1,1,98,0,0.0390226,"\int \left(b \tan ^2(e+f x)\right)^{5/2} \, dx","Int[(b*Tan[e + f*x]^2)^(5/2),x]","\frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^2(e+f x)}}{4 f}-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}-\frac{b^2 \cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}","\frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^2(e+f x)}}{4 f}-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}-\frac{b^2 \cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}",1,"-((b^2*Cot[e + f*x]*Log[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]^2])/f) - (b^2*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^2])/(2*f) + (b^2*Tan[e + f*x]^3*Sqrt[b*Tan[e + f*x]^2])/(4*f)","A",4,3,14,0.2143,1,"{3658, 3473, 3475}"
2,1,61,0,0.0292693,"\int \left(b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[(b*Tan[e + f*x]^2)^(3/2),x]","\frac{b \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}+\frac{b \cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}","\frac{b \tan (e+f x) \sqrt{b \tan ^2(e+f x)}}{2 f}+\frac{b \cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}",1,"(b*Cot[e + f*x]*Log[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]^2])/f + (b*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^2])/(2*f)","A",3,3,14,0.2143,1,"{3658, 3473, 3475}"
3,1,32,0,0.0172923,"\int \sqrt{b \tan ^2(e+f x)} \, dx","Int[Sqrt[b*Tan[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}","-\frac{\cot (e+f x) \sqrt{b \tan ^2(e+f x)} \log (\cos (e+f x))}{f}",1,"-((Cot[e + f*x]*Log[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]^2])/f)","A",2,2,14,0.1429,1,"{3658, 3475}"
4,1,31,0,0.0230007,"\int \frac{1}{\sqrt{b \tan ^2(e+f x)}} \, dx","Int[1/Sqrt[b*Tan[e + f*x]^2],x]","\frac{\tan (e+f x) \log (\sin (e+f x))}{f \sqrt{b \tan ^2(e+f x)}}","\frac{\tan (e+f x) \log (\sin (e+f x))}{f \sqrt{b \tan ^2(e+f x)}}",1,"(Log[Sin[e + f*x]]*Tan[e + f*x])/(f*Sqrt[b*Tan[e + f*x]^2])","A",2,2,14,0.1429,1,"{3658, 3475}"
5,1,66,0,0.0368908,"\int \frac{1}{\left(b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[(b*Tan[e + f*x]^2)^(-3/2),x]","-\frac{\cot (e+f x)}{2 b f \sqrt{b \tan ^2(e+f x)}}-\frac{\tan (e+f x) \log (\sin (e+f x))}{b f \sqrt{b \tan ^2(e+f x)}}","-\frac{\cot (e+f x)}{2 b f \sqrt{b \tan ^2(e+f x)}}-\frac{\tan (e+f x) \log (\sin (e+f x))}{b f \sqrt{b \tan ^2(e+f x)}}",1,"-Cot[e + f*x]/(2*b*f*Sqrt[b*Tan[e + f*x]^2]) - (Log[Sin[e + f*x]]*Tan[e + f*x])/(b*f*Sqrt[b*Tan[e + f*x]^2])","A",3,3,14,0.2143,1,"{3658, 3473, 3475}"
6,1,97,0,0.0390056,"\int \frac{1}{\left(b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[(b*Tan[e + f*x]^2)^(-5/2),x]","-\frac{\cot ^3(e+f x)}{4 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\cot (e+f x)}{2 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\tan (e+f x) \log (\sin (e+f x))}{b^2 f \sqrt{b \tan ^2(e+f x)}}","-\frac{\cot ^3(e+f x)}{4 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\cot (e+f x)}{2 b^2 f \sqrt{b \tan ^2(e+f x)}}+\frac{\tan (e+f x) \log (\sin (e+f x))}{b^2 f \sqrt{b \tan ^2(e+f x)}}",1,"Cot[e + f*x]/(2*b^2*f*Sqrt[b*Tan[e + f*x]^2]) - Cot[e + f*x]^3/(4*b^2*f*Sqrt[b*Tan[e + f*x]^2]) + (Log[Sin[e + f*x]]*Tan[e + f*x])/(b^2*f*Sqrt[b*Tan[e + f*x]^2])","A",4,3,14,0.2143,1,"{3658, 3473, 3475}"
7,1,364,0,0.1457767,"\int \left(b \tan ^3(e+f x)\right)^{5/2} \, dx","Int[(b*Tan[e + f*x]^3)^(5/2),x]","\frac{2 b^2 \tan ^5(e+f x) \sqrt{b \tan ^3(e+f x)}}{13 f}-\frac{2 b^2 \tan ^3(e+f x) \sqrt{b \tan ^3(e+f x)}}{9 f}+\frac{2 b^2 \tan (e+f x) \sqrt{b \tan ^3(e+f x)}}{5 f}-\frac{b^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{b^2 \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b^2 \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{2 b^2 \cot (e+f x) \sqrt{b \tan ^3(e+f x)}}{f}","\frac{2 b^2 \tan ^5(e+f x) \sqrt{b \tan ^3(e+f x)}}{13 f}-\frac{2 b^2 \tan ^3(e+f x) \sqrt{b \tan ^3(e+f x)}}{9 f}+\frac{2 b^2 \tan (e+f x) \sqrt{b \tan ^3(e+f x)}}{5 f}-\frac{b^2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b^2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{b^2 \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b^2 \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{2 b^2 \cot (e+f x) \sqrt{b \tan ^3(e+f x)}}{f}",1,"(-2*b^2*Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^3])/f - (b^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (b^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) - (b^2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (b^2*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (2*b^2*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^3])/(5*f) - (2*b^2*Tan[e + f*x]^3*Sqrt[b*Tan[e + f*x]^3])/(9*f) + (2*b^2*Tan[e + f*x]^5*Sqrt[b*Tan[e + f*x]^3])/(13*f)","A",16,10,14,0.7143,1,"{3658, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204}"
8,1,286,0,0.1265221,"\int \left(b \tan ^3(e+f x)\right)^{3/2} \, dx","Int[(b*Tan[e + f*x]^3)^(3/2),x]","\frac{2 b \tan ^2(e+f x) \sqrt{b \tan ^3(e+f x)}}{7 f}-\frac{2 b \sqrt{b \tan ^3(e+f x)}}{3 f}-\frac{b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{b \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}","\frac{2 b \tan ^2(e+f x) \sqrt{b \tan ^3(e+f x)}}{7 f}-\frac{2 b \sqrt{b \tan ^3(e+f x)}}{3 f}-\frac{b \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{b \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{b \sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}",1,"(-2*b*Sqrt[b*Tan[e + f*x]^3])/(3*f) - (b*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (b*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (b*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) - (b*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (2*b*Tan[e + f*x]^2*Sqrt[b*Tan[e + f*x]^3])/(7*f)","A",14,10,14,0.7143,1,"{3658, 3473, 3476, 329, 297, 1162, 617, 204, 1165, 628}"
9,1,255,0,0.1151065,"\int \sqrt{b \tan ^3(e+f x)} \, dx","Int[Sqrt[b*Tan[e + f*x]^3],x]","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{\sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{\sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{2 \cot (e+f x) \sqrt{b \tan ^3(e+f x)}}{f}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \sqrt{b \tan ^3(e+f x)}}{\sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{\sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}-\frac{\sqrt{b \tan ^3(e+f x)} \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \tan ^{\frac{3}{2}}(e+f x)}+\frac{2 \cot (e+f x) \sqrt{b \tan ^3(e+f x)}}{f}",1,"(2*Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^3])/f + (ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) - (ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Sqrt[b*Tan[e + f*x]^3])/(Sqrt[2]*f*Tan[e + f*x]^(3/2)) + (Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2)) - (Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Sqrt[b*Tan[e + f*x]^3])/(2*Sqrt[2]*f*Tan[e + f*x]^(3/2))","A",13,10,14,0.7143,1,"{3658, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204}"
10,1,255,0,0.1146482,"\int \frac{1}{\sqrt{b \tan ^3(e+f x)}} \, dx","Int[1/Sqrt[b*Tan[e + f*x]^3],x]","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \tan (e+f x)}{f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}}","\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \tan (e+f x)}{f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} f \sqrt{b \tan ^3(e+f x)}}",1,"(-2*Tan[e + f*x])/(f*Sqrt[b*Tan[e + f*x]^3]) + (ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*f*Sqrt[b*Tan[e + f*x]^3]) - (ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*f*Sqrt[b*Tan[e + f*x]^3]) - (Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*f*Sqrt[b*Tan[e + f*x]^3]) + (Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*f*Sqrt[b*Tan[e + f*x]^3])","A",13,10,14,0.7143,1,"{3658, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628}"
11,1,298,0,0.1280697,"\int \frac{1}{\left(b \tan ^3(e+f x)\right)^{3/2}} \, dx","Int[(b*Tan[e + f*x]^3)^(-3/2),x]","-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}+\frac{2}{3 b f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^2(e+f x)}{7 b f \sqrt{b \tan ^3(e+f x)}}","-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}+\frac{2}{3 b f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^2(e+f x)}{7 b f \sqrt{b \tan ^3(e+f x)}}",1,"2/(3*b*f*Sqrt[b*Tan[e + f*x]^3]) - (2*Cot[e + f*x]^2)/(7*b*f*Sqrt[b*Tan[e + f*x]^3]) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*b*f*Sqrt[b*Tan[e + f*x]^3]) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*b*f*Sqrt[b*Tan[e + f*x]^3]) - (Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*b*f*Sqrt[b*Tan[e + f*x]^3]) + (Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*b*f*Sqrt[b*Tan[e + f*x]^3])","A",14,10,14,0.7143,1,"{3658, 3474, 3476, 329, 211, 1165, 628, 1162, 617, 204}"
12,1,364,0,0.147612,"\int \frac{1}{\left(b \tan ^3(e+f x)\right)^{5/2}} \, dx","Int[(b*Tan[e + f*x]^3)^(-5/2),x]","-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}","-\frac{\tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (e+f x)}\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{-1}\left(\sqrt{2} \sqrt{\tan (e+f x)}+1\right) \tan ^{\frac{3}{2}}(e+f x)}{\sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \tan (e+f x)}{b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{\tan ^{\frac{3}{2}}(e+f x) \log \left(\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right)}{2 \sqrt{2} b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot ^5(e+f x)}{13 b^2 f \sqrt{b \tan ^3(e+f x)}}+\frac{2 \cot ^3(e+f x)}{9 b^2 f \sqrt{b \tan ^3(e+f x)}}-\frac{2 \cot (e+f x)}{5 b^2 f \sqrt{b \tan ^3(e+f x)}}",1,"(-2*Cot[e + f*x])/(5*b^2*f*Sqrt[b*Tan[e + f*x]^3]) + (2*Cot[e + f*x]^3)/(9*b^2*f*Sqrt[b*Tan[e + f*x]^3]) - (2*Cot[e + f*x]^5)/(13*b^2*f*Sqrt[b*Tan[e + f*x]^3]) + (2*Tan[e + f*x])/(b^2*f*Sqrt[b*Tan[e + f*x]^3]) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*b^2*f*Sqrt[b*Tan[e + f*x]^3]) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Tan[e + f*x]^(3/2))/(Sqrt[2]*b^2*f*Sqrt[b*Tan[e + f*x]^3]) + (Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*b^2*f*Sqrt[b*Tan[e + f*x]^3]) - (Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]*Tan[e + f*x]^(3/2))/(2*Sqrt[2]*b^2*f*Sqrt[b*Tan[e + f*x]^3])","A",16,10,14,0.7143,1,"{3658, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628}"
13,1,182,0,0.0635726,"\int \left(b \tan ^4(e+f x)\right)^{5/2} \, dx","Int[(b*Tan[e + f*x]^4)^(5/2),x]","\frac{b^2 \tan ^7(e+f x) \sqrt{b \tan ^4(e+f x)}}{9 f}-\frac{b^2 \tan ^5(e+f x) \sqrt{b \tan ^4(e+f x)}}{7 f}+\frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^4(e+f x)}}{5 f}-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^4(e+f x)}}{3 f}-b^2 x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}+\frac{b^2 \cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}","\frac{b^2 \tan ^7(e+f x) \sqrt{b \tan ^4(e+f x)}}{9 f}-\frac{b^2 \tan ^5(e+f x) \sqrt{b \tan ^4(e+f x)}}{7 f}+\frac{b^2 \tan ^3(e+f x) \sqrt{b \tan ^4(e+f x)}}{5 f}-\frac{b^2 \tan (e+f x) \sqrt{b \tan ^4(e+f x)}}{3 f}-b^2 x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}+\frac{b^2 \cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}",1,"(b^2*Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/f - b^2*x*Cot[e + f*x]^2*Sqrt[b*Tan[e + f*x]^4] - (b^2*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/(3*f) + (b^2*Tan[e + f*x]^3*Sqrt[b*Tan[e + f*x]^4])/(5*f) - (b^2*Tan[e + f*x]^5*Sqrt[b*Tan[e + f*x]^4])/(7*f) + (b^2*Tan[e + f*x]^7*Sqrt[b*Tan[e + f*x]^4])/(9*f)","A",7,3,14,0.2143,1,"{3658, 3473, 8}"
14,1,110,0,0.0419221,"\int \left(b \tan ^4(e+f x)\right)^{3/2} \, dx","Int[(b*Tan[e + f*x]^4)^(3/2),x]","\frac{b \tan ^3(e+f x) \sqrt{b \tan ^4(e+f x)}}{5 f}-\frac{b \tan (e+f x) \sqrt{b \tan ^4(e+f x)}}{3 f}-b x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}+\frac{b \cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}","\frac{b \tan ^3(e+f x) \sqrt{b \tan ^4(e+f x)}}{5 f}-\frac{b \tan (e+f x) \sqrt{b \tan ^4(e+f x)}}{3 f}-b x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}+\frac{b \cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}",1,"(b*Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/f - b*x*Cot[e + f*x]^2*Sqrt[b*Tan[e + f*x]^4] - (b*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/(3*f) + (b*Tan[e + f*x]^3*Sqrt[b*Tan[e + f*x]^4])/(5*f)","A",5,3,14,0.2143,1,"{3658, 3473, 8}"
15,1,50,0,0.0207209,"\int \sqrt{b \tan ^4(e+f x)} \, dx","Int[Sqrt[b*Tan[e + f*x]^4],x]","\frac{\cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}-x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}","\frac{\cot (e+f x) \sqrt{b \tan ^4(e+f x)}}{f}-x \cot ^2(e+f x) \sqrt{b \tan ^4(e+f x)}",1,"(Cot[e + f*x]*Sqrt[b*Tan[e + f*x]^4])/f - x*Cot[e + f*x]^2*Sqrt[b*Tan[e + f*x]^4]","A",3,3,14,0.2143,1,"{3658, 3473, 8}"
16,1,51,0,0.0211575,"\int \frac{1}{\sqrt{b \tan ^4(e+f x)}} \, dx","Int[1/Sqrt[b*Tan[e + f*x]^4],x]","-\frac{x \tan ^2(e+f x)}{\sqrt{b \tan ^4(e+f x)}}-\frac{\tan (e+f x)}{f \sqrt{b \tan ^4(e+f x)}}","-\frac{x \tan ^2(e+f x)}{\sqrt{b \tan ^4(e+f x)}}-\frac{\tan (e+f x)}{f \sqrt{b \tan ^4(e+f x)}}",1,"-(Tan[e + f*x]/(f*Sqrt[b*Tan[e + f*x]^4])) - (x*Tan[e + f*x]^2)/Sqrt[b*Tan[e + f*x]^4]","A",3,3,14,0.2143,1,"{3658, 3473, 8}"
17,1,119,0,0.0446079,"\int \frac{1}{\left(b \tan ^4(e+f x)\right)^{3/2}} \, dx","Int[(b*Tan[e + f*x]^4)^(-3/2),x]","-\frac{x \tan ^2(e+f x)}{b \sqrt{b \tan ^4(e+f x)}}-\frac{\tan (e+f x)}{b f \sqrt{b \tan ^4(e+f x)}}-\frac{\cot ^3(e+f x)}{5 b f \sqrt{b \tan ^4(e+f x)}}+\frac{\cot (e+f x)}{3 b f \sqrt{b \tan ^4(e+f x)}}","-\frac{x \tan ^2(e+f x)}{b \sqrt{b \tan ^4(e+f x)}}-\frac{\tan (e+f x)}{b f \sqrt{b \tan ^4(e+f x)}}-\frac{\cot ^3(e+f x)}{5 b f \sqrt{b \tan ^4(e+f x)}}+\frac{\cot (e+f x)}{3 b f \sqrt{b \tan ^4(e+f x)}}",1,"Cot[e + f*x]/(3*b*f*Sqrt[b*Tan[e + f*x]^4]) - Cot[e + f*x]^3/(5*b*f*Sqrt[b*Tan[e + f*x]^4]) - Tan[e + f*x]/(b*f*Sqrt[b*Tan[e + f*x]^4]) - (x*Tan[e + f*x]^2)/(b*Sqrt[b*Tan[e + f*x]^4])","A",5,3,14,0.2143,1,"{3658, 3473, 8}"
18,1,183,0,0.0646674,"\int \frac{1}{\left(b \tan ^4(e+f x)\right)^{5/2}} \, dx","Int[(b*Tan[e + f*x]^4)^(-5/2),x]","-\frac{x \tan ^2(e+f x)}{b^2 \sqrt{b \tan ^4(e+f x)}}-\frac{\tan (e+f x)}{b^2 f \sqrt{b \tan ^4(e+f x)}}-\frac{\cot ^7(e+f x)}{9 b^2 f \sqrt{b \tan ^4(e+f x)}}+\frac{\cot ^5(e+f x)}{7 b^2 f \sqrt{b \tan ^4(e+f x)}}-\frac{\cot ^3(e+f x)}{5 b^2 f \sqrt{b \tan ^4(e+f x)}}+\frac{\cot (e+f x)}{3 b^2 f \sqrt{b \tan ^4(e+f x)}}","-\frac{x \tan ^2(e+f x)}{b^2 \sqrt{b \tan ^4(e+f x)}}-\frac{\tan (e+f x)}{b^2 f \sqrt{b \tan ^4(e+f x)}}-\frac{\cot ^7(e+f x)}{9 b^2 f \sqrt{b \tan ^4(e+f x)}}+\frac{\cot ^5(e+f x)}{7 b^2 f \sqrt{b \tan ^4(e+f x)}}-\frac{\cot ^3(e+f x)}{5 b^2 f \sqrt{b \tan ^4(e+f x)}}+\frac{\cot (e+f x)}{3 b^2 f \sqrt{b \tan ^4(e+f x)}}",1,"Cot[e + f*x]/(3*b^2*f*Sqrt[b*Tan[e + f*x]^4]) - Cot[e + f*x]^3/(5*b^2*f*Sqrt[b*Tan[e + f*x]^4]) + Cot[e + f*x]^5/(7*b^2*f*Sqrt[b*Tan[e + f*x]^4]) - Cot[e + f*x]^7/(9*b^2*f*Sqrt[b*Tan[e + f*x]^4]) - Tan[e + f*x]/(b^2*f*Sqrt[b*Tan[e + f*x]^4]) - (x*Tan[e + f*x]^2)/(b^2*Sqrt[b*Tan[e + f*x]^4])","A",7,3,14,0.2143,1,"{3658, 3473, 8}"
19,1,71,0,0.0439371,"\int \left(b \tan ^n(e+f x)\right)^{5/2} \, dx","Int[(b*Tan[e + f*x]^n)^(5/2),x]","\frac{2 b^2 \tan ^{2 n+1}(e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{1}{4} (5 n+2);\frac{1}{4} (5 n+6);-\tan ^2(e+f x)\right)}{f (5 n+2)}","\frac{2 b^2 \tan ^{2 n+1}(e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{1}{4} (5 n+2);\frac{1}{4} (5 n+6);-\tan ^2(e+f x)\right)}{f (5 n+2)}",1,"(2*b^2*Hypergeometric2F1[1, (2 + 5*n)/4, (6 + 5*n)/4, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + 2*n)*Sqrt[b*Tan[e + f*x]^n])/(f*(2 + 5*n))","A",3,3,14,0.2143,1,"{3659, 3476, 364}"
20,1,65,0,0.0447993,"\int \left(b \tan ^n(e+f x)\right)^{3/2} \, dx","Int[(b*Tan[e + f*x]^n)^(3/2),x]","\frac{2 b \tan ^{n+1}(e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{1}{4} (3 n+2);\frac{3 (n+2)}{4};-\tan ^2(e+f x)\right)}{f (3 n+2)}","\frac{2 b \tan ^{n+1}(e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{1}{4} (3 n+2);\frac{3 (n+2)}{4};-\tan ^2(e+f x)\right)}{f (3 n+2)}",1,"(2*b*Hypergeometric2F1[1, (2 + 3*n)/4, (3*(2 + n))/4, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n)*Sqrt[b*Tan[e + f*x]^n])/(f*(2 + 3*n))","A",3,3,14,0.2143,1,"{3659, 3476, 364}"
21,1,56,0,0.0418333,"\int \sqrt{b \tan ^n(e+f x)} \, dx","Int[Sqrt[b*Tan[e + f*x]^n],x]","\frac{2 \tan (e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{n+2}{4};\frac{n+6}{4};-\tan ^2(e+f x)\right)}{f (n+2)}","\frac{2 \tan (e+f x) \sqrt{b \tan ^n(e+f x)} \, _2F_1\left(1,\frac{n+2}{4};\frac{n+6}{4};-\tan ^2(e+f x)\right)}{f (n+2)}",1,"(2*Hypergeometric2F1[1, (2 + n)/4, (6 + n)/4, -Tan[e + f*x]^2]*Tan[e + f*x]*Sqrt[b*Tan[e + f*x]^n])/(f*(2 + n))","A",3,3,14,0.2143,1,"{3659, 3476, 364}"
22,1,62,0,0.0476349,"\int \frac{1}{\sqrt{b \tan ^n(e+f x)}} \, dx","Int[1/Sqrt[b*Tan[e + f*x]^n],x]","\frac{2 \tan (e+f x) \, _2F_1\left(1,\frac{2-n}{4};\frac{6-n}{4};-\tan ^2(e+f x)\right)}{f (2-n) \sqrt{b \tan ^n(e+f x)}}","\frac{2 \tan (e+f x) \, _2F_1\left(1,\frac{2-n}{4};\frac{6-n}{4};-\tan ^2(e+f x)\right)}{f (2-n) \sqrt{b \tan ^n(e+f x)}}",1,"(2*Hypergeometric2F1[1, (2 - n)/4, (6 - n)/4, -Tan[e + f*x]^2]*Tan[e + f*x])/(f*(2 - n)*Sqrt[b*Tan[e + f*x]^n])","A",3,3,14,0.2143,1,"{3659, 3476, 364}"
23,1,71,0,0.0481613,"\int \frac{1}{\left(b \tan ^n(e+f x)\right)^{3/2}} \, dx","Int[(b*Tan[e + f*x]^n)^(-3/2),x]","\frac{2 \tan ^{1-n}(e+f x) \, _2F_1\left(1,\frac{1}{4} (2-3 n);\frac{3 (2-n)}{4};-\tan ^2(e+f x)\right)}{b f (2-3 n) \sqrt{b \tan ^n(e+f x)}}","\frac{2 \tan ^{1-n}(e+f x) \, _2F_1\left(1,\frac{1}{4} (2-3 n);\frac{3 (2-n)}{4};-\tan ^2(e+f x)\right)}{b f (2-3 n) \sqrt{b \tan ^n(e+f x)}}",1,"(2*Hypergeometric2F1[1, (2 - 3*n)/4, (3*(2 - n))/4, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 - n))/(b*f*(2 - 3*n)*Sqrt[b*Tan[e + f*x]^n])","A",3,3,14,0.2143,1,"{3659, 3476, 364}"
24,1,71,0,0.0481893,"\int \frac{1}{\left(b \tan ^n(e+f x)\right)^{5/2}} \, dx","Int[(b*Tan[e + f*x]^n)^(-5/2),x]","\frac{2 \tan ^{1-2 n}(e+f x) \, _2F_1\left(1,\frac{1}{4} (2-5 n);\frac{1}{4} (6-5 n);-\tan ^2(e+f x)\right)}{b^2 f (2-5 n) \sqrt{b \tan ^n(e+f x)}}","\frac{2 \tan ^{1-2 n}(e+f x) \, _2F_1\left(1,\frac{1}{4} (2-5 n);\frac{1}{4} (6-5 n);-\tan ^2(e+f x)\right)}{b^2 f (2-5 n) \sqrt{b \tan ^n(e+f x)}}",1,"(2*Hypergeometric2F1[1, (2 - 5*n)/4, (6 - 5*n)/4, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 - 2*n))/(b^2*f*(2 - 5*n)*Sqrt[b*Tan[e + f*x]^n])","A",3,3,14,0.2143,1,"{3659, 3476, 364}"
25,1,59,0,0.0400539,"\int \left(b \tan ^n(e+f x)\right)^p \, dx","Int[(b*Tan[e + f*x]^n)^p,x]","\frac{\tan (e+f x) \left(b \tan ^n(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)}{f (n p+1)}","\frac{\tan (e+f x) \left(b \tan ^n(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right)}{f (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^n)^p)/(f*(1 + n*p))","A",3,3,12,0.2500,1,"{3659, 3476, 364}"
26,1,59,0,0.0401488,"\int \left(b \tan ^2(e+f x)\right)^p \, dx","Int[(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (2 p+1);\frac{1}{2} (2 p+3);-\tan ^2(e+f x)\right)}{f (2 p+1)}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (2 p+1);\frac{1}{2} (2 p+3);-\tan ^2(e+f x)\right)}{f (2 p+1)}",1,"(Hypergeometric2F1[1, (1 + 2*p)/2, (3 + 2*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + 2*p))","A",3,3,12,0.2500,1,"{3658, 3476, 364}"
27,1,57,0,0.0387346,"\int \left(b \tan ^3(e+f x)\right)^p \, dx","Int[(b*Tan[e + f*x]^3)^p,x]","\frac{\tan (e+f x) \left(b \tan ^3(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (3 p+1);\frac{3 (p+1)}{2};-\tan ^2(e+f x)\right)}{f (3 p+1)}","\frac{\tan (e+f x) \left(b \tan ^3(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (3 p+1);\frac{3 (p+1)}{2};-\tan ^2(e+f x)\right)}{f (3 p+1)}",1,"(Hypergeometric2F1[1, (1 + 3*p)/2, (3*(1 + p))/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^3)^p)/(f*(1 + 3*p))","A",3,3,12,0.2500,1,"{3658, 3476, 364}"
28,1,59,0,0.0379048,"\int \left(b \tan ^4(e+f x)\right)^p \, dx","Int[(b*Tan[e + f*x]^4)^p,x]","\frac{\tan (e+f x) \left(b \tan ^4(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (4 p+1);\frac{1}{2} (4 p+3);-\tan ^2(e+f x)\right)}{f (4 p+1)}","\frac{\tan (e+f x) \left(b \tan ^4(e+f x)\right)^p \, _2F_1\left(1,\frac{1}{2} (4 p+1);\frac{1}{2} (4 p+3);-\tan ^2(e+f x)\right)}{f (4 p+1)}",1,"(Hypergeometric2F1[1, (1 + 4*p)/2, (3 + 4*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^4)^p)/(f*(1 + 4*p))","A",3,3,12,0.2500,1,"{3658, 3476, 364}"
29,1,32,0,0.0187617,"\int \left(b \tan ^n(e+f x)\right)^{\frac{1}{n}} \, dx","Int[(b*Tan[e + f*x]^n)^n^(-1),x]","-\frac{\cot (e+f x) \log (\cos (e+f x)) \left(b \tan ^n(e+f x)\right)^{\frac{1}{n}}}{f}","-\frac{\cot (e+f x) \log (\cos (e+f x)) \left(b \tan ^n(e+f x)\right)^{\frac{1}{n}}}{f}",1,"-((Cot[e + f*x]*Log[Cos[e + f*x]]*(b*Tan[e + f*x]^n)^n^(-1))/f)","A",2,2,14,0.1429,1,"{3659, 3475}"
30,1,70,0,0.0620982,"\int \sin ^5(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2),x]","-\frac{(a-b) \cos ^5(e+f x)}{5 f}+\frac{(2 a-3 b) \cos ^3(e+f x)}{3 f}-\frac{(a-3 b) \cos (e+f x)}{f}+\frac{b \sec (e+f x)}{f}","-\frac{(a-b) \cos ^5(e+f x)}{5 f}+\frac{(2 a-3 b) \cos ^3(e+f x)}{3 f}-\frac{(a-3 b) \cos (e+f x)}{f}+\frac{b \sec (e+f x)}{f}",1,"-(((a - 3*b)*Cos[e + f*x])/f) + ((2*a - 3*b)*Cos[e + f*x]^3)/(3*f) - ((a - b)*Cos[e + f*x]^5)/(5*f) + (b*Sec[e + f*x])/f","A",3,2,21,0.09524,1,"{3664, 448}"
31,1,48,0,0.0470021,"\int \sin ^3(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \cos ^3(e+f x)}{3 f}-\frac{(a-2 b) \cos (e+f x)}{f}+\frac{b \sec (e+f x)}{f}","\frac{(a-b) \cos ^3(e+f x)}{3 f}-\frac{(a-2 b) \cos (e+f x)}{f}+\frac{b \sec (e+f x)}{f}",1,"-(((a - 2*b)*Cos[e + f*x])/f) + ((a - b)*Cos[e + f*x]^3)/(3*f) + (b*Sec[e + f*x])/f","A",3,2,21,0.09524,1,"{3664, 448}"
32,1,28,0,0.0261331,"\int \sin (e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Sin[e + f*x]*(a + b*Tan[e + f*x]^2),x]","\frac{b \sec (e+f x)}{f}-\frac{(a-b) \cos (e+f x)}{f}","\frac{b \sec (e+f x)}{f}-\frac{(a-b) \cos (e+f x)}{f}",1,"-(((a - b)*Cos[e + f*x])/f) + (b*Sec[e + f*x])/f","A",3,2,19,0.1053,1,"{3664, 14}"
33,1,25,0,0.0277562,"\int \csc (e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]*(a + b*Tan[e + f*x]^2),x]","\frac{b \sec (e+f x)}{f}-\frac{a \tanh ^{-1}(\cos (e+f x))}{f}","\frac{b \sec (e+f x)}{f}-\frac{a \tanh ^{-1}(\cos (e+f x))}{f}",1,"-((a*ArcTanh[Cos[e + f*x]])/f) + (b*Sec[e + f*x])/f","A",3,3,19,0.1579,1,"{3664, 388, 207}"
34,1,51,0,0.052528,"\int \csc ^3(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]","-\frac{(a+2 b) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}+\frac{b \sec (e+f x)}{f}","-\frac{(a+2 b) \tanh ^{-1}(\cos (e+f x))}{2 f}-\frac{a \cot (e+f x) \csc (e+f x)}{2 f}+\frac{b \sec (e+f x)}{f}",1,"-((a + 2*b)*ArcTanh[Cos[e + f*x]])/(2*f) - (a*Cot[e + f*x]*Csc[e + f*x])/(2*f) + (b*Sec[e + f*x])/f","A",4,4,21,0.1905,1,"{3664, 455, 388, 207}"
35,1,79,0,0.0701404,"\int \csc ^5(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2),x]","-\frac{3 (a+4 b) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{(5 a+4 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac{a \cot ^3(e+f x) \csc (e+f x)}{4 f}+\frac{b \sec (e+f x)}{f}","-\frac{3 (a+4 b) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{(5 a+4 b) \cot (e+f x) \csc (e+f x)}{8 f}-\frac{a \cot ^3(e+f x) \csc (e+f x)}{4 f}+\frac{b \sec (e+f x)}{f}",1,"(-3*(a + 4*b)*ArcTanh[Cos[e + f*x]])/(8*f) - ((5*a + 4*b)*Cot[e + f*x]*Csc[e + f*x])/(8*f) - (a*Cot[e + f*x]^3*Csc[e + f*x])/(4*f) + (b*Sec[e + f*x])/f","A",5,5,21,0.2381,1,"{3664, 455, 1157, 388, 207}"
36,1,102,0,0.1180237,"\int \sin ^6(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Sin[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]","-\frac{(a-b) \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{(13 a-19 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}-\frac{(11 a-29 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} x (a-7 b)+\frac{b \tan (e+f x)}{f}","-\frac{(a-b) \sin (e+f x) \cos ^5(e+f x)}{6 f}+\frac{(13 a-19 b) \sin (e+f x) \cos ^3(e+f x)}{24 f}-\frac{(11 a-29 b) \sin (e+f x) \cos (e+f x)}{16 f}+\frac{5}{16} x (a-7 b)+\frac{b \tan (e+f x)}{f}",1,"(5*(a - 7*b)*x)/16 - ((11*a - 29*b)*Cos[e + f*x]*Sin[e + f*x])/(16*f) + ((13*a - 19*b)*Cos[e + f*x]^3*Sin[e + f*x])/(24*f) - ((a - b)*Cos[e + f*x]^5*Sin[e + f*x])/(6*f) + (b*Tan[e + f*x])/f","A",6,6,21,0.2857,1,"{3663, 455, 1814, 1157, 388, 203}"
37,1,74,0,0.0733339,"\int \sin ^4(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \sin (e+f x) \cos ^3(e+f x)}{4 f}-\frac{(5 a-9 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} x (a-5 b)+\frac{b \tan (e+f x)}{f}","\frac{(a-b) \sin (e+f x) \cos ^3(e+f x)}{4 f}-\frac{(5 a-9 b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{3}{8} x (a-5 b)+\frac{b \tan (e+f x)}{f}",1,"(3*(a - 5*b)*x)/8 - ((5*a - 9*b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) + ((a - b)*Cos[e + f*x]^3*Sin[e + f*x])/(4*f) + (b*Tan[e + f*x])/f","A",5,5,21,0.2381,1,"{3663, 455, 1157, 388, 203}"
38,1,46,0,0.047587,"\int \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]","-\frac{(a-b) \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} x (a-3 b)+\frac{b \tan (e+f x)}{f}","-\frac{(a-b) \sin (e+f x) \cos (e+f x)}{2 f}+\frac{1}{2} x (a-3 b)+\frac{b \tan (e+f x)}{f}",1,"((a - 3*b)*x)/2 - ((a - b)*Cos[e + f*x]*Sin[e + f*x])/(2*f) + (b*Tan[e + f*x])/f","A",4,4,21,0.1905,1,"{3663, 455, 388, 203}"
39,1,19,0,0.0129236,"\int \left(a+b \tan ^2(e+f x)\right) \, dx","Int[a + b*Tan[e + f*x]^2,x]","a x+\frac{b \tan (e+f x)}{f}-b x","a x+\frac{b \tan (e+f x)}{f}-b x",1,"a*x - b*x + (b*Tan[e + f*x])/f","A",3,2,12,0.1667,1,"{3473, 8}"
40,1,24,0,0.0322951,"\int \csc ^2(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]","\frac{b \tan (e+f x)}{f}-\frac{a \cot (e+f x)}{f}","\frac{b \tan (e+f x)}{f}-\frac{a \cot (e+f x)}{f}",1,"-((a*Cot[e + f*x])/f) + (b*Tan[e + f*x])/f","A",3,2,21,0.09524,1,"{3663, 14}"
41,1,42,0,0.0432361,"\int \csc ^4(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2),x]","-\frac{(a+b) \cot (e+f x)}{f}-\frac{a \cot ^3(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}","-\frac{(a+b) \cot (e+f x)}{f}-\frac{a \cot ^3(e+f x)}{3 f}+\frac{b \tan (e+f x)}{f}",1,"-(((a + b)*Cot[e + f*x])/f) - (a*Cot[e + f*x]^3)/(3*f) + (b*Tan[e + f*x])/f","A",3,2,21,0.09524,1,"{3663, 448}"
42,1,64,0,0.0528497,"\int \csc ^6(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]","-\frac{(2 a+b) \cot ^3(e+f x)}{3 f}-\frac{(a+2 b) \cot (e+f x)}{f}-\frac{a \cot ^5(e+f x)}{5 f}+\frac{b \tan (e+f x)}{f}","-\frac{(2 a+b) \cot ^3(e+f x)}{3 f}-\frac{(a+2 b) \cot (e+f x)}{f}-\frac{a \cot ^5(e+f x)}{5 f}+\frac{b \tan (e+f x)}{f}",1,"-(((a + 2*b)*Cot[e + f*x])/f) - ((2*a + b)*Cot[e + f*x]^3)/(3*f) - (a*Cot[e + f*x]^5)/(5*f) + (b*Tan[e + f*x])/f","A",3,2,21,0.09524,1,"{3663, 448}"
43,1,107,0,0.1070583,"\int \sin ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\left(a^2-6 a b+6 b^2\right) \cos (e+f x)}{f}-\frac{(a-b)^2 \cos ^5(e+f x)}{5 f}+\frac{2 (a-2 b) (a-b) \cos ^3(e+f x)}{3 f}+\frac{2 b (a-2 b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","-\frac{\left(a^2-6 a b+6 b^2\right) \cos (e+f x)}{f}-\frac{(a-b)^2 \cos ^5(e+f x)}{5 f}+\frac{2 (a-2 b) (a-b) \cos ^3(e+f x)}{3 f}+\frac{2 b (a-2 b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-(((a^2 - 6*a*b + 6*b^2)*Cos[e + f*x])/f) + (2*(a - 2*b)*(a - b)*Cos[e + f*x]^3)/(3*f) - ((a - b)^2*Cos[e + f*x]^5)/(5*f) + (2*(a - 2*b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{3664, 448}"
44,1,80,0,0.083143,"\int \sin ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2,x]","\frac{(a-b)^2 \cos ^3(e+f x)}{3 f}-\frac{(a-3 b) (a-b) \cos (e+f x)}{f}+\frac{b (2 a-3 b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","\frac{(a-b)^2 \cos ^3(e+f x)}{3 f}-\frac{(a-3 b) (a-b) \cos (e+f x)}{f}+\frac{b (2 a-3 b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-(((a - 3*b)*(a - b)*Cos[e + f*x])/f) + ((a - b)^2*Cos[e + f*x]^3)/(3*f) + ((2*a - 3*b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{3664, 448}"
45,1,54,0,0.045866,"\int \sin (e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{(a-b)^2 \cos (e+f x)}{f}+\frac{2 b (a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","-\frac{(a-b)^2 \cos (e+f x)}{f}+\frac{2 b (a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-(((a - b)^2*Cos[e + f*x])/f) + (2*(a - b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)","A",3,2,21,0.09524,1,"{3664, 270}"
46,1,52,0,0.0549429,"\int \csc (e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \tanh ^{-1}(\cos (e+f x))}{f}+\frac{b (2 a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","-\frac{a^2 \tanh ^{-1}(\cos (e+f x))}{f}+\frac{b (2 a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-((a^2*ArcTanh[Cos[e + f*x]])/f) + ((2*a - b)*b*Sec[e + f*x])/f + (b^2*Sec[e + f*x]^3)/(3*f)","A",4,3,21,0.1429,1,"{3664, 390, 207}"
47,1,82,0,0.1090354,"\int \csc ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \csc ^2(e+f x) \sec (e+f x)}{2 f}+\frac{a (a+4 b) \sec (e+f x)}{2 f}-\frac{a (a+4 b) \tanh ^{-1}(\cos (e+f x))}{2 f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","-\frac{a^2 \csc ^2(e+f x) \sec (e+f x)}{2 f}+\frac{a (a+4 b) \sec (e+f x)}{2 f}-\frac{a (a+4 b) \tanh ^{-1}(\cos (e+f x))}{2 f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-(a*(a + 4*b)*ArcTanh[Cos[e + f*x]])/(2*f) + (a*(a + 4*b)*Sec[e + f*x])/(2*f) - (a^2*Csc[e + f*x]^2*Sec[e + f*x])/(2*f) + (b^2*Sec[e + f*x]^3)/(3*f)","A",5,5,23,0.2174,1,"{3664, 463, 459, 321, 207}"
48,1,123,0,0.1297213,"\int \csc ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]","\frac{\left(a^2+8 a b+4 b^2\right) \sec (e+f x)}{4 f}-\frac{\left(3 a^2+24 a b+8 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac{a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac{b^2 \sec ^3(e+f x)}{3 f}","\frac{\left(a^2+8 a b+4 b^2\right) \sec (e+f x)}{4 f}-\frac{\left(3 a^2+24 a b+8 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 f}-\frac{a^2 \csc ^4(e+f x) \sec (e+f x)}{4 f}-\frac{a (a+8 b) \cot (e+f x) \csc (e+f x)}{8 f}+\frac{b^2 \sec ^3(e+f x)}{3 f}",1,"-((3*a^2 + 24*a*b + 8*b^2)*ArcTanh[Cos[e + f*x]])/(8*f) - (a*(a + 8*b)*Cot[e + f*x]*Csc[e + f*x])/(8*f) + ((a^2 + 8*a*b + 4*b^2)*Sec[e + f*x])/(4*f) - (a^2*Csc[e + f*x]^4*Sec[e + f*x])/(4*f) + (b^2*Sec[e + f*x]^3)/(3*f)","A",6,5,23,0.2174,1,"{3664, 463, 455, 1153, 207}"
49,1,122,0,0.1310276,"\int \sin ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\left(a^2-10 a b+13 b^2\right) \tan (e+f x)}{4 f}+\frac{1}{8} x \left(3 a^2-30 a b+35 b^2\right)+\frac{(a-b)^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}-\frac{(a-9 b) (a-b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","-\frac{\left(a^2-10 a b+13 b^2\right) \tan (e+f x)}{4 f}+\frac{1}{8} x \left(3 a^2-30 a b+35 b^2\right)+\frac{(a-b)^2 \sin ^4(e+f x) \tan (e+f x)}{4 f}-\frac{(a-9 b) (a-b) \sin (e+f x) \cos (e+f x)}{8 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"((3*a^2 - 30*a*b + 35*b^2)*x)/8 - ((a - 9*b)*(a - b)*Cos[e + f*x]*Sin[e + f*x])/(8*f) - ((a^2 - 10*a*b + 13*b^2)*Tan[e + f*x])/(4*f) + ((a - b)^2*Sin[e + f*x]^4*Tan[e + f*x])/(4*f) + (b^2*Tan[e + f*x]^3)/(3*f)","A",6,5,23,0.2174,1,"{3663, 463, 455, 1153, 203}"
50,1,85,0,0.1074524,"\int \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{(a-5 b) (a-b) \tan (e+f x)}{2 f}+\frac{(a-b)^2 \sin ^2(e+f x) \tan (e+f x)}{2 f}+\frac{1}{2} x (a-5 b) (a-b)+\frac{b^2 \tan ^3(e+f x)}{3 f}","-\frac{(a-5 b) (a-b) \tan (e+f x)}{2 f}+\frac{(a-b)^2 \sin ^2(e+f x) \tan (e+f x)}{2 f}+\frac{1}{2} x (a-5 b) (a-b)+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"((a - 5*b)*(a - b)*x)/2 - ((a - 5*b)*(a - b)*Tan[e + f*x])/(2*f) + ((a - b)^2*Sin[e + f*x]^2*Tan[e + f*x])/(2*f) + (b^2*Tan[e + f*x]^3)/(3*f)","A",5,5,23,0.2174,1,"{3663, 463, 459, 321, 203}"
51,1,46,0,0.0331447,"\int \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[(a + b*Tan[e + f*x]^2)^2,x]","\frac{b (2 a-b) \tan (e+f x)}{f}+x (a-b)^2+\frac{b^2 \tan ^3(e+f x)}{3 f}","\frac{b (2 a-b) \tan (e+f x)}{f}+x (a-b)^2+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(a - b)^2*x + ((2*a - b)*b*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",4,3,14,0.2143,1,"{3661, 390, 203}"
52,1,46,0,0.0544183,"\int \csc ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot (e+f x)}{f}+\frac{2 a b \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","-\frac{a^2 \cot (e+f x)}{f}+\frac{2 a b \tan (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"-((a^2*Cot[e + f*x])/f) + (2*a*b*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{3663, 270}"
53,1,70,0,0.0724582,"\int \csc ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot ^3(e+f x)}{3 f}+\frac{b (2 a+b) \tan (e+f x)}{f}-\frac{a (a+2 b) \cot (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","-\frac{a^2 \cot ^3(e+f x)}{3 f}+\frac{b (2 a+b) \tan (e+f x)}{f}-\frac{a (a+2 b) \cot (e+f x)}{f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"-((a*(a + 2*b)*Cot[e + f*x])/f) - (a^2*Cot[e + f*x]^3)/(3*f) + (b*(2*a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{3663, 448}"
54,1,93,0,0.0898972,"\int \csc ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\left(a^2+4 a b+b^2\right) \cot (e+f x)}{f}-\frac{a^2 \cot ^5(e+f x)}{5 f}+\frac{2 b (a+b) \tan (e+f x)}{f}-\frac{2 a (a+b) \cot ^3(e+f x)}{3 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}","-\frac{\left(a^2+4 a b+b^2\right) \cot (e+f x)}{f}-\frac{a^2 \cot ^5(e+f x)}{5 f}+\frac{2 b (a+b) \tan (e+f x)}{f}-\frac{2 a (a+b) \cot ^3(e+f x)}{3 f}+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"-(((a^2 + 4*a*b + b^2)*Cot[e + f*x])/f) - (2*a*(a + b)*Cot[e + f*x]^3)/(3*f) - (a^2*Cot[e + f*x]^5)/(5*f) + (2*b*(a + b)*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",3,2,23,0.08696,1,"{3663, 448}"
55,1,117,0,0.1840312,"\int \frac{\sin ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2),x]","-\frac{a^2 \cos (e+f x)}{f (a-b)^3}-\frac{a^2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{f (a-b)^{7/2}}-\frac{\cos ^5(e+f x)}{5 f (a-b)}+\frac{(2 a-b) \cos ^3(e+f x)}{3 f (a-b)^2}","-\frac{a^2 \cos (e+f x)}{f (a-b)^3}-\frac{a^2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{f (a-b)^{7/2}}-\frac{\cos ^5(e+f x)}{5 f (a-b)}+\frac{(2 a-b) \cos ^3(e+f x)}{3 f (a-b)^2}",1,"-((a^2*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/((a - b)^(7/2)*f)) - (a^2*Cos[e + f*x])/((a - b)^3*f) + ((2*a - b)*Cos[e + f*x]^3)/(3*(a - b)^2*f) - Cos[e + f*x]^5/(5*(a - b)*f)","A",4,3,23,0.1304,1,"{3664, 461, 205}"
56,1,84,0,0.1225526,"\int \frac{\sin ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2),x]","\frac{\cos ^3(e+f x)}{3 f (a-b)}-\frac{a \cos (e+f x)}{f (a-b)^2}-\frac{a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}","\frac{\cos ^3(e+f x)}{3 f (a-b)}-\frac{a \cos (e+f x)}{f (a-b)^2}-\frac{a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"-((a*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/((a - b)^(5/2)*f)) - (a*Cos[e + f*x])/((a - b)^2*f) + Cos[e + f*x]^3/(3*(a - b)*f)","A",4,4,23,0.1739,1,"{3664, 453, 325, 205}"
57,1,60,0,0.0550307,"\int \frac{\sin (e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]/(a + b*Tan[e + f*x]^2),x]","-\frac{\cos (e+f x)}{f (a-b)}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}","-\frac{\cos (e+f x)}{f (a-b)}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}",1,"-((Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/((a - b)^(3/2)*f)) - Cos[e + f*x]/((a - b)*f)","A",3,3,21,0.1429,1,"{3664, 325, 205}"
58,1,60,0,0.0702532,"\int \frac{\csc (e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]/(a + b*Tan[e + f*x]^2),x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{a f \sqrt{a-b}}-\frac{\tanh ^{-1}(\cos (e+f x))}{a f}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{a f \sqrt{a-b}}-\frac{\tanh ^{-1}(\cos (e+f x))}{a f}",1,"-((Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(a*Sqrt[a - b]*f)) - ArcTanh[Cos[e + f*x]]/(a*f)","A",4,4,21,0.1905,1,"{3664, 391, 207, 205}"
59,1,89,0,0.1026022,"\int \frac{\csc ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2),x]","-\frac{(a-2 b) \tanh ^{-1}(\cos (e+f x))}{2 a^2 f}-\frac{\sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{a^2 f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f}","-\frac{(a-2 b) \tanh ^{-1}(\cos (e+f x))}{2 a^2 f}-\frac{\sqrt{b} \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{a^2 f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f}",1,"-((Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(a^2*f)) - ((a - 2*b)*ArcTanh[Cos[e + f*x]])/(2*a^2*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f)","A",5,5,23,0.2174,1,"{3664, 471, 522, 207, 205}"
60,1,130,0,0.1756322,"\int \frac{\csc ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2),x]","-\frac{\left(3 a^2-12 a b+8 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 a^3 f}-\frac{(5 a-4 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f}-\frac{\sqrt{b} (a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{a^3 f}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f}","-\frac{\left(3 a^2-12 a b+8 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 a^3 f}-\frac{(5 a-4 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f}-\frac{\sqrt{b} (a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{a^3 f}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f}",1,"-(((a - b)^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(a^3*f)) - ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[Cos[e + f*x]])/(8*a^3*f) - ((5*a - 4*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f)","A",6,6,23,0.2609,1,"{3664, 470, 527, 522, 207, 205}"
61,1,178,0,0.2927283,"\int \frac{\sin ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]","-\frac{\left(11 a^2-4 a b+b^2\right) \sin (e+f x) \cos (e+f x)}{16 f (a-b)^3}+\frac{x \left(15 a^2 b+5 a^3-5 a b^2+b^3\right)}{16 (a-b)^4}-\frac{a^{5/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{f (a-b)^4}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 f (a-b)}+\frac{(3 a-b) \sin (e+f x) \cos ^3(e+f x)}{8 f (a-b)^2}","-\frac{\left(11 a^2-4 a b+b^2\right) \sin (e+f x) \cos (e+f x)}{16 f (a-b)^3}+\frac{x \left(15 a^2 b+5 a^3-5 a b^2+b^3\right)}{16 (a-b)^4}-\frac{a^{5/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{f (a-b)^4}+\frac{\sin ^3(e+f x) \cos ^3(e+f x)}{6 f (a-b)}+\frac{(3 a-b) \sin (e+f x) \cos ^3(e+f x)}{8 f (a-b)^2}",1,"((5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*x)/(16*(a - b)^4) - (a^(5/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)^4*f) - ((11*a^2 - 4*a*b + b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*(a - b)^3*f) + ((3*a - b)*Cos[e + f*x]^3*Sin[e + f*x])/(8*(a - b)^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*(a - b)*f)","A",7,7,23,0.3043,1,"{3663, 470, 578, 527, 522, 203, 205}"
62,1,129,0,0.1515446,"\int \frac{\sin ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]","\frac{x \left(3 a^2+6 a b-b^2\right)}{8 (a-b)^3}-\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{f (a-b)^3}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b)}-\frac{(5 a-b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2}","\frac{x \left(3 a^2+6 a b-b^2\right)}{8 (a-b)^3}-\frac{a^{3/2} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{f (a-b)^3}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b)}-\frac{(5 a-b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2}",1,"((3*a^2 + 6*a*b - b^2)*x)/(8*(a - b)^3) - (a^(3/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)^3*f) - ((5*a - b)*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f)","A",6,6,23,0.2609,1,"{3663, 470, 527, 522, 203, 205}"
63,1,82,0,0.0968373,"\int \frac{\sin ^2(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2),x]","-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{f (a-b)^2}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b)}+\frac{x (a+b)}{2 (a-b)^2}","-\frac{\sqrt{a} \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{f (a-b)^2}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b)}+\frac{x (a+b)}{2 (a-b)^2}",1,"((a + b)*x)/(2*(a - b)^2) - (Sqrt[a]*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)^2*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f)","A",5,5,23,0.2174,1,"{3663, 471, 522, 203, 205}"
64,1,50,0,0.0747476,"\int \frac{1}{a+b \tan ^2(e+f x)} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-1),x]","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a} f (a-b)}","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a} f (a-b)}",1,"x/(a - b) - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*f)","A",3,3,14,0.2143,1,"{3660, 3675, 205}"
65,1,48,0,0.0616512,"\int \frac{\csc ^2(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2),x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{\cot (e+f x)}{a f}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{\cot (e+f x)}{a f}",1,"-((Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(3/2)*f)) - Cot[e + f*x]/(a*f)","A",3,3,23,0.1304,1,"{3663, 325, 205}"
66,1,76,0,0.0902362,"\int \frac{\csc ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]","-\frac{\sqrt{b} (a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{(a-b) \cot (e+f x)}{a^2 f}-\frac{\cot ^3(e+f x)}{3 a f}","-\frac{\sqrt{b} (a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{(a-b) \cot (e+f x)}{a^2 f}-\frac{\cot ^3(e+f x)}{3 a f}",1,"-(((a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(5/2)*f)) - ((a - b)*Cot[e + f*x])/(a^2*f) - Cot[e + f*x]^3/(3*a*f)","A",4,4,23,0.1739,1,"{3663, 453, 325, 205}"
67,1,105,0,0.115417,"\int \frac{\csc ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]","-\frac{\sqrt{b} (a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{7/2} f}-\frac{(2 a-b) \cot ^3(e+f x)}{3 a^2 f}-\frac{(a-b)^2 \cot (e+f x)}{a^3 f}-\frac{\cot ^5(e+f x)}{5 a f}","-\frac{\sqrt{b} (a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{7/2} f}-\frac{(2 a-b) \cot ^3(e+f x)}{3 a^2 f}-\frac{(a-b)^2 \cot (e+f x)}{a^3 f}-\frac{\cot ^5(e+f x)}{5 a f}",1,"-(((a - b)^2*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(7/2)*f)) - ((a - b)^2*Cot[e + f*x])/(a^3*f) - ((2*a - b)*Cot[e + f*x]^3)/(3*a^2*f) - Cot[e + f*x]^5/(5*a*f)","A",4,3,23,0.1304,1,"{3663, 461, 205}"
