + d*x)*PolyLog[3, I*E^(I*(e + f*x))] + I*d*PolyLog[4, I*E^(I*(e + f*x))])))/f^4)","A",1
2,1,151,157,0.1455227,"\int (c+d x)^2 (a+a \sec (e+f x)) \, dx","Integrate[(c + d*x)^2*(a + a*Sec[e + f*x]),x]","a \left(\frac{2 i d \left(f (c+d x) \text{Li}_2\left(-i e^{i (e+f x)}\right)+i d \text{Li}_3\left(-i e^{i (e+f x)}\right)\right)}{f^3}+\frac{2 d \left(d \text{Li}_3\left(i e^{i (e+f x)}\right)-i f (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)\right)}{f^3}-\frac{2 i (c+d x)^2 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{(c+d x)^3}{3 d}\right)","\frac{2 i a d (c+d x) \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{2 i a d (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}-\frac{2 i a (c+d x)^2 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{a (c+d x)^3}{3 d}-\frac{2 a d^2 \text{Li}_3\left(-i e^{i (e+f x)}\right)}{f^3}+\frac{2 a d^2 \text{Li}_3\left(i e^{i (e+f x)}\right)}{f^3}",1,"a*((c + d*x)^3/(3*d) - ((2*I)*(c + d*x)^2*ArcTan[E^(I*(e + f*x))])/f + ((2*I)*d*(f*(c + d*x)*PolyLog[2, (-I)*E^(I*(e + f*x))] + I*d*PolyLog[3, (-I)*E^(I*(e + f*x))]))/f^3 + (2*d*((-I)*f*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))] + d*PolyLog[3, I*E^(I*(e + f*x))]))/f^3)","A",1
3,1,87,93,0.058053,"\int (c+d x) (a+a \sec (e+f x)) \, dx","Integrate[(c + d*x)*(a + a*Sec[e + f*x]),x]","\frac{a \left(f \left(f x (2 c+d x)-4 i (c+d x) \tan ^{-1}\left(e^{i (e+f x)}\right)\right)+2 i d \text{Li}_2\left(-i e^{i (e+f x)}\right)-2 i d \text{Li}_2\left(i e^{i (e+f x)}\right)\right)}{2 f^2}","-\frac{2 i a (c+d x) \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{a (c+d x)^2}{2 d}+\frac{i a d \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{i a d \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}",1,"(a*(f*(f*x*(2*c + d*x) - (4*I)*(c + d*x)*ArcTan[E^(I*(e + f*x))]) + (2*I)*d*PolyLog[2, (-I)*E^(I*(e + f*x))] - (2*I)*d*PolyLog[2, I*E^(I*(e + f*x))]))/(2*f^2)","A",1
4,0,0,21,6.4847564,"\int \frac{a+a \sec (e+f x)}{c+d x} \, dx","Integrate[(a + a*Sec[e + f*x])/(c + d*x),x]","\int \frac{a+a \sec (e+f x)}{c+d x} \, dx","\text{Int}\left(\frac{a \sec (e+f x)+a}{c+d x},x\right)",0,"Integrate[(a + a*Sec[e + f*x])/(c + d*x), x]","A",-1
5,0,0,21,6.4698398,"\int \frac{a+a \sec (e+f x)}{(c+d x)^2} \, dx","Integrate[(a + a*Sec[e + f*x])/(c + d*x)^2,x]","\int \frac{a+a \sec (e+f x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{a \sec (e+f x)+a}{(c+d x)^2},x\right)",0,"Integrate[(a + a*Sec[e + f*x])/(c + d*x)^2, x]","A",-1
6,1,811,371,9.0865429,"\int (c+d x)^3 (a+a \sec (e+f x))^2 \, dx","Integrate[(c + d*x)^3*(a + a*Sec[e + f*x])^2,x]","\frac{1}{16} a^2 (\cos (e+f x)+1)^2 \sec ^4\left(\frac{1}{2} (e+f x)\right) \left(\frac{4 \sin \left(\frac{f x}{2}\right) (c+d x)^3}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{4 \sin \left(\frac{f x}{2}\right) (c+d x)^3}{f \left(\cos \left(\frac{e}{2}\right)+\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)+\sin \left(\frac{1}{2} (e+f x)\right)\right)}+x \left(4 c^3+6 d x c^2+4 d^2 x^2 c+d^3 x^3\right)-\frac{2 i \left(2 f^3 x^3 d^3+8 f^3 x^3 \tan ^{-1}(\cos (e+f x)+i \sin (e+f x)) d^3+6 i f^2 x^2 \log (\cos (2 (e+f x))+i \sin (2 (e+f x))+1) d^3+6 f x \text{Li}_2(-\cos (2 (e+f x))-i \sin (2 (e+f x))) d^3+24 i f x \text{Li}_3(i \cos (e+f x)-\sin (e+f x)) d^3-24 i f x \text{Li}_3(\sin (e+f x)-i \cos (e+f x)) d^3+3 i \text{Li}_3(-\cos (2 (e+f x))-i \sin (2 (e+f x))) d^3-24 \text{Li}_4(i \cos (e+f x)-\sin (e+f x)) d^3+24 \text{Li}_4(\sin (e+f x)-i \cos (e+f x)) d^3+2 i f^3 x^3 \tan (e) d^3+6 c f^3 x^2 d^2+24 c f^3 x^2 \tan ^{-1}(\cos (e+f x)+i \sin (e+f x)) d^2+12 i c f^2 x \log (\cos (2 (e+f x))+i \sin (2 (e+f x))+1) d^2+6 c f \text{Li}_2(-\cos (2 (e+f x))-i \sin (2 (e+f x))) d^2+24 i c f \text{Li}_3(i \cos (e+f x)-\sin (e+f x)) d^2-24 i c f \text{Li}_3(\sin (e+f x)-i \cos (e+f x)) d^2+6 i c f^3 x^2 \tan (e) d^2+6 c^2 f^3 x d+24 c^2 f^3 x \tan ^{-1}(\cos (e+f x)+i \sin (e+f x)) d+6 i c^2 f^2 \log (\cos (2 (e+f x))+i \sin (2 (e+f x))+1) d+12 f^2 (c+d x)^2 \text{Li}_2(i \cos (e+f x)-\sin (e+f x)) d-12 f^2 (c+d x)^2 \text{Li}_2(\sin (e+f x)-i \cos (e+f x)) d+6 i c^2 f^3 x \tan (e) d+8 c^3 f^3 \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))\right)}{f^4}\right)","-\frac{3 i a^2 d^2 (c+d x) \text{Li}_2\left(-e^{2 i (e+f x)}\right)}{f^3}-\frac{12 a^2 d^2 (c+d x) \text{Li}_3\left(-i e^{i (e+f x)}\right)}{f^3}+\frac{12 a^2 d^2 (c+d x) \text{Li}_3\left(i e^{i (e+f x)}\right)}{f^3}+\frac{6 i a^2 d (c+d x)^2 \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{6 i a^2 d (c+d x)^2 \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}+\frac{3 a^2 d (c+d x)^2 \log \left(1+e^{2 i (e+f x)}\right)}{f^2}-\frac{4 i a^2 (c+d x)^3 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{a^2 (c+d x)^3 \tan (e+f x)}{f}-\frac{i a^2 (c+d x)^3}{f}+\frac{a^2 (c+d x)^4}{4 d}+\frac{3 a^2 d^3 \text{Li}_3\left(-e^{2 i (e+f x)}\right)}{2 f^4}-\frac{12 i a^2 d^3 \text{Li}_4\left(-i e^{i (e+f x)}\right)}{f^4}+\frac{12 i a^2 d^3 \text{Li}_4\left(i e^{i (e+f x)}\right)}{f^4}",1,"(a^2*(1 + Cos[e + f*x])^2*Sec[(e + f*x)/2]^4*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + (4*(c + d*x)^3*Sin[(f*x)/2])/(f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) + (4*(c + d*x)^3*Sin[(f*x)/2])/(f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - ((2*I)*(6*c^2*d*f^3*x + 6*c*d^2*f^3*x^2 + 2*d^3*f^3*x^3 + 8*c^3*f^3*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + 24*c^2*d*f^3*x*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + 24*c*d^2*f^3*x^2*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + 8*d^3*f^3*x^3*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + (6*I)*c^2*d*f^2*Log[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + (12*I)*c*d^2*f^2*x*Log[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + (6*I)*d^3*f^2*x^2*Log[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + 12*d*f^2*(c + d*x)^2*PolyLog[2, I*Cos[e + f*x] - Sin[e + f*x]] - 12*d*f^2*(c + d*x)^2*PolyLog[2, (-I)*Cos[e + f*x] + Sin[e + f*x]] + 6*c*d^2*f*PolyLog[2, -Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]] + 6*d^3*f*x*PolyLog[2, -Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]] + (24*I)*c*d^2*f*PolyLog[3, I*Cos[e + f*x] - Sin[e + f*x]] + (24*I)*d^3*f*x*PolyLog[3, I*Cos[e + f*x] - Sin[e + f*x]] - (24*I)*c*d^2*f*PolyLog[3, (-I)*Cos[e + f*x] + Sin[e + f*x]] - (24*I)*d^3*f*x*PolyLog[3, (-I)*Cos[e + f*x] + Sin[e + f*x]] + (3*I)*d^3*PolyLog[3, -Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]] - 24*d^3*PolyLog[4, I*Cos[e + f*x] - Sin[e + f*x]] + 24*d^3*PolyLog[4, (-I)*Cos[e + f*x] + Sin[e + f*x]] + (6*I)*c^2*d*f^3*x*Tan[e] + (6*I)*c*d^2*f^3*x^2*Tan[e] + (2*I)*d^3*f^3*x^3*Tan[e]))/f^4))/16","B",0
7,1,505,262,5.8810257,"\int (c+d x)^2 (a+a \sec (e+f x))^2 \, dx","Integrate[(c + d*x)^2*(a + a*Sec[e + f*x])^2,x]","\frac{1}{12} a^2 (\cos (e+f x)+1)^2 \sec ^4\left(\frac{1}{2} (e+f x)\right) \left(-\frac{3 i \left(4 c^2 f^2 \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))+2 i c d f^2 x \tan (e)+8 c d f^2 x \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))+4 d f (c+d x) \text{Li}_2(i \cos (e+f x)-\sin (e+f x))-4 d f (c+d x) \text{Li}_2(\sin (e+f x)-i \cos (e+f x))+2 i c d f \log (i \sin (2 (e+f x))+\cos (2 (e+f x))+1)+2 c d f^2 x+i d^2 f^2 x^2 \tan (e)+4 d^2 f^2 x^2 \tan ^{-1}(\cos (e+f x)+i \sin (e+f x))+d^2 \text{Li}_2(-\cos (2 (e+f x))-i \sin (2 (e+f x)))+4 i d^2 \text{Li}_3(i \cos (e+f x)-\sin (e+f x))-4 i d^2 \text{Li}_3(\sin (e+f x)-i \cos (e+f x))+2 i d^2 f x \log (i \sin (2 (e+f x))+\cos (2 (e+f x))+1)+d^2 f^2 x^2\right)}{f^3}+x \left(3 c^2+3 c d x+d^2 x^2\right)+\frac{3 (c+d x)^2 \sin \left(\frac{f x}{2}\right)}{f \left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{3 (c+d x)^2 \sin \left(\frac{f x}{2}\right)}{f \left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}\right)","\frac{4 i a^2 d (c+d x) \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{4 i a^2 d (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}+\frac{2 a^2 d (c+d x) \log \left(1+e^{2 i (e+f x)}\right)}{f^2}-\frac{4 i a^2 (c+d x)^2 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{a^2 (c+d x)^2 \tan (e+f x)}{f}-\frac{i a^2 (c+d x)^2}{f}+\frac{a^2 (c+d x)^3}{3 d}-\frac{i a^2 d^2 \text{Li}_2\left(-e^{2 i (e+f x)}\right)}{f^3}-\frac{4 a^2 d^2 \text{Li}_3\left(-i e^{i (e+f x)}\right)}{f^3}+\frac{4 a^2 d^2 \text{Li}_3\left(i e^{i (e+f x)}\right)}{f^3}",1,"(a^2*(1 + Cos[e + f*x])^2*Sec[(e + f*x)/2]^4*(x*(3*c^2 + 3*c*d*x + d^2*x^2) + (3*(c + d*x)^2*Sin[(f*x)/2])/(f*(Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) + (3*(c + d*x)^2*Sin[(f*x)/2])/(f*(Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - ((3*I)*(2*c*d*f^2*x + d^2*f^2*x^2 + 4*c^2*f^2*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + 8*c*d*f^2*x*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + 4*d^2*f^2*x^2*ArcTan[Cos[e + f*x] + I*Sin[e + f*x]] + (2*I)*c*d*f*Log[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + (2*I)*d^2*f*x*Log[1 + Cos[2*(e + f*x)] + I*Sin[2*(e + f*x)]] + 4*d*f*(c + d*x)*PolyLog[2, I*Cos[e + f*x] - Sin[e + f*x]] - 4*d*f*(c + d*x)*PolyLog[2, (-I)*Cos[e + f*x] + Sin[e + f*x]] + d^2*PolyLog[2, -Cos[2*(e + f*x)] - I*Sin[2*(e + f*x)]] + (4*I)*d^2*PolyLog[3, I*Cos[e + f*x] - Sin[e + f*x]] - (4*I)*d^2*PolyLog[3, (-I)*Cos[e + f*x] + Sin[e + f*x]] + (2*I)*c*d*f^2*x*Tan[e] + I*d^2*f^2*x^2*Tan[e]))/f^3))/12","A",0
8,1,330,134,5.6054478,"\int (c+d x) (a+a \sec (e+f x))^2 \, dx","Integrate[(c + d*x)*(a + a*Sec[e + f*x])^2,x]","\frac{a^2 (\cos (e+f x)+1)^2 \sec ^4\left(\frac{1}{2} (e+f x)\right) \left(f x (2 c f+2 d \tan (e)+d f x)+\frac{2 f (c+d x) \sin \left(\frac{f x}{2}\right)}{\left(\cos \left(\frac{e}{2}\right)-\sin \left(\frac{e}{2}\right)\right) \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)}+\frac{2 f (c+d x) \sin \left(\frac{f x}{2}\right)}{\left(\sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right)\right) \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}+8 c f \tanh ^{-1}\left(\cos (e) \tan \left(\frac{f x}{2}\right)+\sin (e)\right)-\frac{4 d \csc (e) \left(i \text{Li}_2\left(-e^{i \left(f x-\tan ^{-1}(\cot (e))\right)}\right)-i \text{Li}_2\left(e^{i \left(f x-\tan ^{-1}(\cot (e))\right)}\right)+\left(f x-\tan ^{-1}(\cot (e))\right) \left(\log \left(1-e^{i \left(f x-\tan ^{-1}(\cot (e))\right)}\right)-\log \left(1+e^{i \left(f x-\tan ^{-1}(\cot (e))\right)}\right)\right)\right)}{\sqrt{\csc ^2(e)}}-2 d f x \tan (e)+2 d (f x \tan (e)+\log (\cos (e+f x)))+8 d \tan ^{-1}(\cot (e)) \tanh ^{-1}\left(\cos (e) \tan \left(\frac{f x}{2}\right)+\sin (e)\right)\right)}{8 f^2}","-\frac{4 i a^2 (c+d x) \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{a^2 (c+d x) \tan (e+f x)}{f}+\frac{a^2 (c+d x)^2}{2 d}+\frac{2 i a^2 d \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{2 i a^2 d \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}+\frac{a^2 d \log (\cos (e+f x))}{f^2}",1,"(a^2*(1 + Cos[e + f*x])^2*Sec[(e + f*x)/2]^4*(8*c*f*ArcTanh[Sin[e] + Cos[e]*Tan[(f*x)/2]] + 8*d*ArcTan[Cot[e]]*ArcTanh[Sin[e] + Cos[e]*Tan[(f*x)/2]] - (4*d*Csc[e]*((f*x - ArcTan[Cot[e]])*(Log[1 - E^(I*(f*x - ArcTan[Cot[e]]))] - Log[1 + E^(I*(f*x - ArcTan[Cot[e]]))]) + I*PolyLog[2, -E^(I*(f*x - ArcTan[Cot[e]]))] - I*PolyLog[2, E^(I*(f*x - ArcTan[Cot[e]]))]))/Sqrt[Csc[e]^2] + (2*f*(c + d*x)*Sin[(f*x)/2])/((Cos[e/2] - Sin[e/2])*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])) + (2*f*(c + d*x)*Sin[(f*x)/2])/((Cos[e/2] + Sin[e/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])) - 2*d*f*x*Tan[e] + f*x*(2*c*f + d*f*x + 2*d*Tan[e]) + 2*d*(Log[Cos[e + f*x]] + f*x*Tan[e])))/(8*f^2)","B",0
9,-1,0,23,0,"\int \frac{(a+a \sec (e+f x))^2}{c+d x} \, dx","Integrate[(a + a*Sec[e + f*x])^2/(c + d*x),x]","\text{\$Aborted}","\text{Int}\left(\frac{(a \sec (e+f x)+a)^2}{c+d x},x\right)",0,"$Aborted","F",-1
10,0,0,23,29.1239659,"\int \frac{(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx","Integrate[(a + a*Sec[e + f*x])^2/(c + d*x)^2,x]","\int \frac{(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx","\text{Int}\left(\frac{(a \sec (e+f x)+a)^2}{(c+d x)^2},x\right)",0,"Integrate[(a + a*Sec[e + f*x])^2/(c + d*x)^2, x]","A",-1
11,1,216,152,1.8012689,"\int \frac{(c+d x)^3}{a+a \sec (e+f x)} \, dx","Integrate[(c + d*x)^3/(a + a*Sec[e + f*x]),x]","\frac{\cos \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \left(x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right) \cos \left(\frac{1}{2} (e+f x)\right)+\frac{8 \cos \left(\frac{1}{2} (e+f x)\right) \left(-6 i d^2 f (c+d x) \text{Li}_2\left(-e^{-i (e+f x)}\right)-\frac{i f^3 (c+d x)^3}{1+e^{i e}}-3 d f^2 (c+d x)^2 \log \left(1+e^{-i (e+f x)}\right)-6 d^3 \text{Li}_3\left(-e^{-i (e+f x)}\right)\right)}{f^4}-\frac{4 \sec \left(\frac{e}{2}\right) (c+d x)^3 \sin \left(\frac{f x}{2}\right)}{f}\right)}{2 a (\sec (e+f x)+1)}","\frac{12 i d^2 (c+d x) \text{Li}_2\left(-e^{i (e+f x)}\right)}{a f^3}-\frac{6 d (c+d x)^2 \log \left(1+e^{i (e+f x)}\right)}{a f^2}-\frac{(c+d x)^3 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{a f}+\frac{i (c+d x)^3}{a f}+\frac{(c+d x)^4}{4 a d}-\frac{12 d^3 \text{Li}_3\left(-e^{i (e+f x)}\right)}{a f^4}",1,"(Cos[(e + f*x)/2]*Sec[e + f*x]*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*Cos[(e + f*x)/2] + (8*Cos[(e + f*x)/2]*(((-I)*f^3*(c + d*x)^3)/(1 + E^(I*e)) - 3*d*f^2*(c + d*x)^2*Log[1 + E^((-I)*(e + f*x))] - (6*I)*d^2*f*(c + d*x)*PolyLog[2, -E^((-I)*(e + f*x))] - 6*d^3*PolyLog[3, -E^((-I)*(e + f*x))]))/f^4 - (4*(c + d*x)^3*Sec[e/2]*Sin[(f*x)/2])/f))/(2*a*(1 + Sec[e + f*x]))","A",1
12,1,528,119,6.4904511,"\int \frac{(c+d x)^2}{a+a \sec (e+f x)} \, dx","Integrate[(c + d*x)^2/(a + a*Sec[e + f*x]),x]","\frac{2 x \left(3 c^2+3 c d x+d^2 x^2\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \sec (e+f x)}{3 (a \sec (e+f x)+a)}-\frac{2 \sec \left(\frac{e}{2}\right) \cos \left(\frac{e}{2}+\frac{f x}{2}\right) \sec (e+f x) \left(c^2 \sin \left(\frac{f x}{2}\right)+2 c d x \sin \left(\frac{f x}{2}\right)+d^2 x^2 \sin \left(\frac{f x}{2}\right)\right)}{f (a \sec (e+f x)+a)}-\frac{8 c d \sec \left(\frac{e}{2}\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \sec (e+f x) \left(\frac{1}{2} f x \sin \left(\frac{e}{2}\right)+\cos \left(\frac{e}{2}\right) \log \left(\cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right)-\sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)\right)\right)}{f^2 \left(\sin ^2\left(\frac{e}{2}\right)+\cos ^2\left(\frac{e}{2}\right)\right) (a \sec (e+f x)+a)}-\frac{8 d^2 \csc \left(\frac{e}{2}\right) \sec \left(\frac{e}{2}\right) \cos ^2\left(\frac{e}{2}+\frac{f x}{2}\right) \sec (e+f x) \left(\frac{1}{4} f^2 x^2 e^{-i \tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)}-\frac{\cot \left(\frac{e}{2}\right) \left(i \text{Li}_2\left(e^{2 i \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right)}\right)+\frac{1}{2} i f x \left(-2 \tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)-\pi \right)-2 \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right) \log \left(1-e^{2 i \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right)}\right)-2 \tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right) \log \left(\sin \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right)\right)-\pi  \log \left(1+e^{-i f x}\right)+\pi  \log \left(\cos \left(\frac{f x}{2}\right)\right)\right)}{\sqrt{\cot ^2\left(\frac{e}{2}\right)+1}}\right)}{f^3 \sqrt{\csc ^2\left(\frac{e}{2}\right) \left(\sin ^2\left(\frac{e}{2}\right)+\cos ^2\left(\frac{e}{2}\right)\right)} (a \sec (e+f x)+a)}","-\frac{4 d (c+d x) \log \left(1+e^{i (e+f x)}\right)}{a f^2}-\frac{(c+d x)^2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{a f}+\frac{i (c+d x)^2}{a f}+\frac{(c+d x)^3}{3 a d}+\frac{4 i d^2 \text{Li}_2\left(-e^{i (e+f x)}\right)}{a f^3}",1,"(2*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Cos[e/2 + (f*x)/2]^2*Sec[e + f*x])/(3*(a + a*Sec[e + f*x])) - (8*c*d*Cos[e/2 + (f*x)/2]^2*Sec[e/2]*Sec[e + f*x]*(Cos[e/2]*Log[Cos[e/2]*Cos[(f*x)/2] - Sin[e/2]*Sin[(f*x)/2]] + (f*x*Sin[e/2])/2))/(f^2*(a + a*Sec[e + f*x])*(Cos[e/2]^2 + Sin[e/2]^2)) - (8*d^2*Cos[e/2 + (f*x)/2]^2*Csc[e/2]*((f^2*x^2)/(4*E^(I*ArcTan[Cot[e/2]])) - (Cot[e/2]*((I/2)*f*x*(-Pi - 2*ArcTan[Cot[e/2]]) - Pi*Log[1 + E^((-I)*f*x)] - 2*((f*x)/2 - ArcTan[Cot[e/2]])*Log[1 - E^((2*I)*((f*x)/2 - ArcTan[Cot[e/2]]))] + Pi*Log[Cos[(f*x)/2]] - 2*ArcTan[Cot[e/2]]*Log[Sin[(f*x)/2 - ArcTan[Cot[e/2]]]] + I*PolyLog[2, E^((2*I)*((f*x)/2 - ArcTan[Cot[e/2]]))]))/Sqrt[1 + Cot[e/2]^2])*Sec[e/2]*Sec[e + f*x])/(f^3*(a + a*Sec[e + f*x])*Sqrt[Csc[e/2]^2*(Cos[e/2]^2 + Sin[e/2]^2)]) - (2*Cos[e/2 + (f*x)/2]*Sec[e/2]*Sec[e + f*x]*(c^2*Sin[(f*x)/2] + 2*c*d*x*Sin[(f*x)/2] + d^2*x^2*Sin[(f*x)/2]))/(f*(a + a*Sec[e + f*x]))","B",0