68,1,204,0,0.3093522,"\int \frac{\sin ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\left(5 a^2+10 a b-b^2\right) \cos (e+f x)}{5 f (a-b)^4}-\frac{b \left(5 a^2+2 b^2\right) \sec (e+f x)}{10 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)}+\frac{(10 a-3 b) \cos ^3(e+f x)}{15 f (a-b)^3}-\frac{\cos ^5(e+f x)}{5 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}-\frac{a \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 f (a-b)^{9/2}}","-\frac{\left(5 a^2+10 a b-b^2\right) \cos (e+f x)}{5 f (a-b)^4}-\frac{b \left(5 a^2+2 b^2\right) \sec (e+f x)}{10 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)}+\frac{(10 a-3 b) \cos ^3(e+f x)}{15 f (a-b)^3}-\frac{\cos ^5(e+f x)}{5 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}-\frac{a \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 f (a-b)^{9/2}}",1,"-(a*Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*(a - b)^(9/2)*f) - ((5*a^2 + 10*a*b - b^2)*Cos[e + f*x])/(5*(a - b)^4*f) + ((10*a - 3*b)*Cos[e + f*x]^3)/(15*(a - b)^3*f) - Cos[e + f*x]^5/(5*(a - b)*f*(a - b + b*Sec[e + f*x]^2)) - (b*(5*a^2 + 2*b^2)*Sec[e + f*x])/(10*(a - b)^4*f*(a - b + b*Sec[e + f*x]^2))","A",6,5,23,0.2174,1,"{3664, 462, 456, 1261, 205}"
69,1,133,0,0.1804494,"\int \frac{\sin ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\cos ^3(e+f x)}{3 f (a-b)^2}-\frac{(a+b) \cos (e+f x)}{f (a-b)^3}-\frac{a b \sec (e+f x)}{2 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 f (a-b)^{7/2}}","\frac{\cos ^3(e+f x)}{3 f (a-b)^2}-\frac{(a+b) \cos (e+f x)}{f (a-b)^3}-\frac{a b \sec (e+f x)}{2 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 f (a-b)^{7/2}}",1,"-(Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*(a - b)^(7/2)*f) - ((a + b)*Cos[e + f*x])/((a - b)^3*f) + Cos[e + f*x]^3/(3*(a - b)^2*f) - (a*b*Sec[e + f*x])/(2*(a - b)^3*f*(a - b + b*Sec[e + f*x]^2))","A",5,4,23,0.1739,1,"{3664, 456, 1261, 205}"
70,1,101,0,0.0731347,"\int \frac{\sin (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{3 \cos (e+f x)}{2 f (a-b)^2}+\frac{\cos (e+f x)}{2 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 f (a-b)^{5/2}}","-\frac{3 \cos (e+f x)}{2 f (a-b)^2}+\frac{\cos (e+f x)}{2 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 f (a-b)^{5/2}}",1,"(-3*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*(a - b)^(5/2)*f) - (3*Cos[e + f*x])/(2*(a - b)^2*f) + Cos[e + f*x]/(2*(a - b)*f*(a - b + b*Sec[e + f*x]^2))","A",4,4,21,0.1905,1,"{3664, 290, 325, 205}"
71,1,110,0,0.1275081,"\int \frac{\csc (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 a^2 f (a-b)^{3/2}}-\frac{\tanh ^{-1}(\cos (e+f x))}{a^2 f}-\frac{b \sec (e+f x)}{2 a f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}","-\frac{\sqrt{b} (3 a-2 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 a^2 f (a-b)^{3/2}}-\frac{\tanh ^{-1}(\cos (e+f x))}{a^2 f}-\frac{b \sec (e+f x)}{2 a f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}",1,"-((3*a - 2*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*a^2*(a - b)^(3/2)*f) - ArcTanh[Cos[e + f*x]]/(a^2*f) - (b*Sec[e + f*x])/(2*a*(a - b)*f*(a - b + b*Sec[e + f*x]^2))","A",5,5,21,0.2381,1,"{3664, 414, 522, 207, 205}"
72,1,147,0,0.1814543,"\int \frac{\csc ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{b \sec (e+f x)}{a^2 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{(a-4 b) \tanh ^{-1}(\cos (e+f x))}{2 a^3 f}-\frac{\sqrt{b} (3 a-4 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 a^3 f \sqrt{a-b}}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left(a+b \sec ^2(e+f x)-b\right)}","-\frac{b \sec (e+f x)}{a^2 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{(a-4 b) \tanh ^{-1}(\cos (e+f x))}{2 a^3 f}-\frac{\sqrt{b} (3 a-4 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 a^3 f \sqrt{a-b}}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left(a+b \sec ^2(e+f x)-b\right)}",1,"-((3*a - 4*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*a^3*Sqrt[a - b]*f) - ((a - 4*b)*ArcTanh[Cos[e + f*x]])/(2*a^3*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*(a - b + b*Sec[e + f*x]^2)) - (b*Sec[e + f*x])/(a^2*f*(a - b + b*Sec[e + f*x]^2))","A",6,6,23,0.2609,1,"{3664, 471, 527, 522, 207, 205}"
73,1,210,0,0.2754534,"\int \frac{\csc ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{3 \left(a^2-8 a b+8 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 a^4 f}-\frac{3 b (3 a-4 b) \sec (e+f x)}{8 a^3 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{3 \sqrt{b} (a-2 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 a^4 f}-\frac{(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left(a+b \sec ^2(e+f x)-b\right)}","-\frac{3 \left(a^2-8 a b+8 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 a^4 f}-\frac{3 b (3 a-4 b) \sec (e+f x)}{8 a^3 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{3 \sqrt{b} (a-2 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{2 a^4 f}-\frac{(5 a-6 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left(a+b \sec ^2(e+f x)-b\right)}",1,"(-3*(a - 2*b)*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(2*a^4*f) - (3*(a^2 - 8*a*b + 8*b^2)*ArcTanh[Cos[e + f*x]])/(8*a^4*f) - ((5*a - 6*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*(a - b + b*Sec[e + f*x]^2)) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f*(a - b + b*Sec[e + f*x]^2)) - (3*(3*a - 4*b)*b*Sec[e + f*x])/(8*a^3*f*(a - b + b*Sec[e + f*x]^2))","A",7,6,23,0.2609,1,"{3664, 470, 527, 522, 207, 205}"
74,1,196,0,0.249952,"\int \frac{\sin ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]","\frac{3 x \left(a^2+6 a b+b^2\right)}{8 (a-b)^4}-\frac{3 \sqrt{a} \sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 f (a-b)^4}-\frac{3 b (3 a+b) \tan (e+f x)}{8 f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{(5 a+b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}","\frac{3 x \left(a^2+6 a b+b^2\right)}{8 (a-b)^4}-\frac{3 \sqrt{a} \sqrt{b} (a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 f (a-b)^4}-\frac{3 b (3 a+b) \tan (e+f x)}{8 f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{(5 a+b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}",1,"(3*(a^2 + 6*a*b + b^2)*x)/(8*(a - b)^4) - (3*Sqrt[a]*Sqrt[b]*(a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*(a - b)^4*f) - ((5*a + b)*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f*(a + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)) - (3*b*(3*a + b)*Tan[e + f*x])/(8*(a - b)^3*f*(a + b*Tan[e + f*x]^2))","A",7,6,23,0.2609,1,"{3663, 470, 527, 522, 203, 205}"
75,1,138,0,0.155567,"\int \frac{\sin ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 \sqrt{a} f (a-b)^3}-\frac{b \tan (e+f x)}{f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x (a+3 b)}{2 (a-b)^3}","-\frac{\sqrt{b} (3 a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 \sqrt{a} f (a-b)^3}-\frac{b \tan (e+f x)}{f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x (a+3 b)}{2 (a-b)^3}",1,"((a + 3*b)*x)/(2*(a - b)^3) - (Sqrt[b]*(3*a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*Sqrt[a]*(a - b)^3*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f*(a + b*Tan[e + f*x]^2)) - (b*Tan[e + f*x])/((a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{3663, 471, 527, 522, 203, 205}"
76,1,97,0,0.083395,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-2),x]","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{3/2} f (a-b)^2}-\frac{b \tan (e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{3/2} f (a-b)^2}-\frac{b \tan (e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}",1,"x/(a - b)^2 - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*f) - (b*Tan[e + f*x])/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",5,5,14,0.3571,1,"{3661, 414, 522, 203, 205}"
77,1,82,0,0.0741918,"\int \frac{\csc ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{5/2} f}-\frac{3 \cot (e+f x)}{2 a^2 f}+\frac{\cot (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)\right)}","-\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{5/2} f}-\frac{3 \cot (e+f x)}{2 a^2 f}+\frac{\cot (e+f x)}{2 a f \left(a+b \tan ^2(e+f x)\right)}",1,"(-3*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(5/2)*f) - (3*Cot[e + f*x])/(2*a^2*f) + Cot[e + f*x]/(2*a*f*(a + b*Tan[e + f*x]^2))","A",4,4,23,0.1739,1,"{3663, 290, 325, 205}"
78,1,116,0,0.1458639,"\int \frac{\csc ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{\sqrt{b} (3 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{7/2} f}-\frac{b (a-b) \tan (e+f x)}{2 a^3 f \left(a+b \tan ^2(e+f x)\right)}-\frac{(a-2 b) \cot (e+f x)}{a^3 f}-\frac{\cot ^3(e+f x)}{3 a^2 f}","-\frac{\sqrt{b} (3 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{7/2} f}-\frac{b (a-b) \tan (e+f x)}{2 a^3 f \left(a+b \tan ^2(e+f x)\right)}-\frac{(a-2 b) \cot (e+f x)}{a^3 f}-\frac{\cot ^3(e+f x)}{3 a^2 f}",1,"-((3*a - 5*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(7/2)*f) - ((a - 2*b)*Cot[e + f*x])/(a^3*f) - Cot[e + f*x]^3/(3*a^2*f) - ((a - b)*b*Tan[e + f*x])/(2*a^3*f*(a + b*Tan[e + f*x]^2))","A",5,4,23,0.1739,1,"{3663, 456, 1261, 205}"
79,1,182,0,0.2231468,"\int \frac{\csc ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{b \left(5 a^2-10 a b+7 b^2\right) \tan (e+f x)}{10 a^4 f \left(a+b \tan ^2(e+f x)\right)}-\frac{\left(5 a^2-20 a b+14 b^2\right) \cot (e+f x)}{5 a^4 f}-\frac{\sqrt{b} (3 a-7 b) (a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{9/2} f}-\frac{(10 a-7 b) \cot ^3(e+f x)}{15 a^3 f}-\frac{\cot ^5(e+f x)}{5 a f \left(a+b \tan ^2(e+f x)\right)}","-\frac{b \left(5 a^2-10 a b+7 b^2\right) \tan (e+f x)}{10 a^4 f \left(a+b \tan ^2(e+f x)\right)}-\frac{\left(5 a^2-20 a b+14 b^2\right) \cot (e+f x)}{5 a^4 f}-\frac{\sqrt{b} (3 a-7 b) (a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{9/2} f}-\frac{(10 a-7 b) \cot ^3(e+f x)}{15 a^3 f}-\frac{\cot ^5(e+f x)}{5 a f \left(a+b \tan ^2(e+f x)\right)}",1,"-((3*a - 7*b)*(a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(9/2)*f) - ((5*a^2 - 20*a*b + 14*b^2)*Cot[e + f*x])/(5*a^4*f) - ((10*a - 7*b)*Cot[e + f*x]^3)/(15*a^3*f) - Cot[e + f*x]^5/(5*a*f*(a + b*Tan[e + f*x]^2)) - (b*(5*a^2 - 10*a*b + 7*b^2)*Tan[e + f*x])/(10*a^4*f*(a + b*Tan[e + f*x]^2))","A",6,5,23,0.2174,1,"{3663, 462, 456, 1261, 205}"
80,1,264,0,0.4105313,"\int \frac{\sin ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{\left(5 a^2+20 a b+2 b^2\right) \cos (e+f x)}{5 f (a-b)^5}-\frac{b \left(35 a^2+40 a b+24 b^2\right) \sec (e+f x)}{40 f (a-b)^5 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{b \left(5 a^2+4 b^2\right) \sec (e+f x)}{20 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{\sqrt{b} \left(15 a^2+40 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 f (a-b)^{11/2}}+\frac{(10 a-b) \cos ^3(e+f x)}{15 f (a-b)^4}-\frac{\cos ^5(e+f x)}{5 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^2}","-\frac{\left(5 a^2+20 a b+2 b^2\right) \cos (e+f x)}{5 f (a-b)^5}-\frac{b \left(35 a^2+40 a b+24 b^2\right) \sec (e+f x)}{40 f (a-b)^5 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{b \left(5 a^2+4 b^2\right) \sec (e+f x)}{20 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{\sqrt{b} \left(15 a^2+40 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 f (a-b)^{11/2}}+\frac{(10 a-b) \cos ^3(e+f x)}{15 f (a-b)^4}-\frac{\cos ^5(e+f x)}{5 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^2}",1,"-(Sqrt[b]*(15*a^2 + 40*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*(a - b)^(11/2)*f) - ((5*a^2 + 20*a*b + 2*b^2)*Cos[e + f*x])/(5*(a - b)^5*f) + ((10*a - b)*Cos[e + f*x]^3)/(15*(a - b)^4*f) - Cos[e + f*x]^5/(5*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^2) - (b*(5*a^2 + 4*b^2)*Sec[e + f*x])/(20*(a - b)^4*f*(a - b + b*Sec[e + f*x]^2)^2) - (b*(35*a^2 + 40*a*b + 24*b^2)*Sec[e + f*x])/(40*(a - b)^5*f*(a - b + b*Sec[e + f*x]^2))","A",7,6,23,0.2609,1,"{3664, 462, 456, 1259, 1261, 205}"
81,1,180,0,0.2463487,"\int \frac{\sin ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\cos ^3(e+f x)}{3 f (a-b)^3}-\frac{(a+2 b) \cos (e+f x)}{f (a-b)^4}-\frac{b (7 a+4 b) \sec (e+f x)}{8 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{a b \sec (e+f x)}{4 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{5 \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 f (a-b)^{9/2}}","\frac{\cos ^3(e+f x)}{3 f (a-b)^3}-\frac{(a+2 b) \cos (e+f x)}{f (a-b)^4}-\frac{b (7 a+4 b) \sec (e+f x)}{8 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{a b \sec (e+f x)}{4 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{5 \sqrt{b} (3 a+4 b) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 f (a-b)^{9/2}}",1,"(-5*Sqrt[b]*(3*a + 4*b)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*(a - b)^(9/2)*f) - ((a + 2*b)*Cos[e + f*x])/((a - b)^4*f) + Cos[e + f*x]^3/(3*(a - b)^3*f) - (a*b*Sec[e + f*x])/(4*(a - b)^3*f*(a - b + b*Sec[e + f*x]^2)^2) - (b*(7*a + 4*b)*Sec[e + f*x])/(8*(a - b)^4*f*(a - b + b*Sec[e + f*x]^2))","A",6,5,23,0.2174,1,"{3664, 456, 1259, 1261, 205}"
82,1,138,0,0.0907196,"\int \frac{\sin (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{15 \cos (e+f x)}{8 f (a-b)^3}+\frac{5 \cos (e+f x)}{8 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)}+\frac{\cos (e+f x)}{4 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 f (a-b)^{7/2}}","-\frac{15 \cos (e+f x)}{8 f (a-b)^3}+\frac{5 \cos (e+f x)}{8 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)}+\frac{\cos (e+f x)}{4 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 f (a-b)^{7/2}}",1,"(-15*Sqrt[b]*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*(a - b)^(7/2)*f) - (15*Cos[e + f*x])/(8*(a - b)^3*f) + Cos[e + f*x]/(4*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^2) + (5*Cos[e + f*x])/(8*(a - b)^2*f*(a - b + b*Sec[e + f*x]^2))","A",5,4,21,0.1905,1,"{3664, 290, 325, 205}"
83,1,166,0,0.2167903,"\int \frac{\csc (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 a^3 f (a-b)^{5/2}}-\frac{b (7 a-4 b) \sec (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{\tanh ^{-1}(\cos (e+f x))}{a^3 f}-\frac{b \sec (e+f x)}{4 a f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^2}","-\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 a^3 f (a-b)^{5/2}}-\frac{b (7 a-4 b) \sec (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)}-\frac{\tanh ^{-1}(\cos (e+f x))}{a^3 f}-\frac{b \sec (e+f x)}{4 a f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^2}",1,"-(Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*a^3*(a - b)^(5/2)*f) - ArcTanh[Cos[e + f*x]]/(a^3*f) - (b*Sec[e + f*x])/(4*a*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^2) - ((7*a - 4*b)*b*Sec[e + f*x])/(8*a^2*(a - b)^2*f*(a - b + b*Sec[e + f*x]^2))","A",6,6,21,0.2857,1,"{3664, 414, 527, 522, 207, 205}"
84,1,205,0,0.2926619,"\int \frac{\csc ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{\sqrt{b} \left(15 a^2-40 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 a^4 f (a-b)^{3/2}}-\frac{b (11 a-12 b) \sec (e+f x)}{8 a^3 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}-\frac{3 b \sec (e+f x)}{4 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{(a-6 b) \tanh ^{-1}(\cos (e+f x))}{2 a^4 f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left(a+b \sec ^2(e+f x)-b\right)^2}","-\frac{\sqrt{b} \left(15 a^2-40 a b+24 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 a^4 f (a-b)^{3/2}}-\frac{b (11 a-12 b) \sec (e+f x)}{8 a^3 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)}-\frac{3 b \sec (e+f x)}{4 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{(a-6 b) \tanh ^{-1}(\cos (e+f x))}{2 a^4 f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left(a+b \sec ^2(e+f x)-b\right)^2}",1,"-(Sqrt[b]*(15*a^2 - 40*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*a^4*(a - b)^(3/2)*f) - ((a - 6*b)*ArcTanh[Cos[e + f*x]])/(2*a^4*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*(a - b + b*Sec[e + f*x]^2)^2) - (3*b*Sec[e + f*x])/(4*a^2*f*(a - b + b*Sec[e + f*x]^2)^2) - ((11*a - 12*b)*b*Sec[e + f*x])/(8*a^3*(a - b)*f*(a - b + b*Sec[e + f*x]^2))","A",7,6,23,0.2609,1,"{3664, 471, 527, 522, 207, 205}"
85,1,259,0,0.3761368,"\int \frac{\csc ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{3 \left(a^2-12 a b+16 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 a^5 f}-\frac{3 \sqrt{b} \left(5 a^2-20 a b+16 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 a^5 f \sqrt{a-b}}-\frac{3 b (a-2 b) \sec (e+f x)}{2 a^4 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{b (7 a-12 b) \sec (e+f x)}{8 a^3 f \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left(a+b \sec ^2(e+f x)-b\right)^2}","-\frac{3 \left(a^2-12 a b+16 b^2\right) \tanh ^{-1}(\cos (e+f x))}{8 a^5 f}-\frac{3 \sqrt{b} \left(5 a^2-20 a b+16 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a-b}}\right)}{8 a^5 f \sqrt{a-b}}-\frac{3 b (a-2 b) \sec (e+f x)}{2 a^4 f \left(a+b \sec ^2(e+f x)-b\right)}-\frac{b (7 a-12 b) \sec (e+f x)}{8 a^3 f \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^2}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left(a+b \sec ^2(e+f x)-b\right)^2}",1,"(-3*Sqrt[b]*(5*a^2 - 20*a*b + 16*b^2)*ArcTan[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b]])/(8*a^5*Sqrt[a - b]*f) - (3*(a^2 - 12*a*b + 16*b^2)*ArcTanh[Cos[e + f*x]])/(8*a^5*f) - ((5*a - 8*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*(a - b + b*Sec[e + f*x]^2)^2) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f*(a - b + b*Sec[e + f*x]^2)^2) - ((7*a - 12*b)*b*Sec[e + f*x])/(8*a^3*f*(a - b + b*Sec[e + f*x]^2)^2) - (3*(a - 2*b)*b*Sec[e + f*x])/(2*a^4*f*(a - b + b*Sec[e + f*x]^2))","A",8,6,23,0.2609,1,"{3664, 470, 527, 522, 207, 205}"
86,1,250,0,0.3304958,"\int \frac{\sin ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{3 \sqrt{b} \left(5 a^2+10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 \sqrt{a} f (a-b)^5}+\frac{3 x \left(a^2+10 a b+5 b^2\right)}{8 (a-b)^5}-\frac{3 b (a+b) \tan (e+f x)}{2 f (a-b)^4 \left(a+b \tan ^2(e+f x)\right)}-\frac{b (7 a+5 b) \tan (e+f x)}{8 f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)^2}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{(5 a+3 b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^2}","-\frac{3 \sqrt{b} \left(5 a^2+10 a b+b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 \sqrt{a} f (a-b)^5}+\frac{3 x \left(a^2+10 a b+5 b^2\right)}{8 (a-b)^5}-\frac{3 b (a+b) \tan (e+f x)}{2 f (a-b)^4 \left(a+b \tan ^2(e+f x)\right)}-\frac{b (7 a+5 b) \tan (e+f x)}{8 f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)^2}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{(5 a+3 b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^2}",1,"(3*(a^2 + 10*a*b + 5*b^2)*x)/(8*(a - b)^5) - (3*Sqrt[b]*(5*a^2 + 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*Sqrt[a]*(a - b)^5*f) - ((5*a + 3*b)*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f*(a + b*Tan[e + f*x]^2)^2) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - (b*(7*a + 5*b)*Tan[e + f*x])/(8*(a - b)^3*f*(a + b*Tan[e + f*x]^2)^2) - (3*b*(a + b)*Tan[e + f*x])/(2*(a - b)^4*f*(a + b*Tan[e + f*x]^2))","A",8,6,23,0.2609,1,"{3663, 470, 527, 522, 203, 205}"
87,1,193,0,0.2458474,"\int \frac{\sin ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{\sqrt{b} \left(15 a^2+10 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{3/2} f (a-b)^4}-\frac{b (11 a+b) \tan (e+f x)}{8 a f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)}-\frac{3 b \tan (e+f x)}{4 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x (a+5 b)}{2 (a-b)^4}","-\frac{\sqrt{b} \left(15 a^2+10 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{3/2} f (a-b)^4}-\frac{b (11 a+b) \tan (e+f x)}{8 a f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)}-\frac{3 b \tan (e+f x)}{4 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x (a+5 b)}{2 (a-b)^4}",1,"((a + 5*b)*x)/(2*(a - b)^4) - (Sqrt[b]*(15*a^2 + 10*a*b - b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(3/2)*(a - b)^4*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - (3*b*Tan[e + f*x])/(4*(a - b)^2*f*(a + b*Tan[e + f*x]^2)^2) - (b*(11*a + b)*Tan[e + f*x])/(8*a*(a - b)^3*f*(a + b*Tan[e + f*x]^2))","A",7,6,23,0.2609,1,"{3663, 471, 527, 522, 203, 205}"
88,1,150,0,0.159388,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-3),x]","-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{5/2} f (a-b)^3}-\frac{b (7 a-3 b) \tan (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \tan (e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}","-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{5/2} f (a-b)^3}-\frac{b (7 a-3 b) \tan (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \tan (e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}",1,"x/(a - b)^3 - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^3*f) - (b*Tan[e + f*x])/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((7*a - 3*b)*b*Tan[e + f*x])/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",6,6,14,0.4286,1,"{3661, 414, 527, 522, 203, 205}"
89,1,112,0,0.0877426,"\int \frac{\csc ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{7/2} f}+\frac{5 \cot (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)\right)}-\frac{15 \cot (e+f x)}{8 a^3 f}+\frac{\cot (e+f x)}{4 a f \left(a+b \tan ^2(e+f x)\right)^2}","-\frac{15 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{7/2} f}+\frac{5 \cot (e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)\right)}-\frac{15 \cot (e+f x)}{8 a^3 f}+\frac{\cot (e+f x)}{4 a f \left(a+b \tan ^2(e+f x)\right)^2}",1,"(-15*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(7/2)*f) - (15*Cot[e + f*x])/(8*a^3*f) + Cot[e + f*x]/(4*a*f*(a + b*Tan[e + f*x]^2)^2) + (5*Cot[e + f*x])/(8*a^2*f*(a + b*Tan[e + f*x]^2))","A",5,4,23,0.1739,1,"{3663, 290, 325, 205}"
90,1,154,0,0.2055482,"\int \frac{\csc ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{5 \sqrt{b} (3 a-7 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{9/2} f}-\frac{b (7 a-11 b) \tan (e+f x)}{8 a^4 f \left(a+b \tan ^2(e+f x)\right)}-\frac{b (a-b) \tan (e+f x)}{4 a^3 f \left(a+b \tan ^2(e+f x)\right)^2}-\frac{(a-3 b) \cot (e+f x)}{a^4 f}-\frac{\cot ^3(e+f x)}{3 a^3 f}","-\frac{5 \sqrt{b} (3 a-7 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{9/2} f}-\frac{b (7 a-11 b) \tan (e+f x)}{8 a^4 f \left(a+b \tan ^2(e+f x)\right)}-\frac{b (a-b) \tan (e+f x)}{4 a^3 f \left(a+b \tan ^2(e+f x)\right)^2}-\frac{(a-3 b) \cot (e+f x)}{a^4 f}-\frac{\cot ^3(e+f x)}{3 a^3 f}",1,"(-5*(3*a - 7*b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(9/2)*f) - ((a - 3*b)*Cot[e + f*x])/(a^4*f) - Cot[e + f*x]^3/(3*a^3*f) - ((a - b)*b*Tan[e + f*x])/(4*a^3*f*(a + b*Tan[e + f*x]^2)^2) - ((7*a - 11*b)*b*Tan[e + f*x])/(8*a^4*f*(a + b*Tan[e + f*x]^2))","A",6,5,23,0.2174,1,"{3663, 456, 1259, 1261, 205}"
91,1,231,0,0.3029114,"\int \frac{\csc ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{\sqrt{b} \left(15 a^2-70 a b+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{11/2} f}-\frac{b \left(35 a^2-110 a b+99 b^2\right) \tan (e+f x)}{40 a^5 f \left(a+b \tan ^2(e+f x)\right)}-\frac{b \left(5 a^2-10 a b+9 b^2\right) \tan (e+f x)}{20 a^4 f \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\left(5 a^2-30 a b+27 b^2\right) \cot (e+f x)}{5 a^5 f}-\frac{(10 a-9 b) \cot ^3(e+f x)}{15 a^4 f}-\frac{\cot ^5(e+f x)}{5 a f \left(a+b \tan ^2(e+f x)\right)^2}","-\frac{\sqrt{b} \left(15 a^2-70 a b+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{11/2} f}-\frac{b \left(35 a^2-110 a b+99 b^2\right) \tan (e+f x)}{40 a^5 f \left(a+b \tan ^2(e+f x)\right)}-\frac{b \left(5 a^2-10 a b+9 b^2\right) \tan (e+f x)}{20 a^4 f \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\left(5 a^2-30 a b+27 b^2\right) \cot (e+f x)}{5 a^5 f}-\frac{(10 a-9 b) \cot ^3(e+f x)}{15 a^4 f}-\frac{\cot ^5(e+f x)}{5 a f \left(a+b \tan ^2(e+f x)\right)^2}",1,"-(Sqrt[b]*(15*a^2 - 70*a*b + 63*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(11/2)*f) - ((5*a^2 - 30*a*b + 27*b^2)*Cot[e + f*x])/(5*a^5*f) - ((10*a - 9*b)*Cot[e + f*x]^3)/(15*a^4*f) - Cot[e + f*x]^5/(5*a*f*(a + b*Tan[e + f*x]^2)^2) - (b*(5*a^2 - 10*a*b + 9*b^2)*Tan[e + f*x])/(20*a^4*f*(a + b*Tan[e + f*x]^2)^2) - (b*(35*a^2 - 110*a*b + 99*b^2)*Tan[e + f*x])/(40*a^5*f*(a + b*Tan[e + f*x]^2))","A",7,6,23,0.2609,1,"{3663, 462, 456, 1259, 1261, 205}"
92,1,161,0,0.1653151,"\int \sin ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{5 f (a-b)}+\frac{2 (5 a-4 b) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{15 f (a-b)^2}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}","-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{5 f (a-b)}+\frac{2 (5 a-4 b) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{15 f (a-b)^2}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/f + (2*(5*a - 4*b)*Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(3/2))/(15*(a - b)^2*f) - (Cos[e + f*x]^5*(a - b + b*Sec[e + f*x]^2)^(3/2))/(5*(a - b)*f)","A",6,6,25,0.2400,1,"{3664, 462, 451, 277, 217, 206}"
93,1,113,0,0.1048134,"\int \sin ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{3 f (a-b)}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{3 f (a-b)}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/f + (Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(3/2))/(3*(a - b)*f)","A",5,5,25,0.2000,1,"{3664, 451, 277, 217, 206}"
94,1,72,0,0.0573515,"\int \sin (e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - (Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/f","A",4,4,23,0.1739,1,"{3664, 277, 217, 206}"
95,1,84,0,0.0873346,"\int \csc (e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}",1,"-((Sqrt[a]*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f","A",6,6,23,0.2609,1,"{3664, 402, 217, 206, 377, 207}"
96,1,127,0,0.1394151,"\int \csc ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 \sqrt{a} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f}","-\frac{(a+b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 \sqrt{a} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f}",1,"-((a + b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*Sqrt[a]*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - (Cot[e + f*x]*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*f)","A",7,7,25,0.2800,1,"{3664, 467, 523, 217, 206, 377, 207}"
97,1,187,0,0.2323471,"\int \csc ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\left(3 a^2+6 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{3/2} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{4 f}-\frac{(3 a+b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 a f}","-\frac{\left(3 a^2+6 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{3/2} f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}-\frac{\cot (e+f x) \csc ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{4 f}-\frac{(3 a+b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 a f}",1,"-((3*a^2 + 6*a*b - b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(3/2)*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f - ((3*a + b)*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(8*a*f) - (Cot[e + f*x]*Csc[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(4*f)","A",8,8,25,0.3200,1,"{3664, 467, 578, 523, 217, 206, 377, 207}"
98,1,189,0,0.2366306,"\int \sin ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\left(3 a^2-12 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{3/2}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}-\frac{(3 a-4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f (a-b)}","\frac{\left(3 a^2-12 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{3/2}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}-\frac{(3 a-4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f (a-b)}",1,"((3*a^2 - 12*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*(a - b)^(3/2)*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - ((3*a - 4*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*(a - b)*f) - (Cos[e + f*x]*Sin[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)","A",8,8,25,0.3200,1,"{3663, 467, 578, 523, 217, 206, 377, 203}"
99,1,128,0,0.1346267,"\int \sin ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Sin[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{(a-2 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f \sqrt{a-b}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}","\frac{(a-2 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f \sqrt{a-b}}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}",1,"((a - 2*b)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*Sqrt[a - b]*f) + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)","A",7,7,25,0.2800,1,"{3663, 467, 523, 217, 206, 377, 203}"
100,1,85,0,0.0543786,"\int \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}",1,"(Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f","A",6,6,16,0.3750,1,"{3661, 402, 217, 206, 377, 203}"
101,1,66,0,0.079029,"\int \csc ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f","A",4,4,25,0.1600,1,"{3663, 277, 217, 206}"
102,1,100,0,0.0954462,"\int \csc ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 a f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}","\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 a f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f - (Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2))/(3*a*f)","A",5,5,25,0.2000,1,"{3663, 451, 277, 217, 206}"
103,1,141,0,0.1282214,"\int \csc ^6(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Csc[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{2 (5 a-b) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{15 a^2 f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{5 a f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}","-\frac{2 (5 a-b) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{15 a^2 f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{5 a f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f - (2*(5*a - b)*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2))/(15*a^2*f) - (Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2))/(5*a*f)","A",6,6,25,0.2400,1,"{3663, 462, 451, 277, 217, 206}"
104,1,227,0,0.2157307,"\int \sin ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b (3 a-7 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f (a-b)}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{5/2}}{5 f (a-b)}+\frac{2 \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{5/2}}{3 f (a-b)}-\frac{(3 a-7 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{3 f (a-b)}+\frac{\sqrt{b} (3 a-7 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}","\frac{b (3 a-7 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f (a-b)}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{5/2}}{5 f (a-b)}+\frac{2 \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{5/2}}{3 f (a-b)}-\frac{(3 a-7 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{3 f (a-b)}+\frac{\sqrt{b} (3 a-7 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}",1,"((3*a - 7*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + ((3*a - 7*b)*b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*(a - b)*f) - ((3*a - 7*b)*Cos[e + f*x]*(a - b + b*Sec[e + f*x]^2)^(3/2))/(3*(a - b)*f) + (2*Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(5/2))/(3*(a - b)*f) - (Cos[e + f*x]^5*(a - b + b*Sec[e + f*x]^2)^(5/2))/(5*(a - b)*f)","A",7,7,25,0.2800,1,"{3664, 462, 453, 277, 195, 217, 206}"
105,1,186,0,0.1645992,"\int \sin ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b (3 a-5 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f (a-b)}+\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{5/2}}{3 f (a-b)}-\frac{(3 a-5 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{3 f (a-b)}+\frac{\sqrt{b} (3 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}","\frac{b (3 a-5 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f (a-b)}+\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{5/2}}{3 f (a-b)}-\frac{(3 a-5 b) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{3 f (a-b)}+\frac{\sqrt{b} (3 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}",1,"((3*a - 5*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + ((3*a - 5*b)*b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*(a - b)*f) - ((3*a - 5*b)*Cos[e + f*x]*(a - b + b*Sec[e + f*x]^2)^(3/2))/(3*(a - b)*f) + (Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(5/2))/(3*(a - b)*f)","A",6,6,25,0.2400,1,"{3664, 453, 277, 195, 217, 206}"
106,1,113,0,0.0819726,"\int \sin (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{3 b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f}-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{f}+\frac{3 \sqrt{b} (a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}","\frac{3 b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f}-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{f}+\frac{3 \sqrt{b} (a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}",1,"(3*(a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (3*b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*f) - (Cos[e + f*x]*(a - b + b*Sec[e + f*x]^2)^(3/2))/f","A",5,5,23,0.2174,1,"{3664, 277, 195, 217, 206}"
107,1,127,0,0.1406755,"\int \csc (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}+\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f}+\frac{\sqrt{b} (3 a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{f}+\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 f}+\frac{\sqrt{b} (3 a-b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}",1,"-((a^(3/2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/f) + ((3*a - b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*f)","A",7,7,23,0.3043,1,"{3664, 416, 523, 217, 206, 377, 207}"
108,1,167,0,0.2055301,"\int \csc ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}-\frac{\sqrt{a} (a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}+\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}-\frac{\cot (e+f x) \csc (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{2 f}","\frac{b \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f}-\frac{\sqrt{a} (a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}+\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}-\frac{\cot (e+f x) \csc (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{2 f}",1,"-(Sqrt[a]*(a + 3*b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (Sqrt[b]*(3*a + b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (b*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/f - (Cot[e + f*x]*Csc[e + f*x]*(a - b + b*Sec[e + f*x]^2)^(3/2))/(2*f)","A",8,8,25,0.3200,1,"{3664, 467, 528, 523, 217, 206, 377, 207}"
109,1,223,0,0.3618104,"\int \csc ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{3 \left(a^2+6 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 \sqrt{a} f}+\frac{3 (a+3 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 f}-\frac{3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 f}+\frac{3 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{4 f}","-\frac{3 \left(a^2+6 a b+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 \sqrt{a} f}+\frac{3 (a+3 b) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 f}-\frac{3 (a+b) \csc ^2(e+f x) \sec (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 f}+\frac{3 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}{4 f}",1,"(-3*(a^2 + 6*a*b + b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*Sqrt[a]*f) + (3*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*f) + (3*(a + 3*b)*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(8*f) - (3*(a + b)*Csc[e + f*x]^2*Sec[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(8*f) - (Cot[e + f*x]*Csc[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(3/2))/(4*f)","A",9,9,25,0.3600,1,"{3664, 467, 577, 582, 523, 217, 206, 377, 207}"
110,1,222,0,0.3215162,"\int \sin ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Sin[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{3 \left(a^2-8 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f \sqrt{a-b}}-\frac{3 (a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{3 (a-2 b) \sin ^2(e+f x) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{3 \sqrt{b} (a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\sin ^3(e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{4 f}","\frac{3 \left(a^2-8 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f \sqrt{a-b}}-\frac{3 (a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{3 (a-2 b) \sin ^2(e+f x) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{3 \sqrt{b} (a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\sin ^3(e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{4 f}",1,"(3*(a^2 - 8*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*Sqrt[a - b]*f) + (3*(a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) - (3*(a - 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*f) + (3*(a - 2*b)*Sin[e + f*x]^2*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*f) - (Cos[e + f*x]*Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2))/(4*f)","A",9,9,25,0.3600,1,"{3663, 467, 577, 582, 523, 217, 206, 377, 203}"
111,1,165,0,0.2017761,"\int \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{(a-4 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}+\frac{\sqrt{b} (3 a-4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\sin (e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{2 f}","\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{(a-4 b) \sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}+\frac{\sqrt{b} (3 a-4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\sin (e+f x) \cos (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{2 f}",1,"((a - 4*b)*Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + ((3*a - 4*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f - (Cos[e + f*x]*Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2))/(2*f)","A",8,8,25,0.3200,1,"{3663, 467, 528, 523, 217, 206, 377, 203}"
112,1,125,0,0.099861,"\int \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}","\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}",1,"((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + ((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)","A",7,7,16,0.4375,1,"{3661, 416, 523, 217, 206, 377, 203}"
113,1,100,0,0.0988543,"\int \csc ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{3 b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{f}","\frac{3 b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{f}",1,"(3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (3*b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f) - (Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2))/f","A",5,5,25,0.2000,1,"{3663, 277, 195, 217, 206}"
114,1,162,0,0.1366026,"\int \csc ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b (3 a+2 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}+\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a+2 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 a f}","\frac{b (3 a+2 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}+\frac{\sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a+2 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 a f}",1,"(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*(3*a + 2*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*a*f) - ((3*a + 2*b)*Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2))/(3*a*f) - (Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(5/2))/(3*a*f)","A",6,6,25,0.2400,1,"{3663, 453, 277, 195, 217, 206}"
115,1,196,0,0.1705118,"\int \csc ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b (3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}+\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 a f}-\frac{2 \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a+4 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 a f}","\frac{b (3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}+\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 a f}-\frac{2 \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{3 a f}-\frac{(3 a+4 b) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 a f}",1,"(Sqrt[b]*(3*a + 4*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*(3*a + 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*a*f) - ((3*a + 4*b)*Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2))/(3*a*f) - (2*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(5/2))/(3*a*f) - (Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(5/2))/(5*a*f)","A",7,7,25,0.2800,1,"{3663, 462, 453, 277, 195, 217, 206}"
116,1,144,0,0.1441107,"\int \frac{\sin ^5(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\left(15 a^2-10 a b+3 b^2\right) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{15 f (a-b)^3}-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{5 f (a-b)}+\frac{2 (5 a-3 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{15 f (a-b)^2}","-\frac{\left(15 a^2-10 a b+3 b^2\right) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{15 f (a-b)^3}-\frac{\cos ^5(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{5 f (a-b)}+\frac{2 (5 a-3 b) \cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{15 f (a-b)^2}",1,"-((15*a^2 - 10*a*b + 3*b^2)*Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(15*(a - b)^3*f) + (2*(5*a - 3*b)*Cos[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(15*(a - b)^2*f) - (Cos[e + f*x]^5*Sqrt[a - b + b*Sec[e + f*x]^2])/(5*(a - b)*f)","A",4,4,25,0.1600,1,"{3664, 462, 453, 264}"
117,1,88,0,0.101122,"\int \frac{\sin ^3(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{3 f (a-b)}-\frac{(3 a-b) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{3 f (a-b)^2}","\frac{\cos ^3(e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{3 f (a-b)}-\frac{(3 a-b) \cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{3 f (a-b)^2}",1,"-((3*a - b)*Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(3*(a - b)^2*f) + (Cos[e + f*x]^3*Sqrt[a - b + b*Sec[e + f*x]^2])/(3*(a - b)*f)","A",3,3,25,0.1200,1,"{3664, 453, 264}"
118,1,37,0,0.0465261,"\int \frac{\sin (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Sin[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f (a-b)}","-\frac{\cos (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{f (a-b)}",1,"-((Cos[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/((a - b)*f))","A",2,2,23,0.08696,1,"{3664, 264}"