13,1,104,67,0.7497181,"\int \frac{c+d x}{a+a \sec (e+f x)} \, dx","Integrate[(c + d*x)/(a + a*Sec[e + f*x]),x]","\frac{\cos \left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \left(\cos \left(\frac{1}{2} (e+f x)\right) \left(f^2 x (2 c+d x)-2 d f x \tan \left(\frac{e}{2}\right)-4 d \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)-2 f \sec \left(\frac{e}{2}\right) (c+d x) \sin \left(\frac{f x}{2}\right)\right)}{a f^2 (\sec (e+f x)+1)}","-\frac{(c+d x) \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{a f}+\frac{(c+d x)^2}{2 a d}-\frac{2 d \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{a f^2}",1,"(Cos[(e + f*x)/2]*Sec[e + f*x]*(-2*f*(c + d*x)*Sec[e/2]*Sin[(f*x)/2] + Cos[(e + f*x)/2]*(f^2*x*(2*c + d*x) - 4*d*Log[Cos[(e + f*x)/2]] - 2*d*f*x*Tan[e/2])))/(a*f^2*(1 + Sec[e + f*x]))","A",1
14,0,0,23,7.1342766,"\int \frac{1}{(c+d x) (a+a \sec (e+f x))} \, dx","Integrate[1/((c + d*x)*(a + a*Sec[e + f*x])),x]","\int \frac{1}{(c+d x) (a+a \sec (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a \sec (e+f x)+a)},x\right)",0,"Integrate[1/((c + d*x)*(a + a*Sec[e + f*x])), x]","A",-1
15,0,0,23,5.2578194,"\int \frac{1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx","Integrate[1/((c + d*x)^2*(a + a*Sec[e + f*x])),x]","\int \frac{1}{(c+d x)^2 (a+a \sec (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a \sec (e+f x)+a)},x\right)",0,"Integrate[1/((c + d*x)^2*(a + a*Sec[e + f*x])), x]","A",-1
16,1,1447,288,7.5535441,"\int \frac{(c+d x)^3}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c + d*x)^3/(a + a*Sec[e + f*x])^2,x]","-\frac{80 c d^2 \csc \left(\frac{e}{2}\right) \left(\frac{1}{4} e^{-i \tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)} f^2 x^2-\frac{\cot \left(\frac{e}{2}\right) \left(\frac{1}{2} i f x \left(-2 \tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)-\pi \right)-\pi  \log \left(1+e^{-i f x}\right)-2 \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right) \log \left(1-e^{2 i \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right)}\right)+\pi  \log \left(\cos \left(\frac{f x}{2}\right)\right)-2 \tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right) \log \left(\sin \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right)\right)+i \text{Li}_2\left(e^{2 i \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right)}\right)\right)}{\sqrt{\cot ^2\left(\frac{e}{2}\right)+1}}\right) \sec \left(\frac{e}{2}\right) \sec ^2(e+f x) \cos ^4\left(\frac{e}{2}+\frac{f x}{2}\right)}{f^3 (\sec (e+f x) a+a)^2 \sqrt{\csc ^2\left(\frac{e}{2}\right) \left(\cos ^2\left(\frac{e}{2}\right)+\sin ^2\left(\frac{e}{2}\right)\right)}}-\frac{40 c^2 d \sec \left(\frac{e}{2}\right) \sec ^2(e+f x) \left(\cos \left(\frac{e}{2}\right) \log \left(\cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right)-\sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)\right)+\frac{1}{2} f x \sin \left(\frac{e}{2}\right)\right) \cos ^4\left(\frac{e}{2}+\frac{f x}{2}\right)}{f^2 (\sec (e+f x) a+a)^2 \left(\cos ^2\left(\frac{e}{2}\right)+\sin ^2\left(\frac{e}{2}\right)\right)}+\frac{16 d^3 \sec \left(\frac{e}{2}\right) \sec ^2(e+f x) \left(\cos \left(\frac{e}{2}\right) \log \left(\cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right)-\sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)\right)+\frac{1}{2} f x \sin \left(\frac{e}{2}\right)\right) \cos ^4\left(\frac{e}{2}+\frac{f x}{2}\right)}{f^4 (\sec (e+f x) a+a)^2 \left(\cos ^2\left(\frac{e}{2}\right)+\sin ^2\left(\frac{e}{2}\right)\right)}-\frac{20 i d^3 e^{-\frac{i e}{2}} \left(f^2 \left(f x-3 i \left(1+e^{i e}\right) \log \left(1+e^{-i (e+f x)}\right)\right) x^2+6 \left(1+e^{i e}\right) f \text{Li}_2\left(-e^{-i (e+f x)}\right) x-6 i \left(1+e^{i e}\right) \text{Li}_3\left(-e^{-i (e+f x)}\right)\right) \sec \left(\frac{e}{2}\right) \sec ^2(e+f x) \cos ^4\left(\frac{e}{2}+\frac{f x}{2}\right)}{3 f^4 (\sec (e+f x) a+a)^2}+\frac{\sec \left(\frac{e}{2}\right) \sec ^2(e+f x) \left(9 d^3 f^3 \cos \left(\frac{f x}{2}\right) x^4+9 d^3 f^3 \cos \left(e+\frac{f x}{2}\right) x^4+3 d^3 f^3 \cos \left(e+\frac{3 f x}{2}\right) x^4+3 d^3 f^3 \cos \left(2 e+\frac{3 f x}{2}\right) x^4+36 c d^2 f^3 \cos \left(\frac{f x}{2}\right) x^3+36 c d^2 f^3 \cos \left(e+\frac{f x}{2}\right) x^3+12 c d^2 f^3 \cos \left(e+\frac{3 f x}{2}\right) x^3+12 c d^2 f^3 \cos \left(2 e+\frac{3 f x}{2}\right) x^3-72 d^3 f^2 \sin \left(\frac{f x}{2}\right) x^3+48 d^3 f^2 \sin \left(e+\frac{f x}{2}\right) x^3-40 d^3 f^2 \sin \left(e+\frac{3 f x}{2}\right) x^3+54 c^2 d f^3 \cos \left(\frac{f x}{2}\right) x^2-24 d^3 f \cos \left(\frac{f x}{2}\right) x^2+54 c^2 d f^3 \cos \left(e+\frac{f x}{2}\right) x^2-24 d^3 f \cos \left(e+\frac{f x}{2}\right) x^2+18 c^2 d f^3 \cos \left(e+\frac{3 f x}{2}\right) x^2+18 c^2 d f^3 \cos \left(2 e+\frac{3 f x}{2}\right) x^2-216 c d^2 f^2 \sin \left(\frac{f x}{2}\right) x^2+144 c d^2 f^2 \sin \left(e+\frac{f x}{2}\right) x^2-120 c d^2 f^2 \sin \left(e+\frac{3 f x}{2}\right) x^2+36 c^3 f^3 \cos \left(\frac{f x}{2}\right) x-48 c d^2 f \cos \left(\frac{f x}{2}\right) x+36 c^3 f^3 \cos \left(e+\frac{f x}{2}\right) x-48 c d^2 f \cos \left(e+\frac{f x}{2}\right) x+12 c^3 f^3 \cos \left(e+\frac{3 f x}{2}\right) x+12 c^3 f^3 \cos \left(2 e+\frac{3 f x}{2}\right) x+96 d^3 \sin \left(\frac{f x}{2}\right) x-216 c^2 d f^2 \sin \left(\frac{f x}{2}\right) x-48 d^3 \sin \left(e+\frac{f x}{2}\right) x+144 c^2 d f^2 \sin \left(e+\frac{f x}{2}\right) x+48 d^3 \sin \left(e+\frac{3 f x}{2}\right) x-120 c^2 d f^2 \sin \left(e+\frac{3 f x}{2}\right) x-24 c^2 d f \cos \left(\frac{f x}{2}\right)-24 c^2 d f \cos \left(e+\frac{f x}{2}\right)+96 c d^2 \sin \left(\frac{f x}{2}\right)-72 c^3 f^2 \sin \left(\frac{f x}{2}\right)-48 c d^2 \sin \left(e+\frac{f x}{2}\right)+48 c^3 f^2 \sin \left(e+\frac{f x}{2}\right)+48 c d^2 \sin \left(e+\frac{3 f x}{2}\right)-40 c^3 f^2 \sin \left(e+\frac{3 f x}{2}\right)\right) \cos \left(\frac{e}{2}+\frac{f x}{2}\right)}{24 f^3 (\sec (e+f x) a+a)^2}","\frac{20 i d^2 (c+d x) \text{Li}_2\left(-e^{i (e+f x)}\right)}{a^2 f^3}+\frac{2 d^2 (c+d x) \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{a^2 f^3}-\frac{10 d (c+d x)^2 \log \left(1+e^{i (e+f x)}\right)}{a^2 f^2}-\frac{d (c+d x)^2 \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{2 a^2 f^2}-\frac{5 (c+d x)^3 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f}+\frac{(c+d x)^3 \tan \left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{6 a^2 f}+\frac{5 i (c+d x)^3}{3 a^2 f}+\frac{(c+d x)^4}{4 a^2 d}-\frac{20 d^3 \text{Li}_3\left(-e^{i (e+f x)}\right)}{a^2 f^4}+\frac{4 d^3 \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{a^2 f^4}",1,"(((-20*I)/3)*d^3*Cos[e/2 + (f*x)/2]^4*(f^2*x^2*(f*x - (3*I)*(1 + E^(I*e))*Log[1 + E^((-I)*(e + f*x))]) + 6*(1 + E^(I*e))*f*x*PolyLog[2, -E^((-I)*(e + f*x))] - (6*I)*(1 + E^(I*e))*PolyLog[3, -E^((-I)*(e + f*x))])*Sec[e/2]*Sec[e + f*x]^2)/(E^((I/2)*e)*f^4*(a + a*Sec[e + f*x])^2) + (16*d^3*Cos[e/2 + (f*x)/2]^4*Sec[e/2]*Sec[e + f*x]^2*(Cos[e/2]*Log[Cos[e/2]*Cos[(f*x)/2] - Sin[e/2]*Sin[(f*x)/2]] + (f*x*Sin[e/2])/2))/(f^4*(a + a*Sec[e + f*x])^2*(Cos[e/2]^2 + Sin[e/2]^2)) - (40*c^2*d*Cos[e/2 + (f*x)/2]^4*Sec[e/2]*Sec[e + f*x]^2*(Cos[e/2]*Log[Cos[e/2]*Cos[(f*x)/2] - Sin[e/2]*Sin[(f*x)/2]] + (f*x*Sin[e/2])/2))/(f^2*(a + a*Sec[e + f*x])^2*(Cos[e/2]^2 + Sin[e/2]^2)) - (80*c*d^2*Cos[e/2 + (f*x)/2]^4*Csc[e/2]*((f^2*x^2)/(4*E^(I*ArcTan[Cot[e/2]])) - (Cot[e/2]*((I/2)*f*x*(-Pi - 2*ArcTan[Cot[e/2]]) - Pi*Log[1 + E^((-I)*f*x)] - 2*((f*x)/2 - ArcTan[Cot[e/2]])*Log[1 - E^((2*I)*((f*x)/2 - ArcTan[Cot[e/2]]))] + Pi*Log[Cos[(f*x)/2]] - 2*ArcTan[Cot[e/2]]*Log[Sin[(f*x)/2 - ArcTan[Cot[e/2]]]] + I*PolyLog[2, E^((2*I)*((f*x)/2 - ArcTan[Cot[e/2]]))]))/Sqrt[1 + Cot[e/2]^2])*Sec[e/2]*Sec[e + f*x]^2)/(f^3*(a + a*Sec[e + f*x])^2*Sqrt[Csc[e/2]^2*(Cos[e/2]^2 + Sin[e/2]^2)]) + (Cos[e/2 + (f*x)/2]*Sec[e/2]*Sec[e + f*x]^2*(-24*c^2*d*f*Cos[(f*x)/2] - 48*c*d^2*f*x*Cos[(f*x)/2] + 36*c^3*f^3*x*Cos[(f*x)/2] - 24*d^3*f*x^2*Cos[(f*x)/2] + 54*c^2*d*f^3*x^2*Cos[(f*x)/2] + 36*c*d^2*f^3*x^3*Cos[(f*x)/2] + 9*d^3*f^3*x^4*Cos[(f*x)/2] - 24*c^2*d*f*Cos[e + (f*x)/2] - 48*c*d^2*f*x*Cos[e + (f*x)/2] + 36*c^3*f^3*x*Cos[e + (f*x)/2] - 24*d^3*f*x^2*Cos[e + (f*x)/2] + 54*c^2*d*f^3*x^2*Cos[e + (f*x)/2] + 36*c*d^2*f^3*x^3*Cos[e + (f*x)/2] + 9*d^3*f^3*x^4*Cos[e + (f*x)/2] + 12*c^3*f^3*x*Cos[e + (3*f*x)/2] + 18*c^2*d*f^3*x^2*Cos[e + (3*f*x)/2] + 12*c*d^2*f^3*x^3*Cos[e + (3*f*x)/2] + 3*d^3*f^3*x^4*Cos[e + (3*f*x)/2] + 12*c^3*f^3*x*Cos[2*e + (3*f*x)/2] + 18*c^2*d*f^3*x^2*Cos[2*e + (3*f*x)/2] + 12*c*d^2*f^3*x^3*Cos[2*e + (3*f*x)/2] + 3*d^3*f^3*x^4*Cos[2*e + (3*f*x)/2] + 96*c*d^2*Sin[(f*x)/2] - 72*c^3*f^2*Sin[(f*x)/2] + 96*d^3*x*Sin[(f*x)/2] - 216*c^2*d*f^2*x*Sin[(f*x)/2] - 216*c*d^2*f^2*x^2*Sin[(f*x)/2] - 72*d^3*f^2*x^3*Sin[(f*x)/2] - 48*c*d^2*Sin[e + (f*x)/2] + 48*c^3*f^2*Sin[e + (f*x)/2] - 48*d^3*x*Sin[e + (f*x)/2] + 144*c^2*d*f^2*x*Sin[e + (f*x)/2] + 144*c*d^2*f^2*x^2*Sin[e + (f*x)/2] + 48*d^3*f^2*x^3*Sin[e + (f*x)/2] + 48*c*d^2*Sin[e + (3*f*x)/2] - 40*c^3*f^2*Sin[e + (3*f*x)/2] + 48*d^3*x*Sin[e + (3*f*x)/2] - 120*c^2*d*f^2*x*Sin[e + (3*f*x)/2] - 120*c*d^2*f^2*x^2*Sin[e + (3*f*x)/2] - 40*d^3*f^2*x^3*Sin[e + (3*f*x)/2]))/(24*f^3*(a + a*Sec[e + f*x])^2)","B",0
17,1,925,229,6.8547605,"\int \frac{(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c + d*x)^2/(a + a*Sec[e + f*x])^2,x]","-\frac{80 d^2 \csc \left(\frac{e}{2}\right) \left(\frac{1}{4} e^{-i \tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)} f^2 x^2-\frac{\cot \left(\frac{e}{2}\right) \left(\frac{1}{2} i f x \left(-2 \tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)-\pi \right)-\pi  \log \left(1+e^{-i f x}\right)-2 \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right) \log \left(1-e^{2 i \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right)}\right)+\pi  \log \left(\cos \left(\frac{f x}{2}\right)\right)-2 \tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right) \log \left(\sin \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right)\right)+i \text{Li}_2\left(e^{2 i \left(\frac{f x}{2}-\tan ^{-1}\left(\cot \left(\frac{e}{2}\right)\right)\right)}\right)\right)}{\sqrt{\cot ^2\left(\frac{e}{2}\right)+1}}\right) \sec \left(\frac{e}{2}\right) \sec ^2(e+f x) \cos ^4\left(\frac{e}{2}+\frac{f x}{2}\right)}{3 f^3 (\sec (e+f x) a+a)^2 \sqrt{\csc ^2\left(\frac{e}{2}\right) \left(\cos ^2\left(\frac{e}{2}\right)+\sin ^2\left(\frac{e}{2}\right)\right)}}-\frac{80 c d \sec \left(\frac{e}{2}\right) \sec ^2(e+f x) \left(\cos \left(\frac{e}{2}\right) \log \left(\cos \left(\frac{e}{2}\right) \cos \left(\frac{f x}{2}\right)-\sin \left(\frac{e}{2}\right) \sin \left(\frac{f x}{2}\right)\right)+\frac{1}{2} f x \sin \left(\frac{e}{2}\right)\right) \cos ^4\left(\frac{e}{2}+\frac{f x}{2}\right)}{3 f^2 (\sec (e+f x) a+a)^2 \left(\cos ^2\left(\frac{e}{2}\right)+\sin ^2\left(\frac{e}{2}\right)\right)}+\frac{\sec \left(\frac{e}{2}\right) \sec ^2(e+f x) \left(3 d^2 x^3 \cos \left(\frac{f x}{2}\right) f^3+9 c d x^2 \cos \left(\frac{f x}{2}\right) f^3+9 c^2 x \cos \left(\frac{f x}{2}\right) f^3+3 d^2 x^3 \cos \left(e+\frac{f x}{2}\right) f^3+9 c d x^2 \cos \left(e+\frac{f x}{2}\right) f^3+9 c^2 x \cos \left(e+\frac{f x}{2}\right) f^3+d^2 x^3 \cos \left(e+\frac{3 f x}{2}\right) f^3+3 c d x^2 \cos \left(e+\frac{3 f x}{2}\right) f^3+3 c^2 x \cos \left(e+\frac{3 f x}{2}\right) f^3+d^2 x^3 \cos \left(2 e+\frac{3 f x}{2}\right) f^3+3 c d x^2 \cos \left(2 e+\frac{3 f x}{2}\right) f^3+3 c^2 x \cos \left(2 e+\frac{3 f x}{2}\right) f^3-18 c^2 \sin \left(\frac{f x}{2}\right) f^2-18 d^2 x^2 \sin \left(\frac{f x}{2}\right) f^2-36 c d x \sin \left(\frac{f x}{2}\right) f^2+12 c^2 \sin \left(e+\frac{f x}{2}\right) f^2+12 d^2 x^2 \sin \left(e+\frac{f x}{2}\right) f^2+24 c d x \sin \left(e+\frac{f x}{2}\right) f^2-10 c^2 \sin \left(e+\frac{3 f x}{2}\right) f^2-10 d^2 x^2 \sin \left(e+\frac{3 f x}{2}\right) f^2-20 c d x \sin \left(e+\frac{3 f x}{2}\right) f^2-4 c d \cos \left(\frac{f x}{2}\right) f-4 d^2 x \cos \left(\frac{f x}{2}\right) f-4 c d \cos \left(e+\frac{f x}{2}\right) f-4 d^2 x \cos \left(e+\frac{f x}{2}\right) f+8 d^2 \sin \left(\frac{f x}{2}\right)-4 d^2 \sin \left(e+\frac{f x}{2}\right)+4 d^2 \sin \left(e+\frac{3 f x}{2}\right)\right) \cos \left(\frac{e}{2}+\frac{f x}{2}\right)}{6 f^3 (\sec (e+f x) a+a)^2}","-\frac{20 d (c+d x) \log \left(1+e^{i (e+f x)}\right)}{3 a^2 f^2}-\frac{d (c+d x) \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f^2}-\frac{5 (c+d x)^2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f}+\frac{(c+d x)^2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{6 a^2 f}+\frac{5 i (c+d x)^2}{3 a^2 f}+\frac{(c+d x)^3}{3 a^2 d}+\frac{20 i d^2 \text{Li}_2\left(-e^{i (e+f x)}\right)}{3 a^2 f^3}+\frac{2 d^2 \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f^3}",1,"(-80*c*d*Cos[e/2 + (f*x)/2]^4*Sec[e/2]*Sec[e + f*x]^2*(Cos[e/2]*Log[Cos[e/2]*Cos[(f*x)/2] - Sin[e/2]*Sin[(f*x)/2]] + (f*x*Sin[e/2])/2))/(3*f^2*(a + a*Sec[e + f*x])^2*(Cos[e/2]^2 + Sin[e/2]^2)) - (80*d^2*Cos[e/2 + (f*x)/2]^4*Csc[e/2]*((f^2*x^2)/(4*E^(I*ArcTan[Cot[e/2]])) - (Cot[e/2]*((I/2)*f*x*(-Pi - 2*ArcTan[Cot[e/2]]) - Pi*Log[1 + E^((-I)*f*x)] - 2*((f*x)/2 - ArcTan[Cot[e/2]])*Log[1 - E^((2*I)*((f*x)/2 - ArcTan[Cot[e/2]]))] + Pi*Log[Cos[(f*x)/2]] - 2*ArcTan[Cot[e/2]]*Log[Sin[(f*x)/2 - ArcTan[Cot[e/2]]]] + I*PolyLog[2, E^((2*I)*((f*x)/2 - ArcTan[Cot[e/2]]))]))/Sqrt[1 + Cot[e/2]^2])*Sec[e/2]*Sec[e + f*x]^2)/(3*f^3*(a + a*Sec[e + f*x])^2*Sqrt[Csc[e/2]^2*(Cos[e/2]^2 + Sin[e/2]^2)]) + (Cos[e/2 + (f*x)/2]*Sec[e/2]*Sec[e + f*x]^2*(-4*c*d*f*Cos[(f*x)/2] - 4*d^2*f*x*Cos[(f*x)/2] + 9*c^2*f^3*x*Cos[(f*x)/2] + 9*c*d*f^3*x^2*Cos[(f*x)/2] + 3*d^2*f^3*x^3*Cos[(f*x)/2] - 4*c*d*f*Cos[e + (f*x)/2] - 4*d^2*f*x*Cos[e + (f*x)/2] + 9*c^2*f^3*x*Cos[e + (f*x)/2] + 9*c*d*f^3*x^2*Cos[e + (f*x)/2] + 3*d^2*f^3*x^3*Cos[e + (f*x)/2] + 3*c^2*f^3*x*Cos[e + (3*f*x)/2] + 3*c*d*f^3*x^2*Cos[e + (3*f*x)/2] + d^2*f^3*x^3*Cos[e + (3*f*x)/2] + 3*c^2*f^3*x*Cos[2*e + (3*f*x)/2] + 3*c*d*f^3*x^2*Cos[2*e + (3*f*x)/2] + d^2*f^3*x^3*Cos[2*e + (3*f*x)/2] + 8*d^2*Sin[(f*x)/2] - 18*c^2*f^2*Sin[(f*x)/2] - 36*c*d*f^2*x*Sin[(f*x)/2] - 18*d^2*f^2*x^2*Sin[(f*x)/2] - 4*d^2*Sin[e + (f*x)/2] + 12*c^2*f^2*Sin[e + (f*x)/2] + 24*c*d*f^2*x*Sin[e + (f*x)/2] + 12*d^2*f^2*x^2*Sin[e + (f*x)/2] + 4*d^2*Sin[e + (3*f*x)/2] - 10*c^2*f^2*Sin[e + (3*f*x)/2] - 20*c*d*f^2*x*Sin[e + (3*f*x)/2] - 10*d^2*f^2*x^2*Sin[e + (3*f*x)/2]))/(6*f^3*(a + a*Sec[e + f*x])^2)","B",0
18,1,172,140,1.6485596,"\int \frac{c+d x}{(a+a \sec (e+f x))^2} \, dx","Integrate[(c + d*x)/(a + a*Sec[e + f*x])^2,x]","\frac{2 \cos \left(\frac{1}{2} (e+f x)\right) \sec ^2(e+f x) \left(\cos ^3\left(\frac{1}{2} (e+f x)\right) \left(3 f^2 x (2 c+d x)-10 d f x \tan \left(\frac{e}{2}\right)-20 d \log \left(\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)+\cos \left(\frac{1}{2} (e+f x)\right) \left(f \tan \left(\frac{e}{2}\right) (c+d x)-d\right)+f \sec \left(\frac{e}{2}\right) (c+d x) \sin \left(\frac{f x}{2}\right)-10 f \sec \left(\frac{e}{2}\right) (c+d x) \sin \left(\frac{f x}{2}\right) \cos ^2\left(\frac{1}{2} (e+f x)\right)\right)}{3 a^2 f^2 (\sec (e+f x)+1)^2}","-\frac{5 (c+d x) \tan \left(\frac{e}{2}+\frac{f x}{2}\right)}{3 a^2 f}+\frac{(c+d x) \tan \left(\frac{e}{2}+\frac{f x}{2}\right) \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{6 a^2 f}+\frac{(c+d x)^2}{2 a^2 d}-\frac{d \sec ^2\left(\frac{e}{2}+\frac{f x}{2}\right)}{6 a^2 f^2}-\frac{10 d \log \left(\cos \left(\frac{e}{2}+\frac{f x}{2}\right)\right)}{3 a^2 f^2}",1,"(2*Cos[(e + f*x)/2]*Sec[e + f*x]^2*(f*(c + d*x)*Sec[e/2]*Sin[(f*x)/2] - 10*f*(c + d*x)*Cos[(e + f*x)/2]^2*Sec[e/2]*Sin[(f*x)/2] + Cos[(e + f*x)/2]^3*(3*f^2*x*(2*c + d*x) - 20*d*Log[Cos[(e + f*x)/2]] - 10*d*f*x*Tan[e/2]) + Cos[(e + f*x)/2]*(-d + f*(c + d*x)*Tan[e/2])))/(3*a^2*f^2*(1 + Sec[e + f*x])^2)","A",1