119,1,42,0,0.0683289,"\int \frac{\csc (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Csc[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{\sqrt{a} f}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{\sqrt{a} f}",1,"-(ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]]/(Sqrt[a]*f))","A",3,3,23,0.1304,1,"{3664, 377, 207}"
120,1,91,0,0.1149375,"\int \frac{\csc ^3(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 a^{3/2} f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 a f}","-\frac{(a-b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 a^{3/2} f}-\frac{\cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{2 a f}",1,"-((a - b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*a^(3/2)*f) - (Cot[e + f*x]*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(2*a*f)","A",5,5,25,0.2000,1,"{3664, 471, 12, 377, 207}"
121,1,143,0,0.1600423,"\int \frac{\csc ^5(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{3 (a-b)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{5/2} f}-\frac{(5 a-3 b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 a^2 f}-\frac{\cot ^3(e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{4 a f}","-\frac{3 (a-b)^2 \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{5/2} f}-\frac{(5 a-3 b) \cot (e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{8 a^2 f}-\frac{\cot ^3(e+f x) \csc (e+f x) \sqrt{a+b \sec ^2(e+f x)-b}}{4 a f}",1,"(-3*(a - b)^2*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(5/2)*f) - ((5*a - 3*b)*Cot[e + f*x]*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(8*a^2*f) - (Cot[e + f*x]^3*Csc[e + f*x]*Sqrt[a - b + b*Sec[e + f*x]^2])/(4*a*f)","A",6,6,25,0.2400,1,"{3664, 470, 527, 12, 377, 207}"
122,1,146,0,0.1666357,"\int \frac{\sin ^4(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{3 a^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{5/2}}+\frac{\sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f (a-b)}-\frac{(5 a-2 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f (a-b)^2}","\frac{3 a^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{5/2}}+\frac{\sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f (a-b)}-\frac{(5 a-2 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f (a-b)^2}",1,"(3*a^2*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*(a - b)^(5/2)*f) - ((5*a - 2*b)*Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*(a - b)^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(4*(a - b)*f)","A",6,6,25,0.2400,1,"{3663, 470, 527, 12, 377, 203}"
123,1,93,0,0.1070258,"\int \frac{\sin ^2(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Sin[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f (a-b)^{3/2}}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f (a-b)}","\frac{a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f (a-b)^{3/2}}-\frac{\sin (e+f x) \cos (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f (a-b)}",1,"(a*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*(a - b)^(3/2)*f) - (Cos[e + f*x]*Sin[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*(a - b)*f)","A",5,5,25,0.2000,1,"{3663, 471, 12, 377, 203}"
124,1,46,0,0.0340249,"\int \frac{1}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[1/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)","A",3,3,16,0.1875,1,"{3661, 377, 203}"
125,1,30,0,0.0696135,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a f}","-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a f}",1,"-((Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(a*f))","A",2,2,25,0.08000,1,"{3663, 264}"
126,1,74,0,0.0900209,"\int \frac{\csc ^4(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{(3 a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}","-\frac{(3 a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}",1,"-((3*a - 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^2*f) - (Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a*f)","A",3,3,25,0.1200,1,"{3663, 453, 264}"
127,1,123,0,0.136504,"\int \frac{\csc ^6(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Csc[e + f*x]^6/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\left(15 a^2-20 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f}-\frac{2 (5 a-2 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f}","-\frac{\left(15 a^2-20 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f}-\frac{2 (5 a-2 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f}",1,"-((15*a^2 - 20*a*b + 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^3*f) - (2*(5*a - 2*b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^2*f) - (Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*a*f)","A",4,4,25,0.1600,1,"{3663, 462, 453, 264}"
128,1,199,0,0.1871865,"\int \frac{\sin ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{2 b \left(15 a^2+10 a b-b^2\right) \sec (e+f x)}{15 f (a-b)^4 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\left(15 a^2+10 a b-b^2\right) \cos (e+f x)}{15 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cos ^5(e+f x)}{5 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}+\frac{2 (5 a-2 b) \cos ^3(e+f x)}{15 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}","-\frac{2 b \left(15 a^2+10 a b-b^2\right) \sec (e+f x)}{15 f (a-b)^4 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\left(15 a^2+10 a b-b^2\right) \cos (e+f x)}{15 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cos ^5(e+f x)}{5 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}+\frac{2 (5 a-2 b) \cos ^3(e+f x)}{15 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-((15*a^2 + 10*a*b - b^2)*Cos[e + f*x])/(15*(a - b)^3*f*Sqrt[a - b + b*Sec[e + f*x]^2]) + (2*(5*a - 2*b)*Cos[e + f*x]^3)/(15*(a - b)^2*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - Cos[e + f*x]^5/(5*(a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - (2*b*(15*a^2 + 10*a*b - b^2)*Sec[e + f*x])/(15*(a - b)^4*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",5,5,25,0.2000,1,"{3664, 462, 453, 271, 191}"
129,1,131,0,0.1343079,"\int \frac{\sin ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{2 b (3 a+b) \sec (e+f x)}{3 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}+\frac{\cos ^3(e+f x)}{3 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{(3 a+b) \cos (e+f x)}{3 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}","-\frac{2 b (3 a+b) \sec (e+f x)}{3 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}+\frac{\cos ^3(e+f x)}{3 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{(3 a+b) \cos (e+f x)}{3 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-((3*a + b)*Cos[e + f*x])/(3*(a - b)^2*f*Sqrt[a - b + b*Sec[e + f*x]^2]) + Cos[e + f*x]^3/(3*(a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - (2*b*(3*a + b)*Sec[e + f*x])/(3*(a - b)^3*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",4,4,25,0.1600,1,"{3664, 453, 271, 191}"
130,1,76,0,0.0638157,"\int \frac{\sin (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{2 b \sec (e+f x)}{f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cos (e+f x)}{f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}","-\frac{2 b \sec (e+f x)}{f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cos (e+f x)}{f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-(Cos[e + f*x]/((a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2])) - (2*b*Sec[e + f*x])/((a - b)^2*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",3,3,23,0.1304,1,"{3664, 271, 191}"
131,1,84,0,0.0965029,"\int \frac{\csc (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{a^{3/2} f}-\frac{b \sec (e+f x)}{a f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{a^{3/2} f}-\frac{b \sec (e+f x)}{a f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-(ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]]/(a^(3/2)*f)) - (b*Sec[e + f*x])/(a*(a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",4,4,23,0.1739,1,"{3664, 382, 377, 207}"
132,1,127,0,0.1577313,"\int \frac{\csc ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{3 b \sec (e+f x)}{2 a^2 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{(a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 a^{5/2} f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a+b \sec ^2(e+f x)-b}}","-\frac{3 b \sec (e+f x)}{2 a^2 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{(a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 a^{5/2} f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \sqrt{a+b \sec ^2(e+f x)-b}}",1,"-((a - 3*b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*a^(5/2)*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - (3*b*Sec[e + f*x])/(2*a^2*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",6,6,25,0.2400,1,"{3664, 471, 527, 12, 377, 207}"
133,1,187,0,0.2360329,"\int \frac{\csc ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{b (13 a-15 b) \sec (e+f x)}{8 a^3 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{3 (a-5 b) (a-b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{7/2} f}-\frac{5 (a-b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \sqrt{a+b \sec ^2(e+f x)-b}}","-\frac{b (13 a-15 b) \sec (e+f x)}{8 a^3 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{3 (a-5 b) (a-b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{7/2} f}-\frac{5 (a-b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \sqrt{a+b \sec ^2(e+f x)-b}}",1,"(-3*(a - 5*b)*(a - b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(7/2)*f) - (5*(a - b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f*Sqrt[a - b + b*Sec[e + f*x]^2]) - ((13*a - 15*b)*b*Sec[e + f*x])/(8*a^3*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",7,6,25,0.2400,1,"{3664, 470, 527, 12, 377, 207}"
134,1,187,0,0.2227874,"\int \frac{\sin ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{3 a (a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{7/2}}-\frac{b (13 a+2 b) \tan (e+f x)}{8 f (a-b)^3 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{5 a \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}","\frac{3 a (a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{7/2}}-\frac{b (13 a+2 b) \tan (e+f x)}{8 f (a-b)^3 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{5 a \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}",1,"(3*a*(a + 4*b)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*(a - b)^(7/2)*f) - (5*a*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - (b*(13*a + 2*b)*Tan[e + f*x])/(8*(a - b)^3*f*Sqrt[a + b*Tan[e + f*x]^2])","A",7,6,25,0.2400,1,"{3663, 470, 527, 12, 377, 203}"
135,1,134,0,0.1574883,"\int \frac{\sin ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f (a-b)^{5/2}}-\frac{3 b \tan (e+f x)}{2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","\frac{(a+2 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f (a-b)^{5/2}}-\frac{3 b \tan (e+f x)}{2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"((a + 2*b)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*(a - b)^(5/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - (3*b*Tan[e + f*x])/(2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",6,6,25,0.2400,1,"{3663, 471, 527, 12, 377, 203}"
136,1,85,0,0.0572698,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-3/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \tan (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \tan (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f) - (b*Tan[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])","A",4,4,16,0.2500,1,"{3661, 382, 377, 203}"
137,1,62,0,0.0979908,"\int \frac{\csc ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{2 b \tan (e+f x)}{a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot (e+f x)}{a f \sqrt{a+b \tan ^2(e+f x)}}","-\frac{2 b \tan (e+f x)}{a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot (e+f x)}{a f \sqrt{a+b \tan ^2(e+f x)}}",1,"-(Cot[e + f*x]/(a*f*Sqrt[a + b*Tan[e + f*x]^2])) - (2*b*Tan[e + f*x])/(a^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",3,3,25,0.1200,1,"{3663, 271, 191}"
138,1,114,0,0.1215582,"\int \frac{\csc ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{2 b (3 a-4 b) \tan (e+f x)}{3 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{3 a f \sqrt{a+b \tan ^2(e+f x)}}","-\frac{2 b (3 a-4 b) \tan (e+f x)}{3 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{(3 a-4 b) \cot (e+f x)}{3 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^3(e+f x)}{3 a f \sqrt{a+b \tan ^2(e+f x)}}",1,"-((3*a - 4*b)*Cot[e + f*x])/(3*a^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^3/(3*a*f*Sqrt[a + b*Tan[e + f*x]^2]) - (2*(3*a - 4*b)*b*Tan[e + f*x])/(3*a^3*f*Sqrt[a + b*Tan[e + f*x]^2])","A",4,4,25,0.1600,1,"{3663, 453, 271, 191}"
139,1,171,0,0.1788636,"\int \frac{\csc ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{2 b \left(15 a^2-40 a b+24 b^2\right) \tan (e+f x)}{15 a^4 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left(15 a^2-40 a b+24 b^2\right) \cot (e+f x)}{15 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{2 (5 a-3 b) \cot ^3(e+f x)}{15 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^5(e+f x)}{5 a f \sqrt{a+b \tan ^2(e+f x)}}","-\frac{2 b \left(15 a^2-40 a b+24 b^2\right) \tan (e+f x)}{15 a^4 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left(15 a^2-40 a b+24 b^2\right) \cot (e+f x)}{15 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{2 (5 a-3 b) \cot ^3(e+f x)}{15 a^2 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{\cot ^5(e+f x)}{5 a f \sqrt{a+b \tan ^2(e+f x)}}",1,"-((15*a^2 - 40*a*b + 24*b^2)*Cot[e + f*x])/(15*a^3*f*Sqrt[a + b*Tan[e + f*x]^2]) - (2*(5*a - 3*b)*Cot[e + f*x]^3)/(15*a^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^5/(5*a*f*Sqrt[a + b*Tan[e + f*x]^2]) - (2*b*(15*a^2 - 40*a*b + 24*b^2)*Tan[e + f*x])/(15*a^4*f*Sqrt[a + b*Tan[e + f*x]^2])","A",5,5,25,0.2000,1,"{3663, 462, 453, 271, 191}"
140,1,248,0,0.2314932,"\int \frac{\sin ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{8 b \left(5 a^2+10 a b+b^2\right) \sec (e+f x)}{15 f (a-b)^5 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{4 b \left(5 a^2+10 a b+b^2\right) \sec (e+f x)}{15 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\left(5 a^2+10 a b+b^2\right) \cos (e+f x)}{5 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\cos ^5(e+f x)}{5 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}+\frac{2 (5 a-b) \cos ^3(e+f x)}{15 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}","-\frac{8 b \left(5 a^2+10 a b+b^2\right) \sec (e+f x)}{15 f (a-b)^5 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{4 b \left(5 a^2+10 a b+b^2\right) \sec (e+f x)}{15 f (a-b)^4 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\left(5 a^2+10 a b+b^2\right) \cos (e+f x)}{5 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\cos ^5(e+f x)}{5 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}+\frac{2 (5 a-b) \cos ^3(e+f x)}{15 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"-((5*a^2 + 10*a*b + b^2)*Cos[e + f*x])/(5*(a - b)^3*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) + (2*(5*a - b)*Cos[e + f*x]^3)/(15*(a - b)^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - Cos[e + f*x]^5/(5*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (4*b*(5*a^2 + 10*a*b + b^2)*Sec[e + f*x])/(15*(a - b)^4*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(5*a^2 + 10*a*b + b^2)*Sec[e + f*x])/(15*(a - b)^5*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",6,6,25,0.2400,1,"{3664, 462, 453, 271, 192, 191}"
141,1,168,0,0.1593059,"\int \frac{\sin ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{8 b (a+b) \sec (e+f x)}{3 f (a-b)^4 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{4 b (a+b) \sec (e+f x)}{3 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}+\frac{\cos ^3(e+f x)}{3 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{(a+b) \cos (e+f x)}{f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}","-\frac{8 b (a+b) \sec (e+f x)}{3 f (a-b)^4 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{4 b (a+b) \sec (e+f x)}{3 f (a-b)^3 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}+\frac{\cos ^3(e+f x)}{3 f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{(a+b) \cos (e+f x)}{f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"-(((a + b)*Cos[e + f*x])/((a - b)^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2))) + Cos[e + f*x]^3/(3*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (4*b*(a + b)*Sec[e + f*x])/(3*(a - b)^3*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (8*b*(a + b)*Sec[e + f*x])/(3*(a - b)^4*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",5,5,25,0.2000,1,"{3664, 453, 271, 192, 191}"
142,1,118,0,0.0776282,"\int \frac{\sin (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{8 b \sec (e+f x)}{3 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{4 b \sec (e+f x)}{3 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\cos (e+f x)}{f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}","-\frac{8 b \sec (e+f x)}{3 f (a-b)^3 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{4 b \sec (e+f x)}{3 f (a-b)^2 \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\cos (e+f x)}{f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"-(Cos[e + f*x]/((a - b)*f*(a - b + b*Sec[e + f*x]^2)^(3/2))) - (4*b*Sec[e + f*x])/(3*(a - b)^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (8*b*Sec[e + f*x])/(3*(a - b)^3*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",4,4,23,0.1739,1,"{3664, 271, 192, 191}"
143,1,136,0,0.1455038,"\int \frac{\csc (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{b (5 a-3 b) \sec (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{a^{5/2} f}-\frac{b \sec (e+f x)}{3 a f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}","-\frac{b (5 a-3 b) \sec (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{a^{5/2} f}-\frac{b \sec (e+f x)}{3 a f (a-b) \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"-(ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]]/(a^(5/2)*f)) - (b*Sec[e + f*x])/(3*a*(a - b)*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - ((5*a - 3*b)*b*Sec[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",6,6,23,0.2609,1,"{3664, 414, 527, 12, 377, 207}"
144,1,177,0,0.20946,"\int \frac{\csc ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{b (13 a-15 b) \sec (e+f x)}{6 a^3 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{5 b \sec (e+f x)}{6 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{(a-5 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 a^{7/2} f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}","-\frac{b (13 a-15 b) \sec (e+f x)}{6 a^3 f (a-b) \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{5 b \sec (e+f x)}{6 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{(a-5 b) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{2 a^{7/2} f}-\frac{\cot (e+f x) \csc (e+f x)}{2 a f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"-((a - 5*b)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(2*a^(7/2)*f) - (Cot[e + f*x]*Csc[e + f*x])/(2*a*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (5*b*Sec[e + f*x])/(6*a^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - ((13*a - 15*b)*b*Sec[e + f*x])/(6*a^3*(a - b)*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",7,6,25,0.2400,1,"{3664, 471, 527, 12, 377, 207}"
145,1,237,0,0.3243431,"\int \frac{\csc ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\left(3 a^2-30 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{9/2} f}-\frac{5 b (11 a-21 b) \sec (e+f x)}{24 a^4 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{b (23 a-35 b) \sec (e+f x)}{24 a^3 f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}","-\frac{\left(3 a^2-30 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a} \sec (e+f x)}{\sqrt{a+b \sec ^2(e+f x)-b}}\right)}{8 a^{9/2} f}-\frac{5 b (11 a-21 b) \sec (e+f x)}{24 a^4 f \sqrt{a+b \sec ^2(e+f x)-b}}-\frac{b (23 a-35 b) \sec (e+f x)}{24 a^3 f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{(5 a-7 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}-\frac{\cot ^3(e+f x) \csc (e+f x)}{4 a f \left(a+b \sec ^2(e+f x)-b\right)^{3/2}}",1,"-((3*a^2 - 30*a*b + 35*b^2)*ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]])/(8*a^(9/2)*f) - ((5*a - 7*b)*Cot[e + f*x]*Csc[e + f*x])/(8*a^2*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (Cot[e + f*x]^3*Csc[e + f*x])/(4*a*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - ((23*a - 35*b)*b*Sec[e + f*x])/(24*a^3*f*(a - b + b*Sec[e + f*x]^2)^(3/2)) - (5*(11*a - 21*b)*b*Sec[e + f*x])/(24*a^4*f*Sqrt[a - b + b*Sec[e + f*x]^2])","A",8,6,25,0.2400,1,"{3664, 470, 527, 12, 377, 207}"
146,1,246,0,0.3334222,"\int \frac{\sin ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\left(3 a^2+24 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{9/2}}-\frac{5 b (11 a+10 b) \tan (e+f x)}{24 f (a-b)^4 \sqrt{a+b \tan ^2(e+f x)}}-\frac{b (23 a+12 b) \tan (e+f x)}{24 f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{(5 a+2 b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^{3/2}}","\frac{\left(3 a^2+24 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 f (a-b)^{9/2}}-\frac{5 b (11 a+10 b) \tan (e+f x)}{24 f (a-b)^4 \sqrt{a+b \tan ^2(e+f x)}}-\frac{b (23 a+12 b) \tan (e+f x)}{24 f (a-b)^3 \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{(5 a+2 b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"((3*a^2 + 24*a*b + 8*b^2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*(a - b)^(9/2)*f) - ((5*a + 2*b)*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) + (Cos[e + f*x]^3*Sin[e + f*x])/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (b*(23*a + 12*b)*Tan[e + f*x])/(24*(a - b)^3*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (5*b*(11*a + 10*b)*Tan[e + f*x])/(24*(a - b)^4*f*Sqrt[a + b*Tan[e + f*x]^2])","A",8,6,25,0.2400,1,"{3663, 470, 527, 12, 377, 203}"
147,1,181,0,0.2088749,"\int \frac{\sin ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Sin[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{(a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f (a-b)^{7/2}}-\frac{b (13 a+2 b) \tan (e+f x)}{6 a f (a-b)^3 \sqrt{a+b \tan ^2(e+f x)}}-\frac{5 b \tan (e+f x)}{6 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","\frac{(a+4 b) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f (a-b)^{7/2}}-\frac{b (13 a+2 b) \tan (e+f x)}{6 a f (a-b)^3 \sqrt{a+b \tan ^2(e+f x)}}-\frac{5 b \tan (e+f x)}{6 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"((a + 4*b)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*(a - b)^(7/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (5*b*Tan[e + f*x])/(6*(a - b)^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (b*(13*a + 2*b)*Tan[e + f*x])/(6*a*(a - b)^3*f*Sqrt[a + b*Tan[e + f*x]^2])","A",7,6,25,0.2400,1,"{3663, 471, 527, 12, 377, 203}"
148,1,134,0,0.1126362,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-5/2),x]","-\frac{b (5 a-2 b) \tan (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \tan (e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{b (5 a-2 b) \tan (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \tan (e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((5*a - 2*b)*b*Tan[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",6,6,16,0.3750,1,"{3661, 414, 527, 12, 377, 203}"
149,1,97,0,0.1055952,"\int \frac{\csc ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{8 b \tan (e+f x)}{3 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{4 b \tan (e+f x)}{3 a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\cot (e+f x)}{a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{8 b \tan (e+f x)}{3 a^3 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{4 b \tan (e+f x)}{3 a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\cot (e+f x)}{a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-(Cot[e + f*x]/(a*f*(a + b*Tan[e + f*x]^2)^(3/2))) - (4*b*Tan[e + f*x])/(3*a^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (8*b*Tan[e + f*x])/(3*a^3*f*Sqrt[a + b*Tan[e + f*x]^2])","A",4,4,25,0.1600,1,"{3663, 271, 192, 191}"
150,1,146,0,0.1490293,"\int \frac{\csc ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{8 b (a-2 b) \tan (e+f x)}{3 a^4 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{4 b (a-2 b) \tan (e+f x)}{3 a^3 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{(a-2 b) \cot (e+f x)}{a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\cot ^3(e+f x)}{3 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{8 b (a-2 b) \tan (e+f x)}{3 a^4 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{4 b (a-2 b) \tan (e+f x)}{3 a^3 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{(a-2 b) \cot (e+f x)}{a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\cot ^3(e+f x)}{3 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-(((a - 2*b)*Cot[e + f*x])/(a^2*f*(a + b*Tan[e + f*x]^2)^(3/2))) - Cot[e + f*x]^3/(3*a*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (4*(a - 2*b)*b*Tan[e + f*x])/(3*a^3*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (8*(a - 2*b)*b*Tan[e + f*x])/(3*a^4*f*Sqrt[a + b*Tan[e + f*x]^2])","A",5,5,25,0.2000,1,"{3663, 453, 271, 192, 191}"
151,1,219,0,0.2288429,"\int \frac{\csc ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Csc[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{8 b \left(5 a^2-20 a b+16 b^2\right) \tan (e+f x)}{15 a^5 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{4 b \left(5 a^2-20 a b+16 b^2\right) \tan (e+f x)}{15 a^4 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\left(5 a^2-20 a b+16 b^2\right) \cot (e+f x)}{5 a^3 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{2 (5 a-4 b) \cot ^3(e+f x)}{15 a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\cot ^5(e+f x)}{5 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{8 b \left(5 a^2-20 a b+16 b^2\right) \tan (e+f x)}{15 a^5 f \sqrt{a+b \tan ^2(e+f x)}}-\frac{4 b \left(5 a^2-20 a b+16 b^2\right) \tan (e+f x)}{15 a^4 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\left(5 a^2-20 a b+16 b^2\right) \cot (e+f x)}{5 a^3 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{2 (5 a-4 b) \cot ^3(e+f x)}{15 a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\cot ^5(e+f x)}{5 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-((5*a^2 - 20*a*b + 16*b^2)*Cot[e + f*x])/(5*a^3*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (2*(5*a - 4*b)*Cot[e + f*x]^3)/(15*a^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - Cot[e + f*x]^5/(5*a*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (4*b*(5*a^2 - 20*a*b + 16*b^2)*Tan[e + f*x])/(15*a^4*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (8*b*(5*a^2 - 20*a*b + 16*b^2)*Tan[e + f*x])/(15*a^5*f*Sqrt[a + b*Tan[e + f*x]^2])","A",6,6,25,0.2400,1,"{3663, 462, 453, 271, 192, 191}"
152,1,92,0,0.1533088,"\int (d \sin (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Sin[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \cos ^2(e+f x)^{p+\frac{1}{2}} \left(b \tan ^2(e+f x)\right)^p (d \sin (e+f x))^m \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);\sin ^2(e+f x)\right)}{f (m+2 p+1)}","\frac{\tan (e+f x) \cos ^2(e+f x)^{p+\frac{1}{2}} \left(b \tan ^2(e+f x)\right)^p (d \sin (e+f x))^m \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);\sin ^2(e+f x)\right)}{f (m+2 p+1)}",1,"((Cos[e + f*x]^2)^(1/2 + p)*Hypergeometric2F1[(1 + 2*p)/2, (1 + m + 2*p)/2, (3 + m + 2*p)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^m*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + m + 2*p))","A",3,3,23,0.1304,1,"{3658, 2602, 2577}"
153,1,121,0,0.1411867,"\int (d \sin (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (d \sin (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};\frac{m+2}{2},-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (m+1)}","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (d \sin (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};\frac{m+2}{2},-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (m+1)}",1,"(AppellF1[(1 + m)/2, (2 + m)/2, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(Sec[e + f*x]^2)^(m/2)*(d*Sin[e + f*x])^m*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + m)*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,25,0.1200,1,"{3667, 511, 510}"
154,1,208,0,0.2228005,"\int \sin ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Sin[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\left(15 a^2-20 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{15 f (a-b)^2}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{p+1}}{5 f (a-b)}+\frac{(10 a-2 b p-7 b) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{p+1}}{15 f (a-b)^2}","-\frac{\left(15 a^2-20 a b (p+1)+4 b^2 \left(p^2+3 p+2\right)\right) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{15 f (a-b)^2}-\frac{\cos ^5(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{p+1}}{5 f (a-b)}+\frac{(10 a-2 b p-7 b) \cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{p+1}}{15 f (a-b)^2}",1,"((10*a - 7*b - 2*b*p)*Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(1 + p))/(15*(a - b)^2*f) - (Cos[e + f*x]^5*(a - b + b*Sec[e + f*x]^2)^(1 + p))/(5*(a - b)*f) - ((15*a^2 - 20*a*b*(1 + p) + 4*b^2*(2 + 3*p + p^2))*Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))]*(a - b + b*Sec[e + f*x]^2)^p)/(15*(a - b)^2*f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p)","A",5,5,23,0.2174,1,"{3664, 462, 453, 365, 364}"
155,1,140,0,0.1113928,"\int \sin ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Sin[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{p+1}}{3 f (a-b)}-\frac{(3 a-2 b (p+1)) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{3 f (a-b)}","\frac{\cos ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^{p+1}}{3 f (a-b)}-\frac{(3 a-2 b (p+1)) \cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{3 f (a-b)}",1,"(Cos[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^(1 + p))/(3*(a - b)*f) - ((3*a - 2*b*(1 + p))*Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))]*(a - b + b*Sec[e + f*x]^2)^p)/(3*(a - b)*f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p)","A",4,4,23,0.1739,1,"{3664, 453, 365, 364}"
156,1,79,0,0.0509318,"\int \sin (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Sin[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{f}","-\frac{\cos (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \sec ^2(e+f x)}{a-b}\right)}{f}",1,"-((Cos[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Sec[e + f*x]^2)/(a - b))]*(a - b + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p))","A",3,3,21,0.1429,1,"{3664, 365, 364}"
157,1,88,0,0.0805196,"\int \csc (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\sec (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a-b}\right)}{f}","-\frac{\sec (e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a-b}\right)}{f}",1,"-((AppellF1[1/2, 1, -p, 3/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/(a - b))]*Sec[e + f*x]*(a - b + b*Sec[e + f*x]^2)^p)/(f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p))","A",3,3,21,0.1429,1,"{3664, 430, 429}"
158,1,92,0,0.1237865,"\int \csc ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\sec ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a-b}\right)}{3 f}","\frac{\sec ^3(e+f x) \left(a+b \sec ^2(e+f x)-b\right)^p \left(\frac{b \sec ^2(e+f x)}{a-b}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};\sec ^2(e+f x),-\frac{b \sec ^2(e+f x)}{a-b}\right)}{3 f}",1,"(AppellF1[3/2, 2, -p, 5/2, Sec[e + f*x]^2, -((b*Sec[e + f*x]^2)/(a - b))]*Sec[e + f*x]^3*(a - b + b*Sec[e + f*x]^2)^p)/(3*f*(1 + (b*Sec[e + f*x]^2)/(a - b))^p)","A",3,3,23,0.1304,1,"{3664, 511, 510}"
159,1,83,0,0.1081844,"\int \sin ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Sin[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{3 f}","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};2,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{3 f}",1,"(AppellF1[3/2, 2, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3663, 511, 510}"
160,1,78,0,0.064155,"\int \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,14,0.2143,1,"{3661, 430, 429}"
161,1,68,0,0.0806688,"\int \csc ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{f}","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"-((Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p))","A",3,3,23,0.1304,1,"{3663, 365, 364}"
162,1,120,0,0.1155665,"\int \csc ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{(3 a-b (1-2 p)) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{3 a f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{3 a f}","-\frac{(3 a-b (1-2 p)) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{3 a f}-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{3 a f}",1,"-(Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(1 + p))/(3*a*f) - ((3*a - b*(1 - 2*p))*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/(3*a*f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",4,4,23,0.1739,1,"{3663, 453, 365, 364}"
163,1,176,0,0.2006368,"\int \csc ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Csc[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\left(15-\frac{b (1-2 p) (10 a-b (3-2 p))}{a^2}\right) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{15 f}-\frac{(10 a-b (3-2 p)) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{15 a^2 f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{5 a f}","-\frac{\left(15 a^2-b (1-2 p) (10 a-b (3-2 p))\right) \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} \, _2F_1\left(-\frac{1}{2},-p;\frac{1}{2};-\frac{b \tan ^2(e+f x)}{a}\right)}{15 a^2 f}-\frac{(10 a-b (3-2 p)) \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{15 a^2 f}-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{5 a f}",1,"-((10*a - b*(3 - 2*p))*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(1 + p))/(15*a^2*f) - (Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(1 + p))/(5*a*f) - ((15 - (b*(10*a - b*(3 - 2*p))*(1 - 2*p))/a^2)*Cot[e + f*x]*Hypergeometric2F1[-1/2, -p, 1/2, -((b*Tan[e + f*x]^2)/a)]*(a + b*Tan[e + f*x]^2)^p)/(15*f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",5,5,23,0.2174,1,"{3663, 462, 453, 365, 364}"
164,1,98,0,0.1811403,"\int (d \sin (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Sin[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) (d \sin (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (m+n p+1);\frac{1}{2} (m+n p+3);\sin ^2(e+f x)\right)}{f (m+n p+1)}","\frac{\tan (e+f x) (d \sin (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (m+n p+1);\frac{1}{2} (m+n p+3);\sin ^2(e+f x)\right)}{f (m+n p+1)}",1,"((Cos[e + f*x]^2)^((1 + n*p)/2)*Hypergeometric2F1[(1 + n*p)/2, (1 + m + n*p)/2, (3 + m + n*p)/2, Sin[e + f*x]^2]*(d*Sin[e + f*x])^m*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + m + n*p))","A",3,3,25,0.1200,1,"{3659, 2602, 2577}"
165,1,63,0,0.1128936,"\int \sin ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Sin[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^3(e+f x) \, _2F_1\left(2,\frac{1}{2} (n p+3);\frac{1}{2} (n p+5);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}","\frac{\tan ^3(e+f x) \, _2F_1\left(2,\frac{1}{2} (n p+3);\frac{1}{2} (n p+5);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}",1,"(Hypergeometric2F1[2, (3 + n*p)/2, (5 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 + n*p))","A",3,3,23,0.1304,1,"{3659, 2591, 364}"
166,1,61,0,0.0483137,"\int \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",3,3,14,0.2143,1,"{3659, 3476, 364}"
167,1,33,0,0.097289,"\int \csc ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Csc[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","-\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}","-\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}",1,"-((Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - n*p)))","A",3,3,23,0.1304,1,"{3659, 2591, 30}"
168,1,69,0,0.116107,"\int \csc ^4(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Csc[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p,x]","-\frac{\cot ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (3-n p)}-\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}","-\frac{\cot ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (3-n p)}-\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}",1,"-((Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - n*p))) - (Cot[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 - n*p))","A",4,3,23,0.1304,1,"{3659, 2591, 14}"
169,1,104,0,0.1276765,"\int \csc ^6(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Csc[e + f*x]^6*(b*(c*Tan[e + f*x])^n)^p,x]","-\frac{\cot ^5(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (5-n p)}-\frac{2 \cot ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (3-n p)}-\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}","-\frac{\cot ^5(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (5-n p)}-\frac{2 \cot ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (3-n p)}-\frac{\cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}",1,"-((Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - n*p))) - (2*Cot[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 - n*p)) - (Cot[e + f*x]^5*(b*(c*Tan[e + f*x])^n)^p)/(f*(5 - n*p))","A",4,3,23,0.1304,1,"{3659, 2591, 270}"
170,1,93,0,0.1420696,"\int \sin ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Sin[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\sin ^3(e+f x) \tan (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+4);\frac{1}{2} (n p+6);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+4)}","\frac{\sin ^3(e+f x) \tan (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+4);\frac{1}{2} (n p+6);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+4)}",1,"((Cos[e + f*x]^2)^((1 + n*p)/2)*Hypergeometric2F1[(1 + n*p)/2, (4 + n*p)/2, (6 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]^3*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(4 + n*p))","A",3,3,23,0.1304,1,"{3659, 2602, 2577}"
171,1,91,0,0.107261,"\int \sin (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Sin[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\sin (e+f x) \tan (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+2)}","\frac{\sin (e+f x) \tan (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+2)}",1,"((Cos[e + f*x]^2)^((1 + n*p)/2)*Hypergeometric2F1[(1 + n*p)/2, (2 + n*p)/2, (4 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 + n*p))","A",3,3,21,0.1429,1,"{3659, 2602, 2577}"
172,1,81,0,0.1239232,"\int \csc (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Csc[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{n p}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+2);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p}","\frac{\sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{n p}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+2);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p}",1,"((Cos[e + f*x]^2)^((1 + n*p)/2)*Hypergeometric2F1[(n*p)/2, (1 + n*p)/2, (2 + n*p)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*n*p)","A",3,3,21,0.1429,1,"{3659, 2601, 2577}"
173,1,92,0,0.1455564,"\int \csc ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Csc[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","-\frac{\csc ^2(e+f x) \sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{1}{2} (n p-2),\frac{1}{2} (n p+1);\frac{n p}{2};\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (2-n p)}","-\frac{\csc ^2(e+f x) \sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \, _2F_1\left(\frac{1}{2} (n p-2),\frac{1}{2} (n p+1);\frac{n p}{2};\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (2-n p)}",1,"-(((Cos[e + f*x]^2)^((1 + n*p)/2)*Csc[e + f*x]^2*Hypergeometric2F1[(-2 + n*p)/2, (1 + n*p)/2, (n*p)/2, Sin[e + f*x]^2]*Sec[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 - n*p)))","A",3,3,23,0.1304,1,"{3659, 2601, 2577}"
174,0,0,0,0.0537551,"\int (d \sin (e+f x))^m \left(a+b \tan ^n(e+f x)\right)^p \, dx","Int[(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^n)^p,x]","\int (d \sin (e+f x))^m \left(a+b \tan ^n(e+f x)\right)^p \, dx","\text{Int}\left((d \sin (e+f x))^m \left(a+b \tan ^n(e+f x)\right)^p,x\right)",0,"Defer[Int][(d*Sin[e + f*x])^m*(a + b*Tan[e + f*x]^n)^p, x]","A",0,0,0,0,-1,"{}"
175,1,99,0,0.1365104,"\int (d \cos (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Cos[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (-m+2 p+1)} \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (-m+2 p+1);\frac{1}{2} (2 p+3);\sin ^2(e+f x)\right)}{f (2 p+1)}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (-m+2 p+1)} \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (-m+2 p+1);\frac{1}{2} (2 p+3);\sin ^2(e+f x)\right)}{f (2 p+1)}",1,"((d*Cos[e + f*x])^m*(Cos[e + f*x]^2)^((1 - m + 2*p)/2)*Hypergeometric2F1[(1 + 2*p)/2, (1 - m + 2*p)/2, (3 + 2*p)/2, Sin[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + 2*p))","A",3,3,23,0.1304,1,"{3658, 2603, 2617}"
176,1,108,0,0.1447849,"\int (d \cos (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Cos[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (d \cos (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{m+2}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}","\frac{\tan (e+f x) \sec ^2(e+f x)^{m/2} (d \cos (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};\frac{m+2}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, (2 + m)/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Cos[e + f*x])^m*(Sec[e + f*x]^2)^(m/2)*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",4,4,25,0.1600,1,"{3669, 3679, 430, 429}"
177,1,101,0,0.1476619,"\int (d \cos (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Cos[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (-m+n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (-m+n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)}{f (n p+1)}","\frac{\tan (e+f x) (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (-m+n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (-m+n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)}{f (n p+1)}",1,"((d*Cos[e + f*x])^m*(Cos[e + f*x]^2)^((1 - m + n*p)/2)*Hypergeometric2F1[(1 + n*p)/2, (1 - m + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",3,3,25,0.1200,1,"{3659, 2603, 2617}"