19,0,0,23,15.2397986,"\int \frac{1}{(c+d x) (a+a \sec (e+f x))^2} \, dx","Integrate[1/((c + d*x)*(a + a*Sec[e + f*x])^2),x]","\int \frac{1}{(c+d x) (a+a \sec (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a \sec (e+f x)+a)^2},x\right)",0,"Integrate[1/((c + d*x)*(a + a*Sec[e + f*x])^2), x]","A",-1
20,0,0,23,17.6083967,"\int \frac{1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx","Integrate[1/((c + d*x)^2*(a + a*Sec[e + f*x])^2),x]","\int \frac{1}{(c+d x)^2 (a+a \sec (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a \sec (e+f x)+a)^2},x\right)",0,"Integrate[1/((c + d*x)^2*(a + a*Sec[e + f*x])^2), x]","A",-1
21,0,0,23,0.8550027,"\int (c+d x)^m (a+a \sec (e+f x))^n \, dx","Integrate[(c + d*x)^m*(a + a*Sec[e + f*x])^n,x]","\int (c+d x)^m (a+a \sec (e+f x))^n \, dx","\text{Int}\left((c+d x)^m (a \sec (e+f x)+a)^n,x\right)",0,"Integrate[(c + d*x)^m*(a + a*Sec[e + f*x])^n, x]","A",-1
22,0,0,21,6.667796,"\int (c+d x)^m (a+a \sec (e+f x)) \, dx","Integrate[(c + d*x)^m*(a + a*Sec[e + f*x]),x]","\int (c+d x)^m (a+a \sec (e+f x)) \, dx","\text{Int}\left((c+d x)^m (a \sec (e+f x)+a),x\right)",0,"Integrate[(c + d*x)^m*(a + a*Sec[e + f*x]), x]","A",-1
23,0,0,23,0.9911241,"\int \frac{(c+d x)^m}{a+a \sec (e+f x)} \, dx","Integrate[(c + d*x)^m/(a + a*Sec[e + f*x]),x]","\int \frac{(c+d x)^m}{a+a \sec (e+f x)} \, dx","\text{Int}\left(\frac{(c+d x)^m}{a \sec (e+f x)+a},x\right)",0,"Integrate[(c + d*x)^m/(a + a*Sec[e + f*x]), x]","A",-1
24,1,365,227,0.4933927,"\int (c+d x)^3 (a+b \sec (e+f x)) \, dx","Integrate[(c + d*x)^3*(a + b*Sec[e + f*x]),x]","\frac{4 a c^3 f^4 x+6 a c^2 d f^4 x^2+4 a c d^2 f^4 x^3+a d^3 f^4 x^4+4 b c^3 f^3 \tanh ^{-1}(\sin (e+f x))-24 i b c^2 d f^3 x \tan ^{-1}\left(e^{i (e+f x)}\right)-24 i b c d^2 f^3 x^2 \tan ^{-1}\left(e^{i (e+f x)}\right)-24 b c d^2 f \text{Li}_3\left(-i e^{i (e+f x)}\right)+24 b c d^2 f \text{Li}_3\left(i e^{i (e+f x)}\right)+12 i b d f^2 (c+d x)^2 \text{Li}_2\left(-i e^{i (e+f x)}\right)-12 i b d f^2 (c+d x)^2 \text{Li}_2\left(i e^{i (e+f x)}\right)-8 i b d^3 f^3 x^3 \tan ^{-1}\left(e^{i (e+f x)}\right)-24 b d^3 f x \text{Li}_3\left(-i e^{i (e+f x)}\right)+24 b d^3 f x \text{Li}_3\left(i e^{i (e+f x)}\right)-24 i b d^3 \text{Li}_4\left(-i e^{i (e+f x)}\right)+24 i b d^3 \text{Li}_4\left(i e^{i (e+f x)}\right)}{4 f^4}","\frac{a (c+d x)^4}{4 d}-\frac{6 b d^2 (c+d x) \text{Li}_3\left(-i e^{i (e+f x)}\right)}{f^3}+\frac{6 b d^2 (c+d x) \text{Li}_3\left(i e^{i (e+f x)}\right)}{f^3}+\frac{3 i b d (c+d x)^2 \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{3 i b d (c+d x)^2 \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}-\frac{2 i b (c+d x)^3 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}-\frac{6 i b d^3 \text{Li}_4\left(-i e^{i (e+f x)}\right)}{f^4}+\frac{6 i b d^3 \text{Li}_4\left(i e^{i (e+f x)}\right)}{f^4}",1,"(4*a*c^3*f^4*x + 6*a*c^2*d*f^4*x^2 + 4*a*c*d^2*f^4*x^3 + a*d^3*f^4*x^4 - (24*I)*b*c^2*d*f^3*x*ArcTan[E^(I*(e + f*x))] - (24*I)*b*c*d^2*f^3*x^2*ArcTan[E^(I*(e + f*x))] - (8*I)*b*d^3*f^3*x^3*ArcTan[E^(I*(e + f*x))] + 4*b*c^3*f^3*ArcTanh[Sin[e + f*x]] + (12*I)*b*d*f^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))] - (12*I)*b*d*f^2*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))] - 24*b*c*d^2*f*PolyLog[3, (-I)*E^(I*(e + f*x))] - 24*b*d^3*f*x*PolyLog[3, (-I)*E^(I*(e + f*x))] + 24*b*c*d^2*f*PolyLog[3, I*E^(I*(e + f*x))] + 24*b*d^3*f*x*PolyLog[3, I*E^(I*(e + f*x))] - (24*I)*b*d^3*PolyLog[4, (-I)*E^(I*(e + f*x))] + (24*I)*b*d^3*PolyLog[4, I*E^(I*(e + f*x))])/(4*f^4)","A",0
25,1,203,157,0.2684476,"\int (c+d x)^2 (a+b \sec (e+f x)) \, dx","Integrate[(c + d*x)^2*(a + b*Sec[e + f*x]),x]","a c^2 x+a c d x^2+\frac{1}{3} a d^2 x^3+\frac{b c^2 \tanh ^{-1}(\sin (e+f x))}{f}+\frac{2 i b d (c+d x) \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{2 i b d (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}-\frac{4 i b c d x \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}-\frac{2 b d^2 \text{Li}_3\left(-i e^{i (e+f x)}\right)}{f^3}+\frac{2 b d^2 \text{Li}_3\left(i e^{i (e+f x)}\right)}{f^3}-\frac{2 i b d^2 x^2 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}","\frac{a (c+d x)^3}{3 d}+\frac{2 i b d (c+d x) \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{2 i b d (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}-\frac{2 i b (c+d x)^2 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}-\frac{2 b d^2 \text{Li}_3\left(-i e^{i (e+f x)}\right)}{f^3}+\frac{2 b d^2 \text{Li}_3\left(i e^{i (e+f x)}\right)}{f^3}",1,"a*c^2*x + a*c*d*x^2 + (a*d^2*x^3)/3 - ((4*I)*b*c*d*x*ArcTan[E^(I*(e + f*x))])/f - ((2*I)*b*d^2*x^2*ArcTan[E^(I*(e + f*x))])/f + (b*c^2*ArcTanh[Sin[e + f*x]])/f + ((2*I)*b*d*(c + d*x)*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - ((2*I)*b*d*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))])/f^2 - (2*b*d^2*PolyLog[3, (-I)*E^(I*(e + f*x))])/f^3 + (2*b*d^2*PolyLog[3, I*E^(I*(e + f*x))])/f^3","A",0
26,1,104,93,0.0132163,"\int (c+d x) (a+b \sec (e+f x)) \, dx","Integrate[(c + d*x)*(a + b*Sec[e + f*x]),x]","a c x+\frac{1}{2} a d x^2+\frac{b c \tanh ^{-1}(\sin (e+f x))}{f}+\frac{i b d \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{i b d \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}-\frac{2 i b d x \tan ^{-1}\left(e^{i e+i f x}\right)}{f}","\frac{a (c+d x)^2}{2 d}-\frac{2 i b (c+d x) \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{i b d \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{i b d \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}",1,"a*c*x + (a*d*x^2)/2 - ((2*I)*b*d*x*ArcTan[E^(I*e + I*f*x)])/f + (b*c*ArcTanh[Sin[e + f*x]])/f + (I*b*d*PolyLog[2, (-I)*E^(I*(e + f*x))])/f^2 - (I*b*d*PolyLog[2, I*E^(I*(e + f*x))])/f^2","A",0
27,0,0,21,1.0031816,"\int \frac{a+b \sec (e+f x)}{c+d x} \, dx","Integrate[(a + b*Sec[e + f*x])/(c + d*x),x]","\int \frac{a+b \sec (e+f x)}{c+d x} \, dx","\text{Int}\left(\frac{a+b \sec (e+f x)}{c+d x},x\right)",0,"Integrate[(a + b*Sec[e + f*x])/(c + d*x), x]","A",-1
28,0,0,21,1.3852215,"\int \frac{a+b \sec (e+f x)}{(c+d x)^2} \, dx","Integrate[(a + b*Sec[e + f*x])/(c + d*x)^2,x]","\int \frac{a+b \sec (e+f x)}{(c+d x)^2} \, dx","\text{Int}\left(\frac{a+b \sec (e+f x)}{(c+d x)^2},x\right)",0,"Integrate[(a + b*Sec[e + f*x])/(c + d*x)^2, x]","A",-1
29,1,646,364,2.5334017,"\int (c+d x)^3 (a+b \sec (e+f x))^2 \, dx","Integrate[(c + d*x)^3*(a + b*Sec[e + f*x])^2,x]","\frac{4 a^2 c^3 f^4 x+6 a^2 c^2 d f^4 x^2+4 a^2 c d^2 f^4 x^3+a^2 d^3 f^4 x^4+8 a b c^3 f^3 \tanh ^{-1}(\sin (e+f x))-48 i a b c^2 d f^3 x \tan ^{-1}\left(e^{i (e+f x)}\right)-48 i a b c d^2 f^3 x^2 \tan ^{-1}\left(e^{i (e+f x)}\right)-48 a b c d^2 f \text{Li}_3\left(-i e^{i (e+f x)}\right)+48 a b c d^2 f \text{Li}_3\left(i e^{i (e+f x)}\right)+24 i a b d f^2 (c+d x)^2 \text{Li}_2\left(-i e^{i (e+f x)}\right)-24 i a b d f^2 (c+d x)^2 \text{Li}_2\left(i e^{i (e+f x)}\right)-16 i a b d^3 f^3 x^3 \tan ^{-1}\left(e^{i (e+f x)}\right)-48 a b d^3 f x \text{Li}_3\left(-i e^{i (e+f x)}\right)+48 a b d^3 f x \text{Li}_3\left(i e^{i (e+f x)}\right)-48 i a b d^3 \text{Li}_4\left(-i e^{i (e+f x)}\right)+48 i a b d^3 \text{Li}_4\left(i e^{i (e+f x)}\right)+4 b^2 c^3 f^3 \tan (e+f x)+12 b^2 c^2 d f^3 x \tan (e+f x)+12 b^2 c^2 d f^2 \log (\cos (e+f x))+12 b^2 c d^2 f^3 x^2 \tan (e+f x)+24 b^2 c d^2 f^2 x \log \left(1+e^{2 i (e+f x)}\right)-12 i b^2 c d^2 f \text{Li}_2\left(-e^{2 i (e+f x)}\right)-12 i b^2 c d^2 f^3 x^2+4 b^2 d^3 f^3 x^3 \tan (e+f x)+12 b^2 d^3 f^2 x^2 \log \left(1+e^{2 i (e+f x)}\right)-12 i b^2 d^3 f x \text{Li}_2\left(-e^{2 i (e+f x)}\right)+6 b^2 d^3 \text{Li}_3\left(-e^{2 i (e+f x)}\right)-4 i b^2 d^3 f^3 x^3}{4 f^4}","\frac{a^2 (c+d x)^4}{4 d}-\frac{12 a b d^2 (c+d x) \text{Li}_3\left(-i e^{i (e+f x)}\right)}{f^3}+\frac{12 a b d^2 (c+d x) \text{Li}_3\left(i e^{i (e+f x)}\right)}{f^3}+\frac{6 i a b d (c+d x)^2 \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{6 i a b d (c+d x)^2 \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}-\frac{4 i a b (c+d x)^3 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}-\frac{12 i a b d^3 \text{Li}_4\left(-i e^{i (e+f x)}\right)}{f^4}+\frac{12 i a b d^3 \text{Li}_4\left(i e^{i (e+f x)}\right)}{f^4}-\frac{3 i b^2 d^2 (c+d x) \text{Li}_2\left(-e^{2 i (e+f x)}\right)}{f^3}+\frac{3 b^2 d (c+d x)^2 \log \left(1+e^{2 i (e+f x)}\right)}{f^2}+\frac{b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac{i b^2 (c+d x)^3}{f}+\frac{3 b^2 d^3 \text{Li}_3\left(-e^{2 i (e+f x)}\right)}{2 f^4}",1,"(4*a^2*c^3*f^4*x - (12*I)*b^2*c*d^2*f^3*x^2 + 6*a^2*c^2*d*f^4*x^2 - (4*I)*b^2*d^3*f^3*x^3 + 4*a^2*c*d^2*f^4*x^3 + a^2*d^3*f^4*x^4 - (48*I)*a*b*c^2*d*f^3*x*ArcTan[E^(I*(e + f*x))] - (48*I)*a*b*c*d^2*f^3*x^2*ArcTan[E^(I*(e + f*x))] - (16*I)*a*b*d^3*f^3*x^3*ArcTan[E^(I*(e + f*x))] + 8*a*b*c^3*f^3*ArcTanh[Sin[e + f*x]] + 24*b^2*c*d^2*f^2*x*Log[1 + E^((2*I)*(e + f*x))] + 12*b^2*d^3*f^2*x^2*Log[1 + E^((2*I)*(e + f*x))] + 12*b^2*c^2*d*f^2*Log[Cos[e + f*x]] + (24*I)*a*b*d*f^2*(c + d*x)^2*PolyLog[2, (-I)*E^(I*(e + f*x))] - (24*I)*a*b*d*f^2*(c + d*x)^2*PolyLog[2, I*E^(I*(e + f*x))] - (12*I)*b^2*c*d^2*f*PolyLog[2, -E^((2*I)*(e + f*x))] - (12*I)*b^2*d^3*f*x*PolyLog[2, -E^((2*I)*(e + f*x))] - 48*a*b*c*d^2*f*PolyLog[3, (-I)*E^(I*(e + f*x))] - 48*a*b*d^3*f*x*PolyLog[3, (-I)*E^(I*(e + f*x))] + 48*a*b*c*d^2*f*PolyLog[3, I*E^(I*(e + f*x))] + 48*a*b*d^3*f*x*PolyLog[3, I*E^(I*(e + f*x))] + 6*b^2*d^3*PolyLog[3, -E^((2*I)*(e + f*x))] - (48*I)*a*b*d^3*PolyLog[4, (-I)*E^(I*(e + f*x))] + (48*I)*a*b*d^3*PolyLog[4, I*E^(I*(e + f*x))] + 4*b^2*c^3*f^3*Tan[e + f*x] + 12*b^2*c^2*d*f^3*x*Tan[e + f*x] + 12*b^2*c*d^2*f^3*x^2*Tan[e + f*x] + 4*b^2*d^3*f^3*x^3*Tan[e + f*x])/(4*f^4)","A",0
30,1,356,257,1.4604453,"\int (c+d x)^2 (a+b \sec (e+f x))^2 \, dx","Integrate[(c + d*x)^2*(a + b*Sec[e + f*x])^2,x]","\frac{3 a^2 c^2 f^3 x+3 a^2 c d f^3 x^2+a^2 d^2 f^3 x^3+6 a b c^2 f^2 \tanh ^{-1}(\sin (e+f x))-24 i a b c d f^2 x \tan ^{-1}\left(e^{i (e+f x)}\right)+12 i a b d f (c+d x) \text{Li}_2\left(-i e^{i (e+f x)}\right)-12 i a b d f (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)-12 i a b d^2 f^2 x^2 \tan ^{-1}\left(e^{i (e+f x)}\right)-12 a b d^2 \text{Li}_3\left(-i e^{i (e+f x)}\right)+12 a b d^2 \text{Li}_3\left(i e^{i (e+f x)}\right)+3 b^2 c^2 f^2 \tan (e+f x)+6 b^2 c d f^2 x \tan (e+f x)+6 b^2 c d f \log (\cos (e+f x))+3 b^2 d^2 f^2 x^2 \tan (e+f x)-3 i b^2 d^2 \text{Li}_2\left(-e^{2 i (e+f x)}\right)+6 b^2 d^2 f x \log \left(1+e^{2 i (e+f x)}\right)-3 i b^2 d^2 f^2 x^2}{3 f^3}","\frac{a^2 (c+d x)^3}{3 d}+\frac{4 i a b d (c+d x) \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{4 i a b d (c+d x) \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}-\frac{4 i a b (c+d x)^2 \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}-\frac{4 a b d^2 \text{Li}_3\left(-i e^{i (e+f x)}\right)}{f^3}+\frac{4 a b d^2 \text{Li}_3\left(i e^{i (e+f x)}\right)}{f^3}+\frac{2 b^2 d (c+d x) \log \left(1+e^{2 i (e+f x)}\right)}{f^2}+\frac{b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac{i b^2 (c+d x)^2}{f}-\frac{i b^2 d^2 \text{Li}_2\left(-e^{2 i (e+f x)}\right)}{f^3}",1,"(3*a^2*c^2*f^3*x - (3*I)*b^2*d^2*f^2*x^2 + 3*a^2*c*d*f^3*x^2 + a^2*d^2*f^3*x^3 - (24*I)*a*b*c*d*f^2*x*ArcTan[E^(I*(e + f*x))] - (12*I)*a*b*d^2*f^2*x^2*ArcTan[E^(I*(e + f*x))] + 6*a*b*c^2*f^2*ArcTanh[Sin[e + f*x]] + 6*b^2*d^2*f*x*Log[1 + E^((2*I)*(e + f*x))] + 6*b^2*c*d*f*Log[Cos[e + f*x]] + (12*I)*a*b*d*f*(c + d*x)*PolyLog[2, (-I)*E^(I*(e + f*x))] - (12*I)*a*b*d*f*(c + d*x)*PolyLog[2, I*E^(I*(e + f*x))] - (3*I)*b^2*d^2*PolyLog[2, -E^((2*I)*(e + f*x))] - 12*a*b*d^2*PolyLog[3, (-I)*E^(I*(e + f*x))] + 12*a*b*d^2*PolyLog[3, I*E^(I*(e + f*x))] + 3*b^2*c^2*f^2*Tan[e + f*x] + 6*b^2*c*d*f^2*x*Tan[e + f*x] + 3*b^2*d^2*f^2*x^2*Tan[e + f*x])/(3*f^3)","A",0
31,1,151,131,0.5265763,"\int (c+d x) (a+b \sec (e+f x))^2 \, dx","Integrate[(c + d*x)*(a + b*Sec[e + f*x])^2,x]","\frac{2 a^2 c f^2 x+a^2 d f^2 x^2+4 a b c f \tanh ^{-1}(\sin (e+f x))+4 i a b d \text{Li}_2\left(-i e^{i (e+f x)}\right)-4 i a b d \text{Li}_2\left(i e^{i (e+f x)}\right)-8 i a b d f x \tan ^{-1}\left(e^{i (e+f x)}\right)+2 b^2 c f \tan (e+f x)+2 b^2 d f x \tan (e+f x)+2 b^2 d \log (\cos (e+f x))}{2 f^2}","\frac{a^2 (c+d x)^2}{2 d}-\frac{4 i a b (c+d x) \tan ^{-1}\left(e^{i (e+f x)}\right)}{f}+\frac{2 i a b d \text{Li}_2\left(-i e^{i (e+f x)}\right)}{f^2}-\frac{2 i a b d \text{Li}_2\left(i e^{i (e+f x)}\right)}{f^2}+\frac{b^2 (c+d x) \tan (e+f x)}{f}+\frac{b^2 d \log (\cos (e+f x))}{f^2}",1,"(2*a^2*c*f^2*x + a^2*d*f^2*x^2 - (8*I)*a*b*d*f*x*ArcTan[E^(I*(e + f*x))] + 4*a*b*c*f*ArcTanh[Sin[e + f*x]] + 2*b^2*d*Log[Cos[e + f*x]] + (4*I)*a*b*d*PolyLog[2, (-I)*E^(I*(e + f*x))] - (4*I)*a*b*d*PolyLog[2, I*E^(I*(e + f*x))] + 2*b^2*c*f*Tan[e + f*x] + 2*b^2*d*f*x*Tan[e + f*x])/(2*f^2)","A",0
32,0,0,23,42.6694641,"\int \frac{(a+b \sec (e+f x))^2}{c+d x} \, dx","Integrate[(a + b*Sec[e + f*x])^2/(c + d*x),x]","\int \frac{(a+b \sec (e+f x))^2}{c+d x} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^2}{c+d x},x\right)",0,"Integrate[(a + b*Sec[e + f*x])^2/(c + d*x), x]","A",-1
33,0,0,23,32.0683431,"\int \frac{(a+b \sec (e+f x))^2}{(c+d x)^2} \, dx","Integrate[(a + b*Sec[e + f*x])^2/(c + d*x)^2,x]","\int \frac{(a+b \sec (e+f x))^2}{(c+d x)^2} \, dx","\text{Int}\left(\frac{(a+b \sec (e+f x))^2}{(c+d x)^2},x\right)",0,"Integrate[(a + b*Sec[e + f*x])^2/(c + d*x)^2, x]","A",-1
34,1,449,526,1.1325743,"\int \frac{(c+d x)^3}{a+b \sec (e+f x)} \, dx","Integrate[(c + d*x)^3/(a + b*Sec[e + f*x]),x]","\frac{\sec (e+f x) (a \cos (e+f x)+b) \left(x \left(4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right)+\frac{4 i b \left(\frac{3 i d \left(f^2 (c+d x)^2 \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)+2 i d f (c+d x) \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)-2 d^2 \text{Li}_4\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)\right)}{f^3}+\frac{3 d \left(2 d \left(f (c+d x) \text{Li}_3\left(\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right)+i d \text{Li}_4\left(\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right)\right)-i f^2 (c+d x)^2 \text{Li}_2\left(\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right)\right)}{f^3}+(c+d x)^3 \log \left(1-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right)-(c+d x)^3 \log \left(1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right)\right)}{f \sqrt{b^2-a^2}}\right)}{4 a (a+b \sec (e+f x))}","\frac{6 i b d^2 (c+d x) \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a f^3 \sqrt{b^2-a^2}}-\frac{6 i b d^2 (c+d x) \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a f^3 \sqrt{b^2-a^2}}+\frac{3 b d (c+d x)^2 \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a f^2 \sqrt{b^2-a^2}}-\frac{3 b d (c+d x)^2 \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a f^2 \sqrt{b^2-a^2}}+\frac{i b (c+d x)^3 \log \left(1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a f \sqrt{b^2-a^2}}-\frac{i b (c+d x)^3 \log \left(1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right)}{a f \sqrt{b^2-a^2}}-\frac{6 b d^3 \text{Li}_4\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a f^4 \sqrt{b^2-a^2}}+\frac{6 b d^3 \text{Li}_4\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a f^4 \sqrt{b^2-a^2}}+\frac{(c+d x)^4}{4 a d}",1,"((b + a*Cos[e + f*x])*(x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + ((4*I)*b*((c + d*x)^3*Log[1 - (a*E^(I*(e + f*x)))/(-b + Sqrt[-a^2 + b^2])] - (c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])] + (3*d*((-I)*f^2*(c + d*x)^2*PolyLog[2, (a*E^(I*(e + f*x)))/(-b + Sqrt[-a^2 + b^2])] + 2*d*(f*(c + d*x)*PolyLog[3, (a*E^(I*(e + f*x)))/(-b + Sqrt[-a^2 + b^2])] + I*d*PolyLog[4, (a*E^(I*(e + f*x)))/(-b + Sqrt[-a^2 + b^2])])))/f^3 + ((3*I)*d*(f^2*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))] + (2*I)*d*f*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))] - 2*d^2*PolyLog[4, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))]))/f^3))/(Sqrt[-a^2 + b^2]*f))*Sec[e + f*x])/(4*a*(a + b*Sec[e + f*x]))","A",1