178,0,0,0,0.1271113,"\int (d \cos (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Cos[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \cos (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","(d \cos (e+f x))^m \left(\frac{\sec (e+f x)}{d}\right)^m \text{Int}\left(\left(\frac{\sec (e+f x)}{d}\right)^{-m} \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"(d*Cos[e + f*x])^m*(Sec[e + f*x]/d)^m*Defer[Int][(a + b*(c*Tan[e + f*x])^n)^p/(Sec[e + f*x]/d)^m, x]","A",0,0,0,0,-1,"{}"
179,1,65,0,0.0350688,"\int \left(a+a \tan ^2(c+d x)\right)^4 \, dx","Int[(a + a*Tan[c + d*x]^2)^4,x]","\frac{a^4 \tan ^7(c+d x)}{7 d}+\frac{3 a^4 \tan ^5(c+d x)}{5 d}+\frac{a^4 \tan ^3(c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}","\frac{a^4 \tan ^7(c+d x)}{7 d}+\frac{3 a^4 \tan ^5(c+d x)}{5 d}+\frac{a^4 \tan ^3(c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}",1,"(a^4*Tan[c + d*x])/d + (a^4*Tan[c + d*x]^3)/d + (3*a^4*Tan[c + d*x]^5)/(5*d) + (a^4*Tan[c + d*x]^7)/(7*d)","A",4,3,14,0.2143,1,"{3657, 12, 3767}"
180,1,50,0,0.0301843,"\int \left(a+a \tan ^2(c+d x)\right)^3 \, dx","Int[(a + a*Tan[c + d*x]^2)^3,x]","\frac{a^3 \tan ^5(c+d x)}{5 d}+\frac{2 a^3 \tan ^3(c+d x)}{3 d}+\frac{a^3 \tan (c+d x)}{d}","\frac{a^3 \tan ^5(c+d x)}{5 d}+\frac{2 a^3 \tan ^3(c+d x)}{3 d}+\frac{a^3 \tan (c+d x)}{d}",1,"(a^3*Tan[c + d*x])/d + (2*a^3*Tan[c + d*x]^3)/(3*d) + (a^3*Tan[c + d*x]^5)/(5*d)","A",4,3,14,0.2143,1,"{3657, 12, 3767}"
181,1,32,0,0.0250517,"\int \left(a+a \tan ^2(c+d x)\right)^2 \, dx","Int[(a + a*Tan[c + d*x]^2)^2,x]","\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}","\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}",1,"(a^2*Tan[c + d*x])/d + (a^2*Tan[c + d*x]^3)/(3*d)","A",4,3,14,0.2143,1,"{3657, 12, 3767}"
182,1,31,0,0.0217207,"\int \frac{1}{a+a \tan ^2(c+d x)} \, dx","Int[(a + a*Tan[c + d*x]^2)^(-1),x]","\frac{\sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x}{2 a}","\frac{\sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x}{2 a}",1,"x/(2*a) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",4,4,14,0.2857,1,"{3657, 12, 2635, 8}"
183,1,55,0,0.0336857,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^2} \, dx","Int[(a + a*Tan[c + d*x]^2)^(-2),x]","\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{3 x}{8 a^2}","\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{3 x}{8 a^2}",1,"(3*x)/(8*a^2) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a^2*d)","A",5,4,14,0.2857,1,"{3657, 12, 2635, 8}"
184,1,79,0,0.0479145,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^3} \, dx","Int[(a + a*Tan[c + d*x]^2)^(-3),x]","\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{16 a^3 d}+\frac{5 x}{16 a^3}","\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{16 a^3 d}+\frac{5 x}{16 a^3}",1,"(5*x)/(16*a^3) + (5*Cos[c + d*x]*Sin[c + d*x])/(16*a^3*d) + (5*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^3*d) + (Cos[c + d*x]^5*Sin[c + d*x])/(6*a^3*d)","A",6,4,14,0.2857,1,"{3657, 12, 2635, 8}"
185,1,74,0,0.0493532,"\int \tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \tan ^4(e+f x)}{4 f}-\frac{(a-b) \tan ^2(e+f x)}{2 f}-\frac{(a-b) \log (\cos (e+f x))}{f}+\frac{b \tan ^6(e+f x)}{6 f}","\frac{(a-b) \tan ^4(e+f x)}{4 f}-\frac{(a-b) \tan ^2(e+f x)}{2 f}-\frac{(a-b) \log (\cos (e+f x))}{f}+\frac{b \tan ^6(e+f x)}{6 f}",1,"-(((a - b)*Log[Cos[e + f*x]])/f) - ((a - b)*Tan[e + f*x]^2)/(2*f) + ((a - b)*Tan[e + f*x]^4)/(4*f) + (b*Tan[e + f*x]^6)/(6*f)","A",4,3,21,0.1429,1,"{3631, 3473, 3475}"
186,1,53,0,0.0387884,"\int \tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \tan ^2(e+f x)}{2 f}+\frac{(a-b) \log (\cos (e+f x))}{f}+\frac{b \tan ^4(e+f x)}{4 f}","\frac{(a-b) \tan ^2(e+f x)}{2 f}+\frac{(a-b) \log (\cos (e+f x))}{f}+\frac{b \tan ^4(e+f x)}{4 f}",1,"((a - b)*Log[Cos[e + f*x]])/f + ((a - b)*Tan[e + f*x]^2)/(2*f) + (b*Tan[e + f*x]^4)/(4*f)","A",3,3,21,0.1429,1,"{3631, 3473, 3475}"
187,1,34,0,0.0221714,"\int \tan (e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Tan[e + f*x]*(a + b*Tan[e + f*x]^2),x]","\frac{b \tan ^2(e+f x)}{2 f}-\frac{(a-b) \log (\cos (e+f x))}{f}","\frac{b \tan ^2(e+f x)}{2 f}-\frac{(a-b) \log (\cos (e+f x))}{f}",1,"-(((a - b)*Log[Cos[e + f*x]])/f) + (b*Tan[e + f*x]^2)/(2*f)","A",2,2,19,0.1053,1,"{3631, 3475}"
188,1,26,0,0.0292719,"\int \cot (e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]*(a + b*Tan[e + f*x]^2),x]","\frac{a \log (\sin (e+f x))}{f}-\frac{b \log (\cos (e+f x))}{f}","\frac{a \log (\sin (e+f x))}{f}-\frac{b \log (\cos (e+f x))}{f}",1,"-((b*Log[Cos[e + f*x]])/f) + (a*Log[Sin[e + f*x]])/f","A",3,2,19,0.1053,1,"{3625, 3475}"
189,1,34,0,0.0322388,"\int \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2),x]","-\frac{(a-b) \log (\sin (e+f x))}{f}-\frac{a \cot ^2(e+f x)}{2 f}","-\frac{(a-b) \log (\sin (e+f x))}{f}-\frac{a \cot ^2(e+f x)}{2 f}",1,"-(a*Cot[e + f*x]^2)/(2*f) - ((a - b)*Log[Sin[e + f*x]])/f","A",3,3,21,0.1429,1,"{3629, 12, 3475}"
190,1,53,0,0.0422992,"\int \cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \cot ^2(e+f x)}{2 f}+\frac{(a-b) \log (\sin (e+f x))}{f}-\frac{a \cot ^4(e+f x)}{4 f}","\frac{(a-b) \cot ^2(e+f x)}{2 f}+\frac{(a-b) \log (\sin (e+f x))}{f}-\frac{a \cot ^4(e+f x)}{4 f}",1,"((a - b)*Cot[e + f*x]^2)/(2*f) - (a*Cot[e + f*x]^4)/(4*f) + ((a - b)*Log[Sin[e + f*x]])/f","A",4,4,21,0.1905,1,"{3629, 12, 3473, 3475}"
191,1,80,0,0.0523921,"\int \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \tan ^5(e+f x)}{5 f}-\frac{(a-b) \tan ^3(e+f x)}{3 f}+\frac{(a-b) \tan (e+f x)}{f}-x (a-b)+\frac{b \tan ^7(e+f x)}{7 f}","\frac{(a-b) \tan ^5(e+f x)}{5 f}-\frac{(a-b) \tan ^3(e+f x)}{3 f}+\frac{(a-b) \tan (e+f x)}{f}-x (a-b)+\frac{b \tan ^7(e+f x)}{7 f}",1,"-((a - b)*x) + ((a - b)*Tan[e + f*x])/f - ((a - b)*Tan[e + f*x]^3)/(3*f) + ((a - b)*Tan[e + f*x]^5)/(5*f) + (b*Tan[e + f*x]^7)/(7*f)","A",5,3,21,0.1429,1,"{3631, 3473, 8}"
192,1,60,0,0.0430068,"\int \tan ^4(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \tan ^3(e+f x)}{3 f}-\frac{(a-b) \tan (e+f x)}{f}+x (a-b)+\frac{b \tan ^5(e+f x)}{5 f}","\frac{(a-b) \tan ^3(e+f x)}{3 f}-\frac{(a-b) \tan (e+f x)}{f}+x (a-b)+\frac{b \tan ^5(e+f x)}{5 f}",1,"(a - b)*x - ((a - b)*Tan[e + f*x])/f + ((a - b)*Tan[e + f*x]^3)/(3*f) + (b*Tan[e + f*x]^5)/(5*f)","A",4,3,21,0.1429,1,"{3631, 3473, 8}"
193,1,40,0,0.0336666,"\int \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \tan (e+f x)}{f}-x (a-b)+\frac{b \tan ^3(e+f x)}{3 f}","\frac{(a-b) \tan (e+f x)}{f}-x (a-b)+\frac{b \tan ^3(e+f x)}{3 f}",1,"-((a - b)*x) + ((a - b)*Tan[e + f*x])/f + (b*Tan[e + f*x]^3)/(3*f)","A",3,3,21,0.1429,1,"{3631, 3473, 8}"
194,1,19,0,0.0119024,"\int \left(a+b \tan ^2(e+f x)\right) \, dx","Int[a + b*Tan[e + f*x]^2,x]","a x+\frac{b \tan (e+f x)}{f}-b x","a x+\frac{b \tan (e+f x)}{f}-b x",1,"a*x - b*x + (b*Tan[e + f*x])/f","A",3,2,12,0.1667,1,"{3473, 8}"
195,1,21,0,0.0253417,"\int \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2),x]","x (-(a-b))-\frac{a \cot (e+f x)}{f}","x (-(a-b))-\frac{a \cot (e+f x)}{f}",1,"-((a - b)*x) - (a*Cot[e + f*x])/f","A",2,2,21,0.09524,1,"{3629, 8}"
196,1,39,0,0.0370697,"\int \cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \cot (e+f x)}{f}+x (a-b)-\frac{a \cot ^3(e+f x)}{3 f}","\frac{(a-b) \cot (e+f x)}{f}+x (a-b)-\frac{a \cot ^3(e+f x)}{3 f}",1,"(a - b)*x + ((a - b)*Cot[e + f*x])/f - (a*Cot[e + f*x]^3)/(3*f)","A",4,4,21,0.1905,1,"{3629, 12, 3473, 8}"
197,1,61,0,0.0456984,"\int \cot ^6(e+f x) \left(a+b \tan ^2(e+f x)\right) \, dx","Int[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2),x]","\frac{(a-b) \cot ^3(e+f x)}{3 f}-\frac{(a-b) \cot (e+f x)}{f}-x (a-b)-\frac{a \cot ^5(e+f x)}{5 f}","\frac{(a-b) \cot ^3(e+f x)}{3 f}-\frac{(a-b) \cot (e+f x)}{f}-x (a-b)-\frac{a \cot ^5(e+f x)}{5 f}",1,"-((a - b)*x) - ((a - b)*Cot[e + f*x])/f + ((a - b)*Cot[e + f*x]^3)/(3*f) - (a*Cot[e + f*x]^5)/(5*f)","A",5,4,21,0.1905,1,"{3629, 12, 3473, 8}"
198,1,105,0,0.1088781,"\int \tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]","\frac{b (2 a-b) \tan ^6(e+f x)}{6 f}+\frac{(a-b)^2 \tan ^4(e+f x)}{4 f}-\frac{(a-b)^2 \tan ^2(e+f x)}{2 f}-\frac{(a-b)^2 \log (\cos (e+f x))}{f}+\frac{b^2 \tan ^8(e+f x)}{8 f}","\frac{b (2 a-b) \tan ^6(e+f x)}{6 f}+\frac{(a-b)^2 \tan ^4(e+f x)}{4 f}-\frac{(a-b)^2 \tan ^2(e+f x)}{2 f}-\frac{(a-b)^2 \log (\cos (e+f x))}{f}+\frac{b^2 \tan ^8(e+f x)}{8 f}",1,"-(((a - b)^2*Log[Cos[e + f*x]])/f) - ((a - b)^2*Tan[e + f*x]^2)/(2*f) + ((a - b)^2*Tan[e + f*x]^4)/(4*f) + ((2*a - b)*b*Tan[e + f*x]^6)/(6*f) + (b^2*Tan[e + f*x]^8)/(8*f)","A",4,3,23,0.1304,1,"{3670, 446, 88}"
199,1,82,0,0.0947573,"\int \tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2,x]","\frac{b (2 a-b) \tan ^4(e+f x)}{4 f}+\frac{(a-b)^2 \tan ^2(e+f x)}{2 f}+\frac{(a-b)^2 \log (\cos (e+f x))}{f}+\frac{b^2 \tan ^6(e+f x)}{6 f}","\frac{b (2 a-b) \tan ^4(e+f x)}{4 f}+\frac{(a-b)^2 \tan ^2(e+f x)}{2 f}+\frac{(a-b)^2 \log (\cos (e+f x))}{f}+\frac{b^2 \tan ^6(e+f x)}{6 f}",1,"((a - b)^2*Log[Cos[e + f*x]])/f + ((a - b)^2*Tan[e + f*x]^2)/(2*f) + ((2*a - b)*b*Tan[e + f*x]^4)/(4*f) + (b^2*Tan[e + f*x]^6)/(6*f)","A",4,3,23,0.1304,1,"{3670, 446, 77}"
200,1,62,0,0.0604605,"\int \tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^2,x]","\frac{b (a-b) \tan ^2(e+f x)}{2 f}+\frac{\left(a+b \tan ^2(e+f x)\right)^2}{4 f}-\frac{(a-b)^2 \log (\cos (e+f x))}{f}","\frac{b (a-b) \tan ^2(e+f x)}{2 f}+\frac{\left(a+b \tan ^2(e+f x)\right)^2}{4 f}-\frac{(a-b)^2 \log (\cos (e+f x))}{f}",1,"-(((a - b)^2*Log[Cos[e + f*x]])/f) + ((a - b)*b*Tan[e + f*x]^2)/(2*f) + (a + b*Tan[e + f*x]^2)^2/(4*f)","A",4,3,21,0.1429,1,"{3670, 444, 43}"
201,1,51,0,0.0635241,"\int \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^2,x]","\frac{a^2 \log (\tan (e+f x))}{f}+\frac{(a-b)^2 \log (\cos (e+f x))}{f}+\frac{b^2 \tan ^2(e+f x)}{2 f}","\frac{a^2 \log (\tan (e+f x))}{f}+\frac{(a-b)^2 \log (\cos (e+f x))}{f}+\frac{b^2 \tan ^2(e+f x)}{2 f}",1,"((a - b)^2*Log[Cos[e + f*x]])/f + (a^2*Log[Tan[e + f*x]])/f + (b^2*Tan[e + f*x]^2)/(2*f)","A",4,3,21,0.1429,1,"{3670, 446, 72}"
202,1,56,0,0.0816643,"\int \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot ^2(e+f x)}{2 f}-\frac{a (a-2 b) \log (\tan (e+f x))}{f}-\frac{(a-b)^2 \log (\cos (e+f x))}{f}","-\frac{a^2 \cot ^2(e+f x)}{2 f}-\frac{a (a-2 b) \log (\tan (e+f x))}{f}-\frac{(a-b)^2 \log (\cos (e+f x))}{f}",1,"-(a^2*Cot[e + f*x]^2)/(2*f) - ((a - b)^2*Log[Cos[e + f*x]])/f - (a*(a - 2*b)*Log[Tan[e + f*x]])/f","A",4,3,23,0.1304,1,"{3670, 446, 88}"
203,1,76,0,0.0883688,"\int \cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot ^4(e+f x)}{4 f}+\frac{a (a-2 b) \cot ^2(e+f x)}{2 f}+\frac{(a-b)^2 \log (\tan (e+f x))}{f}+\frac{(a-b)^2 \log (\cos (e+f x))}{f}","-\frac{a^2 \cot ^4(e+f x)}{4 f}+\frac{a (a-2 b) \cot ^2(e+f x)}{2 f}+\frac{(a-b)^2 \log (\tan (e+f x))}{f}+\frac{(a-b)^2 \log (\cos (e+f x))}{f}",1,"(a*(a - 2*b)*Cot[e + f*x]^2)/(2*f) - (a^2*Cot[e + f*x]^4)/(4*f) + ((a - b)^2*Log[Cos[e + f*x]])/f + ((a - b)^2*Log[Tan[e + f*x]])/f","A",4,3,23,0.1304,1,"{3670, 446, 88}"
204,1,113,0,0.0881873,"\int \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^2,x]","\frac{b (2 a-b) \tan ^7(e+f x)}{7 f}+\frac{(a-b)^2 \tan ^5(e+f x)}{5 f}-\frac{(a-b)^2 \tan ^3(e+f x)}{3 f}+\frac{(a-b)^2 \tan (e+f x)}{f}-x (a-b)^2+\frac{b^2 \tan ^9(e+f x)}{9 f}","\frac{b (2 a-b) \tan ^7(e+f x)}{7 f}+\frac{(a-b)^2 \tan ^5(e+f x)}{5 f}-\frac{(a-b)^2 \tan ^3(e+f x)}{3 f}+\frac{(a-b)^2 \tan (e+f x)}{f}-x (a-b)^2+\frac{b^2 \tan ^9(e+f x)}{9 f}",1,"-((a - b)^2*x) + ((a - b)^2*Tan[e + f*x])/f - ((a - b)^2*Tan[e + f*x]^3)/(3*f) + ((a - b)^2*Tan[e + f*x]^5)/(5*f) + ((2*a - b)*b*Tan[e + f*x]^7)/(7*f) + (b^2*Tan[e + f*x]^9)/(9*f)","A",4,3,23,0.1304,1,"{3670, 461, 203}"
205,1,91,0,0.0788437,"\int \tan ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2,x]","\frac{b (2 a-b) \tan ^5(e+f x)}{5 f}+\frac{(a-b)^2 \tan ^3(e+f x)}{3 f}-\frac{(a-b)^2 \tan (e+f x)}{f}+x (a-b)^2+\frac{b^2 \tan ^7(e+f x)}{7 f}","\frac{b (2 a-b) \tan ^5(e+f x)}{5 f}+\frac{(a-b)^2 \tan ^3(e+f x)}{3 f}-\frac{(a-b)^2 \tan (e+f x)}{f}+x (a-b)^2+\frac{b^2 \tan ^7(e+f x)}{7 f}",1,"(a - b)^2*x - ((a - b)^2*Tan[e + f*x])/f + ((a - b)^2*Tan[e + f*x]^3)/(3*f) + ((2*a - b)*b*Tan[e + f*x]^5)/(5*f) + (b^2*Tan[e + f*x]^7)/(7*f)","A",4,3,23,0.1304,1,"{3670, 461, 203}"
206,1,69,0,0.0737361,"\int \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2,x]","\frac{b (2 a-b) \tan ^3(e+f x)}{3 f}+\frac{(a-b)^2 \tan (e+f x)}{f}-x (a-b)^2+\frac{b^2 \tan ^5(e+f x)}{5 f}","\frac{b (2 a-b) \tan ^3(e+f x)}{3 f}+\frac{(a-b)^2 \tan (e+f x)}{f}-x (a-b)^2+\frac{b^2 \tan ^5(e+f x)}{5 f}",1,"-((a - b)^2*x) + ((a - b)^2*Tan[e + f*x])/f + ((2*a - b)*b*Tan[e + f*x]^3)/(3*f) + (b^2*Tan[e + f*x]^5)/(5*f)","A",4,3,23,0.1304,1,"{3670, 461, 203}"
207,1,46,0,0.0310278,"\int \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[(a + b*Tan[e + f*x]^2)^2,x]","\frac{b (2 a-b) \tan (e+f x)}{f}+x (a-b)^2+\frac{b^2 \tan ^3(e+f x)}{3 f}","\frac{b (2 a-b) \tan (e+f x)}{f}+x (a-b)^2+\frac{b^2 \tan ^3(e+f x)}{3 f}",1,"(a - b)^2*x + ((2*a - b)*b*Tan[e + f*x])/f + (b^2*Tan[e + f*x]^3)/(3*f)","A",4,3,14,0.2143,1,"{3661, 390, 203}"
208,1,38,0,0.06506,"\int \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot (e+f x)}{f}-x (a-b)^2+\frac{b^2 \tan (e+f x)}{f}","-\frac{a^2 \cot (e+f x)}{f}-x (a-b)^2+\frac{b^2 \tan (e+f x)}{f}",1,"-((a - b)^2*x) - (a^2*Cot[e + f*x])/f + (b^2*Tan[e + f*x])/f","A",4,3,23,0.1304,1,"{3670, 461, 203}"
209,1,44,0,0.0707845,"\int \cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot ^3(e+f x)}{3 f}+\frac{a (a-2 b) \cot (e+f x)}{f}+x (a-b)^2","-\frac{a^2 \cot ^3(e+f x)}{3 f}+\frac{a (a-2 b) \cot (e+f x)}{f}+x (a-b)^2",1,"(a - b)^2*x + (a*(a - 2*b)*Cot[e + f*x])/f - (a^2*Cot[e + f*x]^3)/(3*f)","A",4,3,23,0.1304,1,"{3670, 461, 203}"
210,1,68,0,0.0769123,"\int \cot ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^2 \, dx","Int[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^2 \cot ^5(e+f x)}{5 f}+\frac{a (a-2 b) \cot ^3(e+f x)}{3 f}-\frac{(a-b)^2 \cot (e+f x)}{f}-x (a-b)^2","-\frac{a^2 \cot ^5(e+f x)}{5 f}+\frac{a (a-2 b) \cot ^3(e+f x)}{3 f}-\frac{(a-b)^2 \cot (e+f x)}{f}-x (a-b)^2",1,"-((a - b)^2*x) - ((a - b)^2*Cot[e + f*x])/f + (a*(a - 2*b)*Cot[e + f*x]^3)/(3*f) - (a^2*Cot[e + f*x]^5)/(5*f)","A",4,3,23,0.1304,1,"{3670, 461, 203}"
211,1,71,0,0.0993675,"\int \frac{\tan ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2),x]","-\frac{a^2 \log \left(a+b \tan ^2(e+f x)\right)}{2 b^2 f (a-b)}-\frac{\log (\cos (e+f x))}{f (a-b)}+\frac{\tan ^2(e+f x)}{2 b f}","-\frac{a^2 \log \left(a+b \tan ^2(e+f x)\right)}{2 b^2 f (a-b)}-\frac{\log (\cos (e+f x))}{f (a-b)}+\frac{\tan ^2(e+f x)}{2 b f}",1,"-(Log[Cos[e + f*x]]/((a - b)*f)) - (a^2*Log[a + b*Tan[e + f*x]^2])/(2*(a - b)*b^2*f) + Tan[e + f*x]^2/(2*b*f)","A",4,3,23,0.1304,1,"{3670, 446, 72}"
212,1,50,0,0.0862229,"\int \frac{\tan ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2),x]","\frac{a \log \left(a+b \tan ^2(e+f x)\right)}{2 b f (a-b)}+\frac{\log (\cos (e+f x))}{f (a-b)}","\frac{a \log \left(a+b \tan ^2(e+f x)\right)}{2 b f (a-b)}+\frac{\log (\cos (e+f x))}{f (a-b)}",1,"Log[Cos[e + f*x]]/((a - b)*f) + (a*Log[a + b*Tan[e + f*x]^2])/(2*(a - b)*b*f)","A",4,3,23,0.1304,1,"{3670, 446, 72}"
213,1,36,0,0.0530929,"\int \frac{\tan (e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]/(a + b*Tan[e + f*x]^2),x]","-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)}","-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)}",1,"-Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)*f)","A",5,4,21,0.1905,1,"{3670, 444, 36, 31}"
214,1,64,0,0.0823578,"\int \frac{\cot (e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]/(a + b*Tan[e + f*x]^2),x]","\frac{b \log \left(a+b \tan ^2(e+f x)\right)}{2 a f (a-b)}+\frac{\log (\cos (e+f x))}{f (a-b)}+\frac{\log (\tan (e+f x))}{a f}","\frac{b \log \left(a+b \tan ^2(e+f x)\right)}{2 a f (a-b)}+\frac{\log (\cos (e+f x))}{f (a-b)}+\frac{\log (\tan (e+f x))}{a f}",1,"Log[Cos[e + f*x]]/((a - b)*f) + Log[Tan[e + f*x]]/(a*f) + (b*Log[a + b*Tan[e + f*x]^2])/(2*a*(a - b)*f)","A",4,3,21,0.1429,1,"{3670, 446, 72}"
215,1,89,0,0.1128348,"\int \frac{\cot ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2),x]","-\frac{b^2 \log \left(a+b \tan ^2(e+f x)\right)}{2 a^2 f (a-b)}-\frac{(a+b) \log (\tan (e+f x))}{a^2 f}-\frac{\log (\cos (e+f x))}{f (a-b)}-\frac{\cot ^2(e+f x)}{2 a f}","-\frac{b^2 \log \left(a+b \tan ^2(e+f x)\right)}{2 a^2 f (a-b)}-\frac{(a+b) \log (\tan (e+f x))}{a^2 f}-\frac{\log (\cos (e+f x))}{f (a-b)}-\frac{\cot ^2(e+f x)}{2 a f}",1,"-Cot[e + f*x]^2/(2*a*f) - Log[Cos[e + f*x]]/((a - b)*f) - ((a + b)*Log[Tan[e + f*x]])/(a^2*f) - (b^2*Log[a + b*Tan[e + f*x]^2])/(2*a^2*(a - b)*f)","A",4,3,23,0.1304,1,"{3670, 446, 72}"
216,1,115,0,0.1371241,"\int \frac{\cot ^5(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2),x]","\frac{b^3 \log \left(a+b \tan ^2(e+f x)\right)}{2 a^3 f (a-b)}+\frac{\left(a^2+a b+b^2\right) \log (\tan (e+f x))}{a^3 f}+\frac{(a+b) \cot ^2(e+f x)}{2 a^2 f}+\frac{\log (\cos (e+f x))}{f (a-b)}-\frac{\cot ^4(e+f x)}{4 a f}","\frac{b^3 \log \left(a+b \tan ^2(e+f x)\right)}{2 a^3 f (a-b)}+\frac{\left(a^2+a b+b^2\right) \log (\tan (e+f x))}{a^3 f}+\frac{(a+b) \cot ^2(e+f x)}{2 a^2 f}+\frac{\log (\cos (e+f x))}{f (a-b)}-\frac{\cot ^4(e+f x)}{4 a f}",1,"((a + b)*Cot[e + f*x]^2)/(2*a^2*f) - Cot[e + f*x]^4/(4*a*f) + Log[Cos[e + f*x]]/((a - b)*f) + ((a^2 + a*b + b^2)*Log[Tan[e + f*x]])/(a^3*f) + (b^3*Log[a + b*Tan[e + f*x]^2])/(2*a^3*(a - b)*f)","A",4,3,23,0.1304,1,"{3670, 446, 72}"
217,1,85,0,0.1879956,"\int \frac{\tan ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]","\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{5/2} f (a-b)}-\frac{(a+b) \tan (e+f x)}{b^2 f}-\frac{x}{a-b}+\frac{\tan ^3(e+f x)}{3 b f}","\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{5/2} f (a-b)}-\frac{(a+b) \tan (e+f x)}{b^2 f}-\frac{x}{a-b}+\frac{\tan ^3(e+f x)}{3 b f}",1,"-(x/(a - b)) + (a^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)*b^(5/2)*f) - ((a + b)*Tan[e + f*x])/(b^2*f) + Tan[e + f*x]^3/(3*b*f)","A",6,6,23,0.2609,1,"{3670, 479, 582, 522, 203, 205}"
218,1,63,0,0.1063238,"\int \frac{\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{3/2} f (a-b)}+\frac{x}{a-b}+\frac{\tan (e+f x)}{b f}","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{b^{3/2} f (a-b)}+\frac{x}{a-b}+\frac{\tan (e+f x)}{b f}",1,"x/(a - b) - (a^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)*b^(3/2)*f) + Tan[e + f*x]/(b*f)","A",5,5,23,0.2174,1,"{3670, 479, 522, 203, 205}"
219,1,50,0,0.0792586,"\int \frac{\tan ^2(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2),x]","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{b} f (a-b)}-\frac{x}{a-b}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{b} f (a-b)}-\frac{x}{a-b}",1,"-(x/(a - b)) + (Sqrt[a]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/((a - b)*Sqrt[b]*f)","A",4,4,23,0.1739,1,"{3670, 481, 203, 205}"
220,1,50,0,0.0740666,"\int \frac{1}{a+b \tan ^2(e+f x)} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-1),x]","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a} f (a-b)}","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{\sqrt{a} f (a-b)}",1,"x/(a - b) - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*f)","A",3,3,14,0.2143,1,"{3660, 3675, 205}"
221,1,64,0,0.1106398,"\int \frac{\cot ^2(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2} f (a-b)}-\frac{x}{a-b}-\frac{\cot (e+f x)}{a f}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{3/2} f (a-b)}-\frac{x}{a-b}-\frac{\cot (e+f x)}{a f}",1,"-(x/(a - b)) + (b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(3/2)*(a - b)*f) - Cot[e + f*x]/(a*f)","A",5,5,23,0.2174,1,"{3670, 480, 522, 203, 205}"
222,1,84,0,0.1729405,"\int \frac{\cot ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{5/2} f (a-b)}+\frac{(a+b) \cot (e+f x)}{a^2 f}+\frac{x}{a-b}-\frac{\cot ^3(e+f x)}{3 a f}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{5/2} f (a-b)}+\frac{(a+b) \cot (e+f x)}{a^2 f}+\frac{x}{a-b}-\frac{\cot ^3(e+f x)}{3 a f}",1,"x/(a - b) - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(5/2)*(a - b)*f) + ((a + b)*Cot[e + f*x])/(a^2*f) - Cot[e + f*x]^3/(3*a*f)","A",6,6,23,0.2609,1,"{3670, 480, 583, 522, 203, 205}"
223,1,113,0,0.2412093,"\int \frac{\cot ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{7/2} f (a-b)}-\frac{\left(a^2+a b+b^2\right) \cot (e+f x)}{a^3 f}+\frac{(a+b) \cot ^3(e+f x)}{3 a^2 f}-\frac{x}{a-b}-\frac{\cot ^5(e+f x)}{5 a f}","\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{a^{7/2} f (a-b)}-\frac{\left(a^2+a b+b^2\right) \cot (e+f x)}{a^3 f}+\frac{(a+b) \cot ^3(e+f x)}{3 a^2 f}-\frac{x}{a-b}-\frac{\cot ^5(e+f x)}{5 a f}",1,"-(x/(a - b)) + (b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(a^(7/2)*(a - b)*f) - ((a^2 + a*b + b^2)*Cot[e + f*x])/(a^3*f) + ((a + b)*Cot[e + f*x]^3)/(3*a^2*f) - Cot[e + f*x]^5/(5*a*f)","A",7,6,23,0.2609,1,"{3670, 480, 583, 522, 203, 205}"
224,1,90,0,0.1213053,"\int \frac{\tan ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2,x]","\frac{a^2}{2 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{a (a-2 b) \log \left(a+b \tan ^2(e+f x)\right)}{2 b^2 f (a-b)^2}-\frac{\log (\cos (e+f x))}{f (a-b)^2}","\frac{a^2}{2 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{a (a-2 b) \log \left(a+b \tan ^2(e+f x)\right)}{2 b^2 f (a-b)^2}-\frac{\log (\cos (e+f x))}{f (a-b)^2}",1,"-(Log[Cos[e + f*x]]/((a - b)^2*f)) + (a*(a - 2*b)*Log[a + b*Tan[e + f*x]^2])/(2*(a - b)^2*b^2*f) + a^2/(2*(a - b)*b^2*f*(a + b*Tan[e + f*x]^2))","A",4,3,23,0.1304,1,"{3670, 446, 88}"
225,1,69,0,0.1001464,"\int \frac{\tan ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^2}-\frac{a}{2 b f (a-b) \left(a+b \tan ^2(e+f x)\right)}","\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^2}-\frac{a}{2 b f (a-b) \left(a+b \tan ^2(e+f x)\right)}",1,"Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^2*f) - a/(2*(a - b)*b*f*(a + b*Tan[e + f*x]^2))","A",4,3,23,0.1304,1,"{3670, 446, 77}"
226,1,65,0,0.0723874,"\int \frac{\tan (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]/(a + b*Tan[e + f*x]^2)^2,x]","\frac{1}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^2}","\frac{1}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^2}",1,"-Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^2*f) + 1/(2*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",4,3,21,0.1429,1,"{3670, 444, 44}"
227,1,103,0,0.1205962,"\int \frac{\cot (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^2,x]","\frac{b (2 a-b) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^2 f (a-b)^2}+\frac{\log (\tan (e+f x))}{a^2 f}-\frac{b}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{\log (\cos (e+f x))}{f (a-b)^2}","\frac{b (2 a-b) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^2 f (a-b)^2}+\frac{\log (\tan (e+f x))}{a^2 f}-\frac{b}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{\log (\cos (e+f x))}{f (a-b)^2}",1,"Log[Cos[e + f*x]]/((a - b)^2*f) + Log[Tan[e + f*x]]/(a^2*f) + ((2*a - b)*b*Log[a + b*Tan[e + f*x]^2])/(2*a^2*(a - b)^2*f) - b/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",4,3,21,0.1429,1,"{3670, 446, 72}"
228,1,132,0,0.1556958,"\int \frac{\cot ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^2,x]","\frac{b^2}{2 a^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{b^2 (3 a-2 b) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^3 f (a-b)^2}-\frac{(a+2 b) \log (\tan (e+f x))}{a^3 f}-\frac{\cot ^2(e+f x)}{2 a^2 f}-\frac{\log (\cos (e+f x))}{f (a-b)^2}","\frac{b^2}{2 a^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{b^2 (3 a-2 b) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^3 f (a-b)^2}-\frac{(a+2 b) \log (\tan (e+f x))}{a^3 f}-\frac{\cot ^2(e+f x)}{2 a^2 f}-\frac{\log (\cos (e+f x))}{f (a-b)^2}",1,"-Cot[e + f*x]^2/(2*a^2*f) - Log[Cos[e + f*x]]/((a - b)^2*f) - ((a + 2*b)*Log[Tan[e + f*x]])/(a^3*f) - ((3*a - 2*b)*b^2*Log[a + b*Tan[e + f*x]^2])/(2*a^3*(a - b)^2*f) + b^2/(2*a^2*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",4,3,23,0.1304,1,"{3670, 446, 88}"
229,1,161,0,0.1790647,"\int \frac{\cot ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{b^3}{2 a^3 f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{b^3 (4 a-3 b) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^4 f (a-b)^2}+\frac{\left(a^2+2 a b+3 b^2\right) \log (\tan (e+f x))}{a^4 f}+\frac{(a+2 b) \cot ^2(e+f x)}{2 a^3 f}-\frac{\cot ^4(e+f x)}{4 a^2 f}+\frac{\log (\cos (e+f x))}{f (a-b)^2}","-\frac{b^3}{2 a^3 f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{b^3 (4 a-3 b) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^4 f (a-b)^2}+\frac{\left(a^2+2 a b+3 b^2\right) \log (\tan (e+f x))}{a^4 f}+\frac{(a+2 b) \cot ^2(e+f x)}{2 a^3 f}-\frac{\cot ^4(e+f x)}{4 a^2 f}+\frac{\log (\cos (e+f x))}{f (a-b)^2}",1,"((a + 2*b)*Cot[e + f*x]^2)/(2*a^3*f) - Cot[e + f*x]^4/(4*a^2*f) + Log[Cos[e + f*x]]/((a - b)^2*f) + ((a^2 + 2*a*b + 3*b^2)*Log[Tan[e + f*x]])/(a^4*f) + ((4*a - 3*b)*b^3*Log[a + b*Tan[e + f*x]^2])/(2*a^4*(a - b)^2*f) - b^3/(2*a^3*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",4,3,23,0.1304,1,"{3670, 446, 88}"
230,1,130,0,0.1965967,"\int \frac{\tan ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{a^{3/2} (3 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 b^{5/2} f (a-b)^2}+\frac{(3 a-2 b) \tan (e+f x)}{2 b^2 f (a-b)}-\frac{a \tan ^3(e+f x)}{2 b f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}","-\frac{a^{3/2} (3 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 b^{5/2} f (a-b)^2}+\frac{(3 a-2 b) \tan (e+f x)}{2 b^2 f (a-b)}-\frac{a \tan ^3(e+f x)}{2 b f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}",1,"-(x/(a - b)^2) - (a^(3/2)*(3*a - 5*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*(a - b)^2*b^(5/2)*f) + ((3*a - 2*b)*Tan[e + f*x])/(2*(a - b)*b^2*f) - (a*Tan[e + f*x]^3)/(2*(a - b)*b*f*(a + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{3670, 470, 582, 522, 203, 205}"
231,1,95,0,0.1175027,"\int \frac{\tan ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]","\frac{\sqrt{a} (a-3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 b^{3/2} f (a-b)^2}-\frac{a \tan (e+f x)}{2 b f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}","\frac{\sqrt{a} (a-3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 b^{3/2} f (a-b)^2}-\frac{a \tan (e+f x)}{2 b f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}",1,"x/(a - b)^2 + (Sqrt[a]*(a - 3*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*(a - b)^2*b^(3/2)*f) - (a*Tan[e + f*x])/(2*(a - b)*b*f*(a + b*Tan[e + f*x]^2))","A",5,5,23,0.2174,1,"{3670, 470, 522, 203, 205}"
232,1,90,0,0.1036788,"\int \frac{\tan ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2,x]","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b} f (a-b)^2}+\frac{\tan (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 \sqrt{a} \sqrt{b} f (a-b)^2}+\frac{\tan (e+f x)}{2 f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}",1,"-(x/(a - b)^2) + ((a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*Sqrt[a]*(a - b)^2*Sqrt[b]*f) + Tan[e + f*x]/(2*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",5,5,23,0.2174,1,"{3670, 471, 522, 203, 205}"
233,1,97,0,0.0801249,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-2),x]","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{3/2} f (a-b)^2}-\frac{b \tan (e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{3/2} f (a-b)^2}-\frac{b \tan (e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}",1,"x/(a - b)^2 - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*f) - (b*Tan[e + f*x])/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",5,5,14,0.3571,1,"{3661, 414, 522, 203, 205}"
234,1,128,0,0.1919085,"\int \frac{\cot ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^2,x]","\frac{b^{3/2} (5 a-3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{5/2} f (a-b)^2}-\frac{(2 a-3 b) \cot (e+f x)}{2 a^2 f (a-b)}-\frac{b \cot (e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}","\frac{b^{3/2} (5 a-3 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{5/2} f (a-b)^2}-\frac{(2 a-3 b) \cot (e+f x)}{2 a^2 f (a-b)}-\frac{b \cot (e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}",1,"-(x/(a - b)^2) + ((5*a - 3*b)*b^(3/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(5/2)*(a - b)^2*f) - ((2*a - 3*b)*Cot[e + f*x])/(2*a^2*(a - b)*f) - (b*Cot[e + f*x])/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{3670, 472, 583, 522, 203, 205}"
235,1,169,0,0.2871928,"\int \frac{\cot ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]","-\frac{b^{5/2} (7 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{7/2} f (a-b)^2}+\frac{\left(2 a^2+2 a b-5 b^2\right) \cot (e+f x)}{2 a^3 f (a-b)}-\frac{(2 a-5 b) \cot ^3(e+f x)}{6 a^2 f (a-b)}-\frac{b \cot ^3(e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}","-\frac{b^{5/2} (7 a-5 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{7/2} f (a-b)^2}+\frac{\left(2 a^2+2 a b-5 b^2\right) \cot (e+f x)}{2 a^3 f (a-b)}-\frac{(2 a-5 b) \cot ^3(e+f x)}{6 a^2 f (a-b)}-\frac{b \cot ^3(e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}+\frac{x}{(a-b)^2}",1,"x/(a - b)^2 - ((7*a - 5*b)*b^(5/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(7/2)*(a - b)^2*f) + ((2*a^2 + 2*a*b - 5*b^2)*Cot[e + f*x])/(2*a^3*(a - b)*f) - ((2*a - 5*b)*Cot[e + f*x]^3)/(6*a^2*(a - b)*f) - (b*Cot[e + f*x]^3)/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",7,6,23,0.2609,1,"{3670, 472, 583, 522, 203, 205}"
236,1,218,0,0.3416656,"\int \frac{\cot ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^2} \, dx","Int[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^2,x]","\frac{b^{7/2} (9 a-7 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{9/2} f (a-b)^2}+\frac{\left(2 a^2+2 a b-7 b^2\right) \cot ^3(e+f x)}{6 a^3 f (a-b)}-\frac{\left(2 a^2 b+2 a^3+2 a b^2-7 b^3\right) \cot (e+f x)}{2 a^4 f (a-b)}-\frac{(2 a-7 b) \cot ^5(e+f x)}{10 a^2 f (a-b)}-\frac{b \cot ^5(e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}","\frac{b^{7/2} (9 a-7 b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{2 a^{9/2} f (a-b)^2}+\frac{\left(2 a^2+2 a b-7 b^2\right) \cot ^3(e+f x)}{6 a^3 f (a-b)}-\frac{\left(2 a^2 b+2 a^3+2 a b^2-7 b^3\right) \cot (e+f x)}{2 a^4 f (a-b)}-\frac{(2 a-7 b) \cot ^5(e+f x)}{10 a^2 f (a-b)}-\frac{b \cot ^5(e+f x)}{2 a f (a-b) \left(a+b \tan ^2(e+f x)\right)}-\frac{x}{(a-b)^2}",1,"-(x/(a - b)^2) + ((9*a - 7*b)*b^(7/2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(2*a^(9/2)*(a - b)^2*f) - ((2*a^3 + 2*a^2*b + 2*a*b^2 - 7*b^3)*Cot[e + f*x])/(2*a^4*(a - b)*f) + ((2*a^2 + 2*a*b - 7*b^2)*Cot[e + f*x]^3)/(6*a^3*(a - b)*f) - ((2*a - 7*b)*Cot[e + f*x]^5)/(10*a^2*(a - b)*f) - (b*Cot[e + f*x]^5)/(2*a*(a - b)*f*(a + b*Tan[e + f*x]^2))","A",8,6,23,0.2609,1,"{3670, 472, 583, 522, 203, 205}"
237,1,108,0,0.1524644,"\int \frac{\tan ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]","\frac{a^2}{4 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{a (a-2 b)}{2 b^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^3}","\frac{a^2}{4 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{a (a-2 b)}{2 b^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^3}",1,"-Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^3*f) + a^2/(4*(a - b)*b^2*f*(a + b*Tan[e + f*x]^2)^2) - (a*(a - 2*b))/(2*(a - b)^2*b^2*f*(a + b*Tan[e + f*x]^2))","A",4,3,23,0.1304,1,"{3670, 446, 88}"
238,1,97,0,0.1173498,"\int \frac{\tan ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{a}{4 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{1}{2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^3}","-\frac{a}{4 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{1}{2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^3}",1,"Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^3*f) - a/(4*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^2) - 1/(2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",4,3,23,0.1304,1,"{3670, 446, 77}"
239,1,93,0,0.0881894,"\int \frac{\tan (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]/(a + b*Tan[e + f*x]^2)^3,x]","\frac{1}{2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{1}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^3}","\frac{1}{2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{1}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{\log \left(a \cos ^2(e+f x)+b \sin ^2(e+f x)\right)}{2 f (a-b)^3}",1,"-Log[a*Cos[e + f*x]^2 + b*Sin[e + f*x]^2]/(2*(a - b)^3*f) + 1/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) + 1/(2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",4,3,21,0.1429,1,"{3670, 444, 44}"
240,1,148,0,0.1650315,"\int \frac{\cot (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^3,x]","\frac{b \left(3 a^2-3 a b+b^2\right) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^3 f (a-b)^3}-\frac{b (2 a-b)}{2 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\log (\tan (e+f x))}{a^3 f}-\frac{b}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{\log (\cos (e+f x))}{f (a-b)^3}","\frac{b \left(3 a^2-3 a b+b^2\right) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^3 f (a-b)^3}-\frac{b (2 a-b)}{2 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\log (\tan (e+f x))}{a^3 f}-\frac{b}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{\log (\cos (e+f x))}{f (a-b)^3}",1,"Log[Cos[e + f*x]]/((a - b)^3*f) + Log[Tan[e + f*x]]/(a^3*f) + (b*(3*a^2 - 3*a*b + b^2)*Log[a + b*Tan[e + f*x]^2])/(2*a^3*(a - b)^3*f) - b/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((2*a - b)*b)/(2*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",4,3,21,0.1429,1,"{3670, 446, 72}"
241,1,181,0,0.2135558,"\int \frac{\cot ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^3,x]","\frac{b^2 (3 a-2 b)}{2 a^3 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{b^2}{4 a^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{b^2 \left(6 a^2-8 a b+3 b^2\right) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^4 f (a-b)^3}-\frac{(a+3 b) \log (\tan (e+f x))}{a^4 f}-\frac{\cot ^2(e+f x)}{2 a^3 f}-\frac{\log (\cos (e+f x))}{f (a-b)^3}","\frac{b^2 (3 a-2 b)}{2 a^3 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{b^2}{4 a^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{b^2 \left(6 a^2-8 a b+3 b^2\right) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^4 f (a-b)^3}-\frac{(a+3 b) \log (\tan (e+f x))}{a^4 f}-\frac{\cot ^2(e+f x)}{2 a^3 f}-\frac{\log (\cos (e+f x))}{f (a-b)^3}",1,"-Cot[e + f*x]^2/(2*a^3*f) - Log[Cos[e + f*x]]/((a - b)^3*f) - ((a + 3*b)*Log[Tan[e + f*x]])/(a^4*f) - (b^2*(6*a^2 - 8*a*b + 3*b^2)*Log[a + b*Tan[e + f*x]^2])/(2*a^4*(a - b)^3*f) + b^2/(4*a^2*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) + ((3*a - 2*b)*b^2)/(2*a^3*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",4,3,23,0.1304,1,"{3670, 446, 88}"
242,1,210,0,0.2378391,"\int \frac{\cot ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{b^3 (4 a-3 b)}{2 a^4 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b^3}{4 a^3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{b^3 \left(10 a^2-15 a b+6 b^2\right) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^5 f (a-b)^3}+\frac{\left(a^2+3 a b+6 b^2\right) \log (\tan (e+f x))}{a^5 f}+\frac{(a+3 b) \cot ^2(e+f x)}{2 a^4 f}-\frac{\cot ^4(e+f x)}{4 a^3 f}+\frac{\log (\cos (e+f x))}{f (a-b)^3}","-\frac{b^3 (4 a-3 b)}{2 a^4 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b^3}{4 a^3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{b^3 \left(10 a^2-15 a b+6 b^2\right) \log \left(a+b \tan ^2(e+f x)\right)}{2 a^5 f (a-b)^3}+\frac{\left(a^2+3 a b+6 b^2\right) \log (\tan (e+f x))}{a^5 f}+\frac{(a+3 b) \cot ^2(e+f x)}{2 a^4 f}-\frac{\cot ^4(e+f x)}{4 a^3 f}+\frac{\log (\cos (e+f x))}{f (a-b)^3}",1,"((a + 3*b)*Cot[e + f*x]^2)/(2*a^4*f) - Cot[e + f*x]^4/(4*a^3*f) + Log[Cos[e + f*x]]/((a - b)^3*f) + ((a^2 + 3*a*b + 6*b^2)*Log[Tan[e + f*x]])/(a^5*f) + (b^3*(10*a^2 - 15*a*b + 6*b^2)*Log[a + b*Tan[e + f*x]^2])/(2*a^5*(a - b)^3*f) - b^3/(4*a^3*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((4*a - 3*b)*b^3)/(2*a^4*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",4,3,23,0.1304,1,"{3670, 446, 88}"
243,1,153,0,0.229099,"\int \frac{\tan ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\sqrt{a} \left(3 a^2-10 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 b^{5/2} f (a-b)^3}-\frac{a (3 a-7 b) \tan (e+f x)}{8 b^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{a \tan ^3(e+f x)}{4 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}","\frac{\sqrt{a} \left(3 a^2-10 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 b^{5/2} f (a-b)^3}-\frac{a (3 a-7 b) \tan (e+f x)}{8 b^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{a \tan ^3(e+f x)}{4 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}",1,"-(x/(a - b)^3) + (Sqrt[a]*(3*a^2 - 10*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*(a - b)^3*b^(5/2)*f) - (a*Tan[e + f*x]^3)/(4*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^2) - (a*(3*a - 7*b)*Tan[e + f*x])/(8*(a - b)^2*b^2*f*(a + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{3670, 470, 578, 522, 203, 205}"
244,1,145,0,0.1807629,"\int \frac{\tan ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\left(a^2-6 a b-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 \sqrt{a} b^{3/2} f (a-b)^3}+\frac{(a-5 b) \tan (e+f x)}{8 b f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{a \tan (e+f x)}{4 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}","\frac{\left(a^2-6 a b-3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 \sqrt{a} b^{3/2} f (a-b)^3}+\frac{(a-5 b) \tan (e+f x)}{8 b f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{a \tan (e+f x)}{4 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}",1,"x/(a - b)^3 + ((a^2 - 6*a*b - 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*Sqrt[a]*(a - b)^3*b^(3/2)*f) - (a*Tan[e + f*x])/(4*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^2) + ((a - 5*b)*Tan[e + f*x])/(8*(a - b)^2*b*f*(a + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{3670, 470, 527, 522, 203, 205}"
245,1,144,0,0.1539501,"\int \frac{\tan ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3,x]","\frac{\left(3 a^2+6 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{3/2} \sqrt{b} f (a-b)^3}+\frac{(3 a+b) \tan (e+f x)}{8 a f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\tan (e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}","\frac{\left(3 a^2+6 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{3/2} \sqrt{b} f (a-b)^3}+\frac{(3 a+b) \tan (e+f x)}{8 a f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}+\frac{\tan (e+f x)}{4 f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}",1,"-(x/(a - b)^3) + ((3*a^2 + 6*a*b - b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(3/2)*(a - b)^3*Sqrt[b]*f) + Tan[e + f*x]/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) + ((3*a + b)*Tan[e + f*x])/(8*a*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",6,6,23,0.2609,1,"{3670, 471, 527, 522, 203, 205}"
246,1,150,0,0.1517042,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-3),x]","-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{5/2} f (a-b)^3}-\frac{b (7 a-3 b) \tan (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \tan (e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}","-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{5/2} f (a-b)^3}-\frac{b (7 a-3 b) \tan (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \tan (e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}",1,"x/(a - b)^3 - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^3*f) - (b*Tan[e + f*x])/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((7*a - 3*b)*b*Tan[e + f*x])/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",6,6,14,0.4286,1,"{3661, 414, 527, 522, 203, 205}"
247,1,189,0,0.2903212,"\int \frac{\cot ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^3,x]","\frac{b^{3/2} \left(35 a^2-42 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{7/2} f (a-b)^3}-\frac{\left(8 a^2-27 a b+15 b^2\right) \cot (e+f x)}{8 a^3 f (a-b)^2}-\frac{b (9 a-5 b) \cot (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \cot (e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}","\frac{b^{3/2} \left(35 a^2-42 a b+15 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{7/2} f (a-b)^3}-\frac{\left(8 a^2-27 a b+15 b^2\right) \cot (e+f x)}{8 a^3 f (a-b)^2}-\frac{b (9 a-5 b) \cot (e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \cot (e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}",1,"-(x/(a - b)^3) + (b^(3/2)*(35*a^2 - 42*a*b + 15*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(7/2)*(a - b)^3*f) - ((8*a^2 - 27*a*b + 15*b^2)*Cot[e + f*x])/(8*a^3*(a - b)^2*f) - (b*Cot[e + f*x])/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((9*a - 5*b)*b*Cot[e + f*x])/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",7,7,23,0.3043,1,"{3670, 472, 579, 583, 522, 203, 205}"
248,1,240,0,0.3649251,"\int \frac{\cot ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^3,x]","-\frac{b^{5/2} \left(63 a^2-90 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{9/2} f (a-b)^3}-\frac{\left(8 a^2-55 a b+35 b^2\right) \cot ^3(e+f x)}{24 a^3 f (a-b)^2}+\frac{\left(8 a^2 b+8 a^3-55 a b^2+35 b^3\right) \cot (e+f x)}{8 a^4 f (a-b)^2}-\frac{b (11 a-7 b) \cot ^3(e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \cot ^3(e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}","-\frac{b^{5/2} \left(63 a^2-90 a b+35 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{9/2} f (a-b)^3}-\frac{\left(8 a^2-55 a b+35 b^2\right) \cot ^3(e+f x)}{24 a^3 f (a-b)^2}+\frac{\left(8 a^2 b+8 a^3-55 a b^2+35 b^3\right) \cot (e+f x)}{8 a^4 f (a-b)^2}-\frac{b (11 a-7 b) \cot ^3(e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \cot ^3(e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}+\frac{x}{(a-b)^3}",1,"x/(a - b)^3 - (b^(5/2)*(63*a^2 - 90*a*b + 35*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(9/2)*(a - b)^3*f) + ((8*a^3 + 8*a^2*b - 55*a*b^2 + 35*b^3)*Cot[e + f*x])/(8*a^4*(a - b)^2*f) - ((8*a^2 - 55*a*b + 35*b^2)*Cot[e + f*x]^3)/(24*a^3*(a - b)^2*f) - (b*Cot[e + f*x]^3)/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((11*a - 7*b)*b*Cot[e + f*x]^3)/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",8,7,23,0.3043,1,"{3670, 472, 579, 583, 522, 203, 205}"
249,1,297,0,0.468399,"\int \frac{\cot ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^3} \, dx","Int[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^3,x]","\frac{b^{7/2} \left(99 a^2-154 a b+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{11/2} f (a-b)^3}-\frac{\left(8 a^2-91 a b+63 b^2\right) \cot ^5(e+f x)}{40 a^3 f (a-b)^2}+\frac{\left(8 a^2 b+8 a^3-91 a b^2+63 b^3\right) \cot ^3(e+f x)}{24 a^4 f (a-b)^2}-\frac{\left(8 a^2 b^2+8 a^3 b+8 a^4-91 a b^3+63 b^4\right) \cot (e+f x)}{8 a^5 f (a-b)^2}-\frac{b (13 a-9 b) \cot ^5(e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \cot ^5(e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}","\frac{b^{7/2} \left(99 a^2-154 a b+63 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a}}\right)}{8 a^{11/2} f (a-b)^3}-\frac{\left(8 a^2-91 a b+63 b^2\right) \cot ^5(e+f x)}{40 a^3 f (a-b)^2}+\frac{\left(8 a^2 b+8 a^3-91 a b^2+63 b^3\right) \cot ^3(e+f x)}{24 a^4 f (a-b)^2}-\frac{\left(8 a^2 b^2+8 a^3 b+8 a^4-91 a b^3+63 b^4\right) \cot (e+f x)}{8 a^5 f (a-b)^2}-\frac{b (13 a-9 b) \cot ^5(e+f x)}{8 a^2 f (a-b)^2 \left(a+b \tan ^2(e+f x)\right)}-\frac{b \cot ^5(e+f x)}{4 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^2}-\frac{x}{(a-b)^3}",1,"-(x/(a - b)^3) + (b^(7/2)*(99*a^2 - 154*a*b + 63*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(11/2)*(a - b)^3*f) - ((8*a^4 + 8*a^3*b + 8*a^2*b^2 - 91*a*b^3 + 63*b^4)*Cot[e + f*x])/(8*a^5*(a - b)^2*f) + ((8*a^3 + 8*a^2*b - 91*a*b^2 + 63*b^3)*Cot[e + f*x]^3)/(24*a^4*(a - b)^2*f) - ((8*a^2 - 91*a*b + 63*b^2)*Cot[e + f*x]^5)/(40*a^3*(a - b)^2*f) - (b*Cot[e + f*x]^5)/(4*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - ((13*a - 9*b)*b*Cot[e + f*x]^5)/(8*a^2*(a - b)^2*f*(a + b*Tan[e + f*x]^2))","A",9,7,23,0.3043,1,"{3670, 472, 579, 583, 522, 203, 205}"
250,1,115,0,0.0732737,"\int \left(a+b \tan ^2(c+d x)\right)^4 \, dx","Int[(a + b*Tan[c + d*x]^2)^4,x]","\frac{b^2 \left(6 a^2-4 a b+b^2\right) \tan ^3(c+d x)}{3 d}+\frac{b (2 a-b) \left(2 a^2-2 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^3 (4 a-b) \tan ^5(c+d x)}{5 d}+x (a-b)^4+\frac{b^4 \tan ^7(c+d x)}{7 d}","\frac{b^2 \left(6 a^2-4 a b+b^2\right) \tan ^3(c+d x)}{3 d}+\frac{b (2 a-b) \left(2 a^2-2 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^3 (4 a-b) \tan ^5(c+d x)}{5 d}+x (a-b)^4+\frac{b^4 \tan ^7(c+d x)}{7 d}",1,"(a - b)^4*x + ((2*a - b)*b*(2*a^2 - 2*a*b + b^2)*Tan[c + d*x])/d + (b^2*(6*a^2 - 4*a*b + b^2)*Tan[c + d*x]^3)/(3*d) + ((4*a - b)*b^3*Tan[c + d*x]^5)/(5*d) + (b^4*Tan[c + d*x]^7)/(7*d)","A",4,3,14,0.2143,1,"{3661, 390, 203}"
251,1,77,0,0.048896,"\int \left(a+b \tan ^2(c+d x)\right)^3 \, dx","Int[(a + b*Tan[c + d*x]^2)^3,x]","\frac{b \left(3 a^2-3 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^2 (3 a-b) \tan ^3(c+d x)}{3 d}+x (a-b)^3+\frac{b^3 \tan ^5(c+d x)}{5 d}","\frac{b \left(3 a^2-3 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^2 (3 a-b) \tan ^3(c+d x)}{3 d}+x (a-b)^3+\frac{b^3 \tan ^5(c+d x)}{5 d}",1,"(a - b)^3*x + (b*(3*a^2 - 3*a*b + b^2)*Tan[c + d*x])/d + ((3*a - b)*b^2*Tan[c + d*x]^3)/(3*d) + (b^3*Tan[c + d*x]^5)/(5*d)","A",4,3,14,0.2143,1,"{3661, 390, 203}"
252,1,46,0,0.0312722,"\int \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[(a + b*Tan[c + d*x]^2)^2,x]","\frac{b (2 a-b) \tan (c+d x)}{d}+x (a-b)^2+\frac{b^2 \tan ^3(c+d x)}{3 d}","\frac{b (2 a-b) \tan (c+d x)}{d}+x (a-b)^2+\frac{b^2 \tan ^3(c+d x)}{3 d}",1,"(a - b)^2*x + ((2*a - b)*b*Tan[c + d*x])/d + (b^2*Tan[c + d*x]^3)/(3*d)","A",4,3,14,0.2143,1,"{3661, 390, 203}"
253,1,19,0,0.0129866,"\int \left(a+b \tan ^2(c+d x)\right) \, dx","Int[a + b*Tan[c + d*x]^2,x]","a x+\frac{b \tan (c+d x)}{d}-b x","a x+\frac{b \tan (c+d x)}{d}-b x",1,"a*x - b*x + (b*Tan[c + d*x])/d","A",3,2,12,0.1667,1,"{3473, 8}"
254,1,50,0,0.0736336,"\int \frac{1}{a+b \tan ^2(c+d x)} \, dx","Int[(a + b*Tan[c + d*x]^2)^(-1),x]","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)}","\frac{x}{a-b}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)}",1,"x/(a - b) - (Sqrt[b]*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*d)","A",3,3,14,0.2143,1,"{3660, 3675, 205}"
255,1,97,0,0.0920903,"\int \frac{1}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[(a + b*Tan[c + d*x]^2)^(-2),x]","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^2}-\frac{b \tan (c+d x)}{2 a d (a-b) \left(a+b \tan ^2(c+d x)\right)}+\frac{x}{(a-b)^2}","-\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^2}-\frac{b \tan (c+d x)}{2 a d (a-b) \left(a+b \tan ^2(c+d x)\right)}+\frac{x}{(a-b)^2}",1,"x/(a - b)^2 - ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*d) - (b*Tan[c + d*x])/(2*a*(a - b)*d*(a + b*Tan[c + d*x]^2))","A",5,5,14,0.3571,1,"{3661, 414, 522, 203, 205}"
256,1,150,0,0.1442643,"\int \frac{1}{\left(a+b \tan ^2(c+d x)\right)^3} \, dx","Int[(a + b*Tan[c + d*x]^2)^(-3),x]","-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^3}-\frac{b (7 a-3 b) \tan (c+d x)}{8 a^2 d (a-b)^2 \left(a+b \tan ^2(c+d x)\right)}-\frac{b \tan (c+d x)}{4 a d (a-b) \left(a+b \tan ^2(c+d x)\right)^2}+\frac{x}{(a-b)^3}","-\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^3}-\frac{b (7 a-3 b) \tan (c+d x)}{8 a^2 d (a-b)^2 \left(a+b \tan ^2(c+d x)\right)}-\frac{b \tan (c+d x)}{4 a d (a-b) \left(a+b \tan ^2(c+d x)\right)^2}+\frac{x}{(a-b)^3}",1,"x/(a - b)^3 - (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^3*d) - (b*Tan[c + d*x])/(4*a*(a - b)*d*(a + b*Tan[c + d*x]^2)^2) - ((7*a - 3*b)*b*Tan[c + d*x])/(8*a^2*(a - b)^2*d*(a + b*Tan[c + d*x]^2))","A",6,6,14,0.4286,1,"{3661, 414, 527, 522, 203, 205}"
257,1,54,0,0.1052294,"\int \tan ^4(x) \sqrt{a+a \tan ^2(x)} \, dx","Int[Tan[x]^4*Sqrt[a + a*Tan[x]^2],x]","\frac{1}{4} \tan ^3(x) \sqrt{a \sec ^2(x)}-\frac{3}{8} \tan (x) \sqrt{a \sec ^2(x)}+\frac{3}{8} \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))","\frac{1}{4} \tan ^3(x) \sqrt{a \sec ^2(x)}-\frac{3}{8} \tan (x) \sqrt{a \sec ^2(x)}+\frac{3}{8} \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))",1,"(3*ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2])/8 - (3*Sqrt[a*Sec[x]^2]*Tan[x])/8 + (Sqrt[a*Sec[x]^2]*Tan[x]^3)/4","A",5,4,17,0.2353,1,"{3657, 4125, 2611, 3770}"
258,1,30,0,0.0870712,"\int \tan ^3(x) \sqrt{a+a \tan ^2(x)} \, dx","Int[Tan[x]^3*Sqrt[a + a*Tan[x]^2],x]","\frac{\left(a \sec ^2(x)\right)^{3/2}}{3 a}-\sqrt{a \sec ^2(x)}","\frac{\left(a \sec ^2(x)\right)^{3/2}}{3 a}-\sqrt{a \sec ^2(x)}",1,"-Sqrt[a*Sec[x]^2] + (a*Sec[x]^2)^(3/2)/(3*a)","A",4,3,17,0.1765,1,"{3657, 4124, 43}"
259,1,36,0,0.0926689,"\int \tan ^2(x) \sqrt{a+a \tan ^2(x)} \, dx","Int[Tan[x]^2*Sqrt[a + a*Tan[x]^2],x]","\frac{1}{2} \tan (x) \sqrt{a \sec ^2(x)}-\frac{1}{2} \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))","\frac{1}{2} \tan (x) \sqrt{a \sec ^2(x)}-\frac{1}{2} \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))",1,"-(ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2])/2 + (Sqrt[a*Sec[x]^2]*Tan[x])/2","A",4,4,17,0.2353,1,"{3657, 4125, 2611, 3770}"
260,1,10,0,0.0431939,"\int \tan (x) \sqrt{a+a \tan ^2(x)} \, dx","Int[Tan[x]*Sqrt[a + a*Tan[x]^2],x]","\sqrt{a \sec ^2(x)}","\sqrt{a \sec ^2(x)}",1,"Sqrt[a*Sec[x]^2]","A",3,3,15,0.2000,1,"{3657, 4124, 32}"
261,1,24,0,0.0729868,"\int \cot (x) \sqrt{a+a \tan ^2(x)} \, dx","Int[Cot[x]*Sqrt[a + a*Tan[x]^2],x]","-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)","-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)",1,"-(Sqrt[a]*ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]])","A",4,4,15,0.2667,1,"{3657, 4124, 63, 207}"
262,1,14,0,0.0804953,"\int \cot ^2(x) \sqrt{a+a \tan ^2(x)} \, dx","Int[Cot[x]^2*Sqrt[a + a*Tan[x]^2],x]","-\cot (x) \sqrt{a \sec ^2(x)}","-\cot (x) \sqrt{a \sec ^2(x)}",1,"-(Cot[x]*Sqrt[a*Sec[x]^2])","A",4,4,17,0.2353,1,"{3657, 4125, 2606, 8}"
263,1,45,0,0.089685,"\int \cot ^3(x) \sqrt{a+a \tan ^2(x)} \, dx","Int[Cot[x]^3*Sqrt[a + a*Tan[x]^2],x]","\frac{1}{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)-\frac{1}{2} \cot ^2(x) \sqrt{a \sec ^2(x)}","\frac{1}{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)-\frac{1}{2} \cot ^2(x) \sqrt{a \sec ^2(x)}",1,"(Sqrt[a]*ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]])/2 - (Cot[x]^2*Sqrt[a*Sec[x]^2])/2","A",5,5,17,0.2941,1,"{3657, 4124, 51, 63, 207}"
264,1,34,0,0.0996947,"\int \cot ^4(x) \sqrt{a+a \tan ^2(x)} \, dx","Int[Cot[x]^4*Sqrt[a + a*Tan[x]^2],x]","\cot (x) \sqrt{a \sec ^2(x)}-\frac{1}{3} \cot (x) \csc ^2(x) \sqrt{a \sec ^2(x)}","\cot (x) \sqrt{a \sec ^2(x)}-\frac{1}{3} \cot (x) \csc ^2(x) \sqrt{a \sec ^2(x)}",1,"Cot[x]*Sqrt[a*Sec[x]^2] - (Cot[x]*Csc[x]^2*Sqrt[a*Sec[x]^2])/3","A",4,3,17,0.1765,1,"{3657, 4125, 2606}"
265,1,36,0,0.0349328,"\int \sqrt{a+a \tan ^2(c+d x)} \, dx","Int[Sqrt[a + a*Tan[c + d*x]^2],x]","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right)}{d}","\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right)}{d}",1,"(Sqrt[a]*ArcTanh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a*Sec[c + d*x]^2]])/d","A",4,4,16,0.2500,1,"{3657, 4122, 217, 206}"
266,1,32,0,0.0999202,"\int \tan ^3(x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Int[Tan[x]^3*(a + a*Tan[x]^2)^(3/2),x]","\frac{\left(a \sec ^2(x)\right)^{5/2}}{5 a}-\frac{1}{3} \left(a \sec ^2(x)\right)^{3/2}","\frac{\left(a \sec ^2(x)\right)^{5/2}}{5 a}-\frac{1}{3} \left(a \sec ^2(x)\right)^{3/2}",1,"-(a*Sec[x]^2)^(3/2)/3 + (a*Sec[x]^2)^(5/2)/(5*a)","A",4,3,17,0.1765,1,"{3657, 4124, 43}"
267,1,59,0,0.1196278,"\int \tan ^2(x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Int[Tan[x]^2*(a + a*Tan[x]^2)^(3/2),x]","\frac{1}{4} a \tan (x) \sec ^2(x) \sqrt{a \sec ^2(x)}-\frac{1}{8} a \tan (x) \sqrt{a \sec ^2(x)}-\frac{1}{8} a \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))","\frac{1}{4} a \tan (x) \sec ^2(x) \sqrt{a \sec ^2(x)}-\frac{1}{8} a \tan (x) \sqrt{a \sec ^2(x)}-\frac{1}{8} a \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))",1,"-(a*ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2])/8 - (a*Sqrt[a*Sec[x]^2]*Tan[x])/8 + (a*Sec[x]^2*Sqrt[a*Sec[x]^2]*Tan[x])/4","A",5,5,17,0.2941,1,"{3657, 4125, 2611, 3768, 3770}"
268,1,14,0,0.0511205,"\int \tan (x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Int[Tan[x]*(a + a*Tan[x]^2)^(3/2),x]","\frac{1}{3} \left(a \sec ^2(x)\right)^{3/2}","\frac{1}{3} \left(a \sec ^2(x)\right)^{3/2}",1,"(a*Sec[x]^2)^(3/2)/3","A",3,3,15,0.2000,1,"{3657, 4124, 32}"
269,1,37,0,0.0905474,"\int \cot (x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Int[Cot[x]*(a + a*Tan[x]^2)^(3/2),x]","a \sqrt{a \sec ^2(x)}-a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)","a \sqrt{a \sec ^2(x)}-a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)",1,"-(a^(3/2)*ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]]) + a*Sqrt[a*Sec[x]^2]","A",5,5,15,0.3333,1,"{3657, 4124, 50, 63, 207}"
270,1,33,0,0.1090935,"\int \cot ^2(x) \left(a+a \tan ^2(x)\right)^{3/2} \, dx","Int[Cot[x]^2*(a + a*Tan[x]^2)^(3/2),x]","a \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))-a \cot (x) \sqrt{a \sec ^2(x)}","a \cos (x) \sqrt{a \sec ^2(x)} \tanh ^{-1}(\sin (x))-a \cot (x) \sqrt{a \sec ^2(x)}",1,"a*ArcTanh[Sin[x]]*Cos[x]*Sqrt[a*Sec[x]^2] - a*Cot[x]*Sqrt[a*Sec[x]^2]","A",5,5,17,0.2941,1,"{3657, 4125, 2621, 321, 207}"
271,1,68,0,0.0390246,"\int \left(a+a \tan ^2(c+d x)\right)^{3/2} \, dx","Int[(a + a*Tan[c + d*x]^2)^(3/2),x]","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right)}{2 d}+\frac{a \tan (c+d x) \sqrt{a \sec ^2(c+d x)}}{2 d}","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right)}{2 d}+\frac{a \tan (c+d x) \sqrt{a \sec ^2(c+d x)}}{2 d}",1,"(a^(3/2)*ArcTanh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a*Sec[c + d*x]^2]])/(2*d) + (a*Sqrt[a*Sec[c + d*x]^2]*Tan[c + d*x])/(2*d)","A",5,5,16,0.3125,1,"{3657, 4122, 195, 217, 206}"
272,1,98,0,0.048974,"\int \left(a+a \tan ^2(c+d x)\right)^{5/2} \, dx","Int[(a + a*Tan[c + d*x]^2)^(5/2),x]","\frac{3 a^2 \tan (c+d x) \sqrt{a \sec ^2(c+d x)}}{8 d}+\frac{3 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right)}{8 d}+\frac{a \tan (c+d x) \left(a \sec ^2(c+d x)\right)^{3/2}}{4 d}","\frac{3 a^2 \tan (c+d x) \sqrt{a \sec ^2(c+d x)}}{8 d}+\frac{3 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec ^2(c+d x)}}\right)}{8 d}+\frac{a \tan (c+d x) \left(a \sec ^2(c+d x)\right)^{3/2}}{4 d}",1,"(3*a^(5/2)*ArcTanh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a*Sec[c + d*x]^2]])/(8*d) + (3*a^2*Sqrt[a*Sec[c + d*x]^2]*Tan[c + d*x])/(8*d) + (a*(a*Sec[c + d*x]^2)^(3/2)*Tan[c + d*x])/(4*d)","A",6,5,16,0.3125,1,"{3657, 4122, 195, 217, 206}"
273,1,25,0,0.0922876,"\int \frac{\tan ^3(x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Int[Tan[x]^3/Sqrt[a + a*Tan[x]^2],x]","\frac{\sqrt{a \sec ^2(x)}}{a}+\frac{1}{\sqrt{a \sec ^2(x)}}","\frac{\sqrt{a \sec ^2(x)}}{a}+\frac{1}{\sqrt{a \sec ^2(x)}}",1,"1/Sqrt[a*Sec[x]^2] + Sqrt[a*Sec[x]^2]/a","A",4,3,17,0.1765,1,"{3657, 4124, 43}"
274,1,31,0,0.0929046,"\int \frac{\tan ^2(x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Int[Tan[x]^2/Sqrt[a + a*Tan[x]^2],x]","\frac{\sec (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \sec ^2(x)}}-\frac{\tan (x)}{\sqrt{a \sec ^2(x)}}","\frac{\sec (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \sec ^2(x)}}-\frac{\tan (x)}{\sqrt{a \sec ^2(x)}}",1,"(ArcTanh[Sin[x]]*Sec[x])/Sqrt[a*Sec[x]^2] - Tan[x]/Sqrt[a*Sec[x]^2]","A",5,5,17,0.2941,1,"{3657, 4125, 2592, 321, 206}"
275,1,12,0,0.0464586,"\int \frac{\tan (x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Int[Tan[x]/Sqrt[a + a*Tan[x]^2],x]","-\frac{1}{\sqrt{a \sec ^2(x)}}","-\frac{1}{\sqrt{a \sec ^2(x)}}",1,"-(1/Sqrt[a*Sec[x]^2])","A",3,3,15,0.2000,1,"{3657, 4124, 32}"
276,1,35,0,0.0796105,"\int \frac{\cot (x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Int[Cot[x]/Sqrt[a + a*Tan[x]^2],x]","\frac{1}{\sqrt{a \sec ^2(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}","\frac{1}{\sqrt{a \sec ^2(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"-(ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]]/Sqrt[a]) + 1/Sqrt[a*Sec[x]^2]","A",5,5,15,0.3333,1,"{3657, 4124, 51, 63, 207}"
277,1,31,0,0.0923178,"\int \frac{\cot ^2(x)}{\sqrt{a+a \tan ^2(x)}} \, dx","Int[Cot[x]^2/Sqrt[a + a*Tan[x]^2],x]","-\frac{\csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{\tan (x)}{\sqrt{a \sec ^2(x)}}","-\frac{\csc (x) \sec (x)}{\sqrt{a \sec ^2(x)}}-\frac{\tan (x)}{\sqrt{a \sec ^2(x)}}",1,"-((Csc[x]*Sec[x])/Sqrt[a*Sec[x]^2]) - Tan[x]/Sqrt[a*Sec[x]^2]","A",5,4,17,0.2353,1,"{3657, 4125, 2590, 14}"
278,1,30,0,0.0967719,"\int \frac{\tan ^3(x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Int[Tan[x]^3/(a + a*Tan[x]^2)^(3/2),x]","\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}-\frac{1}{a \sqrt{a \sec ^2(x)}}","\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}-\frac{1}{a \sqrt{a \sec ^2(x)}}",1,"1/(3*(a*Sec[x]^2)^(3/2)) - 1/(a*Sqrt[a*Sec[x]^2])","A",4,3,17,0.1765,1,"{3657, 4124, 43}"
279,1,23,0,0.1010712,"\int \frac{\tan ^2(x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Int[Tan[x]^2/(a + a*Tan[x]^2)^(3/2),x]","\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}}","\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}}",1,"(Sin[x]^2*Tan[x])/(3*a*Sqrt[a*Sec[x]^2])","A",4,4,17,0.2353,1,"{3657, 4125, 2564, 30}"
280,1,14,0,0.0489939,"\int \frac{\tan (x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Int[Tan[x]/(a + a*Tan[x]^2)^(3/2),x]","-\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}","-\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}",1,"-1/(3*(a*Sec[x]^2)^(3/2))","A",3,3,15,0.2000,1,"{3657, 4124, 32}"
281,1,53,0,0.0909338,"\int \frac{\cot (x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Int[Cot[x]/(a + a*Tan[x]^2)^(3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{1}{a \sqrt{a \sec ^2(x)}}+\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sec ^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{1}{a \sqrt{a \sec ^2(x)}}+\frac{1}{3 \left(a \sec ^2(x)\right)^{3/2}}",1,"-(ArcTanh[Sqrt[a*Sec[x]^2]/Sqrt[a]]/a^(3/2)) + 1/(3*(a*Sec[x]^2)^(3/2)) + 1/(a*Sqrt[a*Sec[x]^2])","A",6,5,15,0.3333,1,"{3657, 4124, 51, 63, 207}"
282,1,60,0,0.1147215,"\int \frac{\cot ^2(x)}{\left(a+a \tan ^2(x)\right)^{3/2}} \, dx","Int[Cot[x]^2/(a + a*Tan[x]^2)^(3/2),x]","-\frac{\csc (x) \sec (x)}{a \sqrt{a \sec ^2(x)}}-\frac{2 \tan (x)}{a \sqrt{a \sec ^2(x)}}+\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}}","-\frac{\csc (x) \sec (x)}{a \sqrt{a \sec ^2(x)}}-\frac{2 \tan (x)}{a \sqrt{a \sec ^2(x)}}+\frac{\sin ^2(x) \tan (x)}{3 a \sqrt{a \sec ^2(x)}}",1,"-((Csc[x]*Sec[x])/(a*Sqrt[a*Sec[x]^2])) - (2*Tan[x])/(a*Sqrt[a*Sec[x]^2]) + (Sin[x]^2*Tan[x])/(3*a*Sqrt[a*Sec[x]^2])","A",5,4,17,0.2353,1,"{3657, 4125, 2590, 270}"
283,1,24,0,0.0292873,"\int \frac{1}{\sqrt{a+a \tan ^2(c+d x)}} \, dx","Int[1/Sqrt[a + a*Tan[c + d*x]^2],x]","\frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}}","\frac{\tan (c+d x)}{d \sqrt{a \sec ^2(c+d x)}}",1,"Tan[c + d*x]/(d*Sqrt[a*Sec[c + d*x]^2])","A",3,3,16,0.1875,1,"{3657, 4122, 191}"
284,1,58,0,0.0351514,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^{3/2}} \, dx","Int[(a + a*Tan[c + d*x]^2)^(-3/2),x]","\frac{2 \tan (c+d x)}{3 a d \sqrt{a \sec ^2(c+d x)}}+\frac{\tan (c+d x)}{3 d \left(a \sec ^2(c+d x)\right)^{3/2}}","\frac{2 \tan (c+d x)}{3 a d \sqrt{a \sec ^2(c+d x)}}+\frac{\tan (c+d x)}{3 d \left(a \sec ^2(c+d x)\right)^{3/2}}",1,"Tan[c + d*x]/(3*d*(a*Sec[c + d*x]^2)^(3/2)) + (2*Tan[c + d*x])/(3*a*d*Sqrt[a*Sec[c + d*x]^2])","A",4,4,16,0.2500,1,"{3657, 4122, 192, 191}"
285,1,88,0,0.0436301,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^{5/2}} \, dx","Int[(a + a*Tan[c + d*x]^2)^(-5/2),x]","\frac{8 \tan (c+d x)}{15 a^2 d \sqrt{a \sec ^2(c+d x)}}+\frac{4 \tan (c+d x)}{15 a d \left(a \sec ^2(c+d x)\right)^{3/2}}+\frac{\tan (c+d x)}{5 d \left(a \sec ^2(c+d x)\right)^{5/2}}","\frac{8 \tan (c+d x)}{15 a^2 d \sqrt{a \sec ^2(c+d x)}}+\frac{4 \tan (c+d x)}{15 a d \left(a \sec ^2(c+d x)\right)^{3/2}}+\frac{\tan (c+d x)}{5 d \left(a \sec ^2(c+d x)\right)^{5/2}}",1,"Tan[c + d*x]/(5*d*(a*Sec[c + d*x]^2)^(5/2)) + (4*Tan[c + d*x])/(15*a*d*(a*Sec[c + d*x]^2)^(3/2)) + (8*Tan[c + d*x])/(15*a^2*d*Sqrt[a*Sec[c + d*x]^2])","A",5,4,16,0.2500,1,"{3657, 4122, 192, 191}"
286,1,118,0,0.0532144,"\int \frac{1}{\left(a+a \tan ^2(c+d x)\right)^{7/2}} \, dx","Int[(a + a*Tan[c + d*x]^2)^(-7/2),x]","\frac{16 \tan (c+d x)}{35 a^3 d \sqrt{a \sec ^2(c+d x)}}+\frac{8 \tan (c+d x)}{35 a^2 d \left(a \sec ^2(c+d x)\right)^{3/2}}+\frac{6 \tan (c+d x)}{35 a d \left(a \sec ^2(c+d x)\right)^{5/2}}+\frac{\tan (c+d x)}{7 d \left(a \sec ^2(c+d x)\right)^{7/2}}","\frac{16 \tan (c+d x)}{35 a^3 d \sqrt{a \sec ^2(c+d x)}}+\frac{8 \tan (c+d x)}{35 a^2 d \left(a \sec ^2(c+d x)\right)^{3/2}}+\frac{6 \tan (c+d x)}{35 a d \left(a \sec ^2(c+d x)\right)^{5/2}}+\frac{\tan (c+d x)}{7 d \left(a \sec ^2(c+d x)\right)^{7/2}}",1,"Tan[c + d*x]/(7*d*(a*Sec[c + d*x]^2)^(7/2)) + (6*Tan[c + d*x])/(35*a*d*(a*Sec[c + d*x]^2)^(5/2)) + (8*Tan[c + d*x])/(35*a^2*d*(a*Sec[c + d*x]^2)^(3/2)) + (16*Tan[c + d*x])/(35*a^3*d*Sqrt[a*Sec[c + d*x]^2])","A",6,4,16,0.2500,1,"{3657, 4122, 192, 191}"
287,1,22,0,0.0151697,"\int \left(1+\tan ^2(x)\right)^{3/2} \, dx","Int[(1 + Tan[x]^2)^(3/2),x]","\frac{1}{2} \tan (x) \sqrt{\sec ^2(x)}+\frac{1}{2} \sinh ^{-1}(\tan (x))","\frac{1}{2} \tan (x) \sqrt{\sec ^2(x)}+\frac{1}{2} \sinh ^{-1}(\tan (x))",1,"ArcSinh[Tan[x]]/2 + (Sqrt[Sec[x]^2]*Tan[x])/2","A",4,4,10,0.4000,1,"{3657, 4122, 195, 215}"
288,1,3,0,0.0112337,"\int \sqrt{1+\tan ^2(x)} \, dx","Int[Sqrt[1 + Tan[x]^2],x]","\sinh ^{-1}(\tan (x))","\sinh ^{-1}(\tan (x))",1,"ArcSinh[Tan[x]]","A",3,3,10,0.3000,1,"{3657, 4122, 215}"
289,1,11,0,0.0128562,"\int \frac{1}{\sqrt{1+\tan ^2(x)}} \, dx","Int[1/Sqrt[1 + Tan[x]^2],x]","\frac{\tan (x)}{\sqrt{\sec ^2(x)}}","\frac{\tan (x)}{\sqrt{\sec ^2(x)}}",1,"Tan[x]/Sqrt[Sec[x]^2]","A",3,3,10,0.3000,1,"{3657, 4122, 191}"
290,1,35,0,0.0217918,"\int \left(-1-\tan ^2(x)\right)^{3/2} \, dx","Int[(-1 - Tan[x]^2)^(3/2),x]","\frac{1}{2} \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right)-\frac{1}{2} \tan (x) \sqrt{-\sec ^2(x)}","\frac{1}{2} \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right)-\frac{1}{2} \tan (x) \sqrt{-\sec ^2(x)}",1,"ArcTan[Tan[x]/Sqrt[-Sec[x]^2]]/2 - (Sqrt[-Sec[x]^2]*Tan[x])/2","A",5,5,12,0.4167,1,"{3657, 4122, 195, 217, 203}"
291,1,16,0,0.0173641,"\int \sqrt{-1-\tan ^2(x)} \, dx","Int[Sqrt[-1 - Tan[x]^2],x]","-\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right)","-\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}\right)",1,"-ArcTan[Tan[x]/Sqrt[-Sec[x]^2]]","A",4,4,12,0.3333,1,"{3657, 4122, 217, 203}"
292,1,13,0,0.0198659,"\int \frac{1}{\sqrt{-1-\tan ^2(x)}} \, dx","Int[1/Sqrt[-1 - Tan[x]^2],x]","\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}","\frac{\tan (x)}{\sqrt{-\sec ^2(x)}}",1,"Tan[x]/Sqrt[-Sec[x]^2]","A",3,3,12,0.2500,1,"{3657, 4122, 191}"
293,1,117,0,0.146467,"\int \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b^2 f}-\frac{(a+b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 b^2 f}+\frac{\sqrt{a+b \tan ^2(e+f x)}}{f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}","\frac{\left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b^2 f}-\frac{(a+b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 b^2 f}+\frac{\sqrt{a+b \tan ^2(e+f x)}}{f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"-((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f) + Sqrt[a + b*Tan[e + f*x]^2]/f - ((a + b)*(a + b*Tan[e + f*x]^2)^(3/2))/(3*b^2*f) + (a + b*Tan[e + f*x]^2)^(5/2)/(5*b^2*f)","A",7,6,25,0.2400,1,"{3670, 446, 88, 50, 63, 208}"
294,1,88,0,0.1143222,"\int \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 b f}-\frac{\sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}","\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 b f}-\frac{\sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"(Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f - Sqrt[a + b*Tan[e + f*x]^2]/f + (a + b*Tan[e + f*x]^2)^(3/2)/(3*b*f)","A",6,6,25,0.2400,1,"{3670, 446, 80, 50, 63, 208}"
295,1,62,0,0.0710242,"\int \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a+b \tan ^2(e+f x)}}{f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}","\frac{\sqrt{a+b \tan ^2(e+f x)}}{f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"-((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f) + Sqrt[a + b*Tan[e + f*x]^2]/f","A",5,5,23,0.2174,1,"{3670, 444, 50, 63, 208}"
296,1,74,0,0.1005477,"\int \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{f}","\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{f}",1,"-((Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/f) + (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f","A",7,5,23,0.2174,1,"{3670, 446, 83, 63, 208}"
297,1,115,0,0.1477395,"\int \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 \sqrt{a} f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}","\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 \sqrt{a} f}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}",1,"((2*a - b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*Sqrt[a]*f) - (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f - (Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)","A",8,6,25,0.2400,1,"{3670, 446, 99, 156, 63, 208}"
298,1,163,0,0.2118987,"\int \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\left(8 a^2-4 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{3/2} f}+\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a f}","-\frac{\left(8 a^2-4 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{3/2} f}+\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(4 a-b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a f}",1,"-((8*a^2 - 4*a*b - b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*a^(3/2)*f) + (Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f + ((4*a - b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(8*a*f) - (Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)","A",9,7,25,0.2800,1,"{3670, 446, 99, 151, 156, 63, 208}"
299,1,222,0,0.3372206,"\int \tan ^6(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\left(2 a^2 b+a^3+8 a b^2-16 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{16 b^{5/2} f}-\frac{(a-2 b) (a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac{\tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{6 f}+\frac{(a-6 b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{24 b f}-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}","\frac{\left(2 a^2 b+a^3+8 a b^2-16 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{16 b^{5/2} f}-\frac{(a-2 b) (a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{16 b^2 f}+\frac{\tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{6 f}+\frac{(a-6 b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{24 b f}-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}",1,"-((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + ((a^3 + 2*a^2*b + 8*a*b^2 - 16*b^3)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(16*b^(5/2)*f) - ((a - 2*b)*(a + 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(16*b^2*f) + ((a - 6*b)*Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(24*b*f) + (Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(6*f)","A",9,8,25,0.3200,1,"{3670, 478, 582, 523, 217, 206, 377, 203}"
300,1,169,0,0.2103545,"\int \tan ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\left(a^2+4 a b-8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 b^{3/2} f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 b f}+\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}","-\frac{\left(a^2+4 a b-8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 b^{3/2} f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 b f}+\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}",1,"(Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - ((a^2 + 4*a*b - 8*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*b^(3/2)*f) + ((a - 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*b*f) + (Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)","A",8,8,25,0.3200,1,"{3670, 478, 582, 523, 217, 206, 377, 203}"
301,1,123,0,0.1335924,"\int \tan ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Tan[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{(a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 \sqrt{b} f}","-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{(a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 \sqrt{b} f}",1,"-((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + ((a - 2*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*Sqrt[b]*f) + (Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)","A",7,7,25,0.2800,1,"{3670, 478, 523, 217, 206, 377, 203}"
302,1,85,0,0.0525515,"\int \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}",1,"(Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + (Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f","A",6,6,16,0.3750,1,"{3661, 402, 217, 206, 377, 203}"
303,1,75,0,0.096368,"\int \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}","-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"-((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f","A",5,5,25,0.2000,1,"{3670, 475, 12, 377, 203}"
304,1,117,0,0.1540816,"\int \cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 f}+\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}","\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 f}+\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}",1,"(Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + ((3*a - b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a*f) - (Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*f)","A",6,6,25,0.2400,1,"{3670, 475, 583, 12, 377, 203}"
305,1,167,0,0.2308962,"\int \cot ^6(e+f x) \sqrt{a+b \tan ^2(e+f x)} \, dx","Int[Cot[e + f*x]^6*Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\left(15 a^2-5 a b-2 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 f}+\frac{(5 a-b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a f}","-\frac{\left(15 a^2-5 a b-2 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 f}+\frac{(5 a-b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a f}",1,"-((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) - ((15*a^2 - 5*a*b - 2*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^2*f) + ((5*a - b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a*f) - (Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*f)","A",7,6,25,0.2400,1,"{3670, 475, 583, 12, 377, 203}"
306,1,145,0,0.1749627,"\int \tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\left(a+b \tan ^2(e+f x)\right)^{7/2}}{7 b^2 f}-\frac{(a+b) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b^2 f}+\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 f}+\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}","\frac{\left(a+b \tan ^2(e+f x)\right)^{7/2}}{7 b^2 f}-\frac{(a+b) \left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b^2 f}+\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 f}+\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"-(((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f) + ((a - b)*Sqrt[a + b*Tan[e + f*x]^2])/f + (a + b*Tan[e + f*x]^2)^(3/2)/(3*f) - ((a + b)*(a + b*Tan[e + f*x]^2)^(5/2))/(5*b^2*f) + (a + b*Tan[e + f*x]^2)^(7/2)/(7*b^2*f)","A",8,6,25,0.2400,1,"{3670, 446, 88, 50, 63, 208}"
307,1,116,0,0.1450299,"\int \tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b f}-\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}","\frac{\left(a+b \tan ^2(e+f x)\right)^{5/2}}{5 b f}-\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f - ((a - b)*Sqrt[a + b*Tan[e + f*x]^2])/f - (a + b*Tan[e + f*x]^2)^(3/2)/(3*f) + (a + b*Tan[e + f*x]^2)^(5/2)/(5*b*f)","A",7,6,25,0.2400,1,"{3670, 446, 80, 50, 63, 208}"
308,1,90,0,0.0968939,"\int \tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}","\frac{(a-b) \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"-(((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f) + ((a - b)*Sqrt[a + b*Tan[e + f*x]^2])/f + (a + b*Tan[e + f*x]^2)^(3/2)/(3*f)","A",6,5,23,0.2174,1,"{3670, 444, 50, 63, 208}"
309,1,95,0,0.1311874,"\int \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{b \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{f}+\frac{b \sqrt{a+b \tan ^2(e+f x)}}{f}+\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}",1,"-((a^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/f) + ((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f + (b*Sqrt[a + b*Tan[e + f*x]^2])/f","A",8,6,23,0.2609,1,"{3670, 446, 84, 156, 63, 208}"
310,1,116,0,0.1738932,"\int \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\sqrt{a} (2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{a \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}","\frac{\sqrt{a} (2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 f}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{a \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}",1,"(Sqrt[a]*(2*a - 3*b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*f) - ((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f - (a*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)","A",8,6,25,0.2400,1,"{3670, 446, 98, 156, 63, 208}"
311,1,161,0,0.2222761,"\int \cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\left(8 a^2-12 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 \sqrt{a} f}+\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{a \cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(4 a-5 b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}","-\frac{\left(8 a^2-12 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 \sqrt{a} f}+\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f}-\frac{a \cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(4 a-5 b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}",1,"-((8*a^2 - 12*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*Sqrt[a]*f) + ((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/f + ((4*a - 5*b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(8*f) - (a*Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)","A",9,7,25,0.2800,1,"{3670, 446, 98, 151, 156, 63, 208}"
312,1,294,0,0.4479924,"\int \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\left(3 a^2-56 a b+48 b^2\right) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}-\frac{\left(8 a^2 b+3 a^3-80 a b^2+64 b^3\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac{\left(48 a^2 b^2+8 a^3 b+3 a^4-192 a b^3+128 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{128 b^{5/2} f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}","\frac{\left(3 a^2-56 a b+48 b^2\right) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{192 b f}-\frac{\left(8 a^2 b+3 a^3-80 a b^2+64 b^3\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{128 b^2 f}+\frac{\left(48 a^2 b^2+8 a^3 b+3 a^4-192 a b^3+128 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{128 b^{5/2} f}+\frac{b \tan ^7(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}+\frac{(9 a-8 b) \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{48 f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}",1,"-(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + ((3*a^4 + 8*a^3*b + 48*a^2*b^2 - 192*a*b^3 + 128*b^4)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(128*b^(5/2)*f) - ((3*a^3 + 8*a^2*b - 80*a*b^2 + 64*b^3)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(128*b^2*f) + ((3*a^2 - 56*a*b + 48*b^2)*Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(192*b*f) + ((9*a - 8*b)*Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(48*f) + (b*Tan[e + f*x]^7*Sqrt[a + b*Tan[e + f*x]^2])/(8*f)","A",10,8,25,0.3200,1,"{3670, 477, 582, 523, 217, 206, 377, 203}"
313,1,224,0,0.3549259,"\int \tan ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\left(a^2-10 a b+8 b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{16 b f}-\frac{\left(6 a^2 b+a^3-24 a b^2+16 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{16 b^{3/2} f}+\frac{b \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{6 f}+\frac{(7 a-6 b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{24 f}+\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}","\frac{\left(a^2-10 a b+8 b^2\right) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{16 b f}-\frac{\left(6 a^2 b+a^3-24 a b^2+16 b^3\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{16 b^{3/2} f}+\frac{b \tan ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{6 f}+\frac{(7 a-6 b) \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{24 f}+\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}",1,"((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - ((a^3 + 6*a^2*b - 24*a*b^2 + 16*b^3)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(16*b^(3/2)*f) + ((a^2 - 10*a*b + 8*b^2)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(16*b*f) + ((7*a - 6*b)*Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(24*f) + (b*Tan[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(6*f)","A",9,8,25,0.3200,1,"{3670, 477, 582, 523, 217, 206, 377, 203}"
314,1,172,0,0.2487828,"\int \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\left(3 a^2-12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 \sqrt{b} f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(5 a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}","\frac{\left(3 a^2-12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 \sqrt{b} f}+\frac{b \tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 f}+\frac{(5 a-4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}",1,"-(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*Sqrt[b]*f) + ((5*a - 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*f) + (b*Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(4*f)","A",8,8,25,0.3200,1,"{3670, 477, 582, 523, 217, 206, 377, 203}"
315,1,125,0,0.0949356,"\int \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}","\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}+\frac{b \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 f}+\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 f}",1,"((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + ((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*f) + (b*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*f)","A",7,7,16,0.4375,1,"{3661, 416, 523, 217, 206, 377, 203}"
316,1,114,0,0.138248,"\int \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{a \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{a \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{f}",1,"-(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) + (b^(3/2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f - (a*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/f","A",7,7,25,0.2800,1,"{3670, 474, 523, 217, 206, 377, 203}"
317,1,115,0,0.1733306,"\int \cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{a \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 f}+\frac{(3 a-4 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 f}","\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{a \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 f}+\frac{(3 a-4 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 f}",1,"((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f + ((3*a - 4*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*f) - (a*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*f)","A",6,6,25,0.2400,1,"{3670, 474, 583, 12, 377, 203}"
318,1,165,0,0.2447862,"\int \cot ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^{3/2} \, dx","Int[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\left(15 a^2-20 a b+3 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{a \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 f}+\frac{(5 a-6 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 f}","-\frac{\left(15 a^2-20 a b+3 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a f}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f}-\frac{a \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 f}+\frac{(5 a-6 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 f}",1,"-(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/f) - ((15*a^2 - 20*a*b + 3*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a*f) + ((5*a - 6*b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*f) - (a*Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*f)","A",7,6,25,0.2400,1,"{3670, 474, 583, 12, 377, 203}"
319,1,170,0,0.1788377,"\int \left(a+b \tan ^2(c+d x)\right)^{5/2} \, dx","Int[(a + b*Tan[c + d*x]^2)^(5/2),x]","\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)}}\right)}{8 d}+\frac{(a-b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)}}\right)}{d}+\frac{b \tan (c+d x) \left(a+b \tan ^2(c+d x)\right)^{3/2}}{4 d}+\frac{b (7 a-4 b) \tan (c+d x) \sqrt{a+b \tan ^2(c+d x)}}{8 d}","\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)}}\right)}{8 d}+\frac{(a-b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan (c+d x)}{\sqrt{a+b \tan ^2(c+d x)}}\right)}{d}+\frac{b \tan (c+d x) \left(a+b \tan ^2(c+d x)\right)^{3/2}}{4 d}+\frac{b (7 a-4 b) \tan (c+d x) \sqrt{a+b \tan ^2(c+d x)}}{8 d}",1,"((a - b)^(5/2)*ArcTan[(Sqrt[a - b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^2]])/d + (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^2]])/(8*d) + ((7*a - 4*b)*b*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]^2])/(8*d) + (b*Tan[c + d*x]*(a + b*Tan[c + d*x]^2)^(3/2))/(4*d)","A",8,8,16,0.5000,1,"{3661, 416, 528, 523, 217, 206, 377, 203}"
320,1,95,0,0.1402687,"\int \frac{\tan ^5(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 b^2 f}-\frac{(a+b) \sqrt{a+b \tan ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}","\frac{\left(a+b \tan ^2(e+f x)\right)^{3/2}}{3 b^2 f}-\frac{(a+b) \sqrt{a+b \tan ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f)) - ((a + b)*Sqrt[a + b*Tan[e + f*x]^2])/(b^2*f) + (a + b*Tan[e + f*x]^2)^(3/2)/(3*b^2*f)","A",6,5,25,0.2000,1,"{3670, 446, 88, 63, 208}"
321,1,64,0,0.1111239,"\int \frac{\tan ^3(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\sqrt{a+b \tan ^2(e+f x)}}{b f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}","\frac{\sqrt{a+b \tan ^2(e+f x)}}{b f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}",1,"ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f) + Sqrt[a + b*Tan[e + f*x]^2]/(b*f)","A",5,5,25,0.2000,1,"{3670, 446, 80, 63, 208}"
322,1,41,0,0.063777,"\int \frac{\tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Tan[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f))","A",4,4,23,0.1739,1,"{3670, 444, 63, 208}"
323,1,74,0,0.1083943,"\int \frac{\cot (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Cot[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{\sqrt{a} f}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]/(Sqrt[a]*f)) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f)","A",7,5,23,0.2174,1,"{3670, 446, 86, 63, 208}"
324,1,116,0,0.1626745,"\int \frac{\cot ^3(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{3/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}-\frac{\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}","\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{3/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}-\frac{\cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}",1,"((2*a + b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*a^(3/2)*f) - ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f) - (Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(2*a*f)","A",8,6,25,0.2400,1,"{3670, 446, 103, 156, 63, 208}"
325,1,166,0,0.2136808,"\int \frac{\cot ^5(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^5/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\left(8 a^2+4 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{5/2} f}+\frac{(4 a+3 b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a^2 f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 a f}","-\frac{\left(8 a^2+4 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{5/2} f}+\frac{(4 a+3 b) \cot ^2(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 a^2 f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f \sqrt{a-b}}-\frac{\cot ^4(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 a f}",1,"-((8*a^2 + 4*a*b + 3*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*a^(5/2)*f) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/(Sqrt[a - b]*f) + ((4*a + 3*b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2])/(8*a^2*f) - (Cot[e + f*x]^4*Sqrt[a + b*Tan[e + f*x]^2])/(4*a*f)","A",9,7,25,0.2800,1,"{3670, 446, 103, 151, 156, 63, 208}"