35,1,338,394,0.8069281,"\int \frac{(c+d x)^2}{a+b \sec (e+f x)} \, dx","Integrate[(c + d*x)^2/(a + b*Sec[e + f*x]),x]","\frac{\sec (e+f x) (a \cos (e+f x)+b) \left(x \left(3 c^2+3 c d x+d^2 x^2\right)+\frac{3 i b \left(\frac{2 d \left(d \text{Li}_3\left(\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right)-i f (c+d x) \text{Li}_2\left(\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right)\right)}{f^2}+\frac{2 i d \left(f (c+d x) \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)+i d \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)\right)}{f^2}+(c+d x)^2 \log \left(1-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right)-(c+d x)^2 \log \left(1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right)\right)}{f \sqrt{b^2-a^2}}\right)}{3 a (a+b \sec (e+f x))}","\frac{2 b d (c+d x) \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a f^2 \sqrt{b^2-a^2}}-\frac{2 b d (c+d x) \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a f^2 \sqrt{b^2-a^2}}+\frac{i b (c+d x)^2 \log \left(1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a f \sqrt{b^2-a^2}}-\frac{i b (c+d x)^2 \log \left(1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right)}{a f \sqrt{b^2-a^2}}+\frac{2 i b d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a f^3 \sqrt{b^2-a^2}}-\frac{2 i b d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a f^3 \sqrt{b^2-a^2}}+\frac{(c+d x)^3}{3 a d}",1,"((b + a*Cos[e + f*x])*(x*(3*c^2 + 3*c*d*x + d^2*x^2) + ((3*I)*b*((c + d*x)^2*Log[1 - (a*E^(I*(e + f*x)))/(-b + Sqrt[-a^2 + b^2])] - (c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])] + (2*d*((-I)*f*(c + d*x)*PolyLog[2, (a*E^(I*(e + f*x)))/(-b + Sqrt[-a^2 + b^2])] + d*PolyLog[3, (a*E^(I*(e + f*x)))/(-b + Sqrt[-a^2 + b^2])]))/f^2 + ((2*I)*d*(f*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))] + I*d*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))]))/f^2))/(Sqrt[-a^2 + b^2]*f))*Sec[e + f*x])/(3*a*(a + b*Sec[e + f*x]))","A",1
36,1,214,257,0.4794306,"\int \frac{c+d x}{a+b \sec (e+f x)} \, dx","Integrate[(c + d*x)/(a + b*Sec[e + f*x]),x]","\frac{f \left(2 i b (c+d x) \log \left(1-\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right)-2 i b (c+d x) \log \left(1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right)+f x \sqrt{b^2-a^2} (2 c+d x)\right)+2 b d \text{Li}_2\left(\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}-b}\right)-2 b d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{2 a f^2 \sqrt{b^2-a^2}}","\frac{i b (c+d x) \log \left(1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a f \sqrt{b^2-a^2}}-\frac{i b (c+d x) \log \left(1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right)}{a f \sqrt{b^2-a^2}}+\frac{b d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a f^2 \sqrt{b^2-a^2}}-\frac{b d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a f^2 \sqrt{b^2-a^2}}+\frac{(c+d x)^2}{2 a d}",1,"(f*(Sqrt[-a^2 + b^2]*f*x*(2*c + d*x) + (2*I)*b*(c + d*x)*Log[1 - (a*E^(I*(e + f*x)))/(-b + Sqrt[-a^2 + b^2])] - (2*I)*b*(c + d*x)*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])]) + 2*b*d*PolyLog[2, (a*E^(I*(e + f*x)))/(-b + Sqrt[-a^2 + b^2])] - 2*b*d*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(2*a*Sqrt[-a^2 + b^2]*f^2)","A",1
37,0,0,23,1.5871301,"\int \frac{1}{(c+d x) (a+b \sec (e+f x))} \, dx","Integrate[1/((c + d*x)*(a + b*Sec[e + f*x])),x]","\int \frac{1}{(c+d x) (a+b \sec (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a+b \sec (e+f x))},x\right)",0,"Integrate[1/((c + d*x)*(a + b*Sec[e + f*x])), x]","A",-1
38,0,0,23,11.6916614,"\int \frac{1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx","Integrate[1/((c + d*x)^2*(a + b*Sec[e + f*x])),x]","\int \frac{1}{(c+d x)^2 (a+b \sec (e+f x))} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a+b \sec (e+f x))},x\right)",0,"Integrate[1/((c + d*x)^2*(a + b*Sec[e + f*x])), x]","A",-1
39,1,8176,1523,26.1951074,"\int \frac{(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx","Integrate[(c + d*x)^3/(a + b*Sec[e + f*x])^2,x]","\text{Result too large to show}","\frac{(c+d x)^4}{4 a^2 d}+\frac{2 i b \log \left(\frac{e^{i (e+f x)} a}{b-\sqrt{b^2-a^2}}+1\right) (c+d x)^3}{a^2 \sqrt{b^2-a^2} f}-\frac{i b^3 \log \left(\frac{e^{i (e+f x)} a}{b-\sqrt{b^2-a^2}}+1\right) (c+d x)^3}{a^2 \left(b^2-a^2\right)^{3/2} f}-\frac{2 i b \log \left(\frac{e^{i (e+f x)} a}{b+\sqrt{b^2-a^2}}+1\right) (c+d x)^3}{a^2 \sqrt{b^2-a^2} f}+\frac{i b^3 \log \left(\frac{e^{i (e+f x)} a}{b+\sqrt{b^2-a^2}}+1\right) (c+d x)^3}{a^2 \left(b^2-a^2\right)^{3/2} f}+\frac{b^2 \sin (e+f x) (c+d x)^3}{a \left(a^2-b^2\right) f (b+a \cos (e+f x))}-\frac{i b^2 (c+d x)^3}{a^2 \left(a^2-b^2\right) f}+\frac{3 b^2 d \log \left(\frac{e^{i (e+f x)} a}{b-i \sqrt{a^2-b^2}}+1\right) (c+d x)^2}{a^2 \left(a^2-b^2\right) f^2}+\frac{3 b^2 d \log \left(\frac{e^{i (e+f x)} a}{b+i \sqrt{a^2-b^2}}+1\right) (c+d x)^2}{a^2 \left(a^2-b^2\right) f^2}+\frac{6 b d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right) (c+d x)^2}{a^2 \sqrt{b^2-a^2} f^2}-\frac{3 b^3 d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right) (c+d x)^2}{a^2 \left(b^2-a^2\right)^{3/2} f^2}-\frac{6 b d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right) (c+d x)^2}{a^2 \sqrt{b^2-a^2} f^2}+\frac{3 b^3 d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right) (c+d x)^2}{a^2 \left(b^2-a^2\right)^{3/2} f^2}-\frac{6 i b^2 d^2 \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right) (c+d x)}{a^2 \left(a^2-b^2\right) f^3}-\frac{6 i b^2 d^2 \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right) (c+d x)}{a^2 \left(a^2-b^2\right) f^3}+\frac{12 i b d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right) (c+d x)}{a^2 \sqrt{b^2-a^2} f^3}-\frac{6 i b^3 d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right) (c+d x)}{a^2 \left(b^2-a^2\right)^{3/2} f^3}-\frac{12 i b d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right) (c+d x)}{a^2 \sqrt{b^2-a^2} f^3}+\frac{6 i b^3 d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right) (c+d x)}{a^2 \left(b^2-a^2\right)^{3/2} f^3}+\frac{6 b^2 d^3 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) f^4}+\frac{6 b^2 d^3 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) f^4}-\frac{12 b d^3 \text{Li}_4\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} f^4}+\frac{6 b^3 d^3 \text{Li}_4\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} f^4}+\frac{12 b d^3 \text{Li}_4\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} f^4}-\frac{6 b^3 d^3 \text{Li}_4\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} f^4}",1,"Result too large to show","B",0
40,1,11147,1117,22.4049311,"\int \frac{(c+d x)^2}{(a+b \sec (e+f x))^2} \, dx","Integrate[(c + d*x)^2/(a + b*Sec[e + f*x])^2,x]","\text{Result too large to show}","-\frac{i (c+d x)^2 \log \left(\frac{e^{i (e+f x)} a}{b-\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} f}+\frac{i (c+d x)^2 \log \left(\frac{e^{i (e+f x)} a}{b+\sqrt{b^2-a^2}}+1\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} f}-\frac{2 d (c+d x) \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} f^2}+\frac{2 d (c+d x) \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} f^2}-\frac{2 i d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} f^3}+\frac{2 i d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} f^3}-\frac{i (c+d x)^2 b^2}{a^2 \left(a^2-b^2\right) f}+\frac{2 d (c+d x) \log \left(\frac{e^{i (e+f x)} a}{b-i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) f^2}+\frac{2 d (c+d x) \log \left(\frac{e^{i (e+f x)} a}{b+i \sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) f^2}-\frac{2 i d^2 \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) f^3}-\frac{2 i d^2 \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+i \sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) f^3}+\frac{(c+d x)^2 \sin (e+f x) b^2}{a \left(a^2-b^2\right) f (b+a \cos (e+f x))}+\frac{2 i (c+d x)^2 \log \left(\frac{e^{i (e+f x)} a}{b-\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} f}-\frac{2 i (c+d x)^2 \log \left(\frac{e^{i (e+f x)} a}{b+\sqrt{b^2-a^2}}+1\right) b}{a^2 \sqrt{b^2-a^2} f}+\frac{4 d (c+d x) \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} f^2}-\frac{4 d (c+d x) \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} f^2}+\frac{4 i d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} f^3}-\frac{4 i d^2 \text{Li}_3\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} f^3}+\frac{(c+d x)^3}{3 a^2 d}",1,"Result too large to show","B",0
41,1,1037,582,10.0030328,"\int \frac{c+d x}{(a+b \sec (e+f x))^2} \, dx","Integrate[(c + d*x)/(a + b*Sec[e + f*x])^2,x]","\frac{(b+a \cos (e+f x)) \left(d e \sin (e+f x) b^2-c f \sin (e+f x) b^2-d (e+f x) \sin (e+f x) b^2\right) \sec ^2(e+f x)}{a (b-a) (a+b) f^2 (a+b \sec (e+f x))^2}+\frac{b \cos ^2\left(\frac{1}{2} (e+f x)\right) (b+a \cos (e+f x)) \left(-\frac{2 \left(2 a^2-b^2\right) (d e-c f) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{-a-b}}\right)}{\sqrt{-a-b} \sqrt{a-b}}-b d \log \left(\sec ^2\left(\frac{1}{2} (e+f x)\right)\right)+b d \log \left(-\left((b+a \cos (e+f x)) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)-\frac{i \left(2 a^2-b^2\right) d \left(\log \left(i \tan \left(\frac{1}{2} (e+f x)\right)+1\right) \log \left(\frac{i \left(\sqrt{a+b}-\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{a-b}+i \sqrt{a+b}}\right)-\log \left(1-i \tan \left(\frac{1}{2} (e+f x)\right)\right) \log \left(\frac{\sqrt{a+b}-\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{i \sqrt{a-b}+\sqrt{a+b}}\right)+\log \left(1-i \tan \left(\frac{1}{2} (e+f x)\right)\right) \log \left(\frac{i \left(\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a+b}\right)}{\sqrt{a-b}+i \sqrt{a+b}}\right)-\log \left(i \tan \left(\frac{1}{2} (e+f x)\right)+1\right) \log \left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a+b}}{i \sqrt{a-b}+\sqrt{a+b}}\right)-\text{Li}_2\left(\frac{\sqrt{a-b} \left(1-i \tan \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{a-b}-i \sqrt{a+b}}\right)+\text{Li}_2\left(\frac{\sqrt{a-b} \left(1-i \tan \left(\frac{1}{2} (e+f x)\right)\right)}{\sqrt{a-b}+i \sqrt{a+b}}\right)-\text{Li}_2\left(\frac{\sqrt{a-b} \left(i \tan \left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{a-b}-i \sqrt{a+b}}\right)+\text{Li}_2\left(\frac{\sqrt{a-b} \left(i \tan \left(\frac{1}{2} (e+f x)\right)+1\right)}{\sqrt{a-b}+i \sqrt{a+b}}\right)\right)}{\sqrt{a-b} \sqrt{a+b}}\right) \left(\left(2 a^2-b^2\right) (c f+d x f)+a b d \sin (e+f x)\right) \left(\sqrt{a+b}-\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)\right) \left(\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)+\sqrt{a+b}\right) \sec ^2(e+f x)}{a^2 \left(a^2-b^2\right) f^2 (a+b \sec (e+f x))^2 \left(a b d \sin (e+f x)-\left(2 a^2-b^2\right) \left(d e-c f-i d \log \left(1-i \tan \left(\frac{1}{2} (e+f x)\right)\right)+i d \log \left(i \tan \left(\frac{1}{2} (e+f x)\right)+1\right)\right)\right)}+\frac{(e+f x) (-2 d e+2 c f+d (e+f x)) (b+a \cos (e+f x))^2 \sec ^2(e+f x)}{2 a^2 f^2 (a+b \sec (e+f x))^2}","\frac{2 i b (c+d x) \log \left(1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 f \sqrt{b^2-a^2}}-\frac{2 i b (c+d x) \log \left(1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right)}{a^2 f \sqrt{b^2-a^2}}+\frac{b^2 (c+d x) \sin (e+f x)}{a f \left(a^2-b^2\right) (a \cos (e+f x)+b)}+\frac{2 b d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 f^2 \sqrt{b^2-a^2}}-\frac{2 b d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 f^2 \sqrt{b^2-a^2}}+\frac{b^2 d \log (a \cos (e+f x)+b)}{a^2 f^2 \left(a^2-b^2\right)}-\frac{i b^3 (c+d x) \log \left(1+\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 f \left(b^2-a^2\right)^{3/2}}+\frac{i b^3 (c+d x) \log \left(1+\frac{a e^{i (e+f x)}}{\sqrt{b^2-a^2}+b}\right)}{a^2 f \left(b^2-a^2\right)^{3/2}}-\frac{b^3 d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 f^2 \left(b^2-a^2\right)^{3/2}}+\frac{b^3 d \text{Li}_2\left(-\frac{a e^{i (e+f x)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 f^2 \left(b^2-a^2\right)^{3/2}}+\frac{(c+d x)^2}{2 a^2 d}",1,"((e + f*x)*(-2*d*e + 2*c*f + d*(e + f*x))*(b + a*Cos[e + f*x])^2*Sec[e + f*x]^2)/(2*a^2*f^2*(a + b*Sec[e + f*x])^2) + ((b + a*Cos[e + f*x])*Sec[e + f*x]^2*(b^2*d*e*Sin[e + f*x] - b^2*c*f*Sin[e + f*x] - b^2*d*(e + f*x)*Sin[e + f*x]))/(a*(-a + b)*(a + b)*f^2*(a + b*Sec[e + f*x])^2) + (b*Cos[(e + f*x)/2]^2*(b + a*Cos[e + f*x])*((-2*(2*a^2 - b^2)*(d*e - c*f)*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[-a - b]])/(Sqrt[-a - b]*Sqrt[a - b]) - b*d*Log[Sec[(e + f*x)/2]^2] + b*d*Log[-((b + a*Cos[e + f*x])*Sec[(e + f*x)/2]^2)] - (I*(2*a^2 - b^2)*d*(Log[1 + I*Tan[(e + f*x)/2]]*Log[(I*(Sqrt[a + b] - Sqrt[a - b]*Tan[(e + f*x)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])] - Log[1 - I*Tan[(e + f*x)/2]]*Log[(Sqrt[a + b] - Sqrt[a - b]*Tan[(e + f*x)/2])/(I*Sqrt[a - b] + Sqrt[a + b])] + Log[1 - I*Tan[(e + f*x)/2]]*Log[(I*(Sqrt[a + b] + Sqrt[a - b]*Tan[(e + f*x)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])] - Log[1 + I*Tan[(e + f*x)/2]]*Log[(Sqrt[a + b] + Sqrt[a - b]*Tan[(e + f*x)/2])/(I*Sqrt[a - b] + Sqrt[a + b])] - PolyLog[2, (Sqrt[a - b]*(1 - I*Tan[(e + f*x)/2]))/(Sqrt[a - b] - I*Sqrt[a + b])] + PolyLog[2, (Sqrt[a - b]*(1 - I*Tan[(e + f*x)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])] - PolyLog[2, (Sqrt[a - b]*(1 + I*Tan[(e + f*x)/2]))/(Sqrt[a - b] - I*Sqrt[a + b])] + PolyLog[2, (Sqrt[a - b]*(1 + I*Tan[(e + f*x)/2]))/(Sqrt[a - b] + I*Sqrt[a + b])]))/(Sqrt[a - b]*Sqrt[a + b]))*Sec[e + f*x]^2*((2*a^2 - b^2)*(c*f + d*f*x) + a*b*d*Sin[e + f*x])*(Sqrt[a + b] - Sqrt[a - b]*Tan[(e + f*x)/2])*(Sqrt[a + b] + Sqrt[a - b]*Tan[(e + f*x)/2]))/(a^2*(a^2 - b^2)*f^2*(a + b*Sec[e + f*x])^2*(-((2*a^2 - b^2)*(d*e - c*f - I*d*Log[1 - I*Tan[(e + f*x)/2]] + I*d*Log[1 + I*Tan[(e + f*x)/2]])) + a*b*d*Sin[e + f*x]))","A",0
42,0,0,23,30.1474375,"\int \frac{1}{(c+d x) (a+b \sec (e+f x))^2} \, dx","Integrate[1/((c + d*x)*(a + b*Sec[e + f*x])^2),x]","\int \frac{1}{(c+d x) (a+b \sec (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x) (a+b \sec (e+f x))^2},x\right)",0,"Integrate[1/((c + d*x)*(a + b*Sec[e + f*x])^2), x]","A",-1
43,0,0,23,45.4618734,"\int \frac{1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx","Integrate[1/((c + d*x)^2*(a + b*Sec[e + f*x])^2),x]","\int \frac{1}{(c+d x)^2 (a+b \sec (e+f x))^2} \, dx","\text{Int}\left(\frac{1}{(c+d x)^2 (a+b \sec (e+f x))^2},x\right)",0,"Integrate[1/((c + d*x)^2*(a + b*Sec[e + f*x])^2), x]","A",-1
44,0,0,23,1.7966087,"\int (c+d x)^m (a+b \sec (e+f x))^n \, dx","Integrate[(c + d*x)^m*(a + b*Sec[e + f*x])^n,x]","\int (c+d x)^m (a+b \sec (e+f x))^n \, dx","\text{Int}\left((c+d x)^m (a+b \sec (e+f x))^n,x\right)",0,"Integrate[(c + d*x)^m*(a + b*Sec[e + f*x])^n, x]","A",-1
45,0,0,21,0.5356636,"\int (c+d x)^m (a+b \sec (e+f x)) \, dx","Integrate[(c + d*x)^m*(a + b*Sec[e + f*x]),x]","\int (c+d x)^m (a+b \sec (e+f x)) \, dx","\text{Int}\left((c+d x)^m (a+b \sec (e+f x)),x\right)",0,"Integrate[(c + d*x)^m*(a + b*Sec[e + f*x]), x]","A",-1
46,0,0,23,0.7393337,"\int \frac{(c+d x)^m}{a+b \sec (e+f x)} \, dx","Integrate[(c + d*x)^m/(a + b*Sec[e + f*x]),x]","\int \frac{(c+d x)^m}{a+b \sec (e+f x)} \, dx","\text{Int}\left(\frac{(c+d x)^m}{a+b \sec (e+f x)},x\right)",0,"Integrate[(c + d*x)^m/(a + b*Sec[e + f*x]), x]","A",-1