326,1,177,0,0.2224565,"\int \frac{\tan ^6(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^6/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\left(3 a^2+4 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 b^{5/2} f}-\frac{(3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 b^2 f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 b f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}","\frac{\left(3 a^2+4 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{8 b^{5/2} f}-\frac{(3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{8 b^2 f}+\frac{\tan ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{4 b f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)) + ((3*a^2 + 4*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(8*b^(5/2)*f) - ((3*a + 4*b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(8*b^2*f) + (Tan[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(4*b*f)","A",8,8,25,0.3200,1,"{3670, 479, 582, 523, 217, 206, 377, 203}"
327,1,125,0,0.1391769,"\int \frac{\tan ^4(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 b^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}+\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 b f}","-\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 b^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}+\frac{\tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 b f}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f) - ((a + 2*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*b^(3/2)*f) + (Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*b*f)","A",7,7,25,0.2800,1,"{3670, 479, 523, 217, 206, 377, 203}"
328,1,86,0,0.1043907,"\int \frac{\tan ^2(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Tan[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{\sqrt{b} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{\sqrt{b} f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[b]*f)","A",6,6,25,0.2400,1,"{3670, 483, 217, 206, 377, 203}"
329,1,46,0,0.0307983,"\int \frac{1}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[1/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)","A",3,3,16,0.1875,1,"{3661, 377, 203}"
330,1,78,0,0.1190835,"\int \frac{\cot ^2(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^2/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a f}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}-\frac{\cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a f}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)) - (Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(a*f)","A",5,5,25,0.2000,1,"{3670, 480, 12, 377, 203}"
331,1,120,0,0.1645105,"\int \frac{\cot ^4(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^4/Sqrt[a + b*Tan[e + f*x]^2],x]","\frac{(3 a+2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}","\frac{(3 a+2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}-\frac{\cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a f}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f) + ((3*a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^2*f) - (Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a*f)","A",6,6,25,0.2400,1,"{3670, 480, 583, 12, 377, 203}"
332,1,170,0,0.2459439,"\int \frac{\cot ^6(e+f x)}{\sqrt{a+b \tan ^2(e+f x)}} \, dx","Int[Cot[e + f*x]^6/Sqrt[a + b*Tan[e + f*x]^2],x]","-\frac{\left(15 a^2+10 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f}+\frac{(5 a+4 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f}","-\frac{\left(15 a^2+10 a b+8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f}+\frac{(5 a+4 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^2 f}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f \sqrt{a-b}}-\frac{\cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a f}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(Sqrt[a - b]*f)) - ((15*a^2 + 10*a*b + 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^3*f) + ((5*a + 4*b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^2*f) - (Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*a*f)","A",7,6,25,0.2400,1,"{3670, 480, 583, 12, 377, 203}"
333,1,98,0,0.1693117,"\int \frac{\tan ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{a^2}{b^2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{\sqrt{a+b \tan ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}","\frac{a^2}{b^2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{\sqrt{a+b \tan ^2(e+f x)}}{b^2 f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f)) + a^2/((a - b)*b^2*f*Sqrt[a + b*Tan[e + f*x]^2]) + Sqrt[a + b*Tan[e + f*x]^2]/(b^2*f)","A",6,5,25,0.2000,1,"{3670, 446, 87, 63, 208}"
334,1,73,0,0.1328608,"\int \frac{\tan ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}-\frac{a}{b f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}-\frac{a}{b f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f) - a/((a - b)*b*f*Sqrt[a + b*Tan[e + f*x]^2])","A",5,5,25,0.2000,1,"{3670, 446, 78, 63, 208}"
335,1,69,0,0.0884195,"\int \frac{\tan (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{1}{f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}","\frac{1}{f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f)) + 1/((a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])","A",5,5,23,0.2174,1,"{3670, 444, 51, 63, 208}"
336,1,106,0,0.1492073,"\int \frac{\cot (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{b}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{a^{3/2} f}-\frac{b}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]/(a^(3/2)*f)) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f) - b/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])","A",8,6,23,0.2609,1,"{3670, 446, 85, 156, 63, 208}"
337,1,157,0,0.2463749,"\int \frac{\cot ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{b (a-3 b)}{2 a^2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{5/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}-\frac{\cot ^2(e+f x)}{2 a f \sqrt{a+b \tan ^2(e+f x)}}","-\frac{b (a-3 b)}{2 a^2 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{5/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}-\frac{\cot ^2(e+f x)}{2 a f \sqrt{a+b \tan ^2(e+f x)}}",1,"((2*a + 3*b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*a^(5/2)*f) - ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f) - ((a - 3*b)*b)/(2*a^2*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^2/(2*a*f*Sqrt[a + b*Tan[e + f*x]^2])","A",9,7,25,0.2800,1,"{3670, 446, 103, 152, 156, 63, 208}"
338,1,215,0,0.3464503,"\int \frac{\cot ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{b \left(4 a^2+3 a b-15 b^2\right)}{8 a^3 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left(8 a^2+12 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{7/2} f}+\frac{(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}-\frac{\cot ^4(e+f x)}{4 a f \sqrt{a+b \tan ^2(e+f x)}}","\frac{b \left(4 a^2+3 a b-15 b^2\right)}{8 a^3 f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\left(8 a^2+12 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{7/2} f}+\frac{(4 a+5 b) \cot ^2(e+f x)}{8 a^2 f \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{3/2}}-\frac{\cot ^4(e+f x)}{4 a f \sqrt{a+b \tan ^2(e+f x)}}",1,"-((8*a^2 + 12*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*a^(7/2)*f) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(3/2)*f) + (b*(4*a^2 + 3*a*b - 15*b^2))/(8*a^3*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) + ((4*a + 5*b)*Cot[e + f*x]^2)/(8*a^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - Cot[e + f*x]^4/(4*a*f*Sqrt[a + b*Tan[e + f*x]^2])","A",10,8,25,0.3200,1,"{3670, 446, 103, 151, 152, 156, 63, 208}"
339,1,182,0,0.2508037,"\int \frac{\tan ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{(3 a-b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 b^2 f (a-b)}-\frac{(3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 b^{5/2} f}-\frac{a \tan ^3(e+f x)}{b f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}","\frac{(3 a-b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 b^2 f (a-b)}-\frac{(3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{2 b^{5/2} f}-\frac{a \tan ^3(e+f x)}{b f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f)) - ((3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]])/(2*b^(5/2)*f) - (a*Tan[e + f*x]^3)/((a - b)*b*f*Sqrt[a + b*Tan[e + f*x]^2]) + ((3*a - b)*Tan[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(2*(a - b)*b^2*f)","A",8,8,25,0.3200,1,"{3670, 470, 582, 523, 217, 206, 377, 203}"
340,1,123,0,0.1586171,"\int \frac{\tan ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{b^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{a \tan (e+f x)}{b f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{b^{3/2} f}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{a \tan (e+f x)}{b f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(b^(3/2)*f) - (a*Tan[e + f*x])/((a - b)*b*f*Sqrt[a + b*Tan[e + f*x]^2])","A",7,7,25,0.2800,1,"{3670, 470, 523, 217, 206, 377, 203}"
341,1,81,0,0.1111004,"\int \frac{\tan ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\tan (e+f x)}{f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}","\frac{\tan (e+f x)}{f (a-b) \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f)) + Tan[e + f*x]/((a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])","A",4,4,25,0.1600,1,"{3670, 471, 377, 203}"
342,1,85,0,0.0651576,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-3/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \tan (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \tan (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f) - (b*Tan[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2])","A",4,4,16,0.2500,1,"{3661, 382, 377, 203}"
343,1,128,0,0.1841074,"\int \frac{\cot ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{(a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a^2 f (a-b)}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \cot (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","-\frac{(a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{a^2 f (a-b)}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \cot (e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f)) - (b*Cot[e + f*x])/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - ((a - 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(a^2*(a - b)*f)","A",6,6,25,0.2400,1,"{3670, 472, 583, 12, 377, 203}"
344,1,184,0,0.2594541,"\int \frac{\cot ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(3/2),x]","-\frac{(a-4 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f (a-b)}+\frac{(3 a-4 b) (a+2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \cot ^3(e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","-\frac{(a-4 b) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^2 f (a-b)}+\frac{(3 a-4 b) (a+2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \cot ^3(e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f) - (b*Cot[e + f*x]^3)/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) + ((3*a - 4*b)*(a + 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^3*(a - b)*f) - ((a - 4*b)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^2*(a - b)*f)","A",7,6,25,0.2400,1,"{3670, 472, 583, 12, 377, 203}"
345,1,252,0,0.3683141,"\int \frac{\cot ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{3/2}} \, dx","Int[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(3/2),x]","\frac{\left(5 a^2+4 a b-24 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f (a-b)}-\frac{\left(10 a^2 b+15 a^3+8 a b^2-48 b^3\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 f (a-b)}-\frac{(a-6 b) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^2 f (a-b)}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \cot ^5(e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}","\frac{\left(5 a^2+4 a b-24 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^3 f (a-b)}-\frac{\left(10 a^2 b+15 a^3+8 a b^2-48 b^3\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 f (a-b)}-\frac{(a-6 b) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^2 f (a-b)}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{3/2}}-\frac{b \cot ^5(e+f x)}{a f (a-b) \sqrt{a+b \tan ^2(e+f x)}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(3/2)*f)) - (b*Cot[e + f*x]^5)/(a*(a - b)*f*Sqrt[a + b*Tan[e + f*x]^2]) - ((15*a^3 + 10*a^2*b + 8*a*b^2 - 48*b^3)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^4*(a - b)*f) + ((5*a^2 + 4*a*b - 24*b^2)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^3*(a - b)*f) - ((a - 6*b)*Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*a^2*(a - b)*f)","A",8,6,25,0.2400,1,"{3670, 472, 583, 12, 377, 203}"
346,1,115,0,0.2039175,"\int \frac{\tan ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{a^2}{3 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{a (a-2 b)}{b^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}","\frac{a^2}{3 b^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{a (a-2 b)}{b^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f)) + a^2/(3*(a - b)*b^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (a*(a - 2*b))/((a - b)^2*b^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",6,5,25,0.2000,1,"{3670, 446, 87, 63, 208}"
347,1,103,0,0.1556721,"\int \frac{\tan ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{a}{3 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{1}{f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}","-\frac{a}{3 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{1}{f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f) - a/(3*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^(3/2)) - 1/((a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",6,6,25,0.2400,1,"{3670, 446, 78, 51, 63, 208}"
348,1,99,0,0.1076846,"\int \frac{\tan (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{1}{f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{1}{3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}","\frac{1}{f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{1}{3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f)) + 1/(3*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) + 1/((a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",6,5,23,0.2174,1,"{3670, 444, 51, 63, 208}"
349,1,147,0,0.2137026,"\int \frac{\cot (e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{b (2 a-b)}{a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{b}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}","-\frac{b (2 a-b)}{a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{a^{5/2} f}-\frac{b}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}",1,"-(ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]]/(a^(5/2)*f)) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f) - b/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((2*a - b)*b)/(a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",9,7,23,0.3043,1,"{3670, 446, 85, 152, 156, 63, 208}"
350,1,206,0,0.350195,"\int \frac{\cot ^3(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^3/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{b \left(a^2-8 a b+5 b^2\right)}{2 a^3 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{b (3 a-5 b)}{6 a^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{7/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}-\frac{\cot ^2(e+f x)}{2 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{b \left(a^2-8 a b+5 b^2\right)}{2 a^3 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{b (3 a-5 b)}{6 a^2 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{2 a^{7/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}-\frac{\cot ^2(e+f x)}{2 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"((2*a + 5*b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(2*a^(7/2)*f) - ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f) - ((3*a - 5*b)*b)/(6*a^2*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - Cot[e + f*x]^2/(2*a*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (b*(a^2 - 8*a*b + 5*b^2))/(2*a^3*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",10,7,25,0.2800,1,"{3670, 446, 103, 152, 156, 63, 208}"
351,1,272,0,0.443041,"\int \frac{\cot ^5(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^5/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{b \left(3 a^2 b+4 a^3-50 a b^2+35 b^3\right)}{8 a^4 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{b \left(12 a^2+15 a b-35 b^2\right)}{24 a^3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\left(8 a^2+20 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{9/2} f}+\frac{(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}-\frac{\cot ^4(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}","\frac{b \left(3 a^2 b+4 a^3-50 a b^2+35 b^3\right)}{8 a^4 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{b \left(12 a^2+15 a b-35 b^2\right)}{24 a^3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\left(8 a^2+20 a b+35 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a}}\right)}{8 a^{9/2} f}+\frac{(4 a+7 b) \cot ^2(e+f x)}{8 a^2 f \left(a+b \tan ^2(e+f x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^2(e+f x)}}{\sqrt{a-b}}\right)}{f (a-b)^{5/2}}-\frac{\cot ^4(e+f x)}{4 a f \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-((8*a^2 + 20*a*b + 35*b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/(8*a^(9/2)*f) + ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]]/((a - b)^(5/2)*f) + (b*(12*a^2 + 15*a*b - 35*b^2))/(24*a^3*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) + ((4*a + 7*b)*Cot[e + f*x]^2)/(8*a^2*f*(a + b*Tan[e + f*x]^2)^(3/2)) - Cot[e + f*x]^4/(4*a*f*(a + b*Tan[e + f*x]^2)^(3/2)) + (b*(4*a^3 + 3*a^2*b - 50*a*b^2 + 35*b^3))/(8*a^4*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",11,8,25,0.3200,1,"{3670, 446, 103, 151, 152, 156, 63, 208}"
352,1,171,0,0.2667329,"\int \frac{\tan ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{a (a-2 b) \tan (e+f x)}{b^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{b^{5/2} f}-\frac{a \tan ^3(e+f x)}{3 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}","-\frac{a (a-2 b) \tan (e+f x)}{b^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{b^{5/2} f}-\frac{a \tan ^3(e+f x)}{3 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f)) + ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/(b^(5/2)*f) - (a*Tan[e + f*x]^3)/(3*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^(3/2)) - (a*(a - 2*b)*Tan[e + f*x])/((a - b)^2*b^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",8,8,25,0.3200,1,"{3670, 470, 578, 523, 217, 206, 377, 203}"
353,1,131,0,0.1585527,"\int \frac{\tan ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}+\frac{(a-4 b) \tan (e+f x)}{3 b f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{a \tan (e+f x)}{3 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}+\frac{(a-4 b) \tan (e+f x)}{3 b f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{a \tan (e+f x)}{3 b f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (a*Tan[e + f*x])/(3*(a - b)*b*f*(a + b*Tan[e + f*x]^2)^(3/2)) + ((a - 4*b)*Tan[e + f*x])/(3*(a - b)^2*b*f*Sqrt[a + b*Tan[e + f*x]^2])","A",6,6,25,0.2400,1,"{3670, 470, 527, 12, 377, 203}"
354,1,128,0,0.1526944,"\int \frac{\tan ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Tan[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}+\frac{(2 a+b) \tan (e+f x)}{3 a f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan (e+f x)}{3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}+\frac{(2 a+b) \tan (e+f x)}{3 a f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan (e+f x)}{3 f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f)) + Tan[e + f*x]/(3*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) + ((2*a + b)*Tan[e + f*x])/(3*a*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",6,6,25,0.2400,1,"{3670, 471, 527, 12, 377, 203}"
355,1,134,0,0.1009287,"\int \frac{1}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[(a + b*Tan[e + f*x]^2)^(-5/2),x]","-\frac{b (5 a-2 b) \tan (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \tan (e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{b (5 a-2 b) \tan (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \tan (e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Tan[e + f*x])/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((5*a - 2*b)*b*Tan[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2])","A",6,6,16,0.3750,1,"{3661, 414, 527, 12, 377, 203}"
356,1,186,0,0.2787314,"\int \frac{\cot ^2(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^2/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{(a-4 b) (3 a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{b (7 a-4 b) \cot (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \cot (e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{(a-4 b) (3 a-2 b) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)^2}-\frac{b (7 a-4 b) \cot (e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \cot (e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f)) - (b*Cot[e + f*x])/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((7*a - 4*b)*b*Cot[e + f*x])/(3*a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - ((a - 4*b)*(3*a - 2*b)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^3*(a - b)^2*f)","A",7,7,25,0.2800,1,"{3670, 472, 579, 583, 12, 377, 203}"
357,1,249,0,0.3777119,"\int \frac{\cot ^4(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^4/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\left(a^2-12 a b+8 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)^2}+\frac{(a-2 b) \left(3 a^2+8 a b-8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^4 f (a-b)^2}-\frac{b (3 a-2 b) \cot ^3(e+f x)}{a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \cot ^3(e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{\left(a^2-12 a b+8 b^2\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^3 f (a-b)^2}+\frac{(a-2 b) \left(3 a^2+8 a b-8 b^2\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{3 a^4 f (a-b)^2}-\frac{b (3 a-2 b) \cot ^3(e+f x)}{a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \cot ^3(e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f) - (b*Cot[e + f*x]^3)/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((3*a - 2*b)*b*Cot[e + f*x]^3)/(a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]) + ((a - 2*b)*(3*a^2 + 8*a*b - 8*b^2)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^4*(a - b)^2*f) - ((a^2 - 12*a*b + 8*b^2)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(3*a^3*(a - b)^2*f)","A",8,7,25,0.2800,1,"{3670, 472, 579, 583, 12, 377, 203}"
358,1,327,0,0.4962403,"\int \frac{\cot ^6(e+f x)}{\left(a+b \tan ^2(e+f x)\right)^{5/2}} \, dx","Int[Cot[e + f*x]^6/(a + b*Tan[e + f*x]^2)^(5/2),x]","-\frac{\left(a^2-22 a b+16 b^2\right) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 f (a-b)^2}+\frac{\left(4 a^2 b+5 a^3-88 a b^2+64 b^3\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 f (a-b)^2}-\frac{\left(8 a^2 b^2+10 a^3 b+15 a^4-176 a b^3+128 b^4\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^5 f (a-b)^2}-\frac{b (11 a-8 b) \cot ^5(e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \cot ^5(e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}","-\frac{\left(a^2-22 a b+16 b^2\right) \cot ^5(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{5 a^3 f (a-b)^2}+\frac{\left(4 a^2 b+5 a^3-88 a b^2+64 b^3\right) \cot ^3(e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^4 f (a-b)^2}-\frac{\left(8 a^2 b^2+10 a^3 b+15 a^4-176 a b^3+128 b^4\right) \cot (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{15 a^5 f (a-b)^2}-\frac{b (11 a-8 b) \cot ^5(e+f x)}{3 a^2 f (a-b)^2 \sqrt{a+b \tan ^2(e+f x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right)}{f (a-b)^{5/2}}-\frac{b \cot ^5(e+f x)}{3 a f (a-b) \left(a+b \tan ^2(e+f x)\right)^{3/2}}",1,"-(ArcTan[(Sqrt[a - b]*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]^2]]/((a - b)^(5/2)*f)) - (b*Cot[e + f*x]^5)/(3*a*(a - b)*f*(a + b*Tan[e + f*x]^2)^(3/2)) - ((11*a - 8*b)*b*Cot[e + f*x]^5)/(3*a^2*(a - b)^2*f*Sqrt[a + b*Tan[e + f*x]^2]) - ((15*a^4 + 10*a^3*b + 8*a^2*b^2 - 176*a*b^3 + 128*b^4)*Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^5*(a - b)^2*f) + ((5*a^3 + 4*a^2*b - 88*a*b^2 + 64*b^3)*Cot[e + f*x]^3*Sqrt[a + b*Tan[e + f*x]^2])/(15*a^4*(a - b)^2*f) - ((a^2 - 22*a*b + 16*b^2)*Cot[e + f*x]^5*Sqrt[a + b*Tan[e + f*x]^2])/(5*a^3*(a - b)^2*f)","A",9,7,25,0.2800,1,"{3670, 472, 579, 583, 12, 377, 203}"
359,1,72,0,0.0826926,"\int (d \tan (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Tan[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\tan ^2(e+f x)\right)}{f (m+2 p+1)}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (m+2 p+1);\frac{1}{2} (m+2 p+3);-\tan ^2(e+f x)\right)}{f (m+2 p+1)}",1,"(Hypergeometric2F1[1, (1 + m + 2*p)/2, (3 + m + 2*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(d*Tan[e + f*x])^m*(b*Tan[e + f*x]^2)^p)/(f*(1 + m + 2*p))","A",4,4,23,0.1739,1,"{3578, 20, 3476, 364}"
360,1,100,0,0.1190823,"\int (d \tan (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Tan[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\frac{(d \tan (e+f x))^{m+1} \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};1,-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{d f (m+1)}","\frac{(d \tan (e+f x))^{m+1} \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{m+1}{2};1,-p;\frac{m+3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{d f (m+1)}",1,"(AppellF1[(1 + m)/2, 1, -p, (3 + m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Tan[e + f*x])^(1 + m)*(a + b*Tan[e + f*x]^2)^p)/(d*f*(1 + m)*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,25,0.1200,1,"{3670, 511, 510}"
361,1,129,0,0.1712906,"\int \tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{(a+b) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{2 b^2 f (p+1)}+\frac{\left(a+b \tan ^2(e+f x)\right)^{p+2}}{2 b^2 f (p+2)}-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}","-\frac{(a+b) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{2 b^2 f (p+1)}+\frac{\left(a+b \tan ^2(e+f x)\right)^{p+2}}{2 b^2 f (p+2)}-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}",1,"-((a + b)*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*b^2*f*(1 + p)) - (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p)) + (a + b*Tan[e + f*x]^2)^(2 + p)/(2*b^2*f*(2 + p))","A",5,4,23,0.1739,1,"{3670, 446, 88, 68}"
362,1,95,0,0.1028899,"\int \tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}+\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1}}{2 b f (p+1)}","\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}+\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1}}{2 b f (p+1)}",1,"(a + b*Tan[e + f*x]^2)^(1 + p)/(2*b*f*(1 + p)) + (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p))","A",4,4,23,0.1739,1,"{3670, 446, 80, 68}"
363,1,63,0,0.0669837,"\int \tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}","-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}",1,"-(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p))","A",3,3,21,0.1429,1,"{3670, 444, 68}"
364,1,118,0,0.1147799,"\int \cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}","\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)}{2 a f (p+1)}",1,"(Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p)) - (Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*a*f*(1 + p))","A",5,5,21,0.2381,1,"{3670, 446, 86, 65, 68}"
365,1,158,0,0.1677584,"\int \cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p,x]","\frac{(a-b p) \left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)}{2 a^2 f (p+1)}-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}-\frac{\cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{2 a f}","\frac{(a-b p) \left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)}{2 a^2 f (p+1)}-\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}-\frac{\cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{2 a f}",1,"-(Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*a*f) - (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p)) + ((a - b*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*a^2*f*(1 + p))","A",6,6,23,0.2609,1,"{3670, 446, 103, 156, 65, 68}"
366,1,217,0,0.2532993,"\int \cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\left(2 a^2-2 a b p-b^2 (1-p) p\right) \left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)}{4 a^3 f (p+1)}+\frac{(2 a-b p+b) \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{4 a^2 f}+\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}-\frac{\cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{4 a f}","-\frac{\left(2 a^2-2 a b p-b^2 (1-p) p\right) \left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)}{a}+1\right)}{4 a^3 f (p+1)}+\frac{(2 a-b p+b) \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{4 a^2 f}+\frac{\left(a+b \tan ^2(e+f x)\right)^{p+1} \, _2F_1\left(1,p+1;p+2;\frac{b \tan ^2(e+f x)+a}{a-b}\right)}{2 f (p+1) (a-b)}-\frac{\cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^{p+1}}{4 a f}",1,"((2*a + b - b*p)*Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^(1 + p))/(4*a^2*f) - (Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^(1 + p))/(4*a*f) + (Hypergeometric2F1[1, 1 + p, 2 + p, (a + b*Tan[e + f*x]^2)/(a - b)]*(a + b*Tan[e + f*x]^2)^(1 + p))/(2*(a - b)*f*(1 + p)) - ((2*a^2 - 2*a*b*p - b^2*(1 - p)*p)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + (b*Tan[e + f*x]^2)/a]*(a + b*Tan[e + f*x]^2)^(1 + p))/(4*a^3*f*(1 + p))","A",7,7,23,0.3043,1,"{3670, 446, 103, 151, 156, 65, 68}"
367,1,83,0,0.1002624,"\int \tan ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Tan[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan ^7(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{7}{2};1,-p;\frac{9}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{7 f}","\frac{\tan ^7(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{7}{2};1,-p;\frac{9}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{7 f}",1,"(AppellF1[7/2, 1, -p, 9/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^7*(a + b*Tan[e + f*x]^2)^p)/(7*f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3670, 511, 510}"
368,1,83,0,0.0989361,"\int \tan ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Tan[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{5}{2};1,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{5 f}","\frac{\tan ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{5}{2};1,-p;\frac{7}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{5 f}",1,"(AppellF1[5/2, 1, -p, 7/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p)/(5*f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3670, 511, 510}"
369,1,83,0,0.101501,"\int \tan ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Tan[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};1,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{3 f}","\frac{\tan ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{3}{2};1,-p;\frac{5}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{3 f}",1,"(AppellF1[3/2, 1, -p, 5/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3670, 511, 510}"
370,1,78,0,0.0507798,"\int \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}","\frac{\tan (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1,-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, 1, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,14,0.2143,1,"{3661, 430, 429}"
371,1,79,0,0.0984764,"\int \cot ^2(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]^2*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};1,-p;\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}","-\frac{\cot (e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{1}{2};1,-p;\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"-((AppellF1[-1/2, 1, -p, 1/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Cot[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 + (b*Tan[e + f*x]^2)/a)^p))","A",3,3,23,0.1304,1,"{3670, 511, 510}"
372,1,83,0,0.0992018,"\int \cot ^4(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]^4*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{3}{2};1,-p;-\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{3 f}","-\frac{\cot ^3(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{3}{2};1,-p;-\frac{1}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{3 f}",1,"-(AppellF1[-3/2, 1, -p, -1/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Cot[e + f*x]^3*(a + b*Tan[e + f*x]^2)^p)/(3*f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3670, 511, 510}"
373,1,83,0,0.0994013,"\int \cot ^6(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[Cot[e + f*x]^6*(a + b*Tan[e + f*x]^2)^p,x]","-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{5}{2};1,-p;-\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{5 f}","-\frac{\cot ^5(e+f x) \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(-\frac{5}{2};1,-p;-\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{5 f}",1,"-(AppellF1[-5/2, 1, -p, -3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*Cot[e + f*x]^5*(a + b*Tan[e + f*x]^2)^p)/(5*f*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,23,0.1304,1,"{3670, 511, 510}"
374,1,255,0,0.1472425,"\int \left(a+b \tan ^3(c+d x)\right)^4 \, dx","Int[(a + b*Tan[c + d*x]^3)^4,x]","\frac{b^2 \left(6 a^2-b^2\right) \tan ^5(c+d x)}{5 d}-\frac{b^2 \left(6 a^2-b^2\right) \tan ^3(c+d x)}{3 d}+\frac{2 a b \left(a^2-b^2\right) \tan ^2(c+d x)}{d}+\frac{b^2 \left(6 a^2-b^2\right) \tan (c+d x)}{d}+\frac{4 a b \left(a^2-b^2\right) \log (\cos (c+d x))}{d}+x \left(-6 a^2 b^2+a^4+b^4\right)+\frac{a b^3 \tan ^8(c+d x)}{2 d}-\frac{2 a b^3 \tan ^6(c+d x)}{3 d}+\frac{a b^3 \tan ^4(c+d x)}{d}+\frac{b^4 \tan ^{11}(c+d x)}{11 d}-\frac{b^4 \tan ^9(c+d x)}{9 d}+\frac{b^4 \tan ^7(c+d x)}{7 d}","\frac{b^2 \left(6 a^2-b^2\right) \tan ^5(c+d x)}{5 d}-\frac{b^2 \left(6 a^2-b^2\right) \tan ^3(c+d x)}{3 d}+\frac{2 a b \left(a^2-b^2\right) \tan ^2(c+d x)}{d}+\frac{b^2 \left(6 a^2-b^2\right) \tan (c+d x)}{d}+\frac{4 a b \left(a^2-b^2\right) \log (\cos (c+d x))}{d}+x \left(-6 a^2 b^2+a^4+b^4\right)+\frac{a b^3 \tan ^8(c+d x)}{2 d}-\frac{2 a b^3 \tan ^6(c+d x)}{3 d}+\frac{a b^3 \tan ^4(c+d x)}{d}+\frac{b^4 \tan ^{11}(c+d x)}{11 d}-\frac{b^4 \tan ^9(c+d x)}{9 d}+\frac{b^4 \tan ^7(c+d x)}{7 d}",1,"(a^4 - 6*a^2*b^2 + b^4)*x + (4*a*b*(a^2 - b^2)*Log[Cos[c + d*x]])/d + (b^2*(6*a^2 - b^2)*Tan[c + d*x])/d + (2*a*b*(a^2 - b^2)*Tan[c + d*x]^2)/d - (b^2*(6*a^2 - b^2)*Tan[c + d*x]^3)/(3*d) + (a*b^3*Tan[c + d*x]^4)/d + (b^2*(6*a^2 - b^2)*Tan[c + d*x]^5)/(5*d) - (2*a*b^3*Tan[c + d*x]^6)/(3*d) + (b^4*Tan[c + d*x]^7)/(7*d) + (a*b^3*Tan[c + d*x]^8)/(2*d) - (b^4*Tan[c + d*x]^9)/(9*d) + (b^4*Tan[c + d*x]^11)/(11*d)","A",6,5,14,0.3571,1,"{3661, 1810, 635, 203, 260}"
375,1,168,0,0.0977967,"\int \left(a+b \tan ^3(c+d x)\right)^3 \, dx","Int[(a + b*Tan[c + d*x]^3)^3,x]","\frac{b \left(3 a^2-b^2\right) \tan ^2(c+d x)}{2 d}+\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+a x \left(a^2-3 b^2\right)+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}-\frac{a b^2 \tan ^3(c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^8(c+d x)}{8 d}-\frac{b^3 \tan ^6(c+d x)}{6 d}+\frac{b^3 \tan ^4(c+d x)}{4 d}","\frac{b \left(3 a^2-b^2\right) \tan ^2(c+d x)}{2 d}+\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+a x \left(a^2-3 b^2\right)+\frac{3 a b^2 \tan ^5(c+d x)}{5 d}-\frac{a b^2 \tan ^3(c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}+\frac{b^3 \tan ^8(c+d x)}{8 d}-\frac{b^3 \tan ^6(c+d x)}{6 d}+\frac{b^3 \tan ^4(c+d x)}{4 d}",1,"a*(a^2 - 3*b^2)*x + (b*(3*a^2 - b^2)*Log[Cos[c + d*x]])/d + (3*a*b^2*Tan[c + d*x])/d + (b*(3*a^2 - b^2)*Tan[c + d*x]^2)/(2*d) - (a*b^2*Tan[c + d*x]^3)/d + (b^3*Tan[c + d*x]^4)/(4*d) + (3*a*b^2*Tan[c + d*x]^5)/(5*d) - (b^3*Tan[c + d*x]^6)/(6*d) + (b^3*Tan[c + d*x]^8)/(8*d)","A",6,5,14,0.3571,1,"{3661, 1810, 635, 203, 260}"
376,1,89,0,0.059836,"\int \left(a+b \tan ^3(c+d x)\right)^2 \, dx","Int[(a + b*Tan[c + d*x]^3)^2,x]","x \left(a^2-b^2\right)+\frac{a b \tan ^2(c+d x)}{d}+\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \tan ^5(c+d x)}{5 d}-\frac{b^2 \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan (c+d x)}{d}","x \left(a^2-b^2\right)+\frac{a b \tan ^2(c+d x)}{d}+\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \tan ^5(c+d x)}{5 d}-\frac{b^2 \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan (c+d x)}{d}",1,"(a^2 - b^2)*x + (2*a*b*Log[Cos[c + d*x]])/d + (b^2*Tan[c + d*x])/d + (a*b*Tan[c + d*x]^2)/d - (b^2*Tan[c + d*x]^3)/(3*d) + (b^2*Tan[c + d*x]^5)/(5*d)","A",6,5,14,0.3571,1,"{3661, 1810, 635, 203, 260}"
377,1,32,0,0.0194858,"\int \left(a+b \tan ^3(c+d x)\right) \, dx","Int[a + b*Tan[c + d*x]^3,x]","a x+\frac{b \tan ^2(c+d x)}{2 d}+\frac{b \log (\cos (c+d x))}{d}","a x+\frac{b \tan ^2(c+d x)}{2 d}+\frac{b \log (\cos (c+d x))}{d}",1,"a*x + (b*Log[Cos[c + d*x]])/d + (b*Tan[c + d*x]^2)/(2*d)","A",3,2,12,0.1667,1,"{3473, 3475}"
378,1,256,0,0.3804287,"\int \frac{1}{a+b \tan ^3(c+d x)} \, dx","Int[(a + b*Tan[c + d*x]^3)^(-1),x]","\frac{\sqrt[3]{b} \left(a^{4/3}-b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} d \left(a^2+b^2\right)}-\frac{\sqrt[3]{b} \left(a^{4/3}+b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)}{6 a^{2/3} d \left(a^2+b^2\right)}+\frac{\sqrt[3]{b} \left(a^{4/3}+b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{3 a^{2/3} d \left(a^2+b^2\right)}-\frac{b \log \left(a \cos ^3(c+d x)+b \sin ^3(c+d x)\right)}{3 d \left(a^2+b^2\right)}+\frac{a x}{a^2+b^2}","\frac{\sqrt[3]{b} \left(a^{4/3}-b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{2/3} d \left(a^2+b^2\right)}-\frac{\sqrt[3]{b} \left(a^{4/3}+b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)}{6 a^{2/3} d \left(a^2+b^2\right)}+\frac{\sqrt[3]{b} \left(a^{4/3}+b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{3 a^{2/3} d \left(a^2+b^2\right)}-\frac{b \log \left(a \cos ^3(c+d x)+b \sin ^3(c+d x)\right)}{3 d \left(a^2+b^2\right)}+\frac{a x}{a^2+b^2}",1,"(a*x)/(a^2 + b^2) + (b^(1/3)*(a^(4/3) - b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*(a^2 + b^2)*d) - (b*Log[a*Cos[c + d*x]^3 + b*Sin[c + d*x]^3])/(3*(a^2 + b^2)*d) + (b^(1/3)*(a^(4/3) + b^(4/3))*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(3*a^(2/3)*(a^2 + b^2)*d) - (b^(1/3)*(a^(4/3) + b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(6*a^(2/3)*(a^2 + b^2)*d)","A",14,12,14,0.8571,1,"{3661, 6725, 635, 203, 260, 1871, 1860, 31, 634, 617, 204, 628}"
379,1,558,0,0.7279991,"\int \frac{1}{\left(a+b \tan ^3(c+d x)\right)^2} \, dx","Int[(a + b*Tan[c + d*x]^3)^(-2),x]","\frac{\sqrt[3]{b} \left(a^{4/3}-2 b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} d \left(a^2+b^2\right)}+\frac{\sqrt[3]{b} \left(-2 a^{2/3} b^{4/3}+a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d \left(a^2+b^2\right)^2}+\frac{b (\tan (c+d x) (b-a \tan (c+d x))+a)}{3 a d \left(a^2+b^2\right) \left(a+b \tan ^3(c+d x)\right)}-\frac{\sqrt[3]{b} \left(a^{4/3}+2 b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)}{18 a^{5/3} d \left(a^2+b^2\right)}-\frac{\sqrt[3]{b} \left(2 a^{2/3} b^{4/3}+a^2-b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)}{6 \sqrt[3]{a} d \left(a^2+b^2\right)^2}+\frac{\sqrt[3]{b} \left(a^{4/3}+2 b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{9 a^{5/3} d \left(a^2+b^2\right)}+\frac{\sqrt[3]{b} \left(2 a^{2/3} b^{4/3}+a^2-b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{3 \sqrt[3]{a} d \left(a^2+b^2\right)^2}-\frac{2 a b \log \left(a \cos ^3(c+d x)+b \sin ^3(c+d x)\right)}{3 d \left(a^2+b^2\right)^2}+\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}","\frac{\sqrt[3]{b} \left(a^{4/3}-2 b^{4/3}\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{3 \sqrt{3} a^{5/3} d \left(a^2+b^2\right)}+\frac{\sqrt[3]{b} \left(-2 a^{2/3} b^{4/3}+a^2-b^2\right) \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \tan (c+d x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} \sqrt[3]{a} d \left(a^2+b^2\right)^2}+\frac{b (\tan (c+d x) (b-a \tan (c+d x))+a)}{3 a d \left(a^2+b^2\right) \left(a+b \tan ^3(c+d x)\right)}-\frac{\sqrt[3]{b} \left(a^{4/3}+2 b^{4/3}\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)}{18 a^{5/3} d \left(a^2+b^2\right)}-\frac{\sqrt[3]{b} \left(2 a^{2/3} b^{4/3}+a^2-b^2\right) \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \tan (c+d x)+b^{2/3} \tan ^2(c+d x)\right)}{6 \sqrt[3]{a} d \left(a^2+b^2\right)^2}+\frac{\sqrt[3]{b} \left(a^{4/3}+2 b^{4/3}\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{9 a^{5/3} d \left(a^2+b^2\right)}+\frac{\sqrt[3]{b} \left(2 a^{2/3} b^{4/3}+a^2-b^2\right) \log \left(\sqrt[3]{a}+\sqrt[3]{b} \tan (c+d x)\right)}{3 \sqrt[3]{a} d \left(a^2+b^2\right)^2}-\frac{2 a b \log \left(a \cos ^3(c+d x)+b \sin ^3(c+d x)\right)}{3 d \left(a^2+b^2\right)^2}+\frac{x \left(a^2-b^2\right)}{\left(a^2+b^2\right)^2}",1,"((a^2 - b^2)*x)/(a^2 + b^2)^2 + (b^(1/3)*(a^2 - 2*a^(2/3)*b^(4/3) - b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 + b^2)^2*d) + (b^(1/3)*(a^(4/3) - 2*b^(4/3))*ArcTan[(a^(1/3) - 2*b^(1/3)*Tan[c + d*x])/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)*(a^2 + b^2)*d) - (2*a*b*Log[a*Cos[c + d*x]^3 + b*Sin[c + d*x]^3])/(3*(a^2 + b^2)^2*d) + (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) - b^2)*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(3*a^(1/3)*(a^2 + b^2)^2*d) + (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(1/3) + b^(1/3)*Tan[c + d*x]])/(9*a^(5/3)*(a^2 + b^2)*d) - (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) - b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(6*a^(1/3)*(a^2 + b^2)^2*d) - (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*Tan[c + d*x] + b^(2/3)*Tan[c + d*x]^2])/(18*a^(5/3)*(a^2 + b^2)*d) + (b*(a + Tan[c + d*x]*(b - a*Tan[c + d*x])))/(3*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x]^3))","A",21,13,14,0.9286,1,"{3661, 6725, 635, 203, 260, 1854, 1860, 31, 634, 617, 204, 628, 1871}"
380,1,37,0,0.0612837,"\int \frac{1}{1+\tan ^3(x)} \, dx","Int[(1 + Tan[x]^3)^(-1),x]","\frac{x}{2}-\frac{1}{3} \log \left(\tan ^2(x)-\tan (x)+1\right)+\frac{1}{6} \log (\tan (x)+1)-\frac{1}{2} \log (\cos (x))","\frac{x}{2}-\frac{1}{3} \log \left(\tan ^2(x)-\tan (x)+1\right)+\frac{1}{6} \log (\tan (x)+1)-\frac{1}{2} \log (\cos (x))",1,"x/2 - Log[Cos[x]]/2 + Log[1 + Tan[x]]/6 - Log[1 - Tan[x] + Tan[x]^2]/3","A",7,6,8,0.7500,1,"{3661, 2074, 635, 203, 260, 628}"
381,1,216,0,0.1293352,"\int \left(a+b \tan ^4(c+d x)\right)^4 \, dx","Int[(a + b*Tan[c + d*x]^4)^4,x]","\frac{b^2 \left(6 a^2+4 a b+b^2\right) \tan ^7(c+d x)}{7 d}-\frac{b^2 \left(6 a^2+4 a b+b^2\right) \tan ^5(c+d x)}{5 d}+\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \tan ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^3 (4 a+b) \tan ^{11}(c+d x)}{11 d}-\frac{b^3 (4 a+b) \tan ^9(c+d x)}{9 d}+x (a+b)^4+\frac{b^4 \tan ^{15}(c+d x)}{15 d}-\frac{b^4 \tan ^{13}(c+d x)}{13 d}","\frac{b^2 \left(6 a^2+4 a b+b^2\right) \tan ^7(c+d x)}{7 d}-\frac{b^2 \left(6 a^2+4 a b+b^2\right) \tan ^5(c+d x)}{5 d}+\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \tan ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^3 (4 a+b) \tan ^{11}(c+d x)}{11 d}-\frac{b^3 (4 a+b) \tan ^9(c+d x)}{9 d}+x (a+b)^4+\frac{b^4 \tan ^{15}(c+d x)}{15 d}-\frac{b^4 \tan ^{13}(c+d x)}{13 d}",1,"(a + b)^4*x - (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c + d*x])/d + (b*(2*a + b)*(2*a^2 + 2*a*b + b^2)*Tan[c + d*x]^3)/(3*d) - (b^2*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^5)/(5*d) + (b^2*(6*a^2 + 4*a*b + b^2)*Tan[c + d*x]^7)/(7*d) - (b^3*(4*a + b)*Tan[c + d*x]^9)/(9*d) + (b^3*(4*a + b)*Tan[c + d*x]^11)/(11*d) - (b^4*Tan[c + d*x]^13)/(13*d) + (b^4*Tan[c + d*x]^15)/(15*d)","A",4,3,14,0.2143,1,"{3661, 1154, 203}"
382,1,144,0,0.0824736,"\int \left(a+b \tan ^4(c+d x)\right)^3 \, dx","Int[(a + b*Tan[c + d*x]^4)^3,x]","\frac{b \left(3 a^2+3 a b+b^2\right) \tan ^3(c+d x)}{3 d}-\frac{b \left(3 a^2+3 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^2 (3 a+b) \tan ^7(c+d x)}{7 d}-\frac{b^2 (3 a+b) \tan ^5(c+d x)}{5 d}+x (a+b)^3+\frac{b^3 \tan ^{11}(c+d x)}{11 d}-\frac{b^3 \tan ^9(c+d x)}{9 d}","\frac{b \left(3 a^2+3 a b+b^2\right) \tan ^3(c+d x)}{3 d}-\frac{b \left(3 a^2+3 a b+b^2\right) \tan (c+d x)}{d}+\frac{b^2 (3 a+b) \tan ^7(c+d x)}{7 d}-\frac{b^2 (3 a+b) \tan ^5(c+d x)}{5 d}+x (a+b)^3+\frac{b^3 \tan ^{11}(c+d x)}{11 d}-\frac{b^3 \tan ^9(c+d x)}{9 d}",1,"(a + b)^3*x - (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x])/d + (b*(3*a^2 + 3*a*b + b^2)*Tan[c + d*x]^3)/(3*d) - (b^2*(3*a + b)*Tan[c + d*x]^5)/(5*d) + (b^2*(3*a + b)*Tan[c + d*x]^7)/(7*d) - (b^3*Tan[c + d*x]^9)/(9*d) + (b^3*Tan[c + d*x]^11)/(11*d)","A",4,3,14,0.2143,1,"{3661, 1154, 203}"
383,1,82,0,0.0547545,"\int \left(a+b \tan ^4(c+d x)\right)^2 \, dx","Int[(a + b*Tan[c + d*x]^4)^2,x]","\frac{b (2 a+b) \tan ^3(c+d x)}{3 d}-\frac{b (2 a+b) \tan (c+d x)}{d}+x (a+b)^2+\frac{b^2 \tan ^7(c+d x)}{7 d}-\frac{b^2 \tan ^5(c+d x)}{5 d}","\frac{b (2 a+b) \tan ^3(c+d x)}{3 d}-\frac{b (2 a+b) \tan (c+d x)}{d}+x (a+b)^2+\frac{b^2 \tan ^7(c+d x)}{7 d}-\frac{b^2 \tan ^5(c+d x)}{5 d}",1,"(a + b)^2*x - (b*(2*a + b)*Tan[c + d*x])/d + (b*(2*a + b)*Tan[c + d*x]^3)/(3*d) - (b^2*Tan[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x]^7)/(7*d)","A",4,3,14,0.2143,1,"{3661, 1154, 203}"
384,1,35,0,0.0251603,"\int \left(a+b \tan ^4(c+d x)\right) \, dx","Int[a + b*Tan[c + d*x]^4,x]","a x+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}+b x","a x+\frac{b \tan ^3(c+d x)}{3 d}-\frac{b \tan (c+d x)}{d}+b x",1,"a*x + b*x - (b*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d)","A",4,2,12,0.1667,1,"{3473, 8}"
385,1,302,0,0.321428,"\int \frac{1}{a+b \tan ^4(c+d x)} \, dx","Int[(a + b*Tan[c + d*x]^4)^(-1),x]","\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{x}{a+b}","\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)}+\frac{x}{a+b}",1,"x/(a + b) + ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b)*d) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b)*d) - ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d) + ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)*d)","A",13,9,14,0.6429,1,"{3661, 1171, 203, 1168, 1162, 617, 204, 1165, 628}"
386,1,648,0,0.661279,"\int \frac{1}{\left(a+b \tan ^4(c+d x)\right)^2} \, dx","Int[(a + b*Tan[c + d*x]^4)^(-2),x]","\frac{\sqrt[4]{b} \left(\sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} d (a+b)}+\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} d (a+b)^2}-\frac{\sqrt[4]{b} \left(\sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} d (a+b)^2}-\frac{\sqrt[4]{b} \left(\sqrt{a}+3 \sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{16 \sqrt{2} a^{7/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)^2}+\frac{\sqrt[4]{b} \left(\sqrt{a}+3 \sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{16 \sqrt{2} a^{7/4} d (a+b)}+\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)^2}+\frac{b \tan (c+d x) \left(1-\tan ^2(c+d x)\right)}{4 a d (a+b) \left(a+b \tan ^4(c+d x)\right)}+\frac{x}{(a+b)^2}","\frac{\sqrt[4]{b} \left(\sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{8 \sqrt{2} a^{7/4} d (a+b)}+\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)}{2 \sqrt{2} a^{3/4} d (a+b)^2}-\frac{\sqrt[4]{b} \left(\sqrt{a}-3 \sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{8 \sqrt{2} a^{7/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}+1\right)}{2 \sqrt{2} a^{3/4} d (a+b)^2}-\frac{\sqrt[4]{b} \left(\sqrt{a}+3 \sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{16 \sqrt{2} a^{7/4} d (a+b)}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(-\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)^2}+\frac{\sqrt[4]{b} \left(\sqrt{a}+3 \sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{16 \sqrt{2} a^{7/4} d (a+b)}+\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b}\right) \log \left(\sqrt{2} \sqrt[4]{a} \sqrt[4]{b} \tan (c+d x)+\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}{4 \sqrt{2} a^{3/4} d (a+b)^2}+\frac{b \tan (c+d x) \left(1-\tan ^2(c+d x)\right)}{4 a d (a+b) \left(a+b \tan ^4(c+d x)\right)}+\frac{x}{(a+b)^2}",1,"x/(a + b)^2 + ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b)^2*d) + ((Sqrt[a] - 3*Sqrt[b])*b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(a + b)*d) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b)^2*d) - ((Sqrt[a] - 3*Sqrt[b])*b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Tan[c + d*x])/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(a + b)*d) - ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^2*d) - ((Sqrt[a] + 3*Sqrt[b])*b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(16*Sqrt[2]*a^(7/4)*(a + b)*d) + ((Sqrt[a] + Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(4*Sqrt[2]*a^(3/4)*(a + b)^2*d) + ((Sqrt[a] + 3*Sqrt[b])*b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Tan[c + d*x] + Sqrt[b]*Tan[c + d*x]^2])/(16*Sqrt[2]*a^(7/4)*(a + b)*d) + (b*Tan[c + d*x]*(1 - Tan[c + d*x]^2))/(4*a*(a + b)*d*(a + b*Tan[c + d*x]^4))","A",23,10,14,0.7143,1,"{3661, 1239, 203, 1179, 1168, 1162, 617, 204, 1165, 628}"
387,1,650,0,0.5599156,"\int \sqrt{a+b \tan ^4(c+d x)} \, dx","Int[Sqrt[a + b*Tan[c + d*x]^4],x]","\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a+b \tan ^4(c+d x)}}\right)}{2 d}+\frac{\sqrt{b} \tan (c+d x) \sqrt{a+b \tan ^4(c+d x)}}{d \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}-\frac{\sqrt[4]{b} (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}+\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \sqrt{a+b \tan ^4(c+d x)}}-\frac{\sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{d \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left(\sqrt{a}+\sqrt{b}\right) (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}","\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a+b \tan ^4(c+d x)}}\right)}{2 d}+\frac{\sqrt{b} \tan (c+d x) \sqrt{a+b \tan ^4(c+d x)}}{d \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)}-\frac{\sqrt[4]{b} (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}+\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \sqrt{a+b \tan ^4(c+d x)}}-\frac{\sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{d \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left(\sqrt{a}+\sqrt{b}\right) (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}",1,"(Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^4]])/(2*d) + (Sqrt[b]*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]^4])/(d*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)) - (a^(1/4)*b^(1/4)*EllipticE[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(d*Sqrt[a + b*Tan[c + d*x]^4]) + ((Sqrt[a] - Sqrt[b])*b^(1/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*d*Sqrt[a + b*Tan[c + d*x]^4]) - (b^(1/4)*(a + b)*EllipticF[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[b])*d*Sqrt[a + b*Tan[c + d*x]^4]) + ((Sqrt[a] + Sqrt[b])*(a + b)*EllipticPi[-(Sqrt[a] - Sqrt[b])^2/(4*Sqrt[a]*Sqrt[b]), 2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[b])*b^(1/4)*d*Sqrt[a + b*Tan[c + d*x]^4])","A",8,7,16,0.4375,1,"{3661, 1209, 1198, 220, 1196, 1217, 1707}"
388,1,348,0,0.2242846,"\int \frac{1}{\sqrt{a+b \tan ^4(c+d x)}} \, dx","Int[1/Sqrt[a + b*Tan[c + d*x]^4],x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a+b \tan ^4(c+d x)}}\right)}{2 d \sqrt{a+b}}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left(\sqrt{a}+\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a+b \tan ^4(c+d x)}}\right)}{2 d \sqrt{a+b}}-\frac{\sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}+\frac{\left(\sqrt{a}+\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right) \sqrt{\frac{a+b \tan ^4(c+d x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(c+d x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (c+d x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} d \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(c+d x)}}",1,"ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^4]]/(2*Sqrt[a + b]*d) - (b^(1/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[b])*d*Sqrt[a + b*Tan[c + d*x]^4]) + ((Sqrt[a] + Sqrt[b])*EllipticPi[-(Sqrt[a] - Sqrt[b])^2/(4*Sqrt[a]*Sqrt[b]), 2*ArcTan[(b^(1/4)*Tan[c + d*x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)*Sqrt[(a + b*Tan[c + d*x]^4)/(Sqrt[a] + Sqrt[b]*Tan[c + d*x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[b])*b^(1/4)*d*Sqrt[a + b*Tan[c + d*x]^4])","A",4,4,16,0.2500,1,"{3661, 1217, 220, 1707}"
389,1,103,0,0.206797,"\int \tan ^3(x) \sqrt{a+b \tan ^4(x)} \, dx","Int[Tan[x]^3*Sqrt[a + b*Tan[x]^4],x]","-\frac{1}{4} \left(2-\tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}+\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{4 \sqrt{b}}+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)","-\frac{1}{4} \left(2-\tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}+\frac{(a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{4 \sqrt{b}}+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)",1,"((a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/(4*Sqrt[b]) + (Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/2 - ((2 - Tan[x]^2)*Sqrt[a + b*Tan[x]^4])/4","A",8,7,17,0.4118,1,"{3670, 1252, 815, 844, 217, 206, 725}"
390,1,90,0,0.1179511,"\int \tan (x) \sqrt{a+b \tan ^4(x)} \, dx","Int[Tan[x]*Sqrt[a + b*Tan[x]^4],x]","\frac{1}{2} \sqrt{a+b \tan ^4(x)}-\frac{1}{2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)-\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)","\frac{1}{2} \sqrt{a+b \tan ^4(x)}-\frac{1}{2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)-\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)",1,"-(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/2 - (Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/2 + Sqrt[a + b*Tan[x]^4]/2","A",8,7,15,0.4667,1,"{3670, 1248, 735, 844, 217, 206, 725}"
391,1,102,0,0.1708965,"\int \cot (x) \sqrt{a+b \tan ^4(x)} \, dx","Int[Cot[x]*Sqrt[a + b*Tan[x]^4],x]","\frac{1}{2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)","\frac{1}{2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/2 + (Sqrt[a + b]*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/2 - (Sqrt[a]*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/2","A",11,10,15,0.6667,1,"{3670, 1252, 896, 266, 63, 208, 844, 217, 206, 725}"
392,1,643,0,0.4957301,"\int \tan ^2(x) \sqrt{a+b \tan ^4(x)} \, dx","Int[Tan[x]^2*Sqrt[a + b*Tan[x]^4],x]","\frac{a^{3/4} \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{b} \sqrt{a+b \tan ^4(x)}}-\frac{1}{2} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a+b \tan ^4(x)}}\right)+\frac{1}{3} \tan (x) \sqrt{a+b \tan ^4(x)}-\frac{\sqrt{b} \tan (x) \sqrt{a+b \tan ^4(x)}}{\sqrt{a}+\sqrt{b} \tan ^2(x)}+\frac{\sqrt[4]{b} (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}-\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{a+b \tan ^4(x)}}+\frac{\sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{\sqrt{a+b \tan ^4(x)}}-\frac{\left(\sqrt{a}+\sqrt{b}\right) (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}","\frac{a^{3/4} \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{3 \sqrt[4]{b} \sqrt{a+b \tan ^4(x)}}-\frac{1}{2} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a+b \tan ^4(x)}}\right)+\frac{1}{3} \tan (x) \sqrt{a+b \tan ^4(x)}-\frac{\sqrt{b} \tan (x) \sqrt{a+b \tan ^4(x)}}{\sqrt{a}+\sqrt{b} \tan ^2(x)}+\frac{\sqrt[4]{b} (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}-\frac{\sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{a} \sqrt{a+b \tan ^4(x)}}+\frac{\sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} E\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{\sqrt{a+b \tan ^4(x)}}-\frac{\left(\sqrt{a}+\sqrt{b}\right) (a+b) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}",1,"-(Sqrt[a + b]*ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a + b*Tan[x]^4]])/2 + (Tan[x]*Sqrt[a + b*Tan[x]^4])/3 - (Sqrt[b]*Tan[x]*Sqrt[a + b*Tan[x]^4])/(Sqrt[a] + Sqrt[b]*Tan[x]^2) + (a^(1/4)*b^(1/4)*EllipticE[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/Sqrt[a + b*Tan[x]^4] + (a^(3/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(3*b^(1/4)*Sqrt[a + b*Tan[x]^4]) - ((Sqrt[a] - Sqrt[b])*b^(1/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(2*a^(1/4)*Sqrt[a + b*Tan[x]^4]) + (b^(1/4)*(a + b)*EllipticF[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(2*a^(1/4)*(Sqrt[a] - Sqrt[b])*Sqrt[a + b*Tan[x]^4]) - ((Sqrt[a] + Sqrt[b])*(a + b)*EllipticPi[-(Sqrt[a] - Sqrt[b])^2/(4*Sqrt[a]*Sqrt[b]), 2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[b])*b^(1/4)*Sqrt[a + b*Tan[x]^4])","A",12,9,17,0.5294,1,"{3670, 1336, 195, 220, 1209, 1198, 1196, 1217, 1707}"
393,1,148,0,0.3100197,"\int \tan ^3(x) \left(a+b \tan ^4(x)\right)^{3/2} \, dx","Int[Tan[x]^3*(a + b*Tan[x]^4)^(3/2),x]","\frac{\left(3 a^2+12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{16 \sqrt{b}}-\frac{1}{24} \left(4-3 \tan ^2(x)\right) \left(a+b \tan ^4(x)\right)^{3/2}-\frac{1}{16} \left(8 (a+b)-(3 a+4 b) \tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)","\frac{\left(3 a^2+12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{16 \sqrt{b}}-\frac{1}{24} \left(4-3 \tan ^2(x)\right) \left(a+b \tan ^4(x)\right)^{3/2}-\frac{1}{16} \left(8 (a+b)-(3 a+4 b) \tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)",1,"((3*a^2 + 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/(16*Sqrt[b]) + ((a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/2 - ((8*(a + b) - (3*a + 4*b)*Tan[x]^2)*Sqrt[a + b*Tan[x]^4])/16 - ((4 - 3*Tan[x]^2)*(a + b*Tan[x]^4)^(3/2))/24","A",9,7,17,0.4118,1,"{3670, 1252, 815, 844, 217, 206, 725}"
394,1,126,0,0.2083313,"\int \tan (x) \left(a+b \tan ^4(x)\right)^{3/2} \, dx","Int[Tan[x]*(a + b*Tan[x]^4)^(3/2),x]","\frac{1}{6} \left(a+b \tan ^4(x)\right)^{3/2}+\frac{1}{4} \left(2 (a+b)-b \tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}-\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)","\frac{1}{6} \left(a+b \tan ^4(x)\right)^{3/2}+\frac{1}{4} \left(2 (a+b)-b \tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}-\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)-\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)",1,"-(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/4 - ((a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/2 + ((2*(a + b) - b*Tan[x]^2)*Sqrt[a + b*Tan[x]^4])/4 + (a + b*Tan[x]^4)^(3/2)/6","A",9,8,15,0.5333,1,"{3670, 1248, 735, 815, 844, 217, 206, 725}"
395,1,155,0,0.2668591,"\int \cot (x) \left(a+b \tan ^4(x)\right)^{3/2} \, dx","Int[Cot[x]*(a + b*Tan[x]^4)^(3/2),x]","-\frac{1}{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)+\frac{1}{2} a \sqrt{a+b \tan ^4(x)}-\frac{1}{4} \left(2 (a+b)-b \tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}+\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)","-\frac{1}{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)+\frac{1}{2} a \sqrt{a+b \tan ^4(x)}-\frac{1}{4} \left(2 (a+b)-b \tan ^2(x)\right) \sqrt{a+b \tan ^4(x)}+\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)",1,"(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]])/4 + ((a + b)^(3/2)*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/2 - (a^(3/2)*ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/2 + (a*Sqrt[a + b*Tan[x]^4])/2 - ((2*(a + b) - b*Tan[x]^2)*Sqrt[a + b*Tan[x]^4])/4","A",13,12,15,0.8000,1,"{3670, 1252, 896, 266, 50, 63, 208, 815, 844, 217, 206, 725}"
396,1,74,0,0.1288649,"\int \frac{\tan ^3(x)}{\sqrt{a+b \tan ^4(x)}} \, dx","Int[Tan[x]^3/Sqrt[a + b*Tan[x]^4],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \tan ^2(x)}{\sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{b}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}",1,"ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]]/(2*Sqrt[b]) + ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*Sqrt[a + b])","A",7,6,17,0.3529,1,"{3670, 1252, 844, 217, 206, 725}"
397,1,41,0,0.0679814,"\int \frac{\tan (x)}{\sqrt{a+b \tan ^4(x)}} \, dx","Int[Tan[x]/Sqrt[a + b*Tan[x]^4],x]","-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}","-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}",1,"-ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*Sqrt[a + b])","A",4,4,15,0.2667,1,"{3670, 1248, 725, 206}"
398,1,70,0,0.1580191,"\int \frac{\cot (x)}{\sqrt{a+b \tan ^4(x)}} \, dx","Int[Cot[x]/Sqrt[a + b*Tan[x]^4],x]","\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 \sqrt{a}}","\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 \sqrt{a}}",1,"ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*Sqrt[a + b]) - ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]/(2*Sqrt[a])","A",9,8,15,0.5333,1,"{3670, 1252, 961, 725, 206, 266, 63, 208}"
399,1,291,0,0.2402816,"\int \frac{\tan ^2(x)}{\sqrt{a+b \tan ^4(x)}} \, dx","Int[Tan[x]^2/Sqrt[a + b*Tan[x]^4],x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}-\frac{\left(\sqrt{a}+\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \tan (x)}{\sqrt{a+b \tan ^4(x)}}\right)}{2 \sqrt{a+b}}+\frac{\sqrt[4]{a} \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{2 \sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}-\frac{\left(\sqrt{a}+\sqrt{b}\right) \left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right) \sqrt{\frac{a+b \tan ^4(x)}{\left(\sqrt{a}+\sqrt{b} \tan ^2(x)\right)^2}} \Pi \left(-\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{4 \sqrt{a} \sqrt{b}};2 \tan ^{-1}\left(\frac{\sqrt[4]{b} \tan (x)}{\sqrt[4]{a}}\right)|\frac{1}{2}\right)}{4 \sqrt[4]{a} \sqrt[4]{b} \left(\sqrt{a}-\sqrt{b}\right) \sqrt{a+b \tan ^4(x)}}",1,"-ArcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a + b*Tan[x]^4]]/(2*Sqrt[a + b]) + (a^(1/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(2*(Sqrt[a] - Sqrt[b])*b^(1/4)*Sqrt[a + b*Tan[x]^4]) - ((Sqrt[a] + Sqrt[b])*EllipticPi[-(Sqrt[a] - Sqrt[b])^2/(4*Sqrt[a]*Sqrt[b]), 2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(4*a^(1/4)*(Sqrt[a] - Sqrt[b])*b^(1/4)*Sqrt[a + b*Tan[x]^4])","A",4,4,17,0.2353,1,"{3670, 1320, 220, 1707}"
400,1,71,0,0.162221,"\int \frac{\tan ^3(x)}{\left(a+b \tan ^4(x)\right)^{3/2}} \, dx","Int[Tan[x]^3/(a + b*Tan[x]^4)^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{3/2}}-\frac{1-\tan ^2(x)}{2 (a+b) \sqrt{a+b \tan ^4(x)}}","\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{3/2}}-\frac{1-\tan ^2(x)}{2 (a+b) \sqrt{a+b \tan ^4(x)}}",1,"ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(3/2)) - (1 - Tan[x]^2)/(2*(a + b)*Sqrt[a + b*Tan[x]^4])","A",6,6,17,0.3529,1,"{3670, 1252, 823, 12, 725, 206}"
401,1,74,0,0.1145713,"\int \frac{\tan (x)}{\left(a+b \tan ^4(x)\right)^{3/2}} \, dx","Int[Tan[x]/(a + b*Tan[x]^4)^(3/2),x]","\frac{a+b \tan ^2(x)}{2 a (a+b) \sqrt{a+b \tan ^4(x)}}-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{3/2}}","\frac{a+b \tan ^2(x)}{2 a (a+b) \sqrt{a+b \tan ^4(x)}}-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{3/2}}",1,"-ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(3/2)) + (a + b*Tan[x]^2)/(2*a*(a + b)*Sqrt[a + b*Tan[x]^4])","A",6,6,15,0.4000,1,"{3670, 1248, 741, 12, 725, 206}"
402,1,121,0,0.2141657,"\int \frac{\cot (x)}{\left(a+b \tan ^4(x)\right)^{3/2}} \, dx","Int[Cot[x]/(a + b*Tan[x]^4)^(3/2),x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 a^{3/2}}-\frac{a+b \tan ^2(x)}{2 a (a+b) \sqrt{a+b \tan ^4(x)}}+\frac{1}{2 a \sqrt{a+b \tan ^4(x)}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{3/2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 a^{3/2}}-\frac{a+b \tan ^2(x)}{2 a (a+b) \sqrt{a+b \tan ^4(x)}}+\frac{1}{2 a \sqrt{a+b \tan ^4(x)}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{3/2}}",1,"ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(3/2)) - ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]/(2*a^(3/2)) + 1/(2*a*Sqrt[a + b*Tan[x]^4]) - (a + b*Tan[x]^2)/(2*a*(a + b)*Sqrt[a + b*Tan[x]^4])","A",12,11,15,0.7333,1,"{3670, 1252, 961, 741, 12, 725, 206, 266, 51, 63, 208}"
403,1,109,0,0.2277565,"\int \frac{\tan ^3(x)}{\left(a+b \tan ^4(x)\right)^{5/2}} \, dx","Int[Tan[x]^3/(a + b*Tan[x]^4)^(5/2),x]","-\frac{(b-2 a) \tan ^2(x)+3 a}{6 a (a+b)^2 \sqrt{a+b \tan ^4(x)}}-\frac{1-\tan ^2(x)}{6 (a+b) \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{5/2}}","-\frac{(b-2 a) \tan ^2(x)+3 a}{6 a (a+b)^2 \sqrt{a+b \tan ^4(x)}}-\frac{1-\tan ^2(x)}{6 (a+b) \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{5/2}}",1,"ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(5/2)) - (1 - Tan[x]^2)/(6*(a + b)*(a + b*Tan[x]^4)^(3/2)) - (3*a + (-2*a + b)*Tan[x]^2)/(6*a*(a + b)^2*Sqrt[a + b*Tan[x]^4])","A",7,6,17,0.3529,1,"{3670, 1252, 823, 12, 725, 206}"
404,1,117,0,0.1868702,"\int \frac{\tan (x)}{\left(a+b \tan ^4(x)\right)^{5/2}} \, dx","Int[Tan[x]/(a + b*Tan[x]^4)^(5/2),x]","\frac{3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \tan ^4(x)}}+\frac{a+b \tan ^2(x)}{6 a (a+b) \left(a+b \tan ^4(x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{5/2}}","\frac{3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \tan ^4(x)}}+\frac{a+b \tan ^2(x)}{6 a (a+b) \left(a+b \tan ^4(x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{5/2}}",1,"-ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(5/2)) + (a + b*Tan[x]^2)/(6*a*(a + b)*(a + b*Tan[x]^4)^(3/2)) + (3*a^2 + b*(5*a + 2*b)*Tan[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Tan[x]^4])","A",7,7,15,0.4667,1,"{3670, 1248, 741, 823, 12, 725, 206}"
405,1,183,0,0.3023106,"\int \frac{\cot (x)}{\left(a+b \tan ^4(x)\right)^{5/2}} \, dx","Int[Cot[x]/(a + b*Tan[x]^4)^(5/2),x]","-\frac{3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \tan ^4(x)}}+\frac{1}{2 a^2 \sqrt{a+b \tan ^4(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{a+b \tan ^2(x)}{6 a (a+b) \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{1}{6 a \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{5/2}}","-\frac{3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \tan ^4(x)}}+\frac{1}{2 a^2 \sqrt{a+b \tan ^4(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \tan ^4(x)}}{\sqrt{a}}\right)}{2 a^{5/2}}-\frac{a+b \tan ^2(x)}{6 a (a+b) \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{1}{6 a \left(a+b \tan ^4(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{a-b \tan ^2(x)}{\sqrt{a+b} \sqrt{a+b \tan ^4(x)}}\right)}{2 (a+b)^{5/2}}",1,"ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(2*(a + b)^(5/2)) - ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]/(2*a^(5/2)) + 1/(6*a*(a + b*Tan[x]^4)^(3/2)) - (a + b*Tan[x]^2)/(6*a*(a + b)*(a + b*Tan[x]^4)^(3/2)) + 1/(2*a^2*Sqrt[a + b*Tan[x]^4]) - (3*a^2 + b*(5*a + 2*b)*Tan[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Tan[x]^4])","A",14,12,15,0.8000,1,"{3670, 1252, 961, 741, 823, 12, 725, 206, 266, 51, 63, 208}"
406,1,212,0,0.7127587,"\int (d \tan (e+f x))^m \left(a+b \sqrt{c \tan (e+f x)}\right)^2 \, dx","Int[(d*Tan[e + f*x])^m*(a + b*Sqrt[c*Tan[e + f*x]])^2,x]","\frac{\left(a^2-b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1)}+\frac{\left(a^2+b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1)}+\frac{4 a b (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{c f (2 m+3)}","\frac{\left(a^2-b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1)}+\frac{\left(a^2+b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1)}+\frac{4 a b (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{c f (2 m+3)}",1,"((a^2 - b^2*Sqrt[-c^2])*Hypergeometric2F1[1, 1 + m, 2 + m, -((c*Tan[e + f*x])/Sqrt[-c^2])]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*f*(1 + m)) + ((a^2 + b^2*Sqrt[-c^2])*Hypergeometric2F1[1, 1 + m, 2 + m, (c*Tan[e + f*x])/Sqrt[-c^2]]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*f*(1 + m)) + (4*a*b*Hypergeometric2F1[1, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[e + f*x]^2]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*f*(3 + 2*m))","A",9,5,29,0.1724,1,"{3670, 15, 1831, 364, 1286}"
407,1,121,0,0.3309689,"\int (d \tan (e+f x))^m \left(a+b \sqrt{c \tan (e+f x)}\right) \, dx","Int[(d*Tan[e + f*x])^m*(a + b*Sqrt[c*Tan[e + f*x]]),x]","\frac{a \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(e+f x)\right)}{f (m+1)}+\frac{2 b (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{c f (2 m+3)}","\frac{a \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};-\tan ^2(e+f x)\right)}{f (m+1)}+\frac{2 b (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (2 m+7);-\tan ^2(e+f x)\right)}{c f (2 m+3)}",1,"(a*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(f*(1 + m)) + (2*b*Hypergeometric2F1[1, (3 + 2*m)/4, (7 + 2*m)/4, -Tan[e + f*x]^2]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*f*(3 + 2*m))","A",7,4,27,0.1481,1,"{3670, 15, 1831, 364}"
408,1,460,0,1.282522,"\int \frac{(d \tan (e+f x))^m}{a+b \sqrt{c \tan (e+f x)}} \, dx","Int[(d*Tan[e + f*x])^m/(a + b*Sqrt[c*Tan[e + f*x]]),x]","-\frac{b \left(a^2-b^2 \sqrt{-c^2}\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)}-\frac{b \left(a^2+b^2 \sqrt{-c^2}\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)}+\frac{a \left(a^2-b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)}+\frac{a \left(a^2+b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)}+\frac{b^4 c^2 \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{a f (m+1) \left(a^4+b^4 c^2\right)}","-\frac{b \left(a^2-b^2 \sqrt{-c^2}\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)}-\frac{b \left(a^2+b^2 \sqrt{-c^2}\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)}+\frac{a \left(a^2-b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)}+\frac{a \left(a^2+b^2 \sqrt{-c^2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)}+\frac{b^4 c^2 \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{a f (m+1) \left(a^4+b^4 c^2\right)}",1,"(a*(a^2 - b^2*Sqrt[-c^2])*Hypergeometric2F1[1, 1 + m, 2 + m, -((c*Tan[e + f*x])/Sqrt[-c^2])]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*(a^4 + b^4*c^2)*f*(1 + m)) + (a*(a^2 + b^2*Sqrt[-c^2])*Hypergeometric2F1[1, 1 + m, 2 + m, (c*Tan[e + f*x])/Sqrt[-c^2]]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*(a^4 + b^4*c^2)*f*(1 + m)) + (b^4*c^2*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, -((b*Sqrt[c*Tan[e + f*x]])/a)]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(a*(a^4 + b^4*c^2)*f*(1 + m)) - (b*(a^2 - b^2*Sqrt[-c^2])*Hypergeometric2F1[1, (3 + 2*m)/2, (5 + 2*m)/2, -((c*Tan[e + f*x])/Sqrt[-c^2])]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*(a^4 + b^4*c^2)*f*(3 + 2*m)) - (b*(a^2 + b^2*Sqrt[-c^2])*Hypergeometric2F1[1, (3 + 2*m)/2, (5 + 2*m)/2, (c*Tan[e + f*x])/Sqrt[-c^2]]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*(a^4 + b^4*c^2)*f*(3 + 2*m))","A",14,7,29,0.2414,1,"{3670, 15, 6725, 64, 1831, 1286, 364}"
409,1,617,0,1.5794592,"\int \frac{(d \tan (e+f x))^m}{\left(a+b \sqrt{c \tan (e+f x)}\right)^2} \, dx","Int[(d*Tan[e + f*x])^m/(a + b*Sqrt[c*Tan[e + f*x]])^2,x]","-\frac{2 a b \left(-2 a^2 b^2 \sqrt{-c^2}+a^4-b^4 c^2\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)^2}-\frac{2 a b \left(2 a^2 b^2 \sqrt{-c^2}+a^4-b^4 c^2\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)^2}+\frac{\left(-3 a^4 b^2 \sqrt{-c^2}-3 a^2 b^4 c^2+a^6-b^6 \left(-c^2\right)^{3/2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)^2}+\frac{\left(3 a^4 b^2 \sqrt{-c^2}-3 a^2 b^4 c^2+a^6+b^6 \left(-c^2\right)^{3/2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)^2}+\frac{4 a^2 b^4 c^2 \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{f (m+1) \left(a^4+b^4 c^2\right)^2}+\frac{b^4 c^2 \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(2,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{a^2 f (m+1) \left(a^4+b^4 c^2\right)}","-\frac{2 a b \left(-2 a^2 b^2 \sqrt{-c^2}+a^4-b^4 c^2\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)^2}-\frac{2 a b \left(2 a^2 b^2 \sqrt{-c^2}+a^4-b^4 c^2\right) (c \tan (e+f x))^{3/2} (d \tan (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (2 m+3);\frac{1}{2} (2 m+5);\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{c f (2 m+3) \left(a^4+b^4 c^2\right)^2}+\frac{\left(-3 a^4 b^2 \sqrt{-c^2}-3 a^2 b^4 c^2+a^6-b^6 \left(-c^2\right)^{3/2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;-\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)^2}+\frac{\left(3 a^4 b^2 \sqrt{-c^2}-3 a^2 b^4 c^2+a^6+b^6 \left(-c^2\right)^{3/2}\right) \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,m+1;m+2;\frac{c \tan (e+f x)}{\sqrt{-c^2}}\right)}{2 f (m+1) \left(a^4+b^4 c^2\right)^2}+\frac{4 a^2 b^4 c^2 \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(1,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{f (m+1) \left(a^4+b^4 c^2\right)^2}+\frac{b^4 c^2 \tan (e+f x) (d \tan (e+f x))^m \, _2F_1\left(2,2 (m+1);2 m+3;-\frac{b \sqrt{c \tan (e+f x)}}{a}\right)}{a^2 f (m+1) \left(a^4+b^4 c^2\right)}",1,"((a^6 - 3*a^2*b^4*c^2 - 3*a^4*b^2*Sqrt[-c^2] - b^6*(-c^2)^(3/2))*Hypergeometric2F1[1, 1 + m, 2 + m, -((c*Tan[e + f*x])/Sqrt[-c^2])]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*(a^4 + b^4*c^2)^2*f*(1 + m)) + ((a^6 - 3*a^2*b^4*c^2 + 3*a^4*b^2*Sqrt[-c^2] + b^6*(-c^2)^(3/2))*Hypergeometric2F1[1, 1 + m, 2 + m, (c*Tan[e + f*x])/Sqrt[-c^2]]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(2*(a^4 + b^4*c^2)^2*f*(1 + m)) + (4*a^2*b^4*c^2*Hypergeometric2F1[1, 2*(1 + m), 3 + 2*m, -((b*Sqrt[c*Tan[e + f*x]])/a)]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/((a^4 + b^4*c^2)^2*f*(1 + m)) + (b^4*c^2*Hypergeometric2F1[2, 2*(1 + m), 3 + 2*m, -((b*Sqrt[c*Tan[e + f*x]])/a)]*Tan[e + f*x]*(d*Tan[e + f*x])^m)/(a^2*(a^4 + b^4*c^2)*f*(1 + m)) - (2*a*b*(a^4 - b^4*c^2 - 2*a^2*b^2*Sqrt[-c^2])*Hypergeometric2F1[1, (3 + 2*m)/2, (5 + 2*m)/2, -((c*Tan[e + f*x])/Sqrt[-c^2])]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*(a^4 + b^4*c^2)^2*f*(3 + 2*m)) - (2*a*b*(a^4 - b^4*c^2 + 2*a^2*b^2*Sqrt[-c^2])*Hypergeometric2F1[1, (3 + 2*m)/2, (5 + 2*m)/2, (c*Tan[e + f*x])/Sqrt[-c^2]]*(c*Tan[e + f*x])^(3/2)*(d*Tan[e + f*x])^m)/(c*(a^4 + b^4*c^2)^2*f*(3 + 2*m))","A",15,7,29,0.2414,1,"{3670, 15, 6725, 64, 1831, 1286, 364}"
410,1,74,0,0.0993308,"\int (d \tan (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Tan[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) (d \tan (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(1,\frac{1}{2} (m+n p+1);\frac{1}{2} (m+n p+3);-\tan ^2(e+f x)\right)}{f (m+n p+1)}","\frac{\tan (e+f x) (d \tan (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(1,\frac{1}{2} (m+n p+1);\frac{1}{2} (m+n p+3);-\tan ^2(e+f x)\right)}{f (m+n p+1)}",1,"(Hypergeometric2F1[1, (1 + m + n*p)/2, (3 + m + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(d*Tan[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + m + n*p))","A",4,4,25,0.1600,1,"{3578, 20, 3476, 364}"
411,1,63,0,0.093727,"\int \tan ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Tan[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^3(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+3);\frac{1}{2} (n p+5);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}","\frac{\tan ^3(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+3);\frac{1}{2} (n p+5);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}",1,"(Hypergeometric2F1[1, (3 + n*p)/2, (5 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 + n*p))","A",4,4,23,0.1739,1,"{3659, 16, 3476, 364}"
412,1,61,0,0.0493278,"\int \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",3,3,14,0.2143,1,"{3659, 3476, 364}"
413,1,63,0,0.1097966,"\int \cot ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Cot[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","-\frac{\cot (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-1);\frac{1}{2} (n p+1);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}","-\frac{\cot (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-1);\frac{1}{2} (n p+1);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (1-n p)}",1,"-((Cot[e + f*x]*Hypergeometric2F1[1, (-1 + n*p)/2, (1 + n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - n*p)))","A",4,4,23,0.1739,1,"{3659, 16, 3476, 364}"
414,1,65,0,0.1136953,"\int \cot ^4(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Cot[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p,x]","-\frac{\cot ^3(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-3);\frac{1}{2} (n p-1);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (3-n p)}","-\frac{\cot ^3(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-3);\frac{1}{2} (n p-1);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (3-n p)}",1,"-((Cot[e + f*x]^3*Hypergeometric2F1[1, (-3 + n*p)/2, (-1 + n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 - n*p)))","A",4,4,23,0.1739,1,"{3659, 16, 3476, 364}"
415,1,65,0,0.1117817,"\int \cot ^6(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Cot[e + f*x]^6*(b*(c*Tan[e + f*x])^n)^p,x]","-\frac{\cot ^5(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-5);\frac{1}{2} (n p-3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (5-n p)}","-\frac{\cot ^5(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-5);\frac{1}{2} (n p-3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (5-n p)}",1,"-((Cot[e + f*x]^5*Hypergeometric2F1[1, (-5 + n*p)/2, (-3 + n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(5 - n*p)))","A",4,4,23,0.1739,1,"{3659, 16, 3476, 364}"
416,1,63,0,0.0921519,"\int \tan ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^4(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+4);\frac{1}{2} (n p+6);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+4)}","\frac{\tan ^4(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+4);\frac{1}{2} (n p+6);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+4)}",1,"(Hypergeometric2F1[1, (4 + n*p)/2, (6 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p)/(f*(4 + n*p))","A",4,4,23,0.1739,1,"{3659, 16, 3476, 364}"
417,1,63,0,0.068729,"\int \tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+2)}","\frac{\tan ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+2);\frac{1}{2} (n p+4);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+2)}",1,"(Hypergeometric2F1[1, (2 + n*p)/2, (4 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 + n*p))","A",4,4,21,0.1905,1,"{3659, 16, 3476, 364}"
418,1,50,0,0.0800734,"\int \cot (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Cot[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\, _2F_1\left(1,\frac{n p}{2};\frac{n p}{2}+1;-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p}","\frac{\, _2F_1\left(1,\frac{n p}{2};\frac{n p}{2}+1;-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f n p}",1,"(Hypergeometric2F1[1, (n*p)/2, 1 + (n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*n*p)","A",4,4,21,0.1905,1,"{3659, 16, 3476, 364}"
419,1,62,0,0.1082089,"\int \cot ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Cot[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","-\frac{\cot ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-2);\frac{n p}{2};-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (2-n p)}","-\frac{\cot ^2(e+f x) \, _2F_1\left(1,\frac{1}{2} (n p-2);\frac{n p}{2};-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (2-n p)}",1,"-((Cot[e + f*x]^2*Hypergeometric2F1[1, (-2 + n*p)/2, (n*p)/2, -Tan[e + f*x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(2 - n*p)))","A",4,4,23,0.1739,1,"{3659, 16, 3476, 364}"
420,0,0,0,0.0587434,"\int (d \tan (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Tan[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \tan (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left((d \tan (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Defer[Int][(d*Tan[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
421,1,78,0,0.1220438,"\int (d \cot (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Cot[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \cot (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (-m+2 p+1);\frac{1}{2} (-m+2 p+3);-\tan ^2(e+f x)\right)}{f (-m+2 p+1)}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \cot (e+f x))^m \, _2F_1\left(1,\frac{1}{2} (-m+2 p+1);\frac{1}{2} (-m+2 p+3);-\tan ^2(e+f x)\right)}{f (-m+2 p+1)}",1,"((d*Cot[e + f*x])^m*Hypergeometric2F1[1, (1 - m + 2*p)/2, (3 - m + 2*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 - m + 2*p))","A",4,4,23,0.1739,1,"{3658, 2604, 3476, 364}"
422,1,107,0,0.198803,"\int (d \cot (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Cot[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) (d \cot (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1-m}{2};1,-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (1-m)}","\frac{\tan (e+f x) (d \cot (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1-m}{2};1,-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (1-m)}",1,"(AppellF1[(1 - m)/2, 1, -p, (3 - m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Cot[e + f*x])^m*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 - m)*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",4,4,25,0.1600,1,"{3674, 3670, 511, 510}"
423,1,80,0,0.1426175,"\int (d \cot (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Cot[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) (d \cot (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(1,\frac{1}{2} (-m+n p+1);\frac{1}{2} (-m+n p+3);-\tan ^2(e+f x)\right)}{f (-m+n p+1)}","\frac{\tan (e+f x) (d \cot (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(1,\frac{1}{2} (-m+n p+1);\frac{1}{2} (-m+n p+3);-\tan ^2(e+f x)\right)}{f (-m+n p+1)}",1,"((d*Cot[e + f*x])^m*Hypergeometric2F1[1, (1 - m + n*p)/2, (3 - m + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - m + n*p))","A",4,4,25,0.1600,1,"{3659, 2604, 3476, 364}"
424,0,0,0,0.138833,"\int (d \cot (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Cot[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \cot (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\left(\frac{\tan (e+f x)}{d}\right)^m (d \cot (e+f x))^m \text{Int}\left(\left(\frac{\tan (e+f x)}{d}\right)^{-m} \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"(d*Cot[e + f*x])^m*(Tan[e + f*x]/d)^m*Defer[Int][(a + b*(c*Tan[e + f*x])^n)^p/(Tan[e + f*x]/d)^m, x]","A",0,0,0,0,-1,"{}"
425,1,70,0,0.0513948,"\int \sec ^3(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^3*(a + b*Tan[c + d*x]^2),x]","\frac{(4 a-b) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a-b) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}","\frac{(4 a-b) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a-b) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"((4*a - b)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a - b)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",4,4,21,0.1905,1,"{3676, 385, 199, 206}"
426,1,42,0,0.0342149,"\int \sec (c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + b*Tan[c + d*x]^2),x]","\frac{(2 a-b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(2 a-b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}",1,"((2*a - b)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",3,3,19,0.1579,1,"{3676, 385, 206}"
427,1,28,0,0.0327812,"\int \cos (c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Tan[c + d*x]^2),x]","\frac{(a-b) \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","\frac{(a-b) \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b*ArcTanh[Sin[c + d*x]])/d + ((a - b)*Sin[c + d*x])/d","A",3,3,19,0.1579,1,"{3676, 388, 206}"
428,1,32,0,0.0345442,"\int \cos ^3(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + b*Tan[c + d*x]^2),x]","\frac{a \sin (c+d x)}{d}-\frac{(a-b) \sin ^3(c+d x)}{3 d}","\frac{a \sin (c+d x)}{d}-\frac{(a-b) \sin ^3(c+d x)}{3 d}",1,"(a*Sin[c + d*x])/d - ((a - b)*Sin[c + d*x]^3)/(3*d)","A",2,1,21,0.04762,1,"{3676}"
429,1,54,0,0.0496329,"\int \cos ^5(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + b*Tan[c + d*x]^2),x]","\frac{(a-b) \sin ^5(c+d x)}{5 d}-\frac{(2 a-b) \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}","\frac{(a-b) \sin ^5(c+d x)}{5 d}-\frac{(2 a-b) \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}",1,"(a*Sin[c + d*x])/d - ((2*a - b)*Sin[c + d*x]^3)/(3*d) + ((a - b)*Sin[c + d*x]^5)/(5*d)","A",3,2,21,0.09524,1,"{3676, 373}"
430,1,76,0,0.0586209,"\int \cos ^7(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^7*(a + b*Tan[c + d*x]^2),x]","-\frac{(a-b) \sin ^7(c+d x)}{7 d}+\frac{(3 a-2 b) \sin ^5(c+d x)}{5 d}-\frac{(3 a-b) \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}","-\frac{(a-b) \sin ^7(c+d x)}{7 d}+\frac{(3 a-2 b) \sin ^5(c+d x)}{5 d}-\frac{(3 a-b) \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}",1,"(a*Sin[c + d*x])/d - ((3*a - b)*Sin[c + d*x]^3)/(3*d) + ((3*a - 2*b)*Sin[c + d*x]^5)/(5*d) - ((a - b)*Sin[c + d*x]^7)/(7*d)","A",3,2,21,0.09524,1,"{3676, 373}"
431,1,68,0,0.0505921,"\int \sec ^6(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^6*(a + b*Tan[c + d*x]^2),x]","\frac{(a+2 b) \tan ^5(c+d x)}{5 d}+\frac{(2 a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^7(c+d x)}{7 d}","\frac{(a+2 b) \tan ^5(c+d x)}{5 d}+\frac{(2 a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^7(c+d x)}{7 d}",1,"(a*Tan[c + d*x])/d + ((2*a + b)*Tan[c + d*x]^3)/(3*d) + ((a + 2*b)*Tan[c + d*x]^5)/(5*d) + (b*Tan[c + d*x]^7)/(7*d)","A",3,2,21,0.09524,1,"{3675, 373}"
432,1,46,0,0.0405622,"\int \sec ^4(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^4*(a + b*Tan[c + d*x]^2),x]","\frac{(a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^5(c+d x)}{5 d}","\frac{(a+b) \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \tan ^5(c+d x)}{5 d}",1,"(a*Tan[c + d*x])/d + ((a + b)*Tan[c + d*x]^3)/(3*d) + (b*Tan[c + d*x]^5)/(5*d)","A",3,2,21,0.09524,1,"{3675, 373}"
433,1,28,0,0.0289415,"\int \sec ^2(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + b*Tan[c + d*x]^2),x]","\frac{a \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d}","\frac{a \tan (c+d x)}{d}+\frac{b \tan ^3(c+d x)}{3 d}",1,"(a*Tan[c + d*x])/d + (b*Tan[c + d*x]^3)/(3*d)","A",2,1,21,0.04762,1,"{3675}"
434,1,33,0,0.0385714,"\int \cos ^2(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Tan[c + d*x]^2),x]","\frac{(a-b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a+b)","\frac{(a-b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a+b)",1,"((a + b)*x)/2 + ((a - b)*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",3,3,21,0.1429,1,"{3675, 385, 203}"
435,1,61,0,0.0464012,"\int \cos ^4(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + b*Tan[c + d*x]^2),x]","\frac{(a-b) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{(3 a+b) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (3 a+b)","\frac{(a-b) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{(3 a+b) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (3 a+b)",1,"((3*a + b)*x)/8 + ((3*a + b)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((a - b)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",4,4,21,0.1905,1,"{3675, 385, 199, 203}"
436,1,87,0,0.0551299,"\int \cos ^6(c+d x) \left(a+b \tan ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^6*(a + b*Tan[c + d*x]^2),x]","\frac{(a-b) \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{(5 a+b) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(5 a+b) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x (5 a+b)","\frac{(a-b) \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{(5 a+b) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(5 a+b) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x (5 a+b)",1,"((5*a + b)*x)/16 + ((5*a + b)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((5*a + b)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((a - b)*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",5,4,21,0.1905,1,"{3675, 385, 199, 203}"
437,1,128,0,0.1645689,"\int \sec ^3(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Sec[c + d*x]^3*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\left(8 a^2-4 a b+b^2\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(8 a^2-4 a b+b^2\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b (8 a-3 b) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{b \tan (c+d x) \sec ^5(c+d x) \left(a-(a-b) \sin ^2(c+d x)\right)}{6 d}","\frac{\left(8 a^2-4 a b+b^2\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(8 a^2-4 a b+b^2\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b (8 a-3 b) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{b \tan (c+d x) \sec ^5(c+d x) \left(a-(a-b) \sin ^2(c+d x)\right)}{6 d}",1,"((8*a^2 - 4*a*b + b^2)*ArcTanh[Sin[c + d*x]])/(16*d) + ((8*a^2 - 4*a*b + b^2)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + ((8*a - 3*b)*b*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (b*Sec[c + d*x]^5*(a - (a - b)*Sin[c + d*x]^2)*Tan[c + d*x])/(6*d)","A",5,5,23,0.2174,1,"{3676, 413, 385, 199, 206}"
438,1,96,0,0.0859652,"\int \sec (c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Sec[c + d*x]*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\left(8 a^2-8 a b+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 b (2 a-b) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x) \left(a-(a-b) \sin ^2(c+d x)\right)}{4 d}","\frac{\left(8 a^2-8 a b+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 b (2 a-b) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x) \left(a-(a-b) \sin ^2(c+d x)\right)}{4 d}",1,"((8*a^2 - 8*a*b + 3*b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (3*(2*a - b)*b*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*Sec[c + d*x]^3*(a - (a - b)*Sin[c + d*x]^2)*Tan[c + d*x])/(4*d)","A",4,4,21,0.1905,1,"{3676, 413, 385, 206}"
439,1,62,0,0.0896454,"\int \cos (c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Cos[c + d*x]*(a + b*Tan[c + d*x]^2)^2,x]","\frac{(a-b)^2 \sin (c+d x)}{d}+\frac{b (4 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(a-b)^2 \sin (c+d x)}{d}+\frac{b (4 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d}",1,"((4*a - 3*b)*b*ArcTanh[Sin[c + d*x]])/(2*d) + ((a - b)^2*Sin[c + d*x])/d + (b^2*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,4,21,0.1905,1,"{3676, 390, 385, 206}"
440,1,56,0,0.064285,"\int \cos ^3(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Cos[c + d*x]^3*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\left(a^2-b^2\right) \sin (c+d x)}{d}-\frac{(a-b)^2 \sin ^3(c+d x)}{3 d}+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{\left(a^2-b^2\right) \sin (c+d x)}{d}-\frac{(a-b)^2 \sin ^3(c+d x)}{3 d}+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(b^2*ArcTanh[Sin[c + d*x]])/d + ((a^2 - b^2)*Sin[c + d*x])/d - ((a - b)^2*Sin[c + d*x]^3)/(3*d)","A",4,3,23,0.1304,1,"{3676, 390, 206}"
441,1,57,0,0.059917,"\int \cos ^5(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Cos[c + d*x]^5*(a + b*Tan[c + d*x]^2)^2,x]","\frac{a^2 \sin (c+d x)}{d}+\frac{(a-b)^2 \sin ^5(c+d x)}{5 d}-\frac{2 a (a-b) \sin ^3(c+d x)}{3 d}","\frac{a^2 \sin (c+d x)}{d}+\frac{(a-b)^2 \sin ^5(c+d x)}{5 d}-\frac{2 a (a-b) \sin ^3(c+d x)}{3 d}",1,"(a^2*Sin[c + d*x])/d - (2*a*(a - b)*Sin[c + d*x]^3)/(3*d) + ((a - b)^2*Sin[c + d*x]^5)/(5*d)","A",3,2,23,0.08696,1,"{3676, 194}"
442,1,86,0,0.0804554,"\int \cos ^7(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Cos[c + d*x]^7*(a + b*Tan[c + d*x]^2)^2,x]","\frac{a^2 \sin (c+d x)}{d}-\frac{(a-b)^2 \sin ^7(c+d x)}{7 d}+\frac{(a-b) (3 a-b) \sin ^5(c+d x)}{5 d}-\frac{a (3 a-2 b) \sin ^3(c+d x)}{3 d}","\frac{a^2 \sin (c+d x)}{d}-\frac{(a-b)^2 \sin ^7(c+d x)}{7 d}+\frac{(a-b) (3 a-b) \sin ^5(c+d x)}{5 d}-\frac{a (3 a-2 b) \sin ^3(c+d x)}{3 d}",1,"(a^2*Sin[c + d*x])/d - (a*(3*a - 2*b)*Sin[c + d*x]^3)/(3*d) + ((a - b)*(3*a - b)*Sin[c + d*x]^5)/(5*d) - ((a - b)^2*Sin[c + d*x]^7)/(7*d)","A",3,2,23,0.08696,1,"{3676, 373}"
443,1,114,0,0.0989362,"\int \cos ^9(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Cos[c + d*x]^9*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\left(6 a^2-6 a b+b^2\right) \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin (c+d x)}{d}+\frac{(a-b)^2 \sin ^9(c+d x)}{9 d}-\frac{2 (a-b) (2 a-b) \sin ^7(c+d x)}{7 d}-\frac{2 a (2 a-b) \sin ^3(c+d x)}{3 d}","\frac{\left(6 a^2-6 a b+b^2\right) \sin ^5(c+d x)}{5 d}+\frac{a^2 \sin (c+d x)}{d}+\frac{(a-b)^2 \sin ^9(c+d x)}{9 d}-\frac{2 (a-b) (2 a-b) \sin ^7(c+d x)}{7 d}-\frac{2 a (2 a-b) \sin ^3(c+d x)}{3 d}",1,"(a^2*Sin[c + d*x])/d - (2*a*(2*a - b)*Sin[c + d*x]^3)/(3*d) + ((6*a^2 - 6*a*b + b^2)*Sin[c + d*x]^5)/(5*d) - (2*(a - b)*(2*a - b)*Sin[c + d*x]^7)/(7*d) + ((a - b)^2*Sin[c + d*x]^9)/(9*d)","A",3,2,23,0.08696,1,"{3676, 373}"
444,1,96,0,0.0844828,"\int \sec ^6(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Sec[c + d*x]^6*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\left(a^2+4 a b+b^2\right) \tan ^5(c+d x)}{5 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 b (a+b) \tan ^7(c+d x)}{7 d}+\frac{2 a (a+b) \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^9(c+d x)}{9 d}","\frac{\left(a^2+4 a b+b^2\right) \tan ^5(c+d x)}{5 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 b (a+b) \tan ^7(c+d x)}{7 d}+\frac{2 a (a+b) \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^9(c+d x)}{9 d}",1,"(a^2*Tan[c + d*x])/d + (2*a*(a + b)*Tan[c + d*x]^3)/(3*d) + ((a^2 + 4*a*b + b^2)*Tan[c + d*x]^5)/(5*d) + (2*b*(a + b)*Tan[c + d*x]^7)/(7*d) + (b^2*Tan[c + d*x]^9)/(9*d)","A",3,2,23,0.08696,1,"{3675, 373}"
445,1,74,0,0.0693627,"\int \sec ^4(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Sec[c + d*x]^4*(a + b*Tan[c + d*x]^2)^2,x]","\frac{a^2 \tan (c+d x)}{d}+\frac{b (2 a+b) \tan ^5(c+d x)}{5 d}+\frac{a (a+2 b) \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}","\frac{a^2 \tan (c+d x)}{d}+\frac{b (2 a+b) \tan ^5(c+d x)}{5 d}+\frac{a (a+2 b) \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}",1,"(a^2*Tan[c + d*x])/d + (a*(a + 2*b)*Tan[c + d*x]^3)/(3*d) + (b*(2*a + b)*Tan[c + d*x]^5)/(5*d) + (b^2*Tan[c + d*x]^7)/(7*d)","A",3,2,23,0.08696,1,"{3675, 373}"
446,1,49,0,0.0535059,"\int \sec ^2(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Sec[c + d*x]^2*(a + b*Tan[c + d*x]^2)^2,x]","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^5(c+d x)}{5 d}","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{b^2 \tan ^5(c+d x)}{5 d}",1,"(a^2*Tan[c + d*x])/d + (2*a*b*Tan[c + d*x]^3)/(3*d) + (b^2*Tan[c + d*x]^5)/(5*d)","A",3,2,23,0.08696,1,"{3675, 194}"
447,1,55,0,0.0772035,"\int \cos ^2(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Cos[c + d*x]^2*(a + b*Tan[c + d*x]^2)^2,x]","\frac{(a-b)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a+3 b) (a-b)+\frac{b^2 \tan (c+d x)}{d}","\frac{(a-b)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a+3 b) (a-b)+\frac{b^2 \tan (c+d x)}{d}",1,"((a - b)*(a + 3*b)*x)/2 + ((a - b)^2*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (b^2*Tan[c + d*x])/d","A",5,4,23,0.1739,1,"{3675, 390, 385, 203}"
448,1,87,0,0.0849135,"\int \cos ^4(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Cos[c + d*x]^4*(a + b*Tan[c + d*x]^2)^2,x]","\frac{3 \left(a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2+2 a b+3 b^2\right)+\frac{(a-b) \sin (c+d x) \cos ^3(c+d x) \left(a+b \tan ^2(c+d x)\right)}{4 d}","\frac{3 \left(a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2+2 a b+3 b^2\right)+\frac{(a-b) \sin (c+d x) \cos ^3(c+d x) \left(a+b \tan ^2(c+d x)\right)}{4 d}",1,"((3*a^2 + 2*a*b + 3*b^2)*x)/8 + (3*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((a - b)*Cos[c + d*x]^3*Sin[c + d*x]*(a + b*Tan[c + d*x]^2))/(4*d)","A",4,4,23,0.1739,1,"{3675, 413, 385, 203}"
449,1,122,0,0.131196,"\int \cos ^6(c+d x) \left(a+b \tan ^2(c+d x)\right)^2 \, dx","Int[Cos[c + d*x]^6*(a + b*Tan[c + d*x]^2)^2,x]","\frac{\left(5 a^2+2 a b+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(5 a^2+2 a b+b^2\right)+\frac{(a-b) (5 a+3 b) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(a-b) \sin (c+d x) \cos ^5(c+d x) \left(a+b \tan ^2(c+d x)\right)}{6 d}","\frac{\left(5 a^2+2 a b+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(5 a^2+2 a b+b^2\right)+\frac{(a-b) (5 a+3 b) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(a-b) \sin (c+d x) \cos ^5(c+d x) \left(a+b \tan ^2(c+d x)\right)}{6 d}",1,"((5*a^2 + 2*a*b + b^2)*x)/16 + ((5*a^2 + 2*a*b + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((a - b)*(5*a + 3*b)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + ((a - b)*Cos[c + d*x]^5*Sin[c + d*x]*(a + b*Tan[c + d*x]^2))/(6*d)","A",5,5,23,0.2174,1,"{3675, 413, 385, 199, 203}"
450,1,90,0,0.1434502,"\int \frac{\sec ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Sec[c + d*x]^5/(a + b*Tan[c + d*x]^2),x]","\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^2 d}-\frac{(2 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d}","\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^2 d}-\frac{(2 a-3 b) \tanh ^{-1}(\sin (c+d x))}{2 b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d}",1,"-((2*a - 3*b)*ArcTanh[Sin[c + d*x]])/(2*b^2*d) + ((a - b)^(3/2)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*b*d)","A",5,5,23,0.2174,1,"{3676, 414, 522, 206, 208}"
451,1,59,0,0.0811648,"\int \frac{\sec ^3(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Sec[c + d*x]^3/(a + b*Tan[c + d*x]^2),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{b d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b d}","\frac{\tanh ^{-1}(\sin (c+d x))}{b d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b d}",1,"ArcTanh[Sin[c + d*x]]/(b*d) - (Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*b*d)","A",4,4,23,0.1739,1,"{3676, 391, 206, 208}"
452,1,40,0,0.0461854,"\int \frac{\sec (c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Sec[c + d*x]/(a + b*Tan[c + d*x]^2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a-b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d \sqrt{a-b}}",1,"ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a - b]*d)","A",2,2,21,0.09524,1,"{3676, 208}"
453,1,60,0,0.0797293,"\int \frac{\cos (c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Cos[c + d*x]/(a + b*Tan[c + d*x]^2),x]","\frac{\sin (c+d x)}{d (a-b)}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{3/2}}","\frac{\sin (c+d x)}{d (a-b)}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{3/2}}",1,"-((b*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(3/2)*d)) + Sin[c + d*x]/((a - b)*d)","A",3,3,21,0.1429,1,"{3676, 388, 208}"
454,1,88,0,0.1224638,"\int \frac{\cos ^3(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Cos[c + d*x]^3/(a + b*Tan[c + d*x]^2),x]","\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{5/2}}-\frac{\sin ^3(c+d x)}{3 d (a-b)}+\frac{(a-2 b) \sin (c+d x)}{d (a-b)^2}","\frac{b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{5/2}}-\frac{\sin ^3(c+d x)}{3 d (a-b)}+\frac{(a-2 b) \sin (c+d x)}{d (a-b)^2}",1,"(b^2*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(5/2)*d) + ((a - 2*b)*Sin[c + d*x])/((a - b)^2*d) - Sin[c + d*x]^3/(3*(a - b)*d)","A",4,3,23,0.1304,1,"{3676, 390, 208}"
455,1,126,0,0.1478018,"\int \frac{\cos ^5(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Cos[c + d*x]^5/(a + b*Tan[c + d*x]^2),x]","\frac{\left(a^2-3 a b+3 b^2\right) \sin (c+d x)}{d (a-b)^3}-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{7/2}}+\frac{\sin ^5(c+d x)}{5 d (a-b)}-\frac{(2 a-3 b) \sin ^3(c+d x)}{3 d (a-b)^2}","\frac{\left(a^2-3 a b+3 b^2\right) \sin (c+d x)}{d (a-b)^3}-\frac{b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^{7/2}}+\frac{\sin ^5(c+d x)}{5 d (a-b)}-\frac{(2 a-3 b) \sin ^3(c+d x)}{3 d (a-b)^2}",1,"-((b^3*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^(7/2)*d)) + ((a^2 - 3*a*b + 3*b^2)*Sin[c + d*x])/((a - b)^3*d) - ((2*a - 3*b)*Sin[c + d*x]^3)/(3*(a - b)^2*d) + Sin[c + d*x]^5/(5*(a - b)*d)","A",4,3,23,0.1304,1,"{3676, 390, 208}"
456,1,108,0,0.1097959,"\int \frac{\sec ^8(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Sec[c + d*x]^8/(a + b*Tan[c + d*x]^2),x]","\frac{\left(a^2-3 a b+3 b^2\right) \tan (c+d x)}{b^3 d}-\frac{(a-3 b) \tan ^3(c+d x)}{3 b^2 d}-\frac{(a-b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2} d}+\frac{\tan ^5(c+d x)}{5 b d}","\frac{\left(a^2-3 a b+3 b^2\right) \tan (c+d x)}{b^3 d}-\frac{(a-3 b) \tan ^3(c+d x)}{3 b^2 d}-\frac{(a-b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2} d}+\frac{\tan ^5(c+d x)}{5 b d}",1,"-(((a - b)^3*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(7/2)*d)) + ((a^2 - 3*a*b + 3*b^2)*Tan[c + d*x])/(b^3*d) - ((a - 3*b)*Tan[c + d*x]^3)/(3*b^2*d) + Tan[c + d*x]^5/(5*b*d)","A",4,3,23,0.1304,1,"{3675, 390, 205}"
457,1,77,0,0.091478,"\int \frac{\sec ^6(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Sec[c + d*x]^6/(a + b*Tan[c + d*x]^2),x]","\frac{(a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2} d}-\frac{(a-2 b) \tan (c+d x)}{b^2 d}+\frac{\tan ^3(c+d x)}{3 b d}","\frac{(a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2} d}-\frac{(a-2 b) \tan (c+d x)}{b^2 d}+\frac{\tan ^3(c+d x)}{3 b d}",1,"((a - b)^2*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)*d) - ((a - 2*b)*Tan[c + d*x])/(b^2*d) + Tan[c + d*x]^3/(3*b*d)","A",4,3,23,0.1304,1,"{3675, 390, 205}"
458,1,52,0,0.0673449,"\int \frac{\sec ^4(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Sec[c + d*x]^4/(a + b*Tan[c + d*x]^2),x]","\frac{\tan (c+d x)}{b d}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2} d}","\frac{\tan (c+d x)}{b d}-\frac{(a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2} d}",1,"-(((a - b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*b^(3/2)*d)) + Tan[c + d*x]/(b*d)","A",3,3,23,0.1304,1,"{3675, 388, 205}"
459,1,32,0,0.0531762,"\int \frac{\sec ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b} d}",1,"ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)","A",2,2,23,0.08696,1,"{3675, 205}"
460,1,83,0,0.1036573,"\int \frac{\cos ^2(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Cos[c + d*x]^2/(a + b*Tan[c + d*x]^2),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^2}+\frac{\sin (c+d x) \cos (c+d x)}{2 d (a-b)}+\frac{x (a-3 b)}{2 (a-b)^2}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^2}+\frac{\sin (c+d x) \cos (c+d x)}{2 d (a-b)}+\frac{x (a-3 b)}{2 (a-b)^2}",1,"((a - 3*b)*x)/(2*(a - b)^2) + (b^(3/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^2*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*(a - b)*d)","A",5,5,23,0.2174,1,"{3675, 414, 522, 203, 205}"
461,1,129,0,0.1666066,"\int \frac{\cos ^4(c+d x)}{a+b \tan ^2(c+d x)} \, dx","Int[Cos[c + d*x]^4/(a + b*Tan[c + d*x]^2),x]","\frac{x \left(3 a^2-10 a b+15 b^2\right)}{8 (a-b)^3}-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^3}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 d (a-b)}+\frac{(3 a-7 b) \sin (c+d x) \cos (c+d x)}{8 d (a-b)^2}","\frac{x \left(3 a^2-10 a b+15 b^2\right)}{8 (a-b)^3}-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)^3}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 d (a-b)}+\frac{(3 a-7 b) \sin (c+d x) \cos (c+d x)}{8 d (a-b)^2}",1,"((3*a^2 - 10*a*b + 15*b^2)*x)/(8*(a - b)^3) - (b^(5/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)^3*d) + ((3*a - 7*b)*Cos[c + d*x]*Sin[c + d*x])/(8*(a - b)^2*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*(a - b)*d)","A",6,6,23,0.2609,1,"{3675, 414, 527, 522, 203, 205}"
462,1,167,0,0.2672814,"\int \frac{\sec ^7(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^7/(a + b*Tan[c + d*x]^2)^2,x]","\frac{(4 a+b) (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^3 d}+\frac{(2 a-b) (a-b) \sin (c+d x)}{2 a b^2 d \left(a-(a-b) \sin ^2(c+d x)\right)}-\frac{(4 a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d \left(a-(a-b) \sin ^2(c+d x)\right)}","\frac{(4 a+b) (a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^3 d}+\frac{(2 a-b) (a-b) \sin (c+d x)}{2 a b^2 d \left(a-(a-b) \sin ^2(c+d x)\right)}-\frac{(4 a-5 b) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d \left(a-(a-b) \sin ^2(c+d x)\right)}",1,"-((4*a - 5*b)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) + ((a - b)^(3/2)*(4*a + b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^3*d) + ((a - b)*(2*a - b)*Sin[c + d*x])/(2*a*b^2*d*(a - (a - b)*Sin[c + d*x]^2)) + (Sec[c + d*x]*Tan[c + d*x])/(2*b*d*(a - (a - b)*Sin[c + d*x]^2))","A",6,6,23,0.2609,1,"{3676, 414, 527, 522, 206, 208}"
463,1,109,0,0.1408267,"\int \frac{\sec ^5(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^5/(a + b*Tan[c + d*x]^2)^2,x]","-\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^2 d}-\frac{(a-b) \sin (c+d x)}{2 a b d \left(a-(a-b) \sin ^2(c+d x)\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}","-\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^2 d}-\frac{(a-b) \sin (c+d x)}{2 a b d \left(a-(a-b) \sin ^2(c+d x)\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"ArcTanh[Sin[c + d*x]]/(b^2*d) - (Sqrt[a - b]*(2*a + b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^2*d) - ((a - b)*Sin[c + d*x])/(2*a*b*d*(a - (a - b)*Sin[c + d*x]^2))","A",5,5,23,0.2174,1,"{3676, 414, 522, 206, 208}"
464,1,79,0,0.0795542,"\int \frac{\sec ^3(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^3/(a + b*Tan[c + d*x]^2)^2,x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d \sqrt{a-b}}+\frac{\sin (c+d x)}{2 a d \left(a-(a-b) \sin ^2(c+d x)\right)}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d \sqrt{a-b}}+\frac{\sin (c+d x)}{2 a d \left(a-(a-b) \sin ^2(c+d x)\right)}",1,"ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[a - b]*d) + Sin[c + d*x]/(2*a*d*(a - (a - b)*Sin[c + d*x]^2))","A",3,3,23,0.1304,1,"{3676, 199, 208}"
465,1,94,0,0.0830864,"\int \frac{\sec (c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]/(a + b*Tan[c + d*x]^2)^2,x]","\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{3/2}}-\frac{b \sin (c+d x)}{2 a d (a-b) \left(a-(a-b) \sin ^2(c+d x)\right)}","\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{3/2}}-\frac{b \sin (c+d x)}{2 a d (a-b) \left(a-(a-b) \sin ^2(c+d x)\right)}",1,"((2*a - b)*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(3/2)*d) - (b*Sin[c + d*x])/(2*a*(a - b)*d*(a - (a - b)*Sin[c + d*x]^2))","A",3,3,21,0.1429,1,"{3676, 385, 208}"
466,1,114,0,0.1811113,"\int \frac{\cos (c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]/(a + b*Tan[c + d*x]^2)^2,x]","-\frac{b (4 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{5/2}}+\frac{b^2 \sin (c+d x)}{2 a d (a-b)^2 \left(a-(a-b) \sin ^2(c+d x)\right)}+\frac{\sin (c+d x)}{d (a-b)^2}","-\frac{b (4 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{5/2}}+\frac{b^2 \sin (c+d x)}{2 a d (a-b)^2 \left(a-(a-b) \sin ^2(c+d x)\right)}+\frac{\sin (c+d x)}{d (a-b)^2}",1,"-((4*a - b)*b*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(5/2)*d) + Sin[c + d*x]/((a - b)^2*d) + (b^2*Sin[c + d*x])/(2*a*(a - b)^2*d*(a - (a - b)*Sin[c + d*x]^2))","A",5,4,21,0.1905,1,"{3676, 390, 385, 208}"
467,1,143,0,0.2128857,"\int \frac{\cos ^3(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]^3/(a + b*Tan[c + d*x]^2)^2,x]","\frac{b^2 (6 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{7/2}}-\frac{b^3 \sin (c+d x)}{2 a d (a-b)^3 \left(a-(a-b) \sin ^2(c+d x)\right)}-\frac{\sin ^3(c+d x)}{3 d (a-b)^2}+\frac{(a-3 b) \sin (c+d x)}{d (a-b)^3}","\frac{b^2 (6 a-b) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sin (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^{7/2}}-\frac{b^3 \sin (c+d x)}{2 a d (a-b)^3 \left(a-(a-b) \sin ^2(c+d x)\right)}-\frac{\sin ^3(c+d x)}{3 d (a-b)^2}+\frac{(a-3 b) \sin (c+d x)}{d (a-b)^3}",1,"((6*a - b)*b^2*ArcTanh[(Sqrt[a - b]*Sin[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^(7/2)*d) + ((a - 3*b)*Sin[c + d*x])/((a - b)^3*d) - Sin[c + d*x]^3/(3*(a - b)^2*d) - (b^3*Sin[c + d*x])/(2*a*(a - b)^3*d*(a - (a - b)*Sin[c + d*x]^2))","A",5,4,23,0.1739,1,"{3676, 390, 385, 208}"
468,1,127,0,0.1415593,"\int \frac{\sec ^8(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^8/(a + b*Tan[c + d*x]^2)^2,x]","\frac{(5 a+b) (a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2} d}-\frac{(a-b)^3 \tan (c+d x)}{2 a b^3 d \left(a+b \tan ^2(c+d x)\right)}-\frac{(2 a-3 b) \tan (c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}","\frac{(5 a+b) (a-b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{7/2} d}-\frac{(a-b)^3 \tan (c+d x)}{2 a b^3 d \left(a+b \tan ^2(c+d x)\right)}-\frac{(2 a-3 b) \tan (c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}",1,"((a - b)^2*(5*a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(7/2)*d) - ((2*a - 3*b)*Tan[c + d*x])/(b^3*d) + Tan[c + d*x]^3/(3*b^2*d) - ((a - b)^3*Tan[c + d*x])/(2*a*b^3*d*(a + b*Tan[c + d*x]^2))","A",5,4,23,0.1739,1,"{3675, 390, 385, 205}"
469,1,104,0,0.1356154,"\int \frac{\sec ^6(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^6/(a + b*Tan[c + d*x]^2)^2,x]","-\frac{\left(3 a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2} d}+\frac{(a-b)^2 \tan (c+d x)}{2 a b^2 d \left(a+b \tan ^2(c+d x)\right)}+\frac{\tan (c+d x)}{b^2 d}","-\frac{\left(3 a^2-2 a b-b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{5/2} d}+\frac{(a-b)^2 \tan (c+d x)}{2 a b^2 d \left(a+b \tan ^2(c+d x)\right)}+\frac{\tan (c+d x)}{b^2 d}",1,"-((3*a^2 - 2*a*b - b^2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(5/2)*d) + Tan[c + d*x]/(b^2*d) + ((a - b)^2*Tan[c + d*x])/(2*a*b^2*d*(a + b*Tan[c + d*x]^2))","A",5,4,23,0.1739,1,"{3675, 390, 385, 205}"
470,1,77,0,0.0759769,"\int \frac{\sec ^4(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^4/(a + b*Tan[c + d*x]^2)^2,x]","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2} d}-\frac{(a-b) \tan (c+d x)}{2 a b d \left(a+b \tan ^2(c+d x)\right)}","\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} b^{3/2} d}-\frac{(a-b) \tan (c+d x)}{2 a b d \left(a+b \tan ^2(c+d x)\right)}",1,"((a + b)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(3/2)*d) - ((a - b)*Tan[c + d*x])/(2*a*b*d*(a + b*Tan[c + d*x]^2))","A",3,3,23,0.1304,1,"{3675, 385, 205}"
471,1,66,0,0.0619239,"\int \frac{\sec ^2(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Sec[c + d*x]^2/(a + b*Tan[c + d*x]^2)^2,x]","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b} d}+\frac{\tan (c+d x)}{2 a d \left(a+b \tan ^2(c+d x)\right)}","\frac{\tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} \sqrt{b} d}+\frac{\tan (c+d x)}{2 a d \left(a+b \tan ^2(c+d x)\right)}",1,"ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*d) + Tan[c + d*x]/(2*a*d*(a + b*Tan[c + d*x]^2))","A",3,3,23,0.1304,1,"{3675, 199, 205}"
472,1,148,0,0.1860976,"\int \frac{\cos ^2(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]^2/(a + b*Tan[c + d*x]^2)^2,x]","\frac{b^{3/2} (5 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^3}+\frac{b (a+b) \tan (c+d x)}{2 a d (a-b)^2 \left(a+b \tan ^2(c+d x)\right)}+\frac{\sin (c+d x) \cos (c+d x)}{2 d (a-b) \left(a+b \tan ^2(c+d x)\right)}+\frac{x (a-5 b)}{2 (a-b)^3}","\frac{b^{3/2} (5 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^3}+\frac{b (a+b) \tan (c+d x)}{2 a d (a-b)^2 \left(a+b \tan ^2(c+d x)\right)}+\frac{\sin (c+d x) \cos (c+d x)}{2 d (a-b) \left(a+b \tan ^2(c+d x)\right)}+\frac{x (a-5 b)}{2 (a-b)^3}",1,"((a - 5*b)*x)/(2*(a - b)^3) + ((5*a - b)*b^(3/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^3*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*(a - b)*d*(a + b*Tan[c + d*x]^2)) + (b*(a + b)*Tan[c + d*x])/(2*a*(a - b)^2*d*(a + b*Tan[c + d*x]^2))","A",6,6,23,0.2609,1,"{3675, 414, 527, 522, 203, 205}"
473,1,212,0,0.3010119,"\int \frac{\cos ^4(c+d x)}{\left(a+b \tan ^2(c+d x)\right)^2} \, dx","Int[Cos[c + d*x]^4/(a + b*Tan[c + d*x]^2)^2,x]","-\frac{b^{5/2} (7 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^4}+\frac{x \left(3 a^2-14 a b+35 b^2\right)}{8 (a-b)^4}+\frac{b (a-4 b) (3 a+b) \tan (c+d x)}{8 a d (a-b)^3 \left(a+b \tan ^2(c+d x)\right)}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 d (a-b) \left(a+b \tan ^2(c+d x)\right)}+\frac{3 (a-3 b) \sin (c+d x) \cos (c+d x)}{8 d (a-b)^2 \left(a+b \tan ^2(c+d x)\right)}","-\frac{b^{5/2} (7 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^4}+\frac{x \left(3 a^2-14 a b+35 b^2\right)}{8 (a-b)^4}+\frac{b (a-4 b) (3 a+b) \tan (c+d x)}{8 a d (a-b)^3 \left(a+b \tan ^2(c+d x)\right)}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 d (a-b) \left(a+b \tan ^2(c+d x)\right)}+\frac{3 (a-3 b) \sin (c+d x) \cos (c+d x)}{8 d (a-b)^2 \left(a+b \tan ^2(c+d x)\right)}",1,"((3*a^2 - 14*a*b + 35*b^2)*x)/(8*(a - b)^4) - ((7*a - b)*b^(5/2)*ArcTan[(Sqrt[b]*Tan[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^4*d) + (3*(a - 3*b)*Cos[c + d*x]*Sin[c + d*x])/(8*(a - b)^2*d*(a + b*Tan[c + d*x]^2)) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*(a - b)*d*(a + b*Tan[c + d*x]^2)) + ((a - 4*b)*b*(3*a + b)*Tan[c + d*x])/(8*a*(a - b)^3*d*(a + b*Tan[c + d*x]^2))","A",7,6,23,0.2609,1,"{3675, 414, 527, 522, 203, 205}"
474,1,95,0,0.0981185,"\int (d \sec (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Sec[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+2 p+1)} \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (m+2 p+1);\frac{1}{2} (2 p+3);\sin ^2(e+f x)\right)}{f (2 p+1)}","\frac{\tan (e+f x) \left(b \tan ^2(e+f x)\right)^p (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+2 p+1)} \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (m+2 p+1);\frac{1}{2} (2 p+3);\sin ^2(e+f x)\right)}{f (2 p+1)}",1,"((Cos[e + f*x]^2)^((1 + m + 2*p)/2)*Hypergeometric2F1[(1 + 2*p)/2, (1 + m + 2*p)/2, (3 + 2*p)/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^m*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 + 2*p))","A",2,2,23,0.08696,1,"{3658, 2617}"
475,1,108,0,0.0936662,"\int (d \sec (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Sec[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \sec (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1-\frac{m}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}","\frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \sec (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1}{2};1-\frac{m}{2},-p;\frac{3}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f}",1,"(AppellF1[1/2, 1 - m/2, -p, 3/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Sec[e + f*x])^m*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(Sec[e + f*x]^2)^(m/2)*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",3,3,25,0.1200,1,"{3679, 430, 429}"
476,1,97,0,0.1075139,"\int (d \sec (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Sec[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (m+n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)}{f (n p+1)}","\frac{\tan (e+f x) (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (m+n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (m+n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right)}{f (n p+1)}",1,"((Cos[e + f*x]^2)^((1 + m + n*p)/2)*Hypergeometric2F1[(1 + n*p)/2, (1 + m + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*(d*Sec[e + f*x])^m*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",2,2,25,0.08000,1,"{3659, 2617}"
477,1,99,0,0.1349136,"\int \sec ^6(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]^6*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^5(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+5)}+\frac{2 \tan ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}+\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan ^5(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+5)}+\frac{2 \tan ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}+\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p)) + (2*Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 + n*p)) + (Tan[e + f*x]^5*(b*(c*Tan[e + f*x])^n)^p)/(f*(5 + n*p))","A",4,3,23,0.1304,1,"{3659, 2607, 270}"
478,1,65,0,0.1111858,"\int \sec ^4(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]^4*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}+\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+3)}+\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p)) + (Tan[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p)/(f*(3 + n*p))","A",4,3,23,0.1304,1,"{3659, 2607, 14}"
479,1,31,0,0.0925313,"\int \sec ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan (e+f x) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",3,3,23,0.1304,1,"{3659, 2607, 32}"
480,1,61,0,0.042406,"\int \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan (e+f x) \, _2F_1\left(1,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Hypergeometric2F1[1, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",3,3,14,0.2143,1,"{3659, 3476, 364}"
481,1,61,0,0.1038048,"\int \cos ^2(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Cos[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \, _2F_1\left(2,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan (e+f x) \, _2F_1\left(2,\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);-\tan ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"(Hypergeometric2F1[2, (1 + n*p)/2, (3 + n*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",3,3,23,0.1304,1,"{3659, 2607, 364}"
482,1,93,0,0.0968147,"\int \sec ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \sec ^3(e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+4)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+4);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan (e+f x) \sec ^3(e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+4)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+4);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"((Cos[e + f*x]^2)^((4 + n*p)/2)*Hypergeometric2F1[(1 + n*p)/2, (4 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sec[e + f*x]^3*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",2,2,23,0.08696,1,"{3659, 2617}"
483,1,91,0,0.0566201,"\int \sec (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+2)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+2);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\tan (e+f x) \sec (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (n p+2)} \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (n p+2);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"((Cos[e + f*x]^2)^((2 + n*p)/2)*Hypergeometric2F1[(1 + n*p)/2, (2 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sec[e + f*x]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",2,2,21,0.09524,1,"{3659, 2617}"
484,1,79,0,0.0755335,"\int \cos (e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Cos[e + f*x]*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \, _2F_1\left(\frac{n p}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \, _2F_1\left(\frac{n p}{2},\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"((Cos[e + f*x]^2)^((n*p)/2)*Hypergeometric2F1[(n*p)/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",2,2,21,0.09524,1,"{3659, 2617}"
485,1,82,0,0.0918643,"\int \cos ^3(e+f x) \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[Cos[e + f*x]^3*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \, _2F_1\left(\frac{1}{2} (n p-2),\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}","\frac{\sin (e+f x) \cos ^2(e+f x)^{\frac{n p}{2}} \, _2F_1\left(\frac{1}{2} (n p-2),\frac{1}{2} (n p+1);\frac{1}{2} (n p+3);\sin ^2(e+f x)\right) \left(b (c \tan (e+f x))^n\right)^p}{f (n p+1)}",1,"((Cos[e + f*x]^2)^((n*p)/2)*Hypergeometric2F1[(-2 + n*p)/2, (1 + n*p)/2, (3 + n*p)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))","A",2,2,23,0.08696,1,"{3659, 2617}"
486,0,0,0,0.0564878,"\int (d \sec (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Sec[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \sec (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left((d \sec (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Defer[Int][(d*Sec[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
487,0,0,0,0.0532297,"\int \sec ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \sec ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\sec ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Defer[Int][Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
488,0,0,0,0.0275891,"\int \sec (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \sec (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\sec (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Defer[Int][Sec[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
489,0,0,0,0.0414141,"\int \cos (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[Cos[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \cos (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\cos (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Defer[Int][Cos[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
490,0,0,0,0.053067,"\int \cos ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[Cos[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \cos ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\cos ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Defer[Int][Cos[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
491,1,244,0,0.1909059,"\int \sec ^6(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]^6*(a + b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^5(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{n},-p;\frac{n+5}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{5 f}+\frac{2 \tan ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{3 f}+\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}","\frac{\tan ^5(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{5}{n},-p;\frac{n+5}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{5 f}+\frac{2 \tan ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{3 f}+\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}",1,"(Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p)/(f*(1 + (b*(c*Tan[e + f*x])^n)/a)^p) + (2*Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p)/(3*f*(1 + (b*(c*Tan[e + f*x])^n)/a)^p) + (Hypergeometric2F1[5/n, -p, (5 + n)/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]^5*(a + b*(c*Tan[e + f*x])^n)^p)/(5*f*(1 + (b*(c*Tan[e + f*x])^n)/a)^p)","A",9,6,25,0.2400,1,"{3675, 1893, 246, 245, 365, 364}"
492,1,160,0,0.1303088,"\int \sec ^4(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]^4*(a + b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{3 f}+\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}","\frac{\tan ^3(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{3}{n},-p;\frac{n+3}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{3 f}+\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}",1,"(Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p)/(f*(1 + (b*(c*Tan[e + f*x])^n)/a)^p) + (Hypergeometric2F1[3/n, -p, (3 + n)/n, -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p)/(3*f*(1 + (b*(c*Tan[e + f*x])^n)/a)^p)","A",7,6,25,0.2400,1,"{3675, 1893, 246, 245, 365, 364}"
493,1,75,0,0.0769877,"\int \sec ^2(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[Sec[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}","\frac{\tan (e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \left(\frac{b (c \tan (e+f x))^n}{a}+1\right)^{-p} \, _2F_1\left(\frac{1}{n},-p;1+\frac{1}{n};-\frac{b (c \tan (e+f x))^n}{a}\right)}{f}",1,"(Hypergeometric2F1[n^(-1), -p, 1 + n^(-1), -((b*(c*Tan[e + f*x])^n)/a)]*Tan[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p)/(f*(1 + (b*(c*Tan[e + f*x])^n)/a)^p)","A",3,3,25,0.1200,1,"{3675, 246, 245}"
494,0,0,0,0.0144434,"\int \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Defer[Int][(a + b*(c*Tan[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
495,0,0,0,0.0533736,"\int \cos ^2(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int \cos ^2(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\text{Int}\left(\cos ^2(e+f x) \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"Defer[Int][Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p, x]","A",0,0,0,0,-1,"{}"
496,1,98,0,0.1893252,"\int (d \csc (e+f x))^m \left(b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Csc[e + f*x])^m*(b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \cos ^2(e+f x)^{p+\frac{1}{2}} \left(b \tan ^2(e+f x)\right)^p (d \csc (e+f x))^m \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (-m+2 p+1);\frac{1}{2} (-m+2 p+3);\sin ^2(e+f x)\right)}{f (-m+2 p+1)}","\frac{\tan (e+f x) \cos ^2(e+f x)^{p+\frac{1}{2}} \left(b \tan ^2(e+f x)\right)^p (d \csc (e+f x))^m \, _2F_1\left(\frac{1}{2} (2 p+1),\frac{1}{2} (-m+2 p+1);\frac{1}{2} (-m+2 p+3);\sin ^2(e+f x)\right)}{f (-m+2 p+1)}",1,"((Cos[e + f*x]^2)^(1/2 + p)*(d*Csc[e + f*x])^m*Hypergeometric2F1[(1 + 2*p)/2, (1 - m + 2*p)/2, (3 - m + 2*p)/2, Sin[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 - m + 2*p))","A",4,4,23,0.1739,1,"{3658, 2618, 2602, 2577}"
497,1,127,0,0.1828893,"\int (d \csc (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \, dx","Int[(d*Csc[e + f*x])^m*(a + b*Tan[e + f*x]^2)^p,x]","\frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \csc (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1-m}{2};1-\frac{m}{2},-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (1-m)}","\frac{\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \csc (e+f x))^m \left(a+b \tan ^2(e+f x)\right)^p \left(\frac{b \tan ^2(e+f x)}{a}+1\right)^{-p} F_1\left(\frac{1-m}{2};1-\frac{m}{2},-p;\frac{3-m}{2};-\tan ^2(e+f x),-\frac{b \tan ^2(e+f x)}{a}\right)}{f (1-m)}",1,"(AppellF1[(1 - m)/2, 1 - m/2, -p, (3 - m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^2)/a)]*(d*Csc[e + f*x])^m*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 - m)*(Sec[e + f*x]^2)^(m/2)*(1 + (b*Tan[e + f*x]^2)/a)^p)","A",4,4,25,0.1600,1,"{3681, 3667, 511, 510}"
498,1,104,0,0.2135759,"\int (d \csc (e+f x))^m \left(b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Csc[e + f*x])^m*(b*(c*Tan[e + f*x])^n)^p,x]","\frac{\tan (e+f x) (d \csc (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (-m+n p+1);\frac{1}{2} (-m+n p+3);\sin ^2(e+f x)\right)}{f (-m+n p+1)}","\frac{\tan (e+f x) (d \csc (e+f x))^m \cos ^2(e+f x)^{\frac{1}{2} (n p+1)} \left(b (c \tan (e+f x))^n\right)^p \, _2F_1\left(\frac{1}{2} (n p+1),\frac{1}{2} (-m+n p+1);\frac{1}{2} (-m+n p+3);\sin ^2(e+f x)\right)}{f (-m+n p+1)}",1,"((Cos[e + f*x]^2)^((1 + n*p)/2)*(d*Csc[e + f*x])^m*Hypergeometric2F1[(1 + n*p)/2, (1 - m + n*p)/2, (3 - m + n*p)/2, Sin[e + f*x]^2]*Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - m + n*p))","A",4,4,25,0.1600,1,"{3659, 2618, 2602, 2577}"
499,0,0,0,0.1315392,"\int (d \csc (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","Int[(d*Csc[e + f*x])^m*(a + b*(c*Tan[e + f*x])^n)^p,x]","\int (d \csc (e+f x))^m \left(a+b (c \tan (e+f x))^n\right)^p \, dx","\left(\frac{\sin (e+f x)}{d}\right)^m (d \csc (e+f x))^m \text{Int}\left(\left(\frac{\sin (e+f x)}{d}\right)^{-m} \left(a+b (c \tan (e+f x))^n\right)^p,x\right)",0,"(d*Csc[e + f*x])^m*(Sin[e + f*x]/d)^m*Defer[Int][(a + b*(c*Tan[e + f*x])^n)^p/(Sin[e + f*x]/d)^m, x]","A",0,0,0,0,-1,"{}